Advanced Math Cocepts Evaluation

Assessment and Evaluation Masters

00 Title Pg A&E 0-02-834179-1 10/4/00 2:02 PM Page 1

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ISBN: 0-02-834179-1 AMC Assessment and Evaluation Masters

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SAT and ACT Practice Answer Sheet Masters10 question format.............................................................120 question format.............................................................2

Chapter MaterialsChapter 1 Test, Form 1A...................................................3Chapter 1 Test, Form 1B ..................................................5Chapter 1 Test, Form 1C ...................................................7Chapter 1 Test, Form 2A...................................................9Chapter 1 Test, Form 2B ................................................11Chapter 1 Test, Form 2C ................................................13Chapter 1 Open-Ended Assessment ...............................15Chapter 1 Mid-Chapter Test ............................................16Chapter 1 Quizzes A and B .............................................17Chapter 1 Quizzes C and D .............................................18Chapter 1 SAT and ACT Practice ...................................19Chapter 1 Cumulative Review.........................................21Chapter 1 Answers .....................................................22-28

Chapter 2 Test, Form 1A.................................................29Chapter 2 Test, Form 1B ................................................31Chapter 2 Test, Form 1C .................................................33Chapter 2 Test, Form 2A.................................................35Chapter 2 Test, Form 2B ................................................37Chapter 2 Test, Form 2C ................................................39Chapter 2 Open-Ended Assessment ...............................41Chapter 2 Mid-Chapter Test ............................................42Chapter 2 Quizzes A and B .............................................43Chapter 2 Quizzes C and D .............................................44Chapter 2 SAT and ACT Practice ...................................45Chapter 2 Cumulative Review.........................................47Chapter 2 Answers .....................................................48-54

Chapter 3 Test, Form 1A.................................................55Chapter 3 Test, Form 1B ................................................57Chapter 3 Test, Form 1C .................................................59Chapter 3 Test, Form 2A.................................................61Chapter 3 Test, Form 2B ................................................63Chapter 3 Test, Form 2C ................................................65Chapter 3 Open-Ended Assessment ...............................67Chapter 3 Mid-Chapter Test ............................................68Chapter 3 Quizzes A and B .............................................69Chapter 3 Quizzes C and D .............................................70Chapter 3 SAT and ACT Practice ...................................71Chapter 3 Cumulative Review.........................................73Chapter 3 Answers .....................................................74-80

Chapter 4 Test, Form 1A.................................................81Chapter 4 Test, Form 1B ................................................83Chapter 4 Test, Form 1C .................................................85Chapter 4 Test, Form 2A.................................................87Chapter 4 Test, Form 2B ................................................89Chapter 4 Test, Form 2C ................................................91Chapter 4 Open-Ended Assessment ...............................93Chapter 4 Mid-Chapter Test ............................................94Chapter 4 Quizzes A and B .............................................95Chapter 4 Quizzes C and D .............................................96Chapter 4 SAT and ACT Practice ...................................97

Chapter 4 Cumulative Review.........................................99Chapter 4 Answers .................................................100-106

Chapter 5 Test, Form 1A...............................................107Chapter 5 Test, Form 1B ..............................................109Chapter 5 Test, Form 1C ...............................................111Chapter 5 Test, Form 2A...............................................113Chapter 5 Test, Form 2B ..............................................115Chapter 5 Test, Form 2C ..............................................117Chapter 5 Open-Ended Assessment .............................119Chapter 5 Mid-Chapter Test ..........................................120Chapter 5 Quizzes A and B ...........................................121Chapter 5 Quizzes C and D ...........................................122Chapter 5 SAT and ACT Practice .................................123Chapter 5 Cumulative Review.......................................125Chapter 5 Answers .................................................126-132




Contents

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Chapter 10 Test, Form 1A.............................................237Chapter 10 Test, Form 1B ............................................239Chapter 10 Test, Form 1C .............................................241Chapter 10 Test, Form 2A.............................................243Chapter 10 Test, Form 2B ............................................245Chapter 10 Test, Form 2C ............................................247Chapter 10 Open-Ended Assessment ...........................249Chapter 10 Mid-Chapter Test ........................................250Chapter 10 Quizzes A and B .........................................251Chapter 10 Quizzes C and D .........................................252Chapter 10 SAT and ACT Practice ..............................253Chapter 10 Cumulative Review.....................................255Chapter 10 Answers ...............................................256-262

Chapter 11 Test, Form 1A.............................................263Chapter 11 Test, Form 1B ............................................265Chapter 11 Test, Form 1C .............................................267Chapter 11 Test, Form 2A.............................................269Chapter 11 Test, Form 2B ............................................271Chapter 11 Test, Form 2C ............................................273Chapter 11 Open-Ended Assessment ...........................275Chapter 11 Mid-Chapter Test ........................................276Chapter 11 Quizzes A and B .........................................277Chapter 11 Quizzes C and D .........................................278Chapter 11 SAT and ACT Practice ..............................279Chapter 11 Cumulative Review.....................................281Chapter 11 Answers ...............................................282-288

Chapter 12 Test, Form 1A.............................................289Chapter 12 Test, Form 1B ............................................291Chapter 12 Test, Form 1C .............................................293Chapter 12 Test, Form 2A.............................................295Chapter 12 Test, Form 2B ............................................297Chapter 12 Test, Form 2C ............................................299Chapter 12 Open-Ended Assessment ...........................301Chapter 12 Mid-Chapter Test ........................................302Chapter 12 Quizzes A and B .........................................303Chapter 12 Quizzes C and D .........................................304Chapter 12 SAT and ACT Practice ...............................305Chapter 12 Cumulative Review.....................................307Chapter 12 Answers ...............................................308-314

Chapter 13 Test, Form 1A.............................................315Chapter 13 Test, Form 1B ............................................317Chapter 13 Test, Form 1C .............................................319Chapter 13 Test, Form 2A.............................................321Chapter 13 Test, Form 2B ............................................323Chapter 13 Test, Form 2C ............................................325

Chapter 13 Open-Ended Assessment ............................327Chapter 13 Mid-Chapter Test ........................................328Chapter 13 Quizzes A and B .........................................329Chapter 13 Quizzes C and D .........................................330Chapter 13 SAT and ACT Practice ...............................331Chapter 13 Cumulative Review.....................................333Chapter 13 Answers ...............................................334-340



Unit MaterialsUnit 1 Review ........................................................393-394Unit 1 Test..............................................................395-398Unit 1 Answers.......................................................399-400Unit 2 Review ........................................................401-402Unit 2 Test..............................................................403-406Unit 2 Answers.......................................................407-408Unit 3 Review ........................................................409-410Unit 3 Test..............................................................411-414Unit 3 Answers.......................................................415-418Unit 4 Review ........................................................419-420Unit 4 Test..............................................................421-424Unit 4 Answers.......................................................425-426

Semester MaterialsPrecalculus Course (Chapters 4-15)

Semester One Test..............................................427-431Semester Two Test .............................................432-436Final Test............................................................437-444

Trigonometry Course (Chapters 1-9, 11, and 12)Semester One Test..............................................445-449Semester Two Test .............................................450-454Final Test............................................................455-462

Answers..................................................................463-474

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© Glencoe/McGraw-Hill 1 Advanced Mathematical Concepts

SAT and ACT Practice Answer Sheet(10 Questions)

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Write the letter for the correct answer in the blank at the right ofeach problem.

1. What are the domain and range of the relation {(�1, �2), (1, �2), (3, 1)}? 1. ________Is the relation a function? Choose yes or no.A. D � {�1, 1, 3}; R � {�2, 1}; yes B. D � {�2, 1}; R � {�1, 1, 3}; noC. D � {�1, �2, 3}; R � {�1, 1}; no D. D � {�2, 1}; R � {�2, 1}; yes

2. Given ƒ(x) � x2 � 2x, find ƒ(a � 3). 2. ________A. a2 � 4a � 3 B. a2 � 2a � 15 C. a2 � 8a � 3 D. a2 � 4a � 6

3. Which relation is a function? 3. ________A. B. C. D.

4. If ƒ(x) � �x �x

3� and g(x) � 2x � 1, find ( ƒ � g)(x). 4. ________

A. ��2x2

x��

83x � 3� B. ��2x2

x��

63x � 1� C. ��2x2

x��

53x � 3� D. �2x2

x�

�6x

3� 3�

5. If ƒ(x) � x2 � 1 and g(x) � �1x�, find [ ƒ � g](x). 5. ________

A. x � �1x� B. �x12� C. �

x21� 1� D. �

x12� � 1

6. Find the zero of ƒ(x) � ��23�x � 12. 6. ________A. �18 B. �12 C. 12 D. 18

7. Which equation represents a line perpendicular to the graph of x � 3y � 9? 7. ________A. y � ��13�x � 1 B. y � 3x � 1 C. y � �3x � 1 D. y � ��13�x � 1

8. Find the slope and y-intercept of the graph 2x � 3y � 3 � 0. 8. ________

A. m � �23�, b � �3 B. m � ��23�, b � �23�

C. m � 1, b � 2 D. m � ��23�, b � 1

9. Which is the graph of 3x � 2y � 4? 9. ________A. B. C. D.

10. Line k passes through A(�3, �5) and has a slope of ��13�. What is the 10. ________standard form of the equation for line k?A. �x � 3y � 18 � 0 B. x � 3y � 12 � 0C. x � 3y � 18 � 0 D. x � 3y � 18 � 0

11. Write an equation in slope-intercept form for a line passing through 11. ________A(4, �3) and B(10, 5).

A. y � �43�x � �235� B. y � ��43�x � �73� C. y � �43�x � �23

5� D. y � �43�x � �130�


Chapter 1 Test, Form 1A

NAME _____________________________ DATE _______________ PERIOD ________Chapter

1

003-028 A&E C01-0-02-834179-1 10/12/2000 10:37 AM Page 3 (Black plate)


12. Write an equation in slope-intercept form for a line with a slope of �110� 12. ________

and a y-intercept of �2.A. y � 0.1x � 200 B. y � 0.1x � 2C. y � 0.1x � 20 D. y � 0.1x � 2

13. Write an equation in standard form for a line with an x-intercept of 13. ________2 and a y-intercept of 5.A. 2x � 5y � 25 � 0 B. 5x � 2y � 5 � 0C. 2x � 5y � 5 � 0 D. 5x � 2y � 10 � 0

14. Which of the following describes the graphs of 2x � 5y � 9 and 14. ________10x � 4y � 18?A. parallel B. coinciding C. perpendicular D. none of these

15. Write the standard form of the equation of the line parallel to 15. ________the graph of 2y � 6 � 0 and passing through B(4, �1).A. x � 4 � 0 B. x � 1 � 0 C. y � 1 � 0 D. y � 4 � 0

16. Write an equation of the line perpendicular to the graph 16. ________of x � 2y � 6 � 0 and passing through A(�3, 2).A. 2x � y � 4 � 0 B. x � y � 4 � 0C. x � 2y � 7 � 0 D. 2x � y � 8 � 0

17. The table shows data for vehicles sold by a certain automobile dealer 17. ________during a six-year period. Which equation best models the data in the table?

A. y � x � 945 B. y � 720C. y � 45x � 89,010 D. y � �45x � 945

18. Which function describes the graph? 18. ________A. ƒ(x) � x � 1 B. ƒ(x) � 2x � 1C. ƒ(x) � 2x � 1 D. ƒ(x) � �12� x � 1

19. The cost of renting a car is $125 a day. Write a 19. ________function for the situation where d represents the number of days.A. c(d) � 125d � �12

245� d B. c(d) � 125�d � 1�

C. c(d) � � D. c(d) � �20. Which inequality describes the graph? 20. ________

A. y � x B. x � 1 � yC. 2x � y � 0 D. x � 7

Bonus Find the value of k such that ��23� Bonus: ________

is a zero of the function ƒ(x) � �4x7� k�.

A. ��134� B. ��76� C. �16� D. �3

8�

125d if �d� � d125�d � 1� if �d� d

125d if �d� � d125�d � 1� if �d� d

Chapter 1 Test, Form 1A (continued)


1

Year 1994 1995 1996 1997 1998 1999

Number of Vehicles Sold 720 710 800 840 905 945



Chapter 1 Test, Form 1B

NAME _____________________________ DATE _______________ PERIOD ________


1. What are the domain and range of the relation {(4, �1), (4, 1), (3, 2)}? 1. ________Is the relation a function? Choose yes or no.A. D � {�1, 1, 2}; R � {1, 2}; no B. D � {4, 3, 2}; R � {�1, 1}; yesC. D � {4, 3}; R � {�1, 1, 2}; no D. D � {4, �1, 3, 2}; R � {1}; no

2. Given ƒ(x) � �x2 � 2x, find ƒ(�3). 2. ________A. 3 B. �15 C. 15 D. �10


4. If ƒ(x) � �x �1

2� and g(x) � 3x, find ( ƒ � g)(x). 4. ________

A. ��3x2

x��

62x � 1� B. ��3x2

x��

62x � 1� C. ��3x2

x��

62x � 1� D. �3x2

x�

�6x

2� 1�

5. If ƒ(x) � x2 � 1 and g(x) � x � 2, find [ ƒ � g](x). 5. ________A. x2 � 4 B. x2 � 5 C. x2 � 4x � 4 D. x2 � 4x � 5

6. Find the zero of ƒ(x) � �34�x � 12. 6. ________

A. �12 B. �16 C. 9 D. 16

7. Which equation represents a line perpendicular to the graph 7. ________of 2y � x � 2?A. y � �12�x � 2 B. y � �4x � 2 C. y � �2x � 1 D. y � ��12�x � 2

8. Find the slope and y-intercept of the graph of 5x � 3y � 6 � 0. 8. ________A. m � �12�, b � 4 B. m � �53�, b � 2

C. m � �35�, b � 4 D. m � 2, b � �12�

9. Which is the graph of 4x � 2y � 6? 9. ________

A. B. C. D.

10. Write an equation in standard form for a line with a slope of �12� and 10. ________passing through A(1, 2).A. x � 2y � 3 � 0 B. x � 2y � 3 � 0C. �x � 2y � 3 � 0 D. �x � 2y � 3 � 0

11. Which is an equation for the line passing through B(0, 3) and C(�3, 4)? 11. ________A. x � 3y � 9 � 0 B. 3x � y � 3 � 0C. 3x � y � 3 � 0 D. x � 3y � 9 � 0

Chapter

1



Chapter 1 Test, Form 1B (continued)


112. Write an equation in slope-intercept form for the line passing 12. ________

through A(�2, 1) and having a slope of 0.5.A. y � 0.5x � 2 B. y � 0.5x C. y � 0.5x � 3 D. y � 0.5x � 1

13. Write an equation in standard form for a line with an x-intercept of 13. ________3 and a y-intercept of 6.A. 2y � x � 6 � 0 B. y � 2x � 6 � 0C. y � 2x � 3 � 0 D. 2y � x � 3 � 0

14. Which describes the graphs of 2x � 3y � 9 and 4x � 6y � 18? 14. ________A. parallel B. coinciding C. perpendicular D. none of these

15. Write the standard form of an equation of the line parallel to 15. ________the graph of x � 2y � 6 � 0 and passing through A(�3, 2).A. x � 2y � 1 � 0 B. x � 2y � 1 � 0C. x � 2y � 7 � 0 D. x � 2y � 7 � 0

16. Write an equation of the line perpendicular to the graph 16. ________of 2y � 6 � 0 and passing through C(4, �1).A. x � 4 � 0 B. x � 1 � 0 C. y � 1 � 0 D. y � 4 � 0

17. A laboratory test exposes 100 weeds to a certain herbicide. 17. ________Which equation best models the data in the table?

Week 0 1 2 3 4 5

Number of Weeds Remaining 100 82 64 39 24 0

A. y � 100 B. y � x � 100 C. y � 20x � 5 D. y � �20x � 100

18. Which function describes the graph? 18. ________A. ƒ(x) � x B. ƒ(x) � 2xC. ƒ(x) � x � 2 D. ƒ(x) � �12� x

19. The cost of renting a washer and dryer is $30 for the first month, $55 19. ________for two months, and $20 per month for more than two months, up to one year. Write a function for the situation where m represents the number of months.

A. c(m) � � B. c(m) � �C. c(m) � 20m � 30 D. c(m) � 30(m � 1) � 55(m � 2) � 20(m � 3)

20. Which inequality describes the graph? 20. ________A. y �2xB. ��12� x � 1 yC. y �2x � 1D. y �2x � 1

Bonus For what value of k will the graph of 2x � ky � 6 be Bonus: ________perpendicular to the graph of 6x � 4y � 12?

A. ��43� B. �43� C. �3 D. 3

30 if 0 m 155 if 1 � m 220m if 2 � m

30 if 0 m � 155 if 1 m � 220m if 2 m � 12



Chapter 1 Test, Form 1C


1Write the letter for the correct answer in the blank at the right ofeach problem.

1. What are the domain and range of the relation {(2, 2), (4, 2), (6, 2)}? 1. ________Is the relation a function? Choose yes or no.A. D � {2}; R � {2, 4, 6}; yes B. D � {2}; R � {2, 4, 6}; noC. D � {2, 4, 6}; R � {2}; no D. D � {2, 4, 6}; R � {2}; yes

2. Given ƒ(x) � x2 � 2x, find ƒ(4). 2. ________A. 0 B. �8 C. 8 D. 24


4. If ƒ(x) � x � 3 and g(x) � 2x � 4, find ( ƒ � g)(x). 4. ________A. 3x � 7 B. �x � 7 C. �x � 1 D. 3x � 1

5. If ƒ(x) � x2 � 1 and g(x) � 2x, find [ ƒ � g] (x). 5. ________A. 2x2 � 2 B. 2x2 � 1 C. x2 � 4x � 4 D. 4x2 � 1

6. Find the zero of ƒ(x) � 5x � 2. 6. ________A. �25� B. �2 C. 2 D. ��25�

7. Which equation represents a line perpendicular to the graph of 7. ________2x � y � 2?A. y � ��12�x � 2 B. y � 2x � 2 C. y � �2x � 2 D. y � �12�x � 2

8. Find the slope and y-intercept of the graph of 3x � 2y � 8 � 0. 8. ________A. m � �38�, b � 4 B. m � �53�, b � 2

C. m � �32�, b � 4 D. m � 2, b � �12�

9. Which is the graph of x � y � 1? 9. ________A. B. C. D.

10. Write an equation in standard form for a line with a slope of �1 10. ________passing through C(2, 1).A. x � y � 3 � 0 B. x � y � 3 � 0C. �x � y � 3 � 0 D. x � y � 3 � 0

11. Write an equation in standard form for a line passing through 11. ________A(�2, 3) and B(3, 4).A. 5x � y � 17 � 0 B. x � y � 1 � 0C. x � 5y � 19 � 0 D. x � 5y � 17 � 0



12. Write an equation in slope-intercept form for a line with a slope of 2 12. ________and a y-intercept of 1.A. y � �2x � 1 B. y � 2x � 2 C. y � �12�x � 3 D. y � 2x � 1

13. Write an equation in standard form for a line with a slope of 2 and 13. ________a y-intercept of 3.A. 2x � y � 3 � 0 B. �12�x � y � 3 � 0C. 2x � y � 3 � 0 D. �2x � y � 3 � 0

14. Which of the following describes the graphs of 2x � 3y � 9 and 14. ________6x � 9y � 18?A. parallel B. coincidingC. perpendicular D. none of these

15. Write the standard form of the equation of the line parallel to the 15. ________graph of x � 2y � 6 � 0 and passing through C(0, 1).A. x � 2y � 2 � 0 B. x � 2y � 2 � 0C. 2x � y � 2 � 0 D. 2x � y � 2 � 0

16. Write an equation of the line perpendicular to the graph of x � 3 and 16. ________passing through D(4, �1).A. x � 4 � 0 B. x � 1 � 0 C. y � 1 � 0 D. y � 4 � 0

17. What does the correlation value r for a regression line describe 17. ________about the data?A. It describes the accuracy of the data.B. It describes the domain of the data.C. It describes how closely the data fit the line.D. It describes the range of the data.

18. Which function describes the graph? 18. ________A. ƒ(x) � � x � 1 � B. ƒ(x) � � x � 1 �C. ƒ(x) � � x � � 1 D. ƒ(x) � � x � � 1

19. A canoe rental shop on Lake Carmine charges $10 19. ________for one hour or less or $25 for the day. Write a functionfor the situation where t represents time in hours.

A. c(t) � � B. c(t) � �C. c(t) � 10t D. c(t) � 25 � 10t

20. Which inequality describes the graph? 20. ________A. y � x 2B. 2x � 1 � yC. x � y � 2D. y � x � 2

Bonus For what value of k will the graph of Bonus: ________6x � ky � 6 be perpendicular to the graphof 2x � 6y � 12?A. �12� B. 4 C. 2 D. 5

10 if 0 t � 125 if 1 t � 24

10 if t � 125 if 1 t

Chapter 1 Test, Form 1C (continued)


1




NAME _____________________________ DATE _______________ PERIOD ________

1. State the domain and range of the relation 1. ________________{(�2, �1), (0, 0), (1, 0), (2, 1), (�1, 2)}. Then state whether the relation is a function. Write yes or no.

2. If ƒ(x) � 2x2 � x, find ƒ(x � h). 2. ________________

3. State the domain and range of the 3. ________________relation whose graph is shown. Then state whether the relation is a function. Write yes or no.

Given ƒ(x) � x � 3 and g(x) � �x2

1� 9� , find each function.

4. (ƒ � g)(x) 4. ________________

5. [ g � ƒ ](x) 5. ________________

6. Find the zero of ƒ(x) � 4x � �23�. 6. ________________

Graph each equation.7. 4y � 8 � 0 7.

8. y � ��13�x � 2 8.

9. 3x � 2y � 2 � 0 9.

10. Depreciation A car that sold for $18,600 new in 1993 is 10. _______________valued at $6000 in 1999. Find the slope of the line through the points at (1993, 18,600) and (1999, 6000). What does this slope represent?

Chapter

1



11. Write an equation in slope-intercept form for a line that 11. __________________passes through the point C(�2, 3) and has a slope of �23�.

12. Write an equation in standard form for a line passing 12. __________________through A(2, 1) and B(�4, 3).

13. Determine whether the graphs of 4x � y � 2 � 0 and 13. __________________2y � 8x � 4 are parallel, coinciding, perpendicular, ornone of these.

14. Write the slope-intercept form of the equation of the line 14. __________________that passes through C(2, �3) and is parallel to the graph of 3x � 2y � 6 � 0.

15. Write the standard form of the equation of the line that 15. __________________passes through C(3, 4) and is perpendicular to the line that passes through E(4, 1) and F(�2, 4).

16. The table displays data for a toy store’s sales of a specific 16. __________________toy over a six-month period. Write the prediction equation in slope-intercept form for the best-fit line. Use the points (1, 47) and (6, 32).

Graph each function.17. ƒ(x) � 2� x � 1 � � 2 17.

18. ƒ(x) � � 18.

Graph each inequality.19. �2 � x � 2y � 4 19.

20. y �� x � 1 � � 2 20.

Bonus If ƒ(x) � �x� �� 2� and ( ƒ � g)(x) � � x � , find g(x). Bonus: __________________

x � 3 if x 02x if x � 0



1

Month 1 2 3 4 5 6Number of Toys Sold 47 42 43 38 37 32




NAME _____________________________ DATE _______________ PERIOD ________

1. State the domain and range of the relation {(�3, 1), (�1, 0), 1. __________________(0, 4), (�1, 5)}. Then state whether the relation is a function. Write yes or no.

2. If ƒ(x) � 3x2 � 4, find ƒ(a � 2). 2. __________________

3. State the relation shown in the graph 3. __________________as a set of ordered pairs. Then state whetherthe relation is a function. Write yes or no.

Given ƒ(x) � x2 � 4 and g(x) � �x �

12

�, find each function.

4. �ƒg((xx))

� 4. __________________

5. �g � ƒ(x) 5. __________________

6. Find the zero of ƒ(x) � ��23�x � 8. 6. __________________

Graph each equation.

7. x � 2 � 0 7.

8. y � 3x � 2 8.

9. 2x � 3y � 6 � 0 9.

10. Retail The cost of a typical mountain bike was $330 in 1994 and $550 in 1999. Find the slope of the line through the points at (1994, 330) and (1999, 550). What 10. __________________does this slope represent?

Chapter

1



11. Write an equation in slope-intercept form for a line that 11. __________________passes through the point A(4, 1) and has a slope of ��12�.

12. Write an equation in standard form for a line with an 12. __________________x-intercept of �3 and a y-intercept of 4.

13. Determine whether the graphs of 3x � 2y � 5 � 0 and 13. __________________y � ��23�x � 4 are parallel, coinciding, perpendicular, or

none of these.

14. Write the slope-intercept form of the equation of the 14. __________________line that passes through A(�6, 5) and is parallel to the line x � 3y � 6 � 0.

15. Write the standard form of the equation of the line that 15. __________________passes through B(�2, 3) and is perpendicular to the graph of 2y � 6 � 0.

16. A laboratory tests a new fertilizer by applying it to 100 16. __________________seeds. Write a prediction equation in slope-intercept form for the best-fit line. Use the points (1, 8) and (6, 98).

Week 1 2 3 4 5 6

Number of Sprouts 8 20 42 61 85 98

Graph each function.17. ƒ(x) � �2 x 17.

18. ƒ(x) � �x � 2� 18.

Graph each inequality.19. x � 1 y 19.

20. y � 2x � 1 20.

Bonus If ƒ(x) � �x� +� 2� and (g � ƒ)(x) � x � 1, find g(x). Bonus: __________________



1




NAME _____________________________ DATE _______________ PERIOD ________

1. State the domain and range of the relation {(�1, 0), (0, 2), (2, 3), 1. __________________(0, 4)}. Then state whether the relation is a function. Write yes or no.

2. If ƒ(x) � 2x2 � 1, find ƒ(3). 2. __________________

3. State the relation shown in the graph 3. __________________as a set of ordered pairs. Then state whether the relation is a function. Write yes or no.

Given ƒ(x) � x � 3 and g(x) � x2, find each function.4. ( g � ƒ)(x) 4. __________________

5. [ƒ � g](x) 5. __________________

6. Find the zero of ƒ(x) � 4x � 5. 6. __________________

Graph each equation.7. y � x � 1 7.

8. y � 2x � 2 8.

9. 2y � 4x � 1 9.

10. Appreciation An old coin had a value of $840 in 1991 10. __________________and $1160 in 1999. Find the slope of the line through the points at (1991, 840) and (1999, 1160). What does this slope represent?

Chapter

1



11. Write an equation in slope-intercept form for a line that 11. __________________passes through the point A(0, 5) and has a slope of �12�.

12. Write an equation in standard form for a line passing 12. __________________through B(2, 4) and C(3, 8).

13. Determine whether the graphs of x � 2y � 2 � 0 and 13. __________________�3x � 6y � 5 � 0 are parallel, coinciding, perpendicular, or none of these.

14. Write the slope-intercept form of the equation of the line 14. __________________that passes through D(�2, 3) and is parallel to the graph of 2y � 4 � 0.

15. Write the standard form of the equation of the line that 15. __________________passes through E(2, �2) and is perpendicular to the graph of y � 2x � 3.

16. The table shows the average price of a new home in a 16. __________________certain area over a six-year period. Write the prediction equation in slope-intercept form for the best-fit line. Use the points (1994, 102) and (1999, 125).

Graph each function.17. ƒ(x) � �� x � 17.

18. ƒ(x) � � 18.

Graph each inequality.19. y 3x � 1 19.

20. y � � x � � 1 20.

Bonus What value of k in the equation Bonus: __________________2x � ky � 2 � 0 would result in a slope of �13�?

2 if x 0x if x � 0



1

Year 1994 1995 1996 1997 1998 1999Price (Thousands

102 105 115 114 119 125of Dollars)



Chapter 1 Open-Ended Assessment

NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include allrelevant drawings and justify your answer. You may show yoursolution in more than one way or investigate beyond therequirements of the problem.

1. a. The graphs of the equations y � 2x � 3, 2y � x � 11, and 2y � 4x � 8 form three sides of a parallelogram. Complete the parallelogram by writing an equation for the graph that forms the fourth side. Justify your choice.

b. Explain how to determine whether the parallelogram is a rectangle. Is it a rectangle? Justify your answer.

c. Graph the equations on the same coordinate axes. Is the graph consistent with your conclusion?

2. a. Write a function ƒ(x).

b. If g(x) � 2x2 � 1, does (ƒ � g)(x) � ( g � ƒ)(x)? Justify your answer.

c. Does (ƒ � g)(x) � ( g � ƒ)(x)? Justify your answer.

d. Does (ƒ � g)(x) � ( g � ƒ)(x)? Justify your answer.

e. Does ��ƒg�(x) � ��ƒ

g�(x)?

f. What can you conclude about the commutativity of adding, subtracting, multiplying, or dividing two functions?

3. Write a word problem that uses the composition of two functions. Give an example of the composition and solve. What does the answer mean? Use the domain and range in your explanation.

Chapter

1



1. State the domain and range of the relation {(�3, 0), (0, �2), 1. __________________(1, 1), (0, 3)}. Then state whether the relation is a function. Write yes or no.

2. Find ƒ(�2) for ƒ(x) � 3x2 � 1. 2. __________________

3. If ƒ(x) � 2(x � 1)2 � 3x, find ƒ(a � 1). 3. __________________

Given ƒ(x) � 2x � 1 and g(x) � �21x2� , find each function.

4. ��ƒg�(x) 4. __________________

5. [ƒ � g](x) 5. __________________

Graph each equation.6. x � 2y � 4 6.

7. x � �3y � 6 7.

8. Find the zero of ƒ(x) � 2x � 5. 8. __________________

9. Write an equation in standard form for a line passing 9. __________________through A(�1, 2) and B(3, 8).

10. Sales The initial cost of a new model of calculator was 10. __________________$120. After the calculator had been on the market for two years, its price dropped to $86. Let x represent the number of years the calculator has been on the market, and let y represent the selling price. Write an equation that models the selling price of the calculator after any given number of years.

Mid-Chapter Test (Lessons 1-1 through 1-4)


1


1. State the domain and range of the relation 1. __________________{(2, 3), (3, 3), (4, �2), (5, �3)}. Then statewhether the relation is a function. Write yes or no.

2. State the relation shown in the graph 2. __________________as a set of ordered pairs. Then state whether the graph represents a function. Write yes or no.

3. Evaluate ƒ(�2) if ƒ(x) � 3x2 � 2x. 3. __________________

Given f(x) � 2x � 3 and g(x) � 4x2, find each function.4. (ƒ � g)(x) 4. __________________

5. [ƒ � g](x) 5. __________________

Graph each equation.1. y � ��32�x � 1 1.

2. 4x � 3y � 6 2.

3. Find the zero of the function ƒ(x) � 3x � 2. 3. __________________4. Write an equation in slope-intercept form for the line that

passes through A(9, �2) and has a slope of ��23�. 4. __________________5. Write an equation in standard form for the line that passes

through B(�2, �2) and C(4, 1). 5. __________________

Chapter 1, Quiz B (Lessons 1-3 and 1-4)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter 1, Quiz A (Lessons 1-1 and 1-2)

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

1

Chapter

1


Determine whether the graphs of each pair of equations are parallel, coinciding, perpendicular, or none of these.

1. 3x � y � 7 1. __________________2y � 6x � 4

2. y � �13�x � 1 2. __________________x � 3y � 11

3. Write an equation in standard form for the line that 3. __________________passes through A(�2, 4) and is parallel to the graph of 2x � y � 5 � 0.

4. Write an equation in slope-intercept form for the line that 4. __________________passes through B(3, 1) and is perpendicular to the line through C(�1, 1) and D(1, 7).

5. Demographics The table shows data for the 10-year 5. __________________growth rate of the world population. Predict the growth rate for the year 2010. Use the points (1960, 22.0) and (2000, 12.6).

Graph each function.1. ƒ(x) � �x� � 1 1.

2. ƒ(x) � � 2.

Graph each inequality.3. 2x � y 4 4. �3 � x � 3y � 6

x � 3 if x 03x � 1 if x � 0

Chapter 1, Quiz D (Lessons 1-7 and 1-8)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter 1, Quiz C (Lessons 1-5 and 1-6)

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

1

Chapter

1

Year 1960 1970 1980 1990 2000 2010

Ten-year 22.0 20.2 18.5 15.2 12.6 ?Growth Rate (%)Source: U.S. Bureau of the Census



SAT and ACT PracticeNAME _____________________________ DATE _______________ PERIOD ________

After working each problem, record thecorrect answer on the answer sheetprovided or use your own paper.

Multiple Choice1. If 9 � 9 � 9 � 3 � 3 � t, then t �

A 3B 9C 27D 81E 243

2. If 11! � 39,916,800, then �142!!� �

A 9,979,200B 19,958,400C 39,916,800D 79,833,600E 119,750,000

3. (�3)3 � (16)�12�

� (�1)5 �

A ��72�

B �24C �28D 32E �20

4. Which number is a factor of 15 � 26 � 77?A 4B 9C 36D 55E None of these

5. To dilute a concentrated liquid fabricsoftener, the directions state to mix 3 cups of water with 1 cup ofconcentrated liquid. How many gallonsof water will you need to make 6 gallonsof diluted fabric softener?A 1�12� gallons B 3 gallons

C 2�12� gallons D 4 gallons

E 4�12� gallons

6. If a number between 1 and 2 issquared, the result is a numberbetweenA 0 and 1B 2 and 3C 2 and 4D 1 and 4E None of these

7. Seventy-five percent of 32 is whatpercent of 18?

A 1�13�%

B 13�13�%

C 17�79�%

D 75%

E 133�13�%

8. �23� � �56� � �112� � �78� �

A �153� B �2

549�

C �5254� D �32

14�

E �1225�

9. If 9 and 15 each divide M without aremainder, what is the value of M?

A 30B 45C 90D 135E It cannot be determined from the

information given.

10. 145 32 �

A �53

1

2

4�

B 196C 143

D 75

E 57

Chapter

1



11. A long-distance telephone call costs$1.25 a minute for the first 2 minutesand $0.50 for each minute thereafter.At these rates, how much will a 12-minute telephone call cost?A $6.25B $6.75C $7.25D $7.50E $8.50

12. After has been simplified to a

single fraction in lowest terms, what is the denominator?A 33B 6C 12D 4E 11

13. Which of the following expresses theprime factorization of 24? A 1 � 2 � 2 � 3B 2 � 2 � 2 � 3C 2 � 2 � 3 � 4D 1 � 2 � 2 � 3 � 3E 1 � 2 � 2 � 2 � 3

14. �� 4 � � (�7) � � 2 � � � �6 � � (�3) �A �10B �4C �2D �14E 22

15. Which of the following statements istrue?A 5 � 3 � 4 � 6 � 26B 2 � 3 � 1 � 2 � 4 � 3 � 1C 3 � (5 � 2) � 6 � 5(6 � 4) � �5D 2 � (5 � 1) � 3 � 2 � (4 � 1) � 19E (6 � 22) � 5 � 3 � 2 � (4 � 1) � �33

16. At last night’s basketball game, Ryanscored 16 points, Geoff scored 11 points,and Bruce scored 9 points. Together they scored �37� of their team’s points. What was their team’s final score?A 108B 96C 36D 77E 84

17–18. Quantitative ComparisonA if the quantity in Column A is

greaterB if the quantity in Column B is

greaterC if the two quantities are equalD if the relationship cannot be

determined from the informationgiven

0 y 1

Column A Column B

17.

18.

19. Grid-In What is the sum of thepositive odd factors of 36?

20. Grid-In Write �ab� as a decimal

number if a � �32� and b � �54�.

3�16��2�34�

SAT and ACT Practice (continued)


1

y4 y5

0.01% 10�4



Chapter 1 Cumulative ReviewNAME _____________________________ DATE _______________ PERIOD ________

1. Given that x is an integer and �1 � x � 2, state the relation 1. __________________represented by y � �4x � 1 by listing a set of ordered pairs. Then state whether the relation is a function. Write yes or no.(Lesson 1-1)

2. If ƒ(x) � x2 � 5x, find ƒ(�3). (Lesson 1-1) 2. __________________

3. If ƒ(x) � �x �1

4� and g(x) � x2 � 2, find [ƒ � g](x). (Lesson 1-2) 3. __________________

Graph each equation.4. x � 2y � 2 � 0 (Lesson 1-3) 4.

5. x � 2 � 0 (Lesson 1-3) 5.

6. Write an equation in standard form for the line that passes 6. __________________through A(�2, 0) and B(4, 3). (Lesson 1-4)

7. Write an equation in slope-intercept form for the line 7. __________________perpendicular to the graph of y � ��

13� x � 1 and passing

through C(2, 1). (Lesson 1-5)

8. The table shows the number of students in Anne Smith’s 8. __________________Karate School over a six-year period. Write the prediction equation in slope-intercept form for the best-fit line. Use the points (1994, 10) and (1999, 35). (Lesson 1-6)

9. Graph the function ƒ(x) � � . (Lesson 1-7) 9.

10. Graph the inequality y � � x � � 1. (Lesson 1-8) 10.

�2x if x � 01 if x 0

Chapter

1

Year 1994 1995 1996 1997 1998 1999

Number of Students 10 15 20 25 30 35



Page 3

1. A

2. A

3. C

4. A

5. D

6. A

7. B

8. D

9. C

10. C

11. C

Page 4

12. D

13. D

14. C

15. C

16. A

17. C

18. B

19. C

20. B

Bonus: D

Page 5

1. C

2. B

3. D

4. A

5. D

6. D

7. C

8. B

9. C

10. B

11. D

Page 6

12. A

13. B

14. B

15. C

16. A

17. D

18. D

19. A

20. C

Bonus: D

Chapter 1 Answer KeyForm 1A Form 1B



Page 7

1. D

2. C

3. B

4. A

5. D

6. A

7. D

8. C

9. A

10. B

11. D

Page 8

12. D

13. A

14. A

15. B

16. C

17. C

18. B

19. B

20. A

Bonus: C

Page 9

1. D � {�2, �1, 0, 1, 2},R � {�1, 0, 1, 2}; yes

2.

3.

4. �x �

13

�, x � �3

5. �x2 �

16x

�

6. ��61�

7.

8.

9.

10.

Page 10

11. y � �23

�x � �133�

12. x � 3y � 5 � 0

13. coinciding

14. y � �32

�x � 6

15. 2x � y � 2 � 0

16. y � �3x � 50

17.

18.

19.

20.

Bonus: x2 � 2

Chapter 1 Answer KeyForm 1C Form 2A

2x2 � 4xh � 2h2 �

x � hD � {xx � 1}, R � all reals; no

m � �$2100; average annual rate of depreciation of car’s value


Page 11

1. D � {�3, �1, 0}, R � {0, 1, 4, 5}; no

2. 3a2 � 12a � 8

3. {(2, 2), (0, 2),

4. x3 � 2x2 � 4x � 8

5. �x2 �

12

�

6. �12

7.

8.

9.

10.$44; average annual rate ofincrease in thebike’s cost

Page 12

11. y � � �12

� x � 3

12. 4x � 3y � 12 � 0

13. perpendicular

14. y � �13

�x � 7

15. x � 2 � 0

16. y � 18x � 10

17.

18.

19.

20.

Bonus: x2 � 3

Page 13

1. D � {�1, 0, 2}; R � {0, 2, 3, 4};no

2. 17

3. {(1, 0), (2, 1),

(2, 2)}; no

4. x2 � x � 3

5. x2 � 3

6. �45�

7.

8.

9.

10.$40; average annual rate ofincrease in thecoin’s value

Page 14

11. y � �12

� x � 5

12. 4x � y � 4 � 0

13. parallel

14. y � 3

15. x � 2y � 2 � 0

16. y � �253� x � �

45,5352�

17.

18.

19.

20.

Bonus: 6


Chapter 1 Answer KeyForm 2B Form 2C

(2, 2)}; yes



CHAPTER 1 SCORING RUBRIC

Level Specific Criteria

3 Superior • Shows thorough understanding of the concepts equations of parallel and perpendicular lines and adding, subtracting, multiplying, dividing, and composing functions.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Word problem concerning composition of functions is appropriate and makes sense.

• Goes beyond requirements of some or all problems.

2 Satisfactory, • Shows understanding of the concepts equations of parallel with Minor and perpendicular lines and adding, subtracting, multiplying, Flaws dividing, and composing functions.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Word problem concerning composition of functions is appropriate and makes sense.

• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts Satisfactory, equations of parallel and perpendicular lines and adding, with Serious subtracting, multiplying, dividing, and composing functions.Flaws • May not use appropriate strategies to solve problems.

• Computations are mostly correct.• Written explanations are satisfactory.• Word problem concerning composition of functions is mostly appropriate and sensible.

• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts equations of parallel and perpendicular lines and adding, subtracting, multiplying, dividing, and composing functions.

• May not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are not satisfactory.• Word problem concerning composition of functions is not appropriate or sensible.

• Does not satisfy requirements of problems.

Chapter 1 Answer Key


Open-Ended Assessment


Page 151a. The fourth side must have the same

slope as 2y � x � 11, but it must havea different y-intercept. The slope-intercept form of 2y � x � 11 is y � � �1

2� x � �1

21� .

Let the equation of the fourth side be y � � �1

2�x.

1b. If one angle is a right angle, the parallelogram is a rectangle. The slope-intercept form of y � 2x � 3 isy � 2x � 3. The slopes of y � � �1

2� x and y � 2x � 3 are

negative reciprocals of each other.Hence, the lines are perpendicular,and the parallelogram is a rectangle.

1c.

Yes, the points of intersection arethe vertices of a rectangle.

2a. ƒ (x) � x � 4

2b. ( ƒ � g)(x) � x � 4 � 2x2 � 1� 2x2 � x � 5

(g � ƒ )(x) � 2x2 � 1 � x � 4� 2x2 � x � 5� ( ƒ � g)(x)

2c. ( ƒ � g)(x) � x � 4 � (2x2 � 1)� �2x2 � x � 3

( g � ƒ )(x) � 2x2 � 1 � (x � 4)� 2x2 � x � 3� ( ƒ � g)(x)

2d. ( ƒ � g)(x) � (x� 4)(2x2 � 1)� 2x3 � x � 8x2 � 4� 2x3 � 8x2 � x � 4

( g � ƒ )(x) � ( 2x2 � 1)(x � 4)� 2x3 � 8x2 � x � 4� ( ƒ � g)(x)

2e. ( ƒ g)(x) � �2xx2

��

41

�

(g ƒ )(x) � �2xx2

��

41�

� ( ƒ g)(x)

2f. Addition and multiplication offunctions are commutative, butsubtraction and division are not.

3. Sample answer: Jeanette bought anelectric wok that was originally pricedat $38. The department storeadvertised an immediate rebate of $5as well as a 25% discount on smallappliances. What was the final price ofthe wok? Let x represent the originalprice of the wok, r (x ) represent theprice after the rebate, and d (x ) theprice after the discount. r (d (x )) �$23.50 and d (r (x )) � $24.75. Thedomain of each composition is $38,however, the range of the compositiondiffers with the order of thecomposition.




Mid-Chapter TestPage 16

1. D � {�3, 0, 1}, R � {�2, 0, 1, 3}, no

2. 11

3. 2a2 � 3a � 3

4. 4x3 � 2x2

5. �x12� � 1

6.

7.

8. �25�

9. 3x � 2y � 7 � 0

10. y � �17x � 120

Quiz APage 17

1. D � {2, 3, 4, 5}, R � {�3, �2, 3}; yes

2. {(0, 0), (1,1), (1, �1), (2, 2), (2, �2)}; no

3. 16

4. 8x3 � 12x2

5. 8x2 � 3

Quiz BPage 17

1.

2.

3. x � �23

�

4. y � � �23

�x � 4

5. x � 2y � 2 � 0

Quiz CPage 18

1. parallel

2. none of these

3. 2x � y � 8 � 0

4. y � � �13

� x � 2

5. 10.3%

Quiz DPage 18

1.

2.

3.

4.



SAT/ACT Practice Cumulative Review


Page 19

1. D

2. B

3. B

4. D

5. E

6. D

7. E

8. C

9. E

10. D

Page 20

11. D

12. A

13. B

14. B

15. C

16. E

17. A

18. C

19. 13

20. 1.2

Page 21

1. {(�1, 5), (0, 1), (1, �3), (2, �7)}; yes

2. 24

3. �x2 �

12

�

4.

5.

6. x � 2y � 2 � 0

7. y � 3x � 5

8. y � 5x � 9960

9.

10.





NAME _____________________________ DATE _______________ PERIOD ________


1. Which system is inconsistent? 1. ________A. y � 0.5x B. 2x � y � 8 C. x � y � 0 D. y � �1

x � 2y � 1 4x � y � 5 2x � y x � 2

2. Which system of equations is shown by the graph? 2. ______

A. �3x � y � 1 B. 3x � y � 1x � y � 3 x � y � 3

C. x � y � 3 D. �6x � 3y � �3�x � y � 1 3x � y � 1

3. Solve algebraically. 2x � 7y � 5 3. ________6x � y � 5

A. ��815�, �

54

�� B. (1, �1) C. (6, �1) D. ��34

�, �12

��4. Solve algebraically. 6x � 3y � 8z � 4 4. ________

�x � 9y � 2z � 24x � 6y � 4z � 3

A. ��52

�, �53

�, ��34

�� B. (1, �2, �1) C. ��12

�, �13

�, �14

�� D. �5, �2, �54

��5. Find 2 A � B if A � � � and B � � �. 5. ________

6. Find the values of x and y for which � � � � � is true. 6. ________

A. (�1, �1) B. (�2, �4) C. (5, 1) D. (�1, 4)

7. Find DE if D � � � and E � � �. 7. ________

8. A triangle with vertices A(�3, 4), B(3, 1) and C(�1, �5) is rotated 90° 8. ________counterclockwise about the origin. Find the coordinates of A′, B′, and C′.A. A′(�3, �4) B. A′(3, �4) C. A′(4, 3) D. A′(�4, �3)

B′(3, �1) B′(�3, �1) B′(1, �3) B′(�1, 3)C′(�1, 5) C′(1, 5) C′(�5, 1) C′(5, �1)

9. Find the value of � �. 9. ________

A. 26 B. 22 C. �16 D. 20

�11

�3

2�2

1

130

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10

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6�1

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x � 13x

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61

Chapter

2

A. � � B. � � C. � � D. � ��71

84

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10. Find the inverse of � �, if it exists. 10. ________

A. does not exist B. ��17

�� C. �17

�� D. �17

�� 11. Which product represents the solution to the system? 11. ________

12. Which is the graph of the system y � 0, 2x � 3y � 9, and x � 2 � y? 12. ________A. B. C. D.

13. Find the minimum value of ƒ(x, y) � 2x � y � 2 for the polygonal 13. ________convex set determined by this system of inequalities. x � 1 x � 3 y � 0 �

12

�x � y � 5

A. 0 B. �3 C. 3 D. �1.5

14. Describe the linear programming situation for this system of 14. ________inequalities where you are asked to find the maximum value of ƒ(x, y) � x � y. x � 0 y � 0 6x � 3y � 18 x � 3y � 9A. infeasible B. unboundedC. an optimal solution D. alternate optimal solutions

15. A farmer can plant a combination of two different crops on 20 acres 15. ________of land. Seed costs $120 per acre for crop A and $200 per acre for crop B. Government restrictions limit acreage of crop A to 15 acres, but do not limit crop B. Crop A will take 15 hours of labor per acre at a cost of $5.60 per hour, and crop B will require 10 hours of labor per acre at $5.00 per hour. If the expected income from crop A is $600 per acre and from crop B is $520 per acre, how many acres of crop A should be planted in order to maximize profit?A. 5 acres B. 20 acres C. 15 acres D. infeasible

Bonus The system below forms a polygonal convex set. Bonus: ________x � 0 y � �x, if �6 � x � 0y � 10 2x � 3y � 6, if �12 � x � �6What is the area of the closed figure?

A. 60 units2 B. 52 units2 C. 54 units2 D. 120 units2

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14

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14

11

3�4



2

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217� ��

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217�

A. ��217� �

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73

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14

�y � 7x � 14�x � 4y � 1




NAME _____________________________ DATE _______________ PERIOD ________


1. Which term(s) describe(s) this system? 4x � y � 8 1. ________3x � 2y � �5

A. dependent B. consistent and dependentC. consistent and independent D. inconsistent

2. Which system of equations is shown by the graph? 2. ______A. 3x � y � 8 B. y � x

x � 1 3x � �4yC. y � 2x � 1 D. x � y � 4

x � y � 4 3x � y � 8

3. Solve algebraically. 5x � y � 16 3. ________2x � 3y � 3

A. (4, 4) B. (�1, 3) C. (9, �5) D. (3, �1)

4. Solve algebraically. x � 3y � 3z � 0 4. ________2x � 5y � 5z � 1�x � 5y � 6z � �9

A. (3, 4, �3) B. (3, 0, 1) C. (�2, 4, 3) D. (0, �3, 3)

5. Find A � B if A � � � and B � � �. 5. ________

A. � � B. � � C. � � D. � �6. Find the values of x and y for which � � � � � is true. 6. ________

A. �0, ��35�� B. �0, �53�� C. �0, �35�� D. ��2, �53��7. Find DE if D � [5 2] and E � � �. 7. ________

A. � � B. [57] C. [45 �12] D. [33]

8. A triangle with vertices A(�3, 4), B(3, 1), and C(�1, �5) is translated 8. ________5 units left and 3 units up. Find the coordinates of A′, B′, and C′.A. A′(�2, 7) B. A′(�8, 7) C. A′(�8, 7) D. A′(�8, 1)

B′(8, 4) B′(2, �4) B′(�2, 4) B′(�2, �2)C′(�7, �2) C′(6, 2) C′(�6, �2) C′(�6, �8)

9. Find the value of | |. 9. ________

A. 39 B. 37 C. �37 D. 33

�17

52

45�12

9�6

x � 50

10

3y2x6y

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4

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6

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4

24

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Chapter

2




A. does not B. ��112�� C. ��1

12�� D. ��1

12��

exist

11. Which product represents the solution to the system? 3x � 4y � �2 11. ________2x � 5y � 6

A. � �·� � B. �17�� ·� � C. � �·� � D. �17�� ·� �12. Which is the graph of the system? y � 0 x � y 12. ________

x � 0 x � y � �4A. B. C. D.

13. Find the maximum and minimum values of ƒ(x, y) � 2y � x for the 13. ________polygonal convex set determined by this system of inequalities. x � 1 y � 0 x � 4 � yA. minimum: 1; maximum: 4 B. minimum: 0; maximum: 5C. minimum: 1; maximum: 7 D. minimum: 4; maximum: 8

14. Describe the linear programming situation for this system of 14. ________inequalities.x � 1 y � 0 3x � y � 5A. infeasible B. unboundedC. an optimal solution D. alternate optimal solutions

15. A farm supply store carries 50-lb bags of both grain pellets and 15. ________grain mash for pig feed. It can store 600 bags of pig feed. At leasttwice as many of its customers prefer the mash to the pellets. Thestore buys the pellets for $3.75 per bag and sells them for $6.00. Itbuys the mash for $2.50 per bag and sells it for $4.00. If the storeorders no more than $1400 worth of pig feed, how many bags ofmash should the store order to make the most profit?A. 160 bags B. 320 bags C. 200 bags D. 400 bags

Bonus Find the value of c for which the system is consistent and Bonus: ________dependent.3x � y � 54.5y � c � 9x

A. no value B. �8 C. 9.5 D. 7.5

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25

34

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34

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2




NAME _____________________________ DATE _______________ PERIOD ________


1. Which term(s) describe(s) this system? 3x � 2y � 7 1. ________�9x � 6y � �10

A. dependent B. consistent and dependentC. consistent and independent D. inconsistent

2. Solve by graphing. x � y � �1 2. ________x � 2y � 5

A. B. C. D.

3. Solve algebraically. y � 2x � 7 3. ________y � 5 � x

A. (8, �3) B. (2, 3) C. (4, 1) D. (�2, 3)4. Solve algebraically. x � y � z � 5 4. ________

�2x � y � z � �63x � 3y � 2z � �5

A. (1, 0, 4) B. (0, �1, 6) C. (�1, �2, 4) D. (�3, 0, 2)

5. Find A � B if A � � � and B � � �. 5. ________

6. Find the values of x and y for which � � � � � is true. 6. ________

A. (2, 3) B. (4, 1) C. (�1, 6) D. (1, 4)

7. Find EF if E � � � and F � � �. 7. ________

8. A triangle of vertices A(�3, 4), B(3, 1), and C(�1, �5) is dilated by a 8. ________scale factor of 2. Find the coordinates of A′, B′, C′.A. A′(�3, 4) B. A′(�3, 8) C. A′(�6, 8) D. A′(�1.5, 2)

B′(3, �1) B′(3, 2) B′(6, 2) B′(1.5, 0.5)C′(�1, 5) C′(�1, �10) C′(�2, �10) C′(�0.5, �2.5)

9. Find the value of � �. 9. ________

A. 6 B. �3 C. 14 D. �6

12

54

12

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5 � x2y � 5

y3x

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24

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30

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Chapter

2

A. � � B. � � C. � � D. � �74

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A. � � B. � � C. � � D. � ��54

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11. Which product represents the solution to the system? � � � � � � � � 11. ________

12. Which is the graph of the system x � 0 y � 0 �x � 2y � 4? 12. ________A. B. C. D.

13. Find the maximum value of ƒ(x, y) � y � x � 1 for the polygonal 13. ________convex set determined by this system of inequalities.x � 0 y � 0 2x � y � 4A. 1 B. 8 C. 5 D. �2

14. Which term best describes the linear programming 14. ________situation represented by the graph?A. infeasibleB. unboundedC. an optimal solutionD. alternate optimal solutions

15. Chase Quinn wants to expand his cut-flower business. He has 12 15. ________additional acres on which he intends to plant lilies and gladioli. He can plant at most 7 acres of gladiolus bulbs and no more than 11 acres of lilies. In addition, the number of acres planted to gladioli Gcan be no more than twice the number of acres planted to lilies L. The inequality L � 2G � 10 represents his labor restrictions. If his profits are represented by the function ƒ(L, G) � 300L � 200G, how many acres of lilies should he plant to maximize his profit?A. 1 acre B. 11 acres C. 0 acres D. 9.5 acres

Bonus Solve the system|y|� 2 and|x|� 1 by graphing. Bonus: ________

A. no solution B. C. D.

32

xy

14

12

10

73



2

A. does not exist B. ��13

�� C. ��13

�� D. ��14

�� 17

0�3

10

73

�17

0�3

A. �12

�� B. �12

�� C. � � � � � D. � � � � �3

2�1

14

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14

12

32

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32

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NAME _____________________________ DATE _______________ PERIOD ________

1. Identify the system shown by the 1. __________________graph as consistent and independent, consistent and dependent, or inconsistent.

2. Solve by graphing. 2.6 � y � �3x2x � 6y � 4 � 0

3. Solve algebraically. 4x � y � �3 3. __________________5x � 2y � 1

4. Solve algebraically. �3x � y � z � 2 4. __________________5x � 2y � 4z � 21x � 3y � 7z � �10

Use matrices J, K, and L to evaluate each expression. If the matrixdoes not exist, write impossible.

J � � � K � � � L � � � 5. __________________

6. __________________

5. �2K � L 6. 3K � J 7. JL 7. __________________

8. For what values of x, y, and z is � �� true? 8. __________________

9. A triangle with vertices A(�3, 4), B(5, �2), and C(7, �4) 9. __________________is rotated 90° countrclockwise about the origin. Find the coordinates of A′, B′, and C′.

10. Find the value of � �. 10. __________________

11. If it exists, find A�1 if A � � �. 11. __________________71

�35

1�5

3

�321

47

�1

y0

�4y

2 � 2x�3z � 4x

9z � 6

3�1

2�2

�47

�10

52

�83

4�3

1

�526

Chapter

2



12. Write the matrix product that represents the solution to 12. __________________the system. 4x � 3y � �15x � 2y � 3

Solve each system of inequalities by graphing and name thevertices of each polygonal convex set. Then, find the maximumand minimum values for each set.13. y � 5 13.

x � 33y � 4x � 0ƒ(x, y) � 2x � y �1

14. y � 0 14.0 � x � 4�x � y � 6ƒ(x, y) � 3x � 5y

Solve each problem, if possible. If not possible, state whether the problem is infeasible, has alternate optimal solutions, or isunbounded.15. The BJ Electrical Company needs to hire master electricians 15. __________________

and apprentices for a one-week project. Master electricians receive a salary of $750 per week and apprentices receive $350 per week. As part of its contract, the company has agreed to hire at least 30 workers. The local Building Safety Council recommends that each master electrician allow 3 hours for inspection time during the project. This project should require at least 24 hours of inspection time. How many workers of each type should be hired to meet the safety requirements, but minimize the payroll?

16. A company makes two models of light fixtures, A and B,each of which must be assembled and packed. The time required to assemble model A is 12 minutes, and model B takes 18 minutes. It takes 2 minutes to package model Aand 1 minute to package model B. Each week, 240 hours are available for assembly time and 20 hours for packing.a. If model A sells for $1.50 and model B sells for $1.70, 16a. _________________

how many of each model should be made to obtain the maximum weekly profit?

b. What is the maximum weekly profit? 16b. _________________

Bonus Use a matrix equation to find the value of x for the system.ax � by � cdx � ey � ƒ Bonus: __________________



2




NAME _____________________________ DATE _______________ PERIOD ________

1. Identify the system shown by the 1. __________________graph as consistent and independent, consistent and dependent, or inconsistent.

2. Solve by graphing. 5x � 2y � 8 2.x � y � 3

3. Solve algebraically. 3x � 2y � 1 3. __________________2x � 3y � 18

4. Solve. 2x � y � z � �3 4. __________________y � z � 1 � 0x � y � z � 9

Use matrices D, E, and F to find each sum or product. 5. __________________

D � � � E � � � F � � � 6. __________________

5. E � D 6. 3F 7. DF 7. __________________

8. For what values of x and y is [5 �3x] � [�4x 5y] true? 8. __________________

9. A triangle with vertices A(3, �3), B(1, 4), and C(�2, �1) 9. __________________is reflected over the line with equation y � x. Find the coordinates of A′, B′, and C′.

10. Find the value of | |. 10. __________________

11. Given A � � �, find A�1, if it exists. 11. __________________�6

432

�7�2

35

1�6

54

�23

�416

�302

15

�4

�273

Chapter

2



12. Write the matrix product that represents the solution 12. __________________to the system.3x � 2y � 55x � 4y � 7

Solve each system of inequalities by graphing and name thevertices of each polygonal convex set. Then, find the maximumand minimum values for each function.

13. x � y 13. __________________y � 0x � �4ƒ(x, y) � 7x � y

14. y � 0 14. __________________x � �1y � x � 32x � y � 6ƒ(x, y) � 5x � 3y

Solve each problem, if possible. If not possible, state whether the problem is infeasible, has alternate optimal solutions, or isunbounded.

15. A manufacturer of garden furniture makes a Giverny 15. __________________bench and a Kensington bench. The company needs toproduce at least 15 Giverny benches a day and at least 20 Kensington benches a day. It must also meet the demandfor at least twice as many Kensington benches as Givernybenches. The company can produce no more than 60 benchesa day. If each Kensington sells for $250 and each Givernysells for $325, how many of each kind of bench should beproduced for maximum daily income?

16. André Gagné caters small dinner parties on weekends. 16. __________________Because of an increase in demand for his work, he needs tohire more chefs and waiters. He will have to pay each chef$120 per weekend and each waiter $70 per weekend. Heneeds at least 2 waiters for each chef he hires. Find themaximum amount André will need to spend to hire extrahelp for one weekend.

Bonus Find an equation of the line that passes through Bonus: __________________P(2, 1) and through the intersection of�33

x� � �4y

� � 7 and �5x� � �

23y� � �73�.



2




NAME _____________________________ DATE _______________ PERIOD ________

1. Determine whether the system 1. __________________shown by the graph is consistant and independent, consistent and dependent, or inconsistent.

2. Solve by graphing. 2.x � y � 3x � y � 1


4. Solve algebraically. x � 2y � z � �5 4. __________________3x � 2y � z � 32x � y � 2z � �7

Use matrices A, B, and C to find the sum or product.

A � � � B � � � C � � � 5. __________________

6. __________________

5. A � B 6. �2B 7. AC 7. __________________

8. For what values of x and y is � � � � � true? 8. __________________

9. A triangle with vertices A(1, 2), B(2, �4),and C(�2, 0) 9. __________________is translated 2 units left and 3 units up. Find the coordinates A′, B′, and C′.

10. Find the value of � �. 10. __________________

11. If it exists, find A�1 if A � � �. 11. __________________03

21

2�7

�35

5yx � 7

xy � 3

1�6

25

�30

64

�1

8�5

3

�25

�1

�306

Chapter

2



12. Write the matrix product that represents the solution to the system. 3x � 4y � 5

x � 2y � 10 12. __________________

Solve each system of inequalities by graphing, and name thevertices of each polygonal convex set. Then, find the maximumand minimum values for each function.13. x � 0 13.

y � 0x � y � 5ƒ(x, y) � 5x � 3y

14. x � 0 14.0 � y � 42x � y � 6ƒ(x, y) � 3x � 2y

Solve each problem, if possible. If not possible, state whether the problem is infeasible, has alternate optimal solutions, or isunbounded.15. The members of the junior class at White Mountain High 15. __________________

School are selling ice-cream and frozen yogurt cones in the school cafeteria to raise money for their prom. The students have enough ice cream for 50 cones and enough frozen yogurt for 80 cones. They have 100 cones available. If they plan to sell each ice-cream cone for $2 and each frozen yogurt cone for $1, and they sell all 100 cones, what is the maximum amount they can expect to make?

16. Ginny Dettore custom-sews bridal gowns and bridesmaids’ 16. __________________dresses on a part-time basis. Each dress sells for $200, and each gown sells for $650. It takes her 2 weeks to produce a bridesmaid’s dress and 5 weeks to produce a bridal gown. She accepts orders for at least three times as many bridesmaids’ dresses as she does bridal gowns. In the next 22 weeks, what is the maximum amount of money she can expect to earn?

Bonus Find the coordinates of the vertices of the Bonus: __________________triangle determined by the graphs of 4x � 3y � 1 � 0, 4x � 3y � 17 � 0, and 4x � 9y � 13 � 0.



2




NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving clear, concisesolutions to each problem. Be sure to include all relevantdrawings and justify your answers. You may show your solution inmore than one way or investigate beyond the requirements of theproblem.

1. Stores A and B sell cassettes, CDs, and videotapes. The figures given are in the thousands.

Total Units Total UnitsSales Sold Sales Sold

� �Store A Store B

a. Find A � B and write one or two sentences describing the meaning of the sum.

b. Find B � A and write one or two sentences describing the meaning of the difference.

c. Find 2B. Interpret its meaning, and describe at least one situation in which a merchant may be interested in this product.

d. If C = [2 3 3], find CB and write a word problem to illustrate a possible meaning of the product.

e. Does CB � BC? Why or why not?

2. A builder builds two styles of houses: the Executive, on which he makes a profit of $30,000, and the Suburban, on which he makes a $25,000 profit. His construction crew can complete no more than 10 houses each year, and he wishes to build no more than 6 of the Executive per year.a. How many houses of each style should he build to maximize

profit? Explain why your answer makes sense.

b. If he chooses to limit the number of Executives he builds each year to 8, how many houses of each style should he build if he wishes to maximize profit? Explain your answer.

c. If he keeps the original limit on Executives, but raises the limit on the total number of houses he builds, what would be the effect on the number of each style built to maximize profit? Why?

200150175

$1000$1500$2000

CassettesA � Videotapes

CDs

Chapter

2

� �350250250

$1750$2500$3000

CassettesB � Videotapes

CDs



1. Identify the system shown by the graph as consistent and independent, consistent and dependent, or inconsistent 1. __________________

2. Solve by graphing. x � 2y � 7 2.y � �x � 2


4. Solve algebraically. x � 2y � z � 3 4. __________________2x � 3y � 2z � �1x � 3y � 2z � 1

Use matrices A, B, and C to evaluate each expression. If the matrix does not exist, write impossible.

A � � � B � � � C � � � 5. __________________

6. __________________

5. A � (�B) 6. �4C 7. BC 7. __________________


9. A quadrilateral with vertices A(�3, 5), B(1, 2), C(6, 3) 9. __________________and D(�4, �5) is translated 2 units left and 7 units down. Find the coordinates of A′, B′, C′, and D′.

10. Find the coordinates of a triangle with vertices E(3, 4), 10. __________________F(5, �1), and G(�2, 3) after a rotation of 180°counterclockwise about the origin.

y � 2x � 6�14

�3xy

3y � 1

23

�41

0�3

21

3�5

42

5�1

Chapter 2 Mid-Chapter Test (Lessons 2-1 through 2-4)


2



NAME _____________________________ DATE _______________ PERIOD ________


Chapter

2

Chapter

2 Chapter 2, Quiz B (Lessons 2-3 and 2-4)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter

2

1. Identify the system shown by the graph 1. __________________as consistent and independent, consistent and dependent, or inconsistent.

2. Solve by graphing. y � �54�x � 3 2.x � 2y � 8

Solve each system of equations algebraically.

3. 2x � y � �2 4. 3x � 5y � 21 3. __________________4x � 2y � �4 x � y � 5

4. __________________5. 2x � y � 3z � 8

x � 2y � 2z � 3 5. __________________5x � y � z � 1

Use matrices A, B, and C to evaluate each expression. If thematrix does not exist, write impossible.

A � � � B � � � C � � �1. __________________

2. __________________

1. A � B 2. CB 3. 2B � A 3. __________________


5. Each of the following transformations is applied to a 5a. _________________triangle with vertices A(4, 1), B(�5, �3), and C(�2, 6).Find the coordinates of A′, B′, and C′ in each case. 5b. _________________a. translation right 3 units and down 5 unitsb. reflection over the line y � x 5c. _________________c. rotation of 90° counterclockwise about the origin

2y � xx

x � 62y � 1

�426

35

�8

�32

24

9�8

�64


1. Find the value of � �. 1. __________________

2. Find the multiplicative inverse of � �, if it exists. 2. __________________

3. Solve the system 6x � 4y � 0 and 3x � y � 1 by using a 3. __________________matrix equation.

Use the system x � 0, y � 0, y � 4, 2x � y � 4 and the function f(x, y) � 3x � 2y for Exercises 4 and 5.

4. Solve the system by graphing, and name the vertices of 4.the polygonal convex set.

5. Find the minimum and maximum values of the function. 5. __________________

1. Carpentry Emilio has a small carpentry shop where he 1. __________________makes large and small bookcases. His profit on a large bookcase is $80 and on a small bookcase is $50. It takes Emilio 6 hours to make a large bookcase and 2 hours to make a small one. He can spend only 24 hours each week on his carpentry work. He must make at least two of each size each week. What is his maximum weekly profit?

2. Painting Michael Thomas, the manager of a paint store, 2. __________________is mixing paint for a spring sale. There are 32 units of yellow dye, 54 units of brown dye, and an unlimited supply of base paint available. Mr. Thomas plans to mix as many gallons as possible of Autumn Wheat and Harvest Brown paint. Each gallon of Autumn Wheat requires 4 units of yellow dye and 1 unit of brown dye. Each gallon of Harvest Brown paint requires 1 unit of yellow dye and 6 units of brown dye. Find the maximum number of gallons of paint that Mr. Thomas can mix.

2�4

�48

612

�10

�2

4�3

5

Chapter 2, Quiz D (Lesson 2-7)

NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

2

Chapter

2



Chapter 2 SAT and ACT Practice

NAME _____________________________ DATE _______________ PERIOD ________


1. If x � 3 � 2(1 � x) � 1, then what isthe value of x?A. �8B. �2C. 2D. 5E. 8

2. If �ab� � 0.625, then �ab� is equal to which

of the following?A. 1.60B. 2.67C. 2.70D. 3.33E. 4.25

3. Points A, B, C, and D are arranged on a line in that order. If AC � 13,BD � 12, and AD � 21, then BC � ?A. 12B. 9C. 8D. 4E. 3

4. A group of z people buys x widgets at a price of y dollars each. If the people divide the cost of this purchaseevenly, then how much must each person pay in dollars?A. �

xzy�

B. �yxz�

C. �xyz�

D. xy � zE. xyz

5. If 7a � 2b � 11 and a � 2b � 5, thenwhat is the value of a?A. �2.0B. �0.5C. 1.4D. 2.0E. It cannot be determined from the

information given.

6. In the figure below, if m � n and b � 125, then c � f � ?

A. 50B. 55C. 110D. 130E. 180

7. Twenty-five percent of 28 is what percent of 4?A. 7%B. 57%C. 70%D. 128%E. 175%

8. Using the figure below, which of thefollowing is equal to a � c?

A. 2bB. b � 90C. d � eD. b � d � eE. 180 � (d � e)

9. At what coordinates does the line 3y � 5 � x � 1 intersect the y-axis?A. (0, �2)B. (0, �1)C. �0, �13��D. (�2, 0)E. (�6, 0)

10. �(n3)6

n

2(n4)5� �

A. n9

B. n16

C. n19

D. n36

E. n40

Chapter

2

d°

b°c°

e°

a°



11. If m varies directly as n and �mn� � 5,

then what is the value of m when n � 2.2?A. 0.44B. 2.27C. 4.10D. 8.20E. 11.00

12. If ��x �� x2 � 3x, then ��3�� A. 0B. 6.0C. 18.0D. 3.3E. 9.0

13. How many distinct 3-digit numberscontain only nonzero digits?A. 909B. 899C. 789D. 729E. 504

14. If 2x2 � 5x � 9 � 0, then x could equalwhich of the following?A. �1.12B. 0.76C. 1.54D. 2.63E. 3.71

15. If PA and PB are tangent to circle Oat A and B, respectively, then which ofthe following must be true?

I. PB POII. �APO � �BPOIII. �APB � �AOB � �PAO � �PBOA. I onlyB. II onlyC. I and II onlyD. II and III onlyE. I, II, and III

16. In the picture below, if the ratio of QR to RS is 2:3 and the ratio of PQ toQS is 1:2, then what is the ratio of PQto RS?

A. 5:6B. 7:6C. 5:4D. 11:6E. 11:5

17–18. Quantitive ComparisonA. if the quantity in Column A is

greaterB. if the quantity in Column B is

greaterC. if the two quantities are equalD. if the relationship cannot be

determined for the informationgiven

n is 12% of mColumn A Column B

17.

r5 � tr 1

18.

19. Grid-In If the average of m, m � 3,m � 10, and 2m � 11 is 4, what is themode?

20. Grid-In A radio station schedules aone-hour block of time to cover a localmusic competition. The station runstwelve 30-second advertisements anduses the remaining time to broadcastlive music. What is the value of �aldivveer

mtiussinicgttiimm

ee�?

Chapter 2 SAT and ACT Practice (continued)


2

6m

t2 � t

50n

r10 � r6



Chapter 2 Cumulative Review (Chapters 1–2)

NAME _____________________________ DATE _______________ PERIOD ________

1. State the domain and range of {(�5, 2), (4, 3), (�2, 0), 1. __________________(�5, 1)}. Then state whether the relation is a function. Write yes or no.

2. If ƒ(x) � �4x2 and g(x) � �2x�, find [ g � ƒ](x). 2. __________________

3. Write the slope-intercept form of the equation of the line 3. __________________that passes through the point (�5, 4) and has a slope of �1.

4. Write the standard form of the equation of the line 4. __________________perpendicular to the line y � 3x � 5 that passes through (�3, 7).

Graph each function.

5. ƒ(x) � 5.

6. ƒ(x) � ��x� 6.

7. How many solutions does a consistent and dependentsystem of linear equations have? 7. __________________

Solve each system of equations algebraically.8. 4x � 7y � �2 9. �3x � 2y � 3z � �1 8. __________________

x � 2y � 7 2x � 5y � 3z � �64x � 3y � 3z � 22 9. __________________

10. Find �2A � B if A � � � and B � � �. 10. __________________

11. Find the value of � �. 11. __________________

12. A triangle with vertices A(�2, 5), B(�3, 7) and C(6, �2) 12. __________________is ref lected over the line with equation y = x. Find the coordinates of A′, B′, and C′.

13. Write the matrix product that 13. __________________represents the solution to the system. � �·� � � � �

14. Find the maximum and minimum values of 14. __________________ƒ(x, y) � 3x � y for the polygonal convex setdetermined by x � 1, y � 0, and x � 0.5y � 2.

15. Champion Lumber converts logs into lumber or plywood. 15. __________________In a given week, the total production cannot exceed 800 units, of which 200 units of lumber and 300 units of plywood are required by regular customers. The profit on a unit of lumber is $20 and on a unit of plywood is $30. Find the number of units of lumber and plywood that should be produced to maximize profit.

�51

xy

31

1�1

5�2

37

14

3�2

�85

�2�1

14

�53

�2 if x � �1�23�x � �13� if x �1

Chapter

2


Page 29

1. A

2. B

3. D

4. C

5. B

6. B

7. A

8. D

9. D

Page 30

10. C

11. A

12. B

13. D

14. C

15. C

Bonus: C

Page 31

1. C

2. D

3. D

4. B

5. A

6. B

7. D

8. C

9. B

Page 32

10. A

11. D

12. A

13. C

14. B

15. B

Bonus: A



048-054 A&E C02-0-02-834179-1 10/6/00 2:44 PM Page 48



Page 33

1. D

2. B

3. B

4. A

5. C

6. D

7. D

8. C

9. A

Page 34

10. B

11. D

12. A

13. C

14. A

15. B

Bonus: B

Page 351.

2.

3. ��153�, �

1139��

4. (1.6, �1)

5. � �6. impossible

7. � �8. ��1

2�, 3, ��

32��

9.

10.

11.

Page 36

12.

13.

14.

15.

16a.

16b.

Bonus:

�199

17

�181010

48�29�17

5�1

�8�6

121

Form 1C Form 2A

inconsistent

A′(�4, �3), B′(2, 5),C′(4, 7)

101

��213�� 1

3�3

4�2�5

Vertices: (0, 0), (0, 2),(2, 4), (4, 2), (4, 0); min.: �14; max.: 12

8 master and 22 apprentice

300 A, 600 B

$1470

x � �acee

��

bbdf�

Vertices: (�3.75, 5), (3, 5), (3, �4); min.:�11.5; max.: 11

� ��358�

��318� �

378�

�338�

048-054 A&E C02-0-02-834179-1 10/6/00 2:44 PM Page 49

Page 37

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Page 38

12.�12�� · � �

13.

14.

15.

16. unbounded

Bonus: x � y � 3

Page 39

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Page 40

12.

13. Vertices: (0, 0), (0, 5), (5, 0); min.: 0; max.: 25

14. Vertices: (0, 0), (0, 4), (1, 4), (3, 0); min.: �8; max.: 9

15. $150

16. $2500

(2, �3),Bonus: (�1, 1), (8, 5)

57

�23

4�5



consistent anddependent

(3, �4)

(2, 4, �3)

� ��

� ��5

4�, �3

4��

A′(�3, 3),B ′(4, 1),C ′(�1, �2)

29

� ��41�

�81�

�16

�

��112�

�8�23

27

�655

�1

71

�18

3 �18

1512

�69

�5�410

�1�7�1

consistent andindependent

no solution

(4, 1, �7)

� �� (�5, �1)A′(�1, 5),B′(0, �1), C ′(�4, 3)

11

� �0

�13

�

�12

�

��16

�

9�30

12

�16257

90

�18

�12�8

2

�1610

�6

49

�2

5�5

9

�12

� � � � � �510

�43

2�1

vertices: (0, 0), (�4, 0),(�4, �4); min.: �32;max.: 0

vertices: (�1, 0), (�1, 2), (1, 4), (3, 0); min.: �5; max.: 17

20 Givernybenches and 40 Kensingtonbenches

048-054 A&E C02-0-02-834179-1 10/6/00 2:44 PM Page 50


Chapter 2 Answer KeyCHAPTER 2 SCORING RUBRIC


3 Superior • Shows thorough understanding of the concepts addition, subtraction, and multiplication of matrices, sclar multiplicationof a matrix, and finding the maximum value of a function for apolygonal convex set.

• Uses appropriate strategies to solve problem.• Computations are correct.• Written explanations are exemplary.• Word problem concerning linear inequalities is appropriate and makes sense.

• Goes beyond requirements of some of all problems.

2 Satisfactory, • Shows thorough understanding of the concepts addition, with Minor subtraction, and multiplication of matrices, scalar multiplicationFlaws of a matrix, and finding the maximum value of a function for a

polygonal convex set.• Uses appropriate strategies to solve problem.• Computations are mostly correct.• Written explanations are effective.• Word problem concerning the product of matrices is appropriate and mades sense.


1 Nearly • Shows understanding of the concepts addition, subtraction,Satisfactory, and multiplication of matrices, scalar multiplication of awith Serious matrix, and finding the maximum multiplication value of aFlaws function for a polygonal convex set.

• May not use appropriate strategies to solve problem.• Computations are mostly correct.• Written explanations are mostly correct.• Word problem concerning the product of matrices is mostly appropriate and sensible.


0 Unsatisfactory • Shows little or no understanding of the concepts addition,subtraction, and multiplication of matrices, scalar multiplication of a matrix, and finding the maximum value of a function for a polygonal convex set.

• May not use appropriate strategies to solve problem.• Computations are incorrect.• Written explanations are not satisfactory.• Word problem concerning the product of matrices is not appropriate or sensible.


048-054 A&E C02-0-02-834179-1 10/6/00 2:44 PM Page 51

Page 411a. A � B:

Total UnitsSales Sold

� �A � B gives the total sales and number of units sold for both years. For example, the two stores sold 550,000 cassettes in 1993 and 1994. The combined cassette sales were$1,750,000

1b. A � B:Total UnitsSales Sold

� �B � N give the difference between total sales and number of units sold for the two years. For example, 100 more albums were sold in 1994 than in 1993.

1c. 2B:Total UnitsSales Sold

� �1d.

Total UnitsSales Sold

CB: [$20,000 2200]Mrs. Carl wishes to sell twice asmany cassettes, three times asmany videotapes, and three times asmany CDs in 1995 as in 1994.What isthe total value and the total numberof units of her 1995 goal?

1e. CB � BC because BC is undefined.

2a. Let x � numbers of ExecutivesLet y � numbers of Suburbans.x � y � 10 total number of housesx � 6 number of Executives

Profit � $30,000x � $25,000y. Check(0, 10), (0, 0), (6, 0,) and (6, 4).$30,000(0) � $25,000(10) � $250,000$30,000(6) � $25,000(0) � $180,000$30,000(6) � $25,000(4) � $280,000He should build 6 Executives and 4 Suburbans. This answer makessense because he would be buildingthe maximum number of Executives, which are the most profitable.

2b. He should build 8 Executives and 2 Suburbans because he makes agreater profit on the Executives.

2c. He should increase the number ofSuburbans because he is already atthe limit on Executives.

700500500

$3500$5000$6000

CassettesVideotape

CDs

15010047

$750$1000$1000

CassettesVideotape

CDs

550400425

$2750$4000$5000

CassettesVideotape

CDs


Chapter 2 Answer KeyOpen-Ended Assessment

048-054 A&E C02-0-02-834179-1 10/6/00 2:44 PM Page 52



1.

2.

3. (2, �1)

4. (�2, 1, 3)

5.

6.

7.

8. (1, �5)

9.

10.

Quiz APage 43

1.

2.

3.

4. (2, 3)

5. (�1, 4, 2)

Quiz BPage 43

1. � �

2. � �

3. � �4. (�5, �3)

5a.

5b.

5c.

Quiz CPage 44

1. 33

2. does not exist

3. ��29�, � �13

��

4.

5. min.: �6; max.: 6

Quiz DPage 44

1. $600

2. 14 gallons�15

12104

�17�11

36

�10188

6�6

�48


consistent anddependent

� �21

24

� �� 8� 12

16� 4

012

� �12�7

�1021

�6�3

A′(�5, �2), B′(�1, �5), C′(4, �4), D′(�6, �12)

E′(�3, �4),F′(�5, 1),G′(2, �3)

consistent and independent

infinitely many solutions

A′(�1, 4), B′(3, �5), C ′(�6, �2)

A′(1, 4), B′(�3, �5), C ′(6, �2)

A′(7, �4), B′(�2, �8), C ′(1, 1)

Vertices: (0, 0), (0, 3), (3.5, 3), (2, 0)

048-054 A&E C02-0-02-834179-1 10/6/00 2:44 PM Page 53

Page 451. C

2. A

3. D

4. A

5 C

6. C

7. E

8. C

9. A

10. D

Page 4611. E

12. C

13. D

14. E

15. D

16. A

17. C

18. A

19. 5

20. 54/6, 27/3, or 9/1

Page 471. D: {�5, �2, �4};

R: {0, 1, 2, 3}; no

2. ��2x1

2�

3. v � �x � 1

4. x � 3y � 18 � 0

5.

6.

7. infinitely many

8. (3, 2)

9. (4, �1, 3)

10. � �11. �41

12. A′(5, �2), B′(7, �3),C′(�2, 6)

13.�14

�� · � �14. max.: 6, min.:3

15. 200 lumber, 600 plywood

�51

�31

11

56

1�10

2�1


Chapter 2 Answer KeySAT/ACT Practice Cumulative Review

048-054 A&E C02-0-02-834179-1 10/6/00 2:44 PM Page 54



NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right of each problem.

1. The graph of the equation y � x3 � x is symmetric with respect to 1. ________which of the following?A. the x-axis B. the y-axis C. the origin D. none of these

2. If the graph of a function is symmetric about the origin, which of the 2. ________following must be true?A. ƒ(x) � ƒ(�x) B. ƒ(�x) � �ƒ(x) C. ƒ(x) � �ƒ(x)� D. ƒ(x) � �ƒ(

1x)�

3. The graph of an odd function is symmetric with respect to which 3. ________of the following?A. the x-axis B. the y-axis C. the line y � x D. none of these

4. Given the parent function p(x) � �x�, what transformations 4. ________occur in the graph of p(x) � 2�x � 3� � 4?A. vertical expansion by factor B. vertical compression by factor of

of 2, left 3 units, up 4 units 0.5, down 3 units, left 4 unitsC. vertical expansion by factor D. vertical compression by factor of

of 2, right 3 units, up 4 units 0.5, down 3 units, left 4 units

5. Which of the following results in the graph of ƒ(x) � x2 being expanded 5. ________vertically by a factor of 3 and reflected over the x-axis?A. ƒ(x) � �13�x2 B. ƒ(x) � �3x2 C. ƒ(x) � ��x

12� � 3 D. ƒ(x) � ��13�x2

6. Which of the following represents the graph of ƒ(x) � �x�3 � 1? 6. ________A. B. C. D.

7. Solve �2x � 5� � 7. 7. ________A. x � �1 or x � 6 B. �1 � x � 6C. x � 6 D. x � �1

8. Choose the inequality shown by the graph. 8. ________A. y � 2�x� � 1B. y � �2�x� � 1C. y � 2�x� � 1D. y � �2�x� � 1

9. Find the inverse of ƒ(x) � 2�x� � 3. 9. ________A. ƒ�1(x) � ��x �

23��2 B. ƒ�1(x) � ��x �

23��2

C. ƒ�1(x) � �12��x� � 3 D. ƒ�1(x) � �12��x� � 310. Which graph represents a function whose inverse is also a function? 10. ________

A. B. C. D.

Chapter

3

055-080 A&E C03-0-02-834179 10/4/00 2:31 PM Page 55

11. Describe the end behavior of ƒ(x) � 2x4 � 5x � 1. 11. ________A. x → �, ƒ(x)→ � B. x→ �, ƒ(x) → �

x→ ��, ƒ(x) → �� x → ��, ƒ(x) → �C. x → �, ƒ(x) → �� D. x → �, ƒ(x) → ��

x → ��, ƒ(x) → � x → ��, ƒ(x) → ��

12. Which type of discontinuity, if any, is shown 12. ________in the graph at the right?A. jump B. infiniteC. point D. The graph is continuous.

13. Choose the statement that is true for the graph of ƒ(x) � x3 � 12x. 13. ________A. ƒ(x) increases for x �2. B. ƒ(x) decreases for x �2.C. ƒ(x) increases for x 2. D. ƒ(x) decreases for x 2.

14. Which type of critical point, if any, is present in the graph of 14. ________ƒ(x) � (�x � 4)5 � 1?A. maximum B. minimumC. point of inflection D. none of these

15. Which is true for the graph of ƒ(x) � �x3 � 3x � 2? 15. ________A. relative maximum at (1, 0) B. relative minimum at (1, 0)C. relative maximum at (�1, �4) D. relative minimum at (0, �2)

16. Which is true for the graph of y � �xx2

2��

49�? 16. ________

A. vertical asymptotes at x � �3 B. horizontal asymptotes at y � � 2C. vertical asymptotes at x � �2 D. horizontal asymptote at y � 1

17. Find the equation of the slant asymptote for the graph of 17. ________

y � .

A. y � x � 4 B. y � x � 1 C. y � x D. y � 1

18. Which of the following could be the function 18. ________represented by the graph at the right?A. ƒ(x) � �xx

��

12� B. ƒ(x) � �

((xx

�

�

13))((xx

�

� 23))

�

C. ƒ(x) � �xx��

12� D. ƒ(x) � �

((xx

�

�

31))((xx

�

�

23))

�

19. Chemistry The volume V of a gas varies inversely as pressure P is 19. ________exerted. If V � 3.5 liters when P � 5 atmospheres, find V when P � 8 atmospheres.A. 2.188 liters B. 5.600 liters C. 11.429 liters D. 17.5 liters

20. If y varies jointly as x and the cube of z, and y � 378 when x � 4 and 20. ________z � 3, find y when x � 9 and z � 2.A. y � 283.5 B. y � 84 C. y � 567 D. y � 252

Bonus An even function ƒ has a vertical asymptote at x � 3 and a Bonus: ________maximum at x � 0. Which of the following could be ƒ ?

A. ƒ(x) � �x2 �x

9� B. ƒ(x) � �x �x

3� C. ƒ(x) � �x2x�

2

9� D. ƒ(x) � �x4 �x2

81�

x3 � 4x2 � 2x � 5��x2 � 2




3

055-080 A&E C03-0-02-834179 10/4/00 2:31 PM Page 56



NAME _____________________________ DATE _______________ PERIOD ________


1. The graph of the equation y � x4 � 3x2 is symmetric with respect to 1. ________which of the following?A. the x-axis B. the y-axis C. the origin D. none of these

2. If the graph of a function is symmetric with respect to the y-axis, 2. ________which of the following must be true?A. ƒ(x) � ƒ(�x) B. ƒ(x) � �ƒ(x) C. ƒ(x) ��ƒ(x)� D. ƒ(x) � �ƒ(

1x)�

3. The graph of an even function is symmetric with respect to which of 3. ________the following?A. the x-axis B. the y-axis C. the line y � xD. none of these

4. Given the parent function p(x) � x2, what translations occur in 4. ________the graph of p(x) � (x � 7)2 � 3?A. right 7 units, up 3 units B. down 7 units, left 3 unitsC. left 7 units, up 3 units D. right 7 units, down 3 units

5. Which of the following results in the graph of ƒ(x) � x3 being expanded 5. ________vertically by a factor of 3?A. ƒ(x) � x3 � 3 B. ƒ(x) � �13�x3 C. ƒ(x) � 3x3 D. ƒ(x) � ��13�x3

6. Which of the following represents the graph of ƒ(x) � �x2 � 1�? 6. ________A. B. C. D.

7. Solve �2x � 4� 6. 7. ________A. x �1 or x 5 B. �1 x 5C. x 5 D. x �1

8. Choose the inequality shown by the graph. 8. ________A. y � �x � 2� � 1 B. y � �x � 2� � 1C. y � �x � 2� � 1 D. y � �x � 2� � 1

9. Find the inverse of ƒ(x) � �x �1

2�. 9. ________

A. ƒ�1(x) � �x �1

2� B. ƒ�1(x) � �1x� � 2

C. ƒ�1(x) � x � 2 D. ƒ�1(x) � �1x� � 2

10. Which graph represents a function whose inverse is also a function? 10. ________A. B. C. D.

Chapter

3

B

A

B

A

C

C

B

A

D

D

055-080 A&E C03-0-02-834179 10/4/00 2:31 PM Page 57


11. Which type of discontinuity, if any, is shown 11. ________in the graph at the right?A. jump B. infiniteC. point D. The graph is continuous.

12. Describe the end behavior of ƒ(x) � 2x3 � 5x � 1 12. ________A. x → �, ƒ(x) → � B. x → �, ƒ(x) → �

x → ��, ƒ(x) → �� x → ��, ƒ(x) → �C. x → �, ƒ(x) → �� D. x → �, ƒ(x) → ��

x → ��, ƒ(x) → � x → ��, ƒ(x) → ��

13. Choose the statement that is true for the graph of ƒ(x) � �(x � 2)2. 13. ________A. ƒ(x) increases for x �2. B. ƒ(x) decreases for x �2.C. ƒ(x) increases for x 2. D. ƒ(x) decreases for x 2.

14. Which type of critical point, if any, is present in the graph of 14. ________ƒ(x) � (�x � 4)3?A. maximum B. minimumC. point of inflection D. none of these

15. Which is true for the graph of ƒ(x) � x3 � 3x � 2? 15. ________A. relative maximum at (1, 0) B. relative minimum at (�1, 4)C. relative maximum at (�1, 4) D. relative minimum at (0, 2)


2��

49�? 16. ________

A. vertical asymptotes, x � �3 B. horizontal asymptotes, y � �2C. vertical asymptotes, x � �2 D. horizontal asymptote, y � 0

17. Find the equation of the slant asymptote for the graph of 17. ________y � �3x2

x+

�2x

1� 3�.

A. y � x B. y � 3x � 1 C. y � x � 3 D. y � 3x � 5

18. Which of the following could be the function 18. ________represented by the graph at the right?A. ƒ(x) � �x �

x2� B. ƒ(x) � �(x �

x(x3)

�

(x3�

)2)�

C. ƒ(x) � �x �x

2� D. ƒ(x) � �(x �

x(x3)

�

(x3�

)2)�

19. Chemistry The volume V of a gas varies inversely as pressure P is 19. ________exerted. If V � 4 liters when P � 3.5 atmospheres, find V when P � 2.5 atmospheres.A. 5.6 liters B. 2.188 liters C. 2.857 liters D. 10.0 liters

20. If y varies jointly as x and the cube root of z, and y � 120 when x � 3 20. ________and z � 8, find y when x � 4 and z � 27.A. y � 540 B. y � 240 C. y � 60 D. y � 26�23�

Bonus If ƒ( g(x)) � x and ƒ(x) � 3x � 4, find g(x). Bonus: ________A. g(x) � �x �

34� B. g(x) � �x +

34� C. g(x) � �3

x� � 4 D. g(x) � �3x� � 4



3

055-080 A&E C03-0-02-834179 10/4/00 2:31 PM Page 58



NAME _____________________________ DATE _______________ PERIOD ________


1. The graph of the equation y � x2 � 3 is symmetric with respect to 1. ________which of the following?A. the x-axis B. the y-axis C. the origin D. none of these

2. If the graph of a relation is symmetric about the line y � x and the 2. ________point (a, b) is on the graph, which of the following must also be on the graph?A. (�a, b) B. (a, �b) C. (b, a) D. (�a, �b)

3. The graph of an odd function is symmetric with respect to which 3. ________of the following?A. the x-axis B. the y-axis C. the origin D. none of these

4. Given the parent function p(x) � �x�, what transformation occurs in 4. ________the graph of p(x) � �x� �� 2� � 5?A. right 2 units, up 5 units B. up 2 units, right 5 units C. left 2 units, down 5 units D. down 2 units, left 5 units

5. Which of the following results in the graph of ƒ(x) � x2 being 5. ________expanded vertically by a factor of 4?A. ƒ(x) � x2 � 4 B. ƒ(x) � x2 � 4 C. ƒ(x) � 4x2 D. ƒ(x) � �14�x2

6. Which of the following represents the graph of ƒ(x) � �x3�? 6. ________A. B. C. D.

7. Solve �x � 4� 6. 7. ________A. x �2 B. x 10C. �2 x 10 D. x �2 or x 10

8. Choose the inequality shown by the graph. 8. ________A. y � x2 � 1 B. y � (x � 1)2

C. y � x2 � 1 D. y � (x � 1)2

9. Find the inverse of ƒ(x) � 2x � 4. 9. ________A. ƒ�1(x) � �2x

1� 4� B. ƒ�1(x) � �x �

24�

C. ƒ�1(x) � �2x� � 4 D. ƒ�1(x) � �2

x� � 4

10. Which graph represents a function whose inverse is also a function? 10. ________A. B. C. D.

Chapter

3

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 59


11. Which type of discontinuity, if any, is shown 11. ________in the graph at the right?A. jumpB. infiniteC. pointD. The graph is continuous.

12. Describe the end behavior of ƒ(x) � �x2. 12. ________A. x → ∞, ƒ(x) → ∞ B. x → ∞, ƒ(x) → ∞

x → �∞, ƒ(x) → �∞ x → �∞, ƒ(x) → ∞C. x → ∞, ƒ(x) → �∞ D. x → ∞, ƒ(x) → �∞

x → �∞, ƒ(x) → ∞ x → �∞, ƒ(x) → �∞13. Choose the statement that is true for the graph of ƒ(x) � (x � 1)2. 13. ________

A. ƒ(x) increases for x �1. B. ƒ(x) decreases for x �1.C. ƒ(x) increases for x 1. D. ƒ(x) decreases for x 1.

14. Which type of critical point, if any, is present in the graph 14. ________of ƒ(x) � x3 � 1?A. maximum B. minimumC. point of inflection D. none of these

15. Which is true for the graph of ƒ(x) � x3 � 3x? 15. ________A. relative maximum at (1, �2) B. relative minimum at (�1, 2)C. relative maximum at (�1, 2) D. relative minimum at (0, 0)


2��

94�? 16. ________

A. vertical asymptotes at x � �3 B. horizontal asymptotes at y � �2C. vertical asymptotes at x � �2 D. horizontal asymptote at y � 0

17. What type of asymptote is present in the graph of y � �xx2

��

35�? 17. ________

A. vertical B. slant C. horizontal D. none of these

18. Which of the following could be the function 18. ________represented by the graph?A. ƒ(x) � �x �

14� B. ƒ(x) � �(x �

x2�)(x

2� 4)�

C. ƒ(x) � �xx��

24� D. ƒ(x) � �(x �

x2�)(x

2� 4)�

19. Chemistry The volume V of a gas varies inversely as pressure 19. ________P is exerted. If V � 4 liters when P � 3 atmospheres, find V when P � 7 atmospheres.A. 1.714 liters B. 5.25 liters C. 9.333 liters D. 1.5 liters

20. If y varies inversely as the cube root of x, and y � 12 when x � 8, 20. ________find y when x � 1.A. y � 6144 B. y � 24 C. y � 6 D. y � �12

38�

Bonus The graph of ƒ(x) � �x21� c� has a vertical asymptote at x � 3. Bonus: ________

Find c.A. 9 B. 3 C. �3 D. �9



3

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NAME _____________________________ DATE _______________ PERIOD ________

Determine whether the graph of each equation is symmetric withrespect to the origin, the x-axis, the y-axis, the line y � x, the liney � �x, or none of these.

1. xy � �4 1. __________________

2. x � 5y2 � 2 2. __________________

3. Determine whether the function ƒ(x) � �x2 �x

4� is odd, even, 3. __________________or neither.

4. Describe the transformations relating the graph of 4. __________________y � �2x3 � 4 to its parent function, y � x3.

5. Use transformations of the parent graph p(x) � �1x� 5.to sketch the graph of p(x) � ��

1x�� 1.

6. Graph the inequality y 2x2 � 1. 6.

7. Solve �5 � 2x� � 11. 7. __________________

Find the inverse of each function and state whether the inverse is a function.

8. ƒ(x) � �x �x

2� 8. __________________

9. ƒ(x) � x2 � 4 9. __________________

10. Graph ƒ(x) � x3 � 2 and its inverse. State whether the 10. __________________inverse is a function.

Chapter

3

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 61


Determine whether each function is continuous at the given x-value. If discontinuous, state the type of discontinuity (point,jump, or infinite).

11. ƒ(x) � ; x � 1 11. __________________

12. ƒ(x) � �xx2

��

39�; x � �3 12. __________________

13. Describe the end behavior of y � �3x4 � 2x. 13. __________________

14. Locate and classify the extrema for the graph of 14. __________________y � x4 � 3x2 � 2.

15. The function ƒ(x) � x3 � 3x2 � 3x has a critical point 15. __________________when x � 1. Identify the point as a maximum, a minimum, or a point of inf lection, and state its coordinates.

16. Determine the vertical and horizontal asymptotes for 16. __________________the graph of y � �x3 �

x2

5�x2

4� 6x�.

17. Find the slant asymptote for y � �3x2

x�

�5x

2� 1�. 17. __________________

18. Sketch the graph of y � �x3 �x2

x2�

�1

12x�. 18.

19. If y varies directly as x and inversely as the square root 19. __________________of z, and y � 8 when x � 4 and z � 16, find y when x � 10 and z � 25.

20. Physics The kinetic energy Ek of a moving object, 20. __________________measured in joules, varies jointly as the mass mof the object and the square of the speed v. Find the constant of variation k if Ek is 36 joules, m is 4.5 kilograms,and v is 4 meters per second.

Bonus Given the graph of p(x), Bonus:sketch the graph of y � �2p�12�(x � 2)� � 2.

x2 � 1 if x 1�x3 � 2 if x � 1



3

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 62



NAME _____________________________ DATE _______________ PERIOD ________

Determine whether the graph of each equation is symmetric withrespect to the origin, the x-axis, the y-axis, the line y � x, the liney � �x, or none of these.

1. xy � 2 1. __________________

2. y � 5x3 � 2x 2. __________________

3. Determine whether the function ƒ(x) � �x� is odd, even, 3. __________________or neither.

4. Describe the transformations relating the graph of 4. __________________y � �12�(x � 3)2 to its parent function, y � x2.

5. Use transformations of the parent graph of p(x) � �x� 5.to sketch the graph of p(x) � ��x� � 3.

6. Graph the inequality y � (x � 2)3. 6.

7. Solve �2x � 4� � 10. 7. __________________


8. ƒ(x) � x3 � 4 8. __________________

9. ƒ(x) � (x � 1)2 9. __________________

10. Graph ƒ(x) � x2 � 2 and its inverse. State whether the 10. __________________inverse is a function.

Chapter

3

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 63



11. ƒ(x) � ; x � 1 11. __________________

12. ƒ(x) � �xx2

��

39�; x � �3 12. __________________

13. Describe the end behavior of y � 3x3 � 2x. 13. __________________

14. Locate and classify the extrema for the graph 14. __________________of y � x4 � 3x2.

15. The function ƒ(x) � �x3 � 6x2 � 12x � 7 has a critical 15. __________________point when x � �2. Identify the point as a maximum, a minimum, or a point of inflection, and state its coordinates.

16. Determine the vertical and horizontal asymptotes for the 16. __________________graph of y � �x

x3

2

��

x42�.

17. Find the slant asymptote for y � �2x2

x�

�5x

3� 2�. 17. __________________

18. Sketch the graph of y � �xx2 �

�31x�. 18.

19. If y varies directly as x and inversely as the square of z, 19. __________________and y � 8 when x � 4 and z � 3, find y when x � 6 and z � �2.

20. Geometry The volume V of a sphere varies directly as 20. __________________the cube of the radius r. Find the constant of variation kif V is 288� cubic centimeters and r is 6 centimeters.

Bonus Determine the value of k such that Bonus: __________________

ƒ(x) � is continuous when x � 2.2x2 if x 2x � k if x � 2

x 1 x � 1

ifif

x2 � 1 �x � 3


3 Chapter 3 Test, Form 2B (continued)

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 64



NAME _____________________________ DATE _______________ PERIOD ________

Determine whether the graph of each equation is symmetric with respect to the origin, the x-axis, the y-axis, the line y � x, the line y � �x, or none of these.

1. y � 2�x� 1. __________________

2. y � x3 � x 2. __________________

3. Determine whether the function ƒ(x) � x2 � 2 is odd, even, 3. __________________or neither.

4. Describe the transformation relating the graph of 4. __________________y � (x � 1)2 to its parent function, y � x2.

5. Use transformations of the parent graph p(x) � x3 5.to sketch the graph of p(x) � (x � 2)3 � 1.

6. Graph the inequality y � x2 � 1. 6.

7. Solve �x � 4� � 8. 7. __________________


8. ƒ(x) � x2 8. __________________

9. ƒ(x) � x3 � 1 9. __________________

10. Graph ƒ(x) � �2x � 4 and its inverse. State whether the 10. __________________inverse is a function.

Chapter

3

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 65



11. ƒ(x) � ; x � 0 11. __________________

12. ƒ(x) � �xx2�

�

39

� ; x � �3 12. __________________

13. Describe the end behavior of y � x4 � x2. 13. __________________

14. Locate and classify the extrema for the graph of 14. __________________y � �x4 � 2x2.

15. The function ƒ(x) � x3 � 3x has a critical point when x � 0. 15. __________________Identify the point as a maximum, a minimum, or a point of inflection, and state its coordinates.

16. Determine the vertical and horizontal asymptotes for the 16. __________________graph of y � �

xx2 �

�

245

�.

17. Find the slant asymptote for y � �x2

x�

�x

1� 2�. 17. __________________

18. Sketch the graph of y � �x2x� 4�. 18.

19. If y varies directly as the square of x, and y � 200 19. __________________when x � 5, find y when x � 2.

20. If y varies inversely as the cube root of x, and y � 10 20. __________________when x � 27, find y when x � 8.

Bonus Determine the value of k such that Bonus: __________________ƒ(x) � 3x2 � kx � 4 is an even function.

x2 � 1 if x 0� x if x � 0



3

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 66



NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem.

1. a. Draw the parent graph and each of the three transformations that would result in the graph shown below. Write the equation for each graph and describe the transformation.

b. Draw the parent graph of a polynomial function. Then draw a transformation of the graph. Describe the type of transformation you performed. Use equations as needed to clarify your answer.

c. Sketch the graph of y � �x(x �x3�)(x

4� 4)�. Describe how you

determined the shape of the graph and its critical points.

d. Write the equation of a graph that has a vertical asymptote at x � 2 and point discontinuity at x � �1. Describe your reasoning.

2. The critical points of ƒ(x) � �15�x5 � �14�x4 � �23�x3 � 2 are at x � 0,2, and �1. Determine whether each of these critical points is the location of a maximum, a minimum, or a point of inflection.Explain why x � 3 is not a critical point of the function.

3. Mrs. Custer has 100 bushels of soybeans to sell. The current price for soybeans is $6.00 a bushel. She expects the market price of a bushel to rise in the coming weeks at a rate of $0.10 per week. For each week she waits to sell, she loses 1 bushel due to spoilage.

a. Find a function to express Mrs. Custer’s total income from selling the soybeans. Graph the function and determine when Mrs. Custer should sell the soybeans in order to maximizeher income. Justify your answer by showing that selling ashort time earlier or a short time later would result in lessincome.

b. Suppose Mrs. Custer loses 5 bushels per week due to spoilage. How would this affect her selling decision?

Chapter

3

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 67


Determine whether the graph of each equation issymmetric with respect to the origin, the x-axis, they-axis, the line y � x, the line y � �x, or none of these.

1. x � y2 � 1 1. __________________

2. xy � 8 2. __________________

3. Determine whether the function ƒ(x) � 2(x � 1)2 � 4x is 3. __________________odd, even, or neither.

4. Describe the transformations that relate the graph 4. __________________of p(x) � (0.5x)3 � 1 to its parent function p(x) � x3.

5. Use transformations of the parent graph p(x) � �x� 5.to sketch the graph of p(x) � 2�x � 3�.

6. Graph the inequality y � �x� �� 1�. 6.

7. Solve �x � 4� � 6. 7. __________________


8. ƒ(x) � x3 � 1 8. __________________

9. ƒ(x) � ��1x� 9. __________________

10. Graph ƒ(x) � (x � 3)2 and its inverse. State whether 10. __________________the inverse is a function.



3

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 68

Determine whether the graph of each equation is symmetric with respect to the origin, the x-axis, the y-axis, the line y � x, the line y � �x, or none of these.

1. y � (x � 2)3 2. y � �x14� 1. __________________

2. __________________

3. Determine whether the function ƒ(x) � x3 � 4x is 3. __________________odd, even, or neither.

4. Describe how the graphs of ƒ(x) � �x� and g(x) � �4�x� 4. __________________are related.

5. Use transformations of the parent graph of p(x) � x2 to 5.sketch the graph of p(x) � (x � 1)2 � 2.

1. Graph y �x2 � 1. 1.

2. Solve �x � 3� 4. 2. __________________

3. Find the inverse of ƒ(x) � 3x � 6. Is the inverse a function? 3. __________________

4. Graph ƒ(x) � (x � 1)2 � 2 and its inverse. 4.

5. What is the equation of the line that acts as a line of 5. __________________between the graphs symmetryof ƒ(x) and ƒ�1(x)?


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

3

Chapter

3

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 69

Determine whether the function is continuous at the given x-value. If discontinuous, state the type of discontinuity ( jump, infinite, or point).

1. ƒ(x) � ; x � 0 1. __________________

2. ƒ(x) � �x2

1� 1� ; x � 1 2. __________________

3. Describe the end behavior of y � x4 � 3x � 1. 3. __________________

4. Locate and classify the extrema for the graph of 4. __________________y � 2x3 � 6x.

5. The function ƒ(x) � �x55� � �43�x3 has a critical point at x � 2. 5. __________________

Identify the point as a maximum, a minimum, or a point of inflection, and state its coordinates.

1. Determine the equations of the vertical and horizontal 1. __________________asymptotes, if any, for the graph of y � �x2 �

x5

2

x � 6� .

2. Find the slant asymptote for ƒ(x) � �x2 �

x �4x

2� 3�. 2. __________________

3. Use the parent graph ƒ(x) � �1x� to graph the function 3. __________________g(x) � �x �

23�. Describe the transformation(s) that have

taken place. Identify the new locations of the asymptotes.

Find the constant of variation for each relation and use it to write an equation for each statement. Then solve the equation.

4. If y varies directly as x, and y � 12 when x � 8, find y when 4. __________________x � 14.

5. If y varies jointly as x and the cube of z, and y � 48 when 5. __________________x � 3 and z � 2, find y when x � �2 and z � 5.

x2 + 1 if x 0x if x � 0


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

3

Chapter

3

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 70



NAME _____________________________ DATE _______________ PERIOD ________


Multiple Choice1. If x � y � 4, what is the value of

�y � x� � �x � y�? CA 0B 4C 8D 16E It cannot be determined from the

information given.

2. If 2x � 5 � 4y and 12y � 6x � 15, thenwhich of the following is true? BA 2y � 5B 2x � 5C y � 2x 5D x � 2y 5E 2y � x 5

3. If x � �45�, then 3(x � 1) � 4x is how many fifths? AA 13B 15C 17D 23E 25

4. If x2 � 3y � 10 and �xy� � �1, then

which of the following could be a value of x? BA �5B �2C �1D 2E 3

5. 168:252 as 6: ? CA 4B 5C 9D 18E 36

6. (3.1 � 12.4) � (3.1 � 12.4) � AA �24.8B �6.2C 0D 6.2E 24.8

7. If x � 5z and y � �15z1+ 2�, then which of

the following is equivalent to y? DA �x +

52�

B �31x�

C �x �1

2�

D �3x1� 2�

E None of these

8. If �x �

xy

� � 3 and �z �

zy

� � 5, then which

of the following is the value of �xz�? A

A �12�

B �53�

C �53�

D 2E 15

9. �1 � �12 � � �14 � � �18 � � �1 16� � �3 1

2� � E

A �3�8

6��

B ��86�2��

C ��42��

D �14�

E �3�81�4��

Chapter

3

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 71


10. If 12 and 18 each divide R without aremainder, which of the following couldbe the value of R?A 24B 36C 48D 54E 120

11. If x � y � 5 and x � y � 3, then x2 � y2 �A 2B 4C 8D 15E 16

12. If n 0, then which of the followingstatements are true?I. �

3��n� 0

II. �1� �� n� 1III. �

5n�3� 0

A I onlyB III onlyC I and III onlyD II and III onlyE None of the statements are true.

13. If x* is defined to be x � 2, which of thefollowing is the value of (3* � 4*)*?A 0B 1C 2D 3E 4

14. Which of the following is the total costof 3�12� pounds of apples at $0.56 perpound and 4�12� pounds of bananas at$0.44 per pound?A $2.00B $3.00C $3.94D $4.00E $4.06

15. If a and b are real numbers where a b and a � b 0, which of the following MUST be true?A a 0B b 0C �a� �b�D ab 0E It cannot be determined from the

information given.

16. For all x, (3x3 � 13x2 � 6x � 8)(x � 4) �A 3x4 � 25x3 � 46x2 � 16x � 32B 3x4 � 25x3 � 46x2 � 32x � 32C 3x4 � x3 � 46x2 � 16x � 32D 3x4 � 25x3 � 58x2 � 32x � 32E 3x4 � 25x3 � 58x2 � 16x � 32





Column A Column B

17.

18. (2x � 3)2 4x(x � 3)

19. Grid-In For nonzero real numbers a, b, c, and d, d � 2a, a � 2b, and b � 2c. What is the value of �dc�?

20. Grid-In If � �34�, what is the

value of x?

3��4 � �x �

x1�



3

The larger valueof x in theequation

x2 � 7x � 12 � 0

The smallervalue of x in the

equationx2 � 7x � 12 � 0

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 72


Chapter 3, Cumulative Review (Chapters 1-3)

NAME _____________________________ DATE _______________ PERIOD ________

1. State the domain and range of {(�2, 2), (0, 2), (2, 2)}. Then 1. __________________state whether the relation is a function. Write yes or no.

2. If ƒ(x) � �x �3

1� and g(x) � x � 1, find ƒ( g(x)). 2. __________________

3. Write the standard form of the equation of the line that is 3. __________________parallel to the line with equation y � 2x � 3 and passes through the point at (�1, 5).

4. Graph ƒ(x) � �x � 1�. 4.

Solve each system of equations algebraically.

5. 3x � 5y � 21 5. __________________x � y � 5

6. x � 2y � z � 7 6. __________________3x � y � z � 22x � 3y � 2z � 7

7. How many solutions does a consistent and independent 7. __________________system of linear equations have?

8. Find the value of � �. 8. __________________

9. Determine whether the graph of y � x3 � 2x is symmetric 9. __________________with respect to the origin, the x-axis, the y-axis, or none of these.

10. Describe the transformations relating the graph of 10. __________________y � 2(x � 1)2 to the graph of y � x2.

11. Solve �3x � 6� 9. 11. __________________

12. Find the inverse of the function ƒ(x) � �x �3

2�. 12. __________________

13. State the type of discontinuity ( jump, infinite, or point) 13. __________________that is present in the graph of ƒ(x) � �x�.

14. Determine the horizontal asymptote for the graph of 14. __________________ƒ(x) � �xx

��

35�.

15. If y varies inversely as the square root of x, and y � 20 15. __________________when x � 9, find y when x � 16.

�7�2

35

Chapter

3

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 73


Page 55

1. C

2. B

3. D

4. C

5. B

6. D

7. A

8. D

9. A

10. D

Page 56

11. B

12. B

13. C

14. C

15. A

16. D

17. A

18. B

19. A

20. D

Bonus: D

Page 57

1. B

2. A

3. B

4. A

5. C

6. C

7. B

8. A

9. B

10. D

Page 58

11. C

12. A

13. C

14. C

15. C

16. A

17. D

18. B

19. A

20. B

Bonus: B


055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 74



Page 59

1. B

2. C

3. C

4. C

5. C

6. C

7. C

8. A

9. B

10. D

Page 60

11. A

12. D

13. A

14. C

15. C

16. C

17. B

18. D

19. A

20. B

Bonus: A

Page 61

1. y � x, y � �x

2. x-axis

3. odd

4.

5.

6.

7. x � 8

8. ƒ�1(x) � �12�x

x� ; yes

9. ƒ�1(x) � � �x� �� 4� ;no

10. yes

Page 6211. discontinuous

jump12. discontinuous

infinite13. x→ ∞, y→ �∞,

x→ �∞, y→�∞14. rel. max. at (0, 2);

abs. min. at (�1.22, �0.25)

15. point of inflectionat (1, 1)

16. horizontal at y � 0

17. y � 3x � 1

18.

19. y � 16

20. k � �21�

Bonus:

Form 1C Form 2A

reflected over the x-axis, expandedvertically by factorof 2, translated up 4 units

x � �3 or

vertical at x � 0, 3;

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 75

Page 63

1. y � x, y � �x

2. origin

3. even

4.

5.

6.

7. �3 � x � 7

8.

9.

10. no

Page 64

11. continuous

12.

13.

14.

15.

16.

17. y � 2x � 1

18.

19. y � 27

20. k � �43

��

Bonus: k � 6

Page 65

1. y-axis

2. origin

3. even

4.

5.

6.

7. �4 � x � 12

8. ƒ�1(x) � ��x�; no

9.

10. yes

Page 66

11.

12. continuous

13.

14. abs. max. at (0, 0)

15. point of inflectionat (0, 0)

16. vertical at x � �5;horizontal at y � 0

17. y � x

18.

19. y � 32

20. y � 15

Bonus: k � 0



translated right3 units,compressedvertically by afactor of 0.5

ƒ�1(x) � �3x� �� 4�;

yes

ƒ�1(x) � 1��x� ; no

discontinuous;point

x→ ∞, y→ ∞, x→ ∞, y→�∞

rel. max. at (0, 0);abs. min. at(�1.22, �2.25)

point of inflectionat (0, 2)

vertical at x � 0,1; horizontal at y � 0

translated right1 unit

ƒ�1(x) � �3

x� �� 1�;yes

discontinuous;jump

x→ ∞, y→ ∞, x→ �∞, y→�∞

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 76




3 Superior • Shows thorough understanding of the concepts parentgraph, transformation asymptote, hole in graph, critical point, maximum, minimum, and point of inflection.

• Uses appropriate strategies to solve problems and transform graphs.

• Computations are correct.• Written explanations are exemplary.• Graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.

2 Satisfactory, • Shows understanding of the concepts parent graph, with Minor transformation, asymptote, hole in graph, critical point,Flaws maximum, minimum, and point of inflection.

• Uses appropriate strategies to solve problems andtransform graphs.

• Computations are mostly correct.• Written explanations are effective.• Graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts parentSatisfactory, graph, transformation, asymptote, hole in graph, criticalwith Serious point, maximum, minimum, and point of inflection.Flaws • May not use appropriate strategies to solve problems and

transform graphs.• Computations are mostly correct.• Written explanations are satisfactory.• Graphs are mostly accurate and appropriate.• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts parentgraph, transformation, asymptote, hole in graph, critical point, maximum, minimum, and point of inflection.

• May not use appropriate strategies to solve problems and transform graphs.

• Computations are incorrect.• Written explanations are not satisfactory.• Graphs are not accurate or appropriate.• Does not satisfy requirements of problems.

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 77


Page 671a.

1. parent graph 2. transformation

3. transformation 4. transformation

The parent graph y � x2 is reflectedover the x-axis and translated 5 unitsright and 2 units down.

1b.1. parent graph 2. transformation

3. transformation

The parent graph y � x3 is translated 2 units left and 1 unit up.

1c.

Since x � 4 is a common factor of the numerator and the denominator,the graph has point discontinuity at x � 4. Because y increases ordecreases without bound close to x � �3 and x � 0, there are verticalasymptotes at x � �3 and x � 0.

1d. ( x) � �(x �

x2�)(x

1� 1)

�

The graph ƒ( x) is � �x �

12

�, exept at x � �1, where it is undefined.ƒ( x) � �

x �1

2� is undefined at x � 2.

It approaches �∞ to the left of x � 2 and �∞ to the right.

2. ��1, �16303�� is a maximum. (0, 2) is a

point of inflection. �2, � �1145�� is a

minimum. The value x � 3 is not acritical point because a line drawntangent to the graph at this point isneither horizontal nor vertical. Thegraph is increasing at x � 2.99 and at x � 3.01.

3a. ƒ( x) � total income, x � number of weeks ƒ( x) � (6.00 � 0.10x)(100 � x)

� �0.10x2 � 4x � 600

Examining the graph of the functionreveals a maximum at x � 20. ƒ( 20) � 640, ƒ( 19.9) � 639.999, andƒ( 20.1) � 639.999, so ƒ( 19.9) � ƒ( 20)and ƒ( 20) � ƒ( 20.1), which showsthat the function has a maximum of640 when x � 20.

3b. She would sell earlier because in 20 weeks she would have nosoybeans left.


055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 78


1. x-axis2. y � x, y � �x, origin

3. neither

4.

5.

6.

7. x � �2 or x � 10

8.

9. ƒ�1(x) � ��1x�; yes

10. no

Quiz APage 69

1. none of these

2. y-axis

3. odd

4.

5.

Quiz BPage 69

1.

2. {x|x � �7 or x 1}

3. ƒ�1(x) � �x �3

6� ; yes

4.

5. y � x

Quiz CPage 70

1. discontinuous; jump

2. discontinuous; infinite

3. x →∞, y →∞, x →�∞, y →∞

4. rel. max. (�1, 4); rel. min. (1, �4)

5. min. at (2, �4.27)

Quiz DPage 70

1. vertical at x � 2 and 3, horizontal at y � 1

2. y � x � 2

3.

4. 1.5; y � 1.5x; 21

5. 2; y � 2xz3; �500



translated up 1 unit,expanded horizontallyby a factor of 0.5

ƒ�1(x) � �3

x� �� 1�; yes

g(x) is the reflection of ƒ(x) overthe x-axis and is expandedvertically by a factor of 4

translated right 3 units,expanded vertically by afactor of 2; verticalasymptote at x � 3,horizontal asymptoteunchanged at y � 0

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 79


Page 71

1. C

2. B

3. A

4. B

5. C

6. A

7. D

8. A

9. E

Page 72

10. B

11. D

12. E

13. B

14. C

15. A

16. D

17. A

18. A

19. 8

20. �1

Page 73

1. D � {�2, 0, 2}; R � {2}; yes

2. �x3�

3. 2x � y � �7

4.

5. (2, 3)

6. (2, �1, 3)

7. one

8. 29

9. origin

10.

11. �1 � x � 5

12. ƒ�1(x) � �x3� � 2

13. jump

14. y � 1

15. 15


expanded vertically by afactor of 2, translated right1 unit

055-080 A&E C03-0-02-834179 10/4/00 2:32 PM Page 80



NAME _____________________________ DATE _______________ PERIOD ________


1. Use the Remainder Theorem to find the remainder when 1. ________16x5 � 32x4 � 81x � 162 is divided by x � 2. State whether the binomial is a factor of the polynomial.A. 1348; no B. 0; yes C. �700; yes D. 0; no

2. Solve 3t2 � 24t � �30 by completing the square. 2. ________A. 4 � �6� B. 10, �2 C. 10, 22 D. 4 � �1�4�

3. Use synthetic division to divide 8x4 � 20x3 � 14x2 � 8x � 1 by x � 1. 3. ________A. 8x3 � 28x2 � 14x R11 B. 8x3 � 28x2 � 14x � 6 R7C. 8x3 � 36x2 � 18x � 10 R9 D. 8x3 � 28x2 � 14x � 8

4. Solve �3

x� �� 2� � �6

9�x� �� 1�0�. 4. ________

A. �1, 6 B. ��13 �2

�1�9�3�� C. 1, �6 D. �13 �2�1�4�5��

5. List the possible rational roots of 2x3 � 17x2 � 23x � 42 � 0. 5. ________

A. �1, �2, �3, �6, �7, �14, �21, �42, ��12�, ��32�, ��72�, ��221�

B. �1, �2, �6, �7, �21, �42, ��12�, ��72�, ��221�

C. �1, �2, �3, �6, �7, �14, �21, �42, ��23�, ��72�, ��221�

D. �1, �2, �3, �6, �7, �14, �21, �42, ��12�, ��27�, ��221�

6. Determine the rational roots of 6x3 � 25x2 � 2x � 8 � 0. 6. ________A. �12�, �23�, �4 B. ��12�, �23�, 4 C. ��23�, �12�, 4 D. ��23�, ��12�, 4

7. Solve �2xx� 5� � �4

xx��

21� � � �x

32x��

28x�. 7. ________

A. ��1 �12

�4�3�3�� B. 2, ��32� C. ��

29� D. ��

23�, �2

1�

8. Find the discriminant of 2x2 � 9 � 4x and describe the nature of the 8. ________roots of the equation.A. 56; exactly one real root B. 56; two distinct real rootsC. �56; no real roots D. �56; two distinct real roots

9. Solve � 3x2 � 4 � 0 by using the Quadratic Formula. 9. ________

A. �2 �

32i

� B. ��2�

33�

� C. ��63�� D. 0, �3

4�

10. Determine between which consecutive integers one or more real 10. ________zeros of ƒ(x) � 3x4 � x3 � 2x2 � 4 are located.A. no real zeros B. 0 and 1C. �2 and �3 D. �1 and 0

Chapter

4

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 81


11. Find the value of k so that the remainder of (�kx4 � 146x2 � 32) � 11. ________(x � 4) is 0.A. �34

7� B. 9 C. ��347� D. �9

12. Find the number of possible negative real zeros for 12. ________ƒ(x) � 6 � x4 � 2x2 � 5x3 � 12x.A. 2 or 0 B. 3 or 1 C. 0 D. 1

13. Solve �7�x� �� 2� � 4. 13. ________A. x 1 B. x 2 C. x � 2 D. ��27� � x � 2

14. Decompose �x2

5�

xx�

�

112

� into partial fractions. 14. ________

A. �x �2

3� � �x �3

4� B. �x �3

3� � �x �2

4�

C. �x �3

3� � �x �2

4� D. �x �2

3� � �x �3

4�

15. Solve �52x� � �3x

2�x

4� �x 3�x

2�. 15. ________

A. 0 � x �45� B. 0 x �45� C. x � 0, x � �54� D. x 0, x � �5

4�

16. Approximate the real zeros of ƒ(x) � 2x4 � 3x2 � 2 to the nearest tenth. 16. ________A. �2 B. �1.4 C. �1.5 D. no real zeros

17. Solve �x� �� 4� � �x� � �2�. 17. ________A. �2 B. ��12� C. 2 D. �2

1�

18. Which polynomial function best models the set of data below? 18. ________

A. y � 0.02x4 � 0.25x2 � 0.11x � 0.84B. y � 0.2x4 � 0.25x2 � 0.11x � 0.84C. y � 0.2x4 � 0.25x2 � 0.11x � 0.84D. y � 0.02x4 � 0.25x2 � 0.11x � 0.84

19. Solve �x �6

4� � 2 �13�. 19. ________

A. x 4 B. �25� � x � 4 C. x � �25�, x 4 D. x �52�

20. Find the polynomial equation of least degree with roots �4, 2i, 20. ________and � 2i.A. x3 � 4x2 � 4x � 16 � 0 B. x3 � 4x2 � 4x � 16 � 0C. x3 � 4x2 � 4x � 16 � 0 D. x3 � 4x2 � 4x � 16 � 0

Bonus Find the discriminant of (2 ��3�)x2 � (4 � �3�)x � 1. Bonus: ________A. 11 � 12�3� B. �16 � 9�3� C. �2 � 2�3� D. 27 � 4�3�



4

x �5 �4 �3 �2 �1 0 1 2 3 4 5

f(x) 4 0 0 0 0 1 1 0 0 1 4

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 82



NAME _____________________________ DATE _______________ PERIOD ________


1. Use the Remainder Theorem to find the remainder when 1. ________2x3 � 6x2 � 3x � 1 is divided by x � 1. State whether the binomial is a factor of the polynomial.A. 0; yes B. �2; no C. 10; no D. �1; yes

2. Solve x2 � 20x � 8 by completing the square. 2. ________A. 5 � �2�3� B. 5 � 3�3� C. 10 � 2�2�3� D. 10 � 6�3�

3. Use synthetic division to divide x3 � 5x2 � 5x � 2 by x � 2. 3. ________A. x2 � 7x � 19 R36 B. x2 � 3x � 1C. x2 � 4 D. x2 � 7x � 9 R16

4. Solve �3

2�x� �� 1� � 4 � �1. 4. ________A. 14 B. 13 C. �14 D. �13

5. List the possible rational roots of 4x3 � 5x2 � x � 2 � 0. 5. ________A. �1, ��12�, ��14�, �2 B. �1, ��12�, �2, �4

C. �1, ��12�, ��14� D. �1, ��14�, �2

6. Determine the rational roots of x3 � 4x2 � 6x � 9 � 0. 6. ________A. �3 B. �3 C. 3 D. �3, 9

7. Solve �xx��

21� � �2x

x��

67� � �

x2 �

x �

5x3� 6

�. 7. ________

A. �2 � 3�3� B. �23�, 4 C. �2 � 6�3� D. ��23�, �4

8. Find the discriminant of 5x2 � 8x � 3 � 0 and describe the 8. ________nature of the roots of the equation.A. 4; two distinct real roots B. 0; exactly one real rootC. �76; no real roots D. 124; two distinct real roots

9. Solve �3x2 � 4x � 4 � 0 by using the Quadratic Formula. 9. ________

A. ��2 �32i�2�� B. �23� � 4i�2� C. �2 � 2

3i�2�� D. 2, �6

10. Determine between which consecutive integers one or more 10. ________real zeros of ƒ(x) � �x3 � 2x2 � x � 5 are located.A. 0 and 1 B. 1 and 2 C. �2 and �1 D. at �5

11. Find the value of k so that the remainder of 11. ________(x3 � 3x2 � kx � 24) � (x � 4) is 0.A. �22 B. �10 C. 22 D. 10

Chapter

4

C

D

B

B

A

A

B

D

C

C

D

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 83


12. Find the number of possible negative real zeros for 12. ________ƒ(x) � x3 � 4x2 � 3x � 9.A. 2 or 0 B. 3 or 1 C. 1 D. 0

13. Solve �x� �� 2� � 2 ≥ 7. 13. ________A. �2 x ≤ 79 B. x �2, x � 79C. x � �2 D. x � 79

14. Decompose �2x

�22�

x9�

x2�

35


A. �2x4� 1� � �x �

35� B. �2x

4� 1� � �x �

35�

C. �x �3

5� � �2x4� 1� D. �2x

��3

1� � �x �4

5�

15. Solve �x2 �

33x

� � �xx

�

�

23

� � �1x

�. 15. ________

A. �3 � x � �1 B. �1 � x � 0 C. �3 � x � 0 D. x �3

16. Approximate the real zeros of ƒ(x) � x3 � 5x2 � 2 to the nearest tenth. 16. ________A. �0.5, 0.7, 4.9 B. �0.6, 0.7, 4.9C. � 0.6 D. �0.6, 0.7, 5.0

17. Solve 5 � �x� �� 2� � 8 � �x� �� 7�. 17. ________A. 7 B. 0 C. �7 D. 21


A. y � 0.9x3 � 4.9x2 � 0.5x � 14.4B. y � 0.9x3 � 4.9x2 � 0.5x � 14.4C. y � 1.0x3 � 4.9x2 � 0.9x � 17.5D. y � 0.9x3 � 4.8x2 � 1.2x � 15.0

19. Solve �2x� �x��

11�. 19. ________

A. x 0 B. x � �23�, x 1 C. 0 � x � 1 D. 0 � x � �23�, x 1

20. Find the polynomial equation of least degree with roots �1, 3, and �3i. 20. ________A. x4 � 2x3 � 6x � 9 � 0B. x4 � 2x3 � 6x2 � 18x � 27 � 0C. x4 � 2x3 � 6x2 � 18x � 27 � 0D. x4 � 2x3 � 12x2 � 18x � 27 � 0

Bonus Solve x3 � �1. Bonus: ________

A. �1, �1 �2i�3�� B. 1, �1 C. �1 D. �1, �i



4

x �3 �2 �1 0 1 2 3 4 5 6 7

f(x) �60 �10 10 15 10 0 �5 0 15 50 100

A

D

A

A

B

A

B

D

C

A

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 84



NAME _____________________________ DATE _______________ PERIOD ________


1. Use the Remainder Theorem to find the remainder when 1. ________2x3 � x2 � 3x � 7 is divided by x � 2. State whether the binomial is a factor of the polynomial.A. 25; yes B. �11; no C. 33; no D. �11; yes

2. Solve x2 � 10x � 1575 by completing the square. 2. ________A. 45, �35 B. 5 � 15�7� C. �5 � 15�7� D. 35, �45

3. Use synthetic division to divide x3 � 2x2 � 5x � 1 by x � 1. 3. ________A. x2 � 3x � 8 R�7 B. x2 � x � 4 R�3C. x2 � 3x � 8 R9 D. x2 � x � 4 R5

4. Solve �3

x� �� 1� � 3. 4. ________A. �26 B. 26 C. 64 D. 28

5. List the possible rational roots of 2x4 � x2 � 3 � 0. 5. ________A. �1, �2, �3, ��12�, ��13�, ��23�, ��2

3�

B. �1, �2, �3, �4, �6, �12, ��12�, ��23�

C. �1, �3, ��12�, ��23�

D. �1, �2, �3, ��23�, ��23�

6. Determine the rational roots of 3x3 � 7x2 � x � 2 � 0. 6. ________A. 2, �13� B. �2, ��13� C. 2 D. �2

7. Solve �x �

14

� � �x2 � 3

1x � 4� � �

x �

41

�. 7. ________

A. �2 B. �6 C. 6 D. 2

8. Find the discriminant of 16x2 � 9x � 13 � 0 and describe the nature 8. ________of the roots of the equation.A. 29; two distinct real roots B. 0; exactly one real rootC. �751; no real roots D. 751; two distinct real roots

9. Solve 2x2 � 4x � 7 � 0 by using the Quadratic Formula. 9. ________

A. 1 � i�1�0� B. �1 � �i�

21�0�� C. 1 � �

i�21�0�� D. 1 � 5i

10. Determine between which consecutive integers one or more real 10. ________zeros of ƒ(x) � x3 � x2 � 5 are located.A. 2 and 3 B. 1 and 2 C. �2 and �1 D. �1 and 0

11. Find the value of k so that the remainder of (x3 � 5x2 � 4x � k) � 11. ________(x � 5) is 0.A. �230 B. �20 C. 54 D. 20

Chapter

4

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 85


12. Find the number of possible negative real zeros for 12. ________ƒ(x) � x3 � 2x2 � x � 1.A. 3 B. 2 or 0 C. 3 or 1 D. 1

13. Solve 2 � �x� �� 2� � 11. 13. ________A. x 79 B. x � 79 C. x � 2 D. 2 x 79

14. Decompose �x2

8�

x �

3x2�

24


A. �x �6

1� � �x �2

4� B. �x �2

4� � �x �6

1�

C. �x �2

1� � �x �6

4� D. �x �2

4� � �x �6

1�

15. Solve �x2

1�

43x

� � �8x

� �x�

�10

3�. 15. ________

A. �19 � x � 0 B. x � 0, x 3C. �19 � x � 3 D. �19 � x � 0, x 3

16. Approximate the real zeros of ƒ(x) � 2x3 � 3x2 � 1 to the nearest tenth. 16. ________A. �1.0 B. �1.0, 0.5 C. �1.0, 0.0 D. 0.5

17. Solve �6�x� �� 2� � �4�x� �� 4�. 17. ________A. �12� B. �1 C. 3 D. �3


A. y � 0.2x3 � 0.4x2 � 2.2x � 2.0B. y � 2x3 � 40x2 � 217x � 199C. y � 0.02x3 � 0.40x2 � 2.17x � 1.99D. y � 0.02x3 � 0.40x2 � 2.17x � 1.99

19. Solve 1 � �x �5

1� � �76�. 19. ________

A. 1 � x 31 B. x 31 C. x 1, x � 31 D. x � 1

20. Find a polynomial equation of least degree with 20. ________roots �3, 0, and 3.A. x3 � x2 � 3x � 9 � 0 B. x3 � x2 � 3x � 9 � 0C. x3 � 9x � 0 D. x3 � 9x � 0

Bonus Solve 16x4 � 16x3 � 32x2 � 36x � 9 � 0. Bonus: ________A. ��12�, ��32� B. �12�, ��32� C. ��12�, �32� D. ��32�, 0



4

x 0 2 4 6 8 10 12 14 16 18 20

f(x) 2 5 5 4 2 0 �2 �2 0 5 14

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 86



NAME _____________________________ DATE _______________ PERIOD ________

Solve each equation or inequality.

1. (3x � 2)2 � 121 1. __________________

2. �32�t2 � 6t � ��125� 2. ________________________________

3. 4 � �a �4

2� � �45� 3. _______________________________________________

4. �2�x� �� 5� � 2�2�x� � 1 4. __________________

5. �1�2�b� �� 3� �5�b� �� 2� 5. ______________________

6. �x �

22

� � �2 �

xx

� � �4 �

13x2

� 6. _____________

7. �d� �� 6� � 3 � �d� 7. ______________________________

8. Use the Remainder Theorem to find the remainder when 8. __________________x5 � x3 � x is divided by x � 3. State whether the binomial is a factor of the polynomial.

9. Determine between which consecutive integers the 9. __________________real zeros of ƒ(x) � 4x4 � 4x3 � 25x2 � x � 6 are located.

10. Decompose �2x�

23�x

5�x

1�9

3� into partial fractions. 10. __________________

11. Find the value of k so that the remainder of 11. __________________(x4 � 3x3 � kx2 � 10x � 12) � (x � 3) is 0.

12. Approximate the real zeros of ƒ(x) � 2x4 � 3x2 � 20 to 12. __________________the nearest tenth.

Chapter

4

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 87


13. Use the Upper Bound Theorem to find an integral upper 13. __________________bound and the Lower Bound Theorem to find an integral lower bound of the zeros of ƒ(x) � 2x3 � 4x2 � 2.

14. Write a polynomial function with integral coefficients to 14. __________________model the set of data below.

15. Find the discriminant of 5x � 3x2 � �2 and describe the 15. __________________nature of the roots of the equation.

16. Find the number of possible positive real zeros and the 16. __________________number of possible negative real zeros for ƒ(x) � 2x4 � 7x3 � 5x2 � 28x � 12.

17. List the possible rational roots of 2x3 � 3x2 � 17x � 12 � 0. 17. __________________

18. Determine the rational roots of x3 � 6x2 � 12x � 8 � 0. 18. __________________

19. Write a polynomial equation of least degree with 19. __________________roots �2, 2, �3i, and 3i. How many times does the graph of the related function intersect the x-axis?

20. Francesca jumps upward on a trampoline with an initial 20. __________________velocity of 17 feet per second. The distance d(t) traveled by a free-falling object can be modeled by the formula d(t) � v0t � �12�gt2, where v0 is the initial velocity and grepresents the acceleration due to gravity (32 feet per second squared). Find the maximum height that Francesca will travel above the trampoline on this jump.

Bonus Find ƒ if ƒ is a cubic polynomial function such Bonus: __________________that ƒ(0) � 0 and ƒ(x) is positive only when x 4.

Chapter 4Test, Form 2A (continued)


4

x 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

f(x) 7.3 11.2 12.1 11.2 8.0 6.2 3.5 2.5 2.2 5.7 12.0

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 88



NAME _____________________________ DATE _______________ PERIOD ________

Solve each equation or inequality.1. 8x2 � 5x � 13 � 0 1. __________________

2. x2 � 4x � �13 2. __________________

3. �6q� � 4 � �3q� 3. __________________

4. �3

1�0�x� �� 2� � 3 � �5 4. __________________

5. �2�n� �� 5� � 8 � 11 5. __________________

6. �aa��

43� � �3a

a��

32� � �

a4� 6. __________________

7. �3�x� �� 4� � 7 � 5 7. __________________

8. Use the Remainder Theorem to find the remainder when 8. __________________x3 � 5x2 � 5x � 2 is divided by x � 2. State whether the binomial is a factor of the polynomial.

9. Determine between which consecutive integers the real 9. __________________zeros of ƒ(x) � x3 � x2 � 4x � 2 are located.

10. Decompose �x2

8�

x �

3x1�

74

� into partial fractions. 10. __________________

11. Find the value of k so that the remainder of 11. __________________(x3 � 2x2 � kx � 6) � (x � 2) is 0.

Chapter

4��1

83�, 1

2 � 3i

�1

n � 7

�8 � a � �3, a � �3

no solution

2 and 3; at �1

0; yes

q � ��34

�, q � 0

�x �

34

� � �x �

51

�

�5

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 89


12. Approximate the real zeros of ƒ(x) � 2x4 � x2 � 3 to 12. __________________the nearest tenth.

13. Use the Upper Bound Theorem to find an integral upper 13. __________________bound and the Lower Bound Theorem to find an integral lower bound of the zeros of ƒ(x) � x3 � 3x2 � 2.

14. Write a polynomial function to model the set of data below. 14. __________________

15. Find the discriminant of 4x2 � 12x � �9 and describe 15. __________________the nature of the roots of the equation.

16. Find the number of possible positive real zeros and 16. __________________the number of possible negative real zeros for ƒ(x) � x3 � 4x2 � 3x � 9.

17. List the possible rational roots of 4x3 � 5x2 � x � 2 � 0. 17. __________________

18. Determine the rational roots of x3 � 4x2 � 6x � 9 � 0. 18. __________________

19. Write a polynomial equation of least degree with roots �2, 19. __________________2, �1, and �12�. How many times does the graph of the relatedfunction intersect the x-axis?

20. Belinda is jumping on a trampoline. After 4 jumps, she 20. __________________jumps up with an initial velocity of 17 feet per second.The function d(t) � 17t � 16t2 gives the height in feet of Belinda above the trampoline as a function of time in seconds after the fifth jump. How long after her fifth jump will it take for her to return to the trampoline again?

Bonus Factor x4 � 2x3 � 2x � 1. Bonus: __________________



4

x 4 5 6 7 8 9 10 11 12 13 14

f(x) 7 9 9 8 6 3 1 1 2 8 17

�1.2

Sample answer:upper 3, lower �1

Sample answer: y 0.1x3 � 3.3x2 �23.9x � 45.2

0; 1 real root

1; 2 or 0

�1, �2, ��14

�, ��21�

�3

2x4 � x3 � 9x2 �4x � 4 0; 4

about 1.06 s

(x � 1)(x � 1)3

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 90



NAME _____________________________ DATE _______________ PERIOD ________


1. 4x2 � 4x � 17 � 0 1. __________________

2. x2 � 6x � �72� 2. _______________________________________________

3. �45� 2 � �3x� 3. _________________________________________________

4. �3

y� �� 4� � 3 4. __________________

5. �2�x� �� 5� � 4 9 5. __________________

6. �32y� � 6 �

5y� 6. _________________________________________________

7. �2�x� �� 5� � 7 � �4 7. __________________

8. Use the Remainder Theorem to find the remainder when 8. __________________x3 � 5x2 � 6x � 3 is divided by x � 1. State whether the binomial is a factor of the polynomial.

9. Determine between which consecutive integers the 9. __________________real zeros of ƒ(x) � x3 � x2 � 5 are located.

10. Decompose �1x02

x�

�

44� into partial fractions. 10. __________________

11. Find the value of k so that the remainder of 11. __________________(2x2 � kx � 3) � (x � 1) is zero.

12. Approximate the real zeros of ƒ(x) � x3 � 2x2 � 4x � 5 to 12. __________________the nearest tenth.

Chapter

4

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 91


13. Use the Upper Bound Theorem to find an integral upper 13. __________________bound and the Lower Bound Theorem to find an integral lower bound of the zeros of ƒ(x) � � 2x3 � 4x2 � 1.

14. Write a polynomial function to model the set of data below. 14. __________________

15. Find the discriminant of 2x2 � x � 7 � 0 and describe the 15. __________________nature of the roots of the equation.

16. Find the number of possible positive real zeros and the 16. __________________number of possible negative real zeros for ƒ(x) � x3 � 2x2 � x � 2.


18. Determine the rational roots of 2x3 � 3x2 � 17x � 12 � 0. 18. __________________

19. Write a polynomial equation of least degree with 19. __________________roots �3, �1, and 5. How many times does the graph of the related function intersect the x-axis?

20. What type of polynomial function could be the best model 20. __________________for the set of data below?

Bonus Determine the value of k such that Bonus: __________________ƒ(x) � kx3 � x2 � 7x � 9 has possible rational roots of �1, �3, �9, ��16�, ��13�, ��12�, ��32�, ��92�.



4

x �1 �0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

f(x) 0.5 �3.1 �4.2 �4.4 �2.1 0.8 3.4 4.3 3.6 0.9 �5.4

x �3 �2 �1 0 1 2 3

f(x) 196 25 �2 1 �8 1 130

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 92


Chapter 4, Open-Ended Assessment

NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear,concise solution to each problem. Be sure to include all relevantdrawings and justify your answers. You may show your solution inmore than one way or investigate beyond the requirements of theproblem.

1. Use what you have learned about the discriminant to answer the following.a. Write a polynomial equation with two imaginary roots. Explain

your answer.

b. Write a polynomial equation with two real roots. Explain your answer.

c. Write a polynomial equation with one real root. Explain your answer.

2. Given the function ƒ(x) � 6x5 � 2x4 � 5x3 � 4x2 � x � 4, answer the following.a. How many positive real zeros are possible? Explain.

b. How many negative real zeros are possible? Explain.

c. What are the possible rational zeros? Explain.

d. Is it possible that there are no real zeros? Explain.

e. Are there any real zeros greater than 2? Explain.

f. What term could you add to the above polynomial to increase the number of possible positive real zeros by one? Does the term you added increase the number of possible negative real zeros? How do you know?

g. Write a polynomial equation. Then describe its roots.

3. A 36-foot-tall light pole has a 39-foot-long wire attached to its top.A stake will be driven into the ground to secure the other end of the wire. The distance from the pole to where the stake should be driven is given by the equation 39 � �d�2�� 3�6�2�, where d represents the distance.a. Find d.

b. What relationship was used to write the given equation? What do the values 39, 36, and d represent?

Chapter

4

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 93


1. Determine whether �1 is a root of x4 � 2x3 � 5x � 2 � 0. 1. __________________Explain.

2. Write a polynomial equation of least degree with roots 2. __________________�4, 1, i, and �i. How many times does the graph of the related function intersect the x-axis?

3. Find the complex roots for the equation x2 � 20 � 0. 3. __________________

4. Solve the equation x2 � 6x � 10 � 0 by completing 4. __________________the square.

5. Find the discriminant of 6 � 5x � 6x2. Then solve the 5. __________________equation by using the Quadratic Formula.

6. Use the Remainder Theorem to find the remainder when 6. __________________x3 � 3x2 � 4 is divided by x � 2. State whether the binomial is a factor of the polynomial.


List the possible rational roots of each equation. Then determine the rational roots.

8. x4 � 10x2 � 9 � 0 8. __________________

9. 2x3 � 7x � 2 � 0 9. __________________

10. Find the number of possible positive real zeros and the 10. __________________number of possible negative real zeros for the function ƒ(x) � x3 � x2 � x � 1. Then determine the rational zeros.



4

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 94

1. Determine whether �3 is a root of x3 � 3x2 � x � 1 � 0. 1. __________________Explain.

2. Write a polynomial equation of least degree with 2. __________________roots 3, �1, 2i, and �2i. How many times does the graph of the related function intersect the x-axis?

3. Find the complex roots of the equation �4x4 � 3x2 � 1 � 0. 3. __________________

4. Solve x2 � 10x � 35 � 0 by completing the square. 4. __________________

5. Find the discriminant of 15x2 � 4x � 1 and describe 5. __________________the nature of the roots of the equation. Then solve the equation by using the Quadratic Formula.

Use the Remainder Theorem to find the remainder for each division. State whether the binomial is a factor of the polynomial.

1. (x3 � 6x � 9) � (x � 3) 1. __________________

2. (x4 � 6x2 � 8) � (x � �2�) 2. __________________


4. List the possible rational roots of 3x3 � 4x2 � 5x � 2 � 0. 4. __________________Then determine the rational roots.

5. Find the number of possible positive real zeros and the 5. __________________number of possible negative real zeros for ƒ(x) � 2x3 � 9x2 �3x � 4. Then determine the rational zeros.


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

4

Chapter

4

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 95

1. Determine between which consecutive integers the real 1. __________________zeros of ƒ(x) � x3 � 4x2 � 3x � 5 are located.

2. Approximate the real zeros of ƒ(x) � x4 � 3x3 � 2x � 1 to 2. __________________the nearest tenth.

3. Solve �a �

a2� � �a �

62� � 2. 3. __________________

4. Decompose �4p

p

2

3

�

�

1p32

p�

�

2p12

� into partial fractions. 4. __________________

5. Solve �w2� � �w �

61� �5. 5. __________________


1. �x� �� 3� � 2 1. __________________

2. �3

2�m� �� 1� � �3 2. __________________

3. �3�t�� 7� 7 3. __________________

4. Determine the type of polynomial 4. __________________function that would best fit the scatter plot shown.

5. Write a polynomial function with integral coefficients to 5. __________________model the set of data below.


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

4

Chapter

4

x �1 �0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

f(x) 47.6 0.1 �9.3 �0.3 11.6 18.1 14.6 6.4 0.4 12.2 63.8

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 96



NAME _____________________________ DATE _______________ PERIOD ________


Multiple Choice1. The vertices of a parallelogram are

P(0, 2), Q(3, 0), R(7, 4), and S(4, 6).Find the length of the longer sides. AA 4�2�B �1�3�C �3�7�D �5�3�E None of these

2. A right triangle has vertices A(�5, �5),B(5 � x, �9), and C(�1, �9). Find thevalue of x. DA �10 B �9C �4 D 10E 15

3. �23� � ��34�� 56�� 78�� C

A �214� B ��7

1�

C ��18� D �3

E �81�

4. ��

�5� �

5�5

� � E

A 5 � 5�5�B ��14�(1 � 5�5�)

C ��14� � 5�5�

D �5 �55�5��

E None of these

5. Find the slope of a line perpendicularto 3x � 2y � �7. AA �32� B ��2

3�

C �23� D ��32�

E ��72�

6. If the midpoint of the segment joining points A��12�, 1�15�� and B�x � �23�, �45�� has

coordinates ��152�, 1�, find the value of x.

A �31� B

B ��31�

C 1D �1E None of these

7. If �6x� � 9, then �8x� � AA 12 B 11C �23� D �3

4�

E �136�

8. If a pounds of potatoes serves b adults,how many adults can be served with cpounds of potatoes? BA �ab

c�

B �bac�

C �abc�

D �acb�

E It cannot be determined from theinformation given.

9. Which are the coordinates of P, Q, R,and S if lines PQ and RS are neitherparallel nor perpendicular? DA P(4, 3), Q(2, 1), R(0, 5), S(�2, 3)B P(6, 0), Q(�1, 0), R(2, 8), S(2, 5)C P(5, 4), Q(7, 2), R(1, 3), S(�1, 1)D P(8, 13), Q(3, 10), R(11, 5), S(6, 3)E P(26, 18), Q(10, 6), R(�13, 25),

S(�17, 22)

10. What is the length of the line segmentwhose endpoints are represented bythe points C(�6, �9) and D(8, �3)? AA 2�5�8� B 4�1�0�C 2�3�7� D 4�5�8�E 2�1�0�

Chapter

4

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 97


11. If x# means 4(x � 2)2, find the value of(3#)#.A 8 B 10C 12 D 16E 36

12. Each number below is the product oftwo consecutive positive integers. Forwhich of these is the greater of thetwo consecutive integers an eveninteger?A 6B 20C 42D 56E 72

13. For which equation is the sum of theroots the greatest?A (x � 6)2 � 4B (x � 2)2 � 9C (x � 5)2 � 16D (x � 8)2 � 25E x2 � 36

14. If �a1� � �a

1� � 12, then 3a �

A �16� B �14�

C �13� D �15�

E �21�

15. For which value of k are the points A(0, �5), B(6, k), and C(�4, �13)collinear?A 7B �3C 1D �32�

E �23�

16. Find the midpoint of the segmentwith endpoints at (a � b, c) and (2a, �3c).

A ��3a2� b�, �c�

B ��3a2� b�, 2c�

C ��32a�, �c�

D (4c, b � a)E (a � b, �2c)





Column A Column B

17. 0 � y � x � 1

18. x � �3

19. Grid-In If the slope of line AB is �23�

and lines AB and CD are parallel,what is the value of x if thecoordinates of C and D are (0, �3) and (x, 1), respectively?

20. Grid-In A parallelogram has verticesat A(1, 3), B(3, 5), C(4, 2), and D(2, 0).What is the x-coordinate of the pointat which the diagonals bisect eachother?



4

x � y �1x� � �y1�

�x2 �

x �

6x3� 9� �x

x2

��

39�

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 98

C(1, �3)



NAME _____________________________ DATE _______________ PERIOD ________

1. State the domain and range of the relation {(�2, 5), (3, �2), 1. __________________(0, 5)}. Then state whether the relation is a function. Write yes or no.

2. Find [ƒ � g](x) if ƒ(x) � x � 5 and g(x) � 3x2. 2. __________________

3. Graph y 3|x| � 2. 3.

4. Solve the system of equations. 2x � y � z � 0 4. __________________x � y � z � 6x � 2y � z � 3

5. The coordinates of the vertices of �ABC are A(1, �1), 5. __________________B(2, 2), and C(3, 1). Find the coordinates of the vertices of the image of �ABC after a 270� counterclockwise rotation about the origin.

6. Gabriel works no more than 15 hours per week during the 6. __________________school year. He is paid $12 per hour for tutoring math and $9 per hour for working at the grocery store. He does not want to tutor for more than 8 hours per week. What are Gabriel’s maximum earnings?

7. Determine whether the graph of y � �x42� is symmetric to 7. __________________

the x-axis, the y-axis, the line y � x, the line y � �x, or none of these.

8. Graph y � 4 � �3

x�� 2� using the graph of the function y � x3. 8.

9. Describe the end behavior of y � �4x7 � 3x3 � 5. 9. __________________

10. Determine the slant asymptote for ƒ(x) � �x2 �

x �3x

1� 2�. 10. __________________

11. Solve �x2 ��43xx� 32� � �x �

28� � �x �

34�. 11. __________________

12. Solve 43x � x3 � x4 � 10 � 21x2. 12. __________________

Chapter

4D {�2, 0, 3},R {�2, 5}; yes

3x2 � 5

(2, �1, 3)

A(�1, �1), B(2, �2),

x →∞, y→�∞,x→�∞, y →∞

y x � 4

y-axis

$159

2

�5, 2, 2 � �3�

081-099 A&E C04-0-02-83417 10/4/00 2:34 PM Page 99

Page 81

1. B

2. A

3. B

4. A

5. A

6. B

7. D

8. C

9. B

10. A

Page 82

11. D

12. C

13. D

14. A

15. C

16. B

17. D

18. A

19. C

20. C

Bonus: D

Page 83

1. C

2. D

3. B

4. B

5. A

6. A

7. B

8. D

9. C

10. B

11. D

Page 8412. A

13. D

14. A

15. A

16. B

17. A

18. B

19. D

20. C

Bonus: A



100-106 A&E C04-0-02-83417 10/4/00 2:36 PM Page 100

Sample answer: y � x3 � 19x2 �116x � 213



Page 85

1. B

2. A

3. D

4. D

5. C

6. D

7. B

8. C

9. C

10. B

11. B

Page 86

12. C

13. B

14. A

15. D

16. B

17. C

18. D

19. A

20. D

Bonus: B

Page 87

1. �3, �133�

2. 2 � i

3.

4. �29

�

5. �14

� � b � �57

�

6.

7. no solution

8. 273; no

9.

10. ��2x

5� 1� � �

x �4

3�

11. 2

12. �1.6

Page 88

13. lower �1, upper 2

14.

15. 49; 2 real

16. 3 or 1; 1

17.

18. 2

19. x4 � 5x2 � 36 � 0; 2

20. about 4.5 ft

Bonus: ƒ(x) � x3 � 4x2

Form 1C Form 2A

a � ��143�,

a � �2

x � �3; �2 � x � 2;x � 3

0 and 1,�1 and 0; at �2, at 3

�1, �2, �3, �4,

�6, �12, � �12

�, � �23�

100-106 A&E C04-0-02-83417 10/4/00 2:36 PM Page 101



Page 89

1. ��183�, 1

2. 2 � 3i

3. q � ��34

�, q � 0

4. �1

5. n 7

6. a � �3

7. no solution

8. 0; yes

9. 2 and 3; at �1

10. �x �

34

� � �x �

51

�

11. �5

Page 90

12. �1.2

13. upper 3, lower �1

14.

15. 0; 1 real root

16. 1; 2 or 0

17. �1, �2, ��14

�, ��21�

18. �3

19.

20. about 1.06 s

Bonus:

Page 91

1. �1 �23�2��

2. 3 � �5�2

2��

3. 0 � x � �25�

4. 23

5. ��52

� � x � 10

6. y � ��1183�, y � 0

7. 2

8. �1; no

9. 1 and 2

10. �x �

62

� � �x �

42

�

11. 1

12. 3.5

Page 92

13. upper 3; lower �1

14.

15.�55; 2 complex roots

16. 2 or 0; 1

17.

18. �4, 1, �23�

19.

20. quartic

Bonus: 6

Sample answer:y � 0.1x3 � 3.3x2

� 23.9x � 45.2

�8 � a � �3,

Sample answer:

2x4 � x3 �9x2 � 4x �4 � 0; 4

ƒ(x) �(x � 1)(x � 1)3

Sample answer:y � �1.0x3 �4.1x2 � 0.3x � 4.6

�1, �2, �3, �4,

�6, �12, � �12

�, � �23�

x3 � x2 � 17x �15 � 0; 3

100-106 A&E C04-0-02-83417 10/4/00 2:36 PM Page 102




3 Superior • Shows thorough understanding of the concepts positive, negative, and real roots; rational equations; and radical equations.

• Uses appropriate strategies to solve problems and finds the number of roots.

• Computations are correct.• Written explanations are exemplary.• Goes beyond requirements of some or all problems.

2 Satisfactory, • Shows understanding of the concepts positive, negative, with Minor and real roots; rational equations; and radical Flaws equations.

• Uses appropriate strategies to solve problems and finds the number of roots.

• Computations are mostly correct.• Written explanations are effective.• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts positive, Satisfactory, negative, and real roots; rational equations; and radical with Serious equations.Flaws • May not use appropriate strategies to solve problems

and find the number of roots.• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts positive,negative, and real roots; rational equations; and radical equations.

• May not use appropriate strategies to solve problems and find the number of roots.

• Computations are incorrect.• Written explanations are not satisfactory.• Does not satisfy requirements of problems.

100-106 A&E C04-0-02-83417 10/4/00 2:36 PM Page 103

Page 93

1a. Sample answer: x2 � 2x � 2 � 0.The discriminant of the equation is�4, which is less than 0.Therefore, the equation has noreal roots.

1b. Sample answer: x2 � 2x � 2 � 0.The discriminant of the equation is12, which is greater than 0.Therefore, the equation has tworeal roots.

1c. Sample answer: x2 � 2x � 1 � 0.The discriminant of the equation is0. Therefore, the equation hasexactly one real root.

2a. The number of possible positivereal zeros is 3 or 1 because thereare three sign changes in ƒ( x).

2b. The number of possible negativereal zeros is 2 or 0 because thereare two sign changes in ƒ(�x).

2c. The possible rational zeros are ��

61�, ��1

3�, ��1

2�, ��2

3�, ��4

3�, �1,

�2, and �4 .

2d. No, it has five zeros, and complexroots are in pairs.

2e. No, because there are no signchanges in the quotient andremainder when the polynomial isdivided by x � 2.

2f. Sample answer: add �x6. Addingthe term does not change thenumber of possible negative realzeros because the number of signchanges in ƒ( �x) does notincrease.

2g. y � 2x2 � 3x � 2; The equation hasno positive real roots. There maybe two negative real roots.Possible rational roots are ��

21�, �1, and �2,

because the equation has nopositive real roots.

3a. 15 ft3b. the Pythagorean Theorem; the

hypotenuse and the legs of a righttriangle



100-106 A&E C04-0-02-83417 10/4/00 2:36 PM Page 104



1. Yes, becauseƒ(�1) � 0

2.

3. �2�5�i

4. �3 � i

5. 169; �23

�, ��23�

6. 0; yes

7. �5

8. �1, �3, �9; �1, �3

9. �1, ��12

�, �2; �2

10. 2 or 0; 1; �1

Quiz APage 95

1.

2. x4 � 2x3 � x2 � 8x� 12 � 0; 2

3. � �12

� i

4. �5 � i �1�0�

5. �44; 2 imaginary roots; �2 �

1i5�1�1��

Quiz BPage 95

1. 18; no

2. 0; yes

3. �5

4.

5. 2 or 0; 1; ��12

�, 1, 4

Quiz CPage 96

1. �5 and �4; �1 and0; 1 and 2

2. �0.9, 2.8

3. 4 � 2�3�

4. �p6� � �

p �5

2� � �

p �7

1�

5. �1 � w � 0, �25

� � w � 1

Quiz DPage 96

1. 7

2. �13

3. t � 14

4. cubic

5.


x4 � 3x3 � 3x2 �3x � 4 � 0; 2

No, because ƒ(�3) � �2.

��31�, ��2

3�, �1, �2;

�1, �13

�, 2 Sample answer:ƒ(x) � 4x4 � 24x3 �35x2 � 6x � 9

100-106 A&E C04-0-02-83417 10/4/00 2:37 PM Page 105

Page 97

1. A

2. D

3. C

4. E

5. D

6. B

7. A

8. B

9. D

10. A

Page 98

11. D

12. D

13. A

14. E

15. A

16. A

17. A

18. C

19. 6

20. 2.5

Page 99

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.



C(1, �3)

D � {�2, 0, 3},R � {�2, 5}; yes

3x2 � 5

(2, �1, 3)

A(�1, �1), B(2, �2),

x →∞, y→�∞,x→�∞, y →∞

y � x � 4

y-axis

$159

2

�5, 2, 2 � �3�

100-106 A&E C04-0-02-83417 10/4/00 2:37 PM Page 106



NAME _____________________________ DATE _______________ PERIOD ________


1. Change 128.433° to degrees, minutes, and seconds. 1. ________A. 128° 25′ 58″ B. 128° 25′ 59″ C. 128° 25′ 92″ D. 128° 26′ 00″

2. Write 43° 18′ 35″ as a decimal to the nearest thousandth of a degree. 2. ________A. 43.306° B. 43.308° C. 43.309° D. 43.310°

3. Give the angle measure represented by 3.25 rotations clockwise. 3. ________A. �1170° B. �90° C. 90° D. 1170°

4. Identify all coterminal angles between �360° and 360° 4. ________for the angle �420°.A. �60° and 300° B. �30° and 330°C. 30° and �330° D. 60° and �300°

5. Find the measure of the reference angle for 1046°. 5. ________A. �56° B. 56° C. 34° D. �34°

6. Find the value of the tangent for �A. 6. ________A. �2�

25�� B. ��2

5��

C. �23� D. ��35��

7. Find the value of the secant for �R. 7. ________A. ��5

7�0�� B. �3�14

1�4��

C. ��35�� D. ��3

1�4��

8. Which of the following is equal to csc θ ? 8. ________A. �sin

1θ� B. �co

1s θ� C. �ta

1n θ� D. �se

1c θ�

9. If cot θ � 0.85, find tan θ . 9. ________A. 0.588 B. 0.85 C. 1.176 D. 1.7

10. Find cos (�270°). 10. ________A. undefined B. �1 C. 1 D. 0

11. Find the exact value of sec 300°. 11. ________A. �2 B. ��2�

33�� C. 2 D. �2�

33��

12. Find the value of csc θ for angle θ in standard position if 12. ________the point at (5, �2) lies on its terminal side.A. ��2

2�9�� B. ��2�29

2�9�� C. ��52�9�� D. �5�

292�9��

13. Suppose θ is an angle in standard position whose terminal side 13. ________lies in Quadrant II. If sin θ � �11

23� , find the value of sec θ .

A. ��153� B. ��15

3� C. ��152� D. �1

123�

Chapter

5

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 107


For Exercises 14 and 15, refer to the figure. The angle of elevation from the end of the shadow to the top of the building is 63° and the distance is 220 feet.

14. Find the height of the building to the nearest foot. 14. ________A. 100 ft B. 196 ftC. 432 ft D. 112 ft

15. Find the length of the shadow to the nearest foot. 15. ________A. 100 ft B. 196 ftC. 432 ft D. 112 ft

16. If 0° ≤ x ≤ 360°, solve the equation sec x � �2. 16. ________A. 150° and 210° B. 210° and 330°C. 120° and 240° D. 240° and 300°

17. Assuming an angle in Quadrant I, evaluate csc �cot�1 �43��. 17. ________

A. �35� B. �53� C. �45� D. �54�

18. Given the triangle at the right, find B to the 18. ________nearest tenth of a degree if b � 10 and c � 14.A. 44.4° B. 35.5°C. 54.5° D. 45.6°

For Exercises 19 and 20, round answers to the nearest tenth.19. In �ABC, A � 27° 35′, B � 78° 23′, and c � 19. Find a. 19. ________

A. 8.6 B. 9.2 C. 12.8 D. 19.4

20. If A � 42.2°, B � 13.6°, and a � 41.3, find the area of �ABC. 20. ________A. 138.8 units2 B. 493.8 units2 C. 327.4 units2 D. 246.9 units2

21. Determine the number of possible solutions if A � 62°, a � 4, 21. ________and b � 6.A. none B. one C. two D. three

22. Determine the greatest possible value for B if A � 30°, a � 5, 22. ________and b � 8.A. 23.1° B. 53.1° C. 126.9° D. 96.9°

For Exercises 23-25, round answers to the nearest tenth.23. In �ABC, A � 47°, b � 12, and c � 8. Find a. 23. ________

A. 6.3 B. 8.7 C. 8.8 D. 18.4

24. In �ABC, a � 7.8, b � 4.2, and c � 3.9. Find B. 24. ________A. 15.1° B. 148.7° C. 78.9° D. 16.2°

25. If a � 22, b � 14, and c � 30, find the area of �ABC. 25. ________A. 33 units2 B. 121.0 units2 C. 130.2 units2 D. 143.8 units2

Bonus The terminal side of an angle θ in standard position Bonus: ________coincides with the line 4x � y � 0 in Quadrant II. Find sec θto the nearest thousandth.

A. �0.243 B. �4.123 C. 0.243 D. 4.123



5

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 108



NAME _____________________________ DATE _______________ PERIOD ________


1. Change 110.23° to degrees, minutes, and seconds. 1. ________A. 110° 13′ 00″ B. 110° 13′ 8″ C. 110° 13′ 28″ D. 110° 13′ 48″

2. Write 24° 38′ 42″ as a decimal to the nearest thousandth of a degree. 2. ________A. 24.645° B. 24.646° C. 24.647° D. 24.648°

3. Give the angle measure represented by 1.75 rotations counterclockwise. 3. ________A. �630° B. �90° C. 90° D. 630°

4. Identify the coterminal angle between �360° and 360° for the angle �120°. 4. ________A. �240° B. 60° C. 240° D. 300°

5. Find the measure of the reference angle for 295°. 5. ________A. 25° B. �65° C. �25° D. 65°

6. Find the value of the cosine for �A. 6. ________

A. �12� B. ��23��

C. ��33�� D. 2

7. Find the value of the cosecant for �R. 7. ________

A. ��26�� B. ��3

6��

C. ��31�5�� D. ��2

1�0��

8. Which of the following is equal to sec �? 8. ________A. �sin

1�

� B. �co1s �� C. �ta

1n �� D. �se

1c ��

9. If tan � � 0.25, find cot �. 9. ________A. 0.25 B. 4 C. 0.5 D. 14

10. Find cot (�180°). 10. ________A. undefined B. �1 C. 1 D. 0

11. Find the exact value of tan 240°. 11. ________A. ��3� B. ��

33�� C. �3� D. ��

33��

12. Find the value of sec � for angle � in standard position if the 12. ________point at (�2, �4) lies on its terminal side.A. ��2

5�� B. �5� C. ��25�� D. ��5�

13. Suppose � is an angle in standard position whose terminal side 13. ________lies in Quadrant III. If sin � � ��11

23�, find the value of cot �.

A. ��153� B. ��15

3� C. �152� D. �1

123�

Chapter

5

D

A

D

C

D

B

D

B

B

A

C

D

C

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 109


For Exercises 14 and 15, refer to the figure. The angleof elevation from the end of the shadow to the top ofthe building is 56° and the distance is 120 feet.14. Find the height of the building to the nearest foot. 14. ________

A. 99 ft B. 67 ftC. 178 ft D. 81 ft


16. If 0° � x � 360°, solve the equation tan x � �1. 16. ________A. 135° and 315° B. 45° and 225° C. 45° and 315° D. 225° and 315°

17. Assuming an angle in Quadrant I, evaluate tan �cos�1 �45��. 17. ________

A. �34� B. �53� C. �45� D. �54�

18. Given the triangle at the right, find A to the nearest 18. ________tenth of a degree if b � 10 and c � 14.A. 44.4° B. 35.5°C. 54.5° D. 45.6°

For Exercises 19 and 20, round answers to the nearest tenth.19. In �ABC, A � 41° 15′, B � 107° 39′, and c � 19. Find b. 19. ________

A. 10.0 B. 24.3 C. 35.1 D. 54.6

20. If A � 52.6°, B � 49.8°, and a � 33.8, find the area of �ABC. 20. ________A. 117.9 units2 B. 338.2 units2 C. 536.4 units2 D. 1072.8 units2

21. Determine the number of possible solutions if A � 62°, a � 7, and b � 6. 21. ________A. none B. one C. two D. three

For Exercises 22–25, round answers to the nearest tenth.22. Determine the least possible value for B if A � 30°, a � 5, 22. ________

and b � 8.A. 23.1° B. 53.1° C. 126.9° D. 96.9°

23. In �ABC, B � 52°, a � 14, and c � 9. Find b. 23. ________A. 8.2 B. 11.0 C. 11.1 D. 18.4

24. In �ABC, a � 7.8, b � 4.2, and c � 3.9. Find A. 24. ________A. 15.1° B. 78.9° C. 148.7° D. 16.2°

25. If a � 32, b � 26, and c � 40, find the area of �ABC. 25. ________A. 49 units2 B. 121.0 units2 C. 298.6 units2 D. 415.2 units2

Bonus The terminal side of an angle � in standard position Bonus: ________coincides with the line 2x � y � 0 in Quadrant III.Find cos � to the nearest ten-thousandth.

A. �0.4472 B. 0.4472 C. �0.8944 D. 0.8944



5

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 110



NAME _____________________________ DATE _______________ PERIOD ________


1. Change 36.3� to degrees, minutes, and seconds. 1. ________A. 36� 18′ 00″ B. 36� 18′ 16″ C. 36� 18′ 24″ D. 36� 18′ 28″

2. Write 21� 44′ 3″ as a decimal to the nearest thousandth of a degree. 2. ________A. 21.741� B. 21.742� C. 21.743� D. 21.744�

3. Give the angle measure represented by 0.5 rotation clockwise. 3. ________A. �180� B. �90� C. 90� D. 180�

4. Identify the coterminal angle between 0� and 360� for the angle 480�. 4. ________A. 30� B. 60� C. 120� D. 240�

5. Find the measure of the reference angle for 235�. 5. ________A. �125� B. 55� C. 25� D. �55�

6. Find the value of the cosine for �A. 6. ________A. ��1�

51�9�� B. ��1

1�21�9��

C. �152� D. �15

2�

7. Find the value of the cotangent for �R. 7. ________A. �43� B. �4

3�

C. �45� D. �45�

8. Which of the following is equal to cot �? 8. ________A. �sin

1�

� B. �co1s �� C. �se

1c �� D. �tan

1�

�

9. If cos � � 0.5, find sec �. 9. ________A. 0.25 B. 0.5 C. 1 D. 2

10. Find tan 180�. 10. ________A. undefined B. �1 C. 1 D. 0

11. Find the exact value of cos 135�. 11. ________A. �1 B. ��2

2�� C. 1 D. ��22��

12. Find the value of csc � for angle � in standard position if the point at 12. ________(3, �1) lies on its terminal side.A. ��1�0� B. ��1

1�00�� C. ��3

1�0�� D. �3

13. Suppose � is an angle in standard position whose terminal side lies 13. ________in Quadrant II. If cos � � ��11

23�, find the value of tan �.

A. ��152� B. ��15

3� C. ��152� D. �1

123�

Chapter

5

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 111


For Exercises 14 and 15, refer to the figure. The angle of elevationfrom the end of the shadow to the top of the building is 70� andthe distance is 180 feet.14. Find the height of the building to the nearest foot. 14. ________

A. 62 ft B. 66 ftC. 169 ft D. 495 ft


16. If 0� � x � 360�, solve the equation sin x � ��23��. 16. ________

A. 120� and 240� B. 240� and 300�C. 210� and 330� D. 150� and 210�

17. Assuming an angle in Quadrant I, evaluate cos �tan�1 �43��. 17. ________

A. �35� B. �53� C. �45� D. �45�

18. Given the triangle at the right, find B to the 18. ________nearest tenth of a degree if b � 8 and c � 12.A. 33.7� B. 41.8�C. 48.2� D. 56.3�

For Exercises 19 and 20, round answers to the nearest tenth.19. In �ABC, A � 102� 12�, B � 23� 21�, and c � 19.8. Find a. 19. ________

A. 8.0 B. 23.8 C. 48.8 D. 64.4

20. If A � 32.2�, b � 21.5, and c � 11.3, find the area of �ABC. 20. ________A. 129.5 units2 B. 102.8 units2 C. 64.7 units2 D. 32.6 units2

21. Determine the number of possible solutions if A � 48�, a � 5, and b � 6. 21. ________A. none B. one C. two D. three

22. Determine the least possible value for B if A � 20�, 22. ________a � 7, and b � 11.A. 12.6� B. 32.5� C. 147.5� D. 96.9�

For Exercises 23–25, round answers to the nearest tenth.23. In �ABC, A � 52�, b � 9, and c � 14. Find a. 23. ________

A. 6.3 B. 8.7 C. 8.8 D. 11.0

24. In �ABC, a � 2.4, b � 8.2, and c � 10.1. Find B. 24. ________A. 15.1� B. 21.7� C. 28.9� D. 33.3�

25. If a � 12, b � 30, and c � 22, find the area of �ABC. 25. ________A. 33.7 units2 B. 93.4 units2 C. 113.1 units2 D. 143.8 units2

Bonus The terminal side of an angle � in standard position Bonus: ________coincides with the line y � �12�x in Quadrant I. Find sin � to the nearest thousandth.

A. 0.233 B. 0.447 C. 0.508 D. 0.693



5

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 112



NAME _____________________________ DATE _______________ PERIOD ________

1. Change 225.639� to degrees, minutes, and seconds. 1. __________________

2. Write 23� 16′ 25″ as a decimal to the nearest thousandth of a degree. 2. ____________________

3. State the angle measure represented by 2.4 rotations clockwise. 3. ____________________

4. Identify all coterminal angles between �360� and 360� for the 4. ____________________angle �540�.

5. Find the measure of the reference angle for 562�. 5. ____________________

6. Find the value of the sine for �A. 6. ____________________

7. Find the value of the cotangent for �A. 7. ____________________

8. Find the value of the secant for �A. 8. ____________________

9. If csc � � �2, find sin �. 9. ____________________

10. Find sin (�270�). 10. ____________________

11. Find the exact value of cot 330�. 11. ____________________

12. Find the exact value of sec � for angle � in standard position if 12. ____________________the point at (�3, 2) lies on its terminal side.

13. Suppose � is an angle in standard position whose terminal side 13. ____________________lies in Quadrant IV. If cos � � �11

23�, find the value of csc �.

Chapter

5

For Exercises 6–8, refer to the figure.

Exercises 6–8

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 113


For Exercises 14 and 15, refer to the figure. The angle of elevationfrom the far side of the pool to the top of the waterfall is 75�, andthe distance is 185 feet.14. Find the height of the waterfall 14. __________________

to the nearest foot.

15. Find the width across the pool 15. __________________to the nearest foot.

16. If 0� � x � 360�, solve cot x � ��3�. 16. __________________

17. Assuming an angle in Quadrant I, evaluate sec �tan�1 �34��. 17. __________________

18. Given triangle at the right, 18. __________________find B to the nearest tenth of a degree if a � 8 and b � 20.

For Exercises 19 and 20, round answers to the nearest tenth.19. In �ABC, A � 47� 15′, B � 58� 33′, and c � 23. Find a. 19. __________________

20. If A � 37.2�, B � 17.9�, and a � 22.3, find the area of �ABC. 20. __________________

21. Determine the number of possible solutions if A � 47�, 21. __________________a � 5, and b � 4.

22. Determine the least possible value for c if A � 30�, 22. __________________a � 5, and b � 8.

For Exercises 23-25, round answers to the nearest tenth.23. In �ABC, A � 118�, b � 8, and c � 6. Find a. 23. __________________

24. In �ABC, a � 9, b � 5, and c � 12. Find B. 24. __________________

25. If a � 12, b � 24, and c � 30, find the area of �ABC. 25. __________________

Bonus The terminal side of an angle � in standard Bonus: __________________position coincides with the line 3x � y � 0 in Quadrant II. Find csc � to the nearest thousandth.



5

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 114



NAME _____________________________ DATE _______________ PERIOD ________

1. Change 124.63° to degrees, minutes, and seconds. 1. __________________

2. Write 48° 32′ 15″ as a decimal to the nearest thousandth of 2. __________________a degree.

3. State the angle measure represented by 1.25 rotations 3. __________________clockwise.

4. Identify all coterminal angles between �360° and 360° for 4. __________________the angle 630°.

5. Find the measure of the reference angle for 310°. 5. __________________

6. Find the value of the cosine for �A. 6. __________________

7. Find the value of the cosecant for �A. 7. __________________

8. Find the value of the cotangent for �A. 8. __________________

9. If sec � � �4, find cos �. 9. __________________

10. Find tan (�180°). 10. __________________

11. Find the exact value of sec 240°. 11. __________________

12. Find the exact value of sec � for angle � in standard 12. __________________position if the point at (�4, 5) lies on its terminal side.

13. Suppose � is an angle in standard position whose terminal 13. __________________side lies in Quadrant IV. If cos � � �11

23�, find the value of cot �.

Chapter

5124° 37′ 48″

48.538°

�450°

�90° and 270°

50°

��73�3��

�47�

��43�3��

��14

�

0

�2

��44�1��

��152�

Exercises 6–8

For Exercises 6–8, refer to the figure.

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 115

186 feet

135 feet

210° and 330°

�1132�

56.9°

10.7

164.9 units2

one

9.9

10.5

129.8°

154.7 units2

�0.3162




5For Exercises 14 and 15, refer to the figure. The angle of elevationfrom the far side of the pool to the top of the waterfall is 54°, andthe distance is 230 feet.14. Find the height of the waterfall 14. __________________



16. If 0° � x � 360°, solve the equation csc x � �2. 16. __________________

17. Assuming an angle in Quadrant I, evaluate cos �cot�1 �152��. 17. __________________

18. Given the triangle at the right, 18. __________________find B to the nearest tenth of a degree if a � 12 and c � 22.

For Exercises 19 and 20, round answers to the nearest tenth.19. In �ABC, A � 42°, B � 68°, and c � 15. Find a. 19. __________________

20. If A � 27.2°, B � 67.4°, and a � 12.8, find the area of �ABC. 20. __________________

21. Determine the number of possible solutions if A � 110°, 21. __________________a � 5, and b � 4.

For Exercises 22–25, round answers to the nearest tenth.22. Determine the greatest possible value for c if A � 30°, 22. __________________

a � 5, and b � 8.

23. In �ABC, A � 59°, b � 12, and c � 4. Find a. 23. __________________

24. In �ABC, a � 4, b � 11, and c � 8. Find B. 24. __________________

25. If a � 21, b � 15, and c � 28, find the area of � ABC. 25. __________________

Bonus The terminal side of an angle � in standard Bonus: _________________position coincides with the line 3x � y � 0 in Quadrant III. Find cos � to the nearest ten-thousandth.

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 116



NAME _____________________________ DATE _______________ PERIOD ________


2. Write 75� 30′ as a decimal to the nearest thousandth of a 2. __________________degree.

3. State the angle measure represented by 1.5 rotations 3. __________________counterclockwise.

4. Identify a coterminal angle between 0� and 360� for the 4. __________________angle �225�.

5. Find the measure of the reference angle for 235�. 5. __________________

6. Find the value of the sine for �A. 6. __________________


8. Find the value of the secant for �A. 8. __________________

9. If tan � � �3, find cot �. 9. __________________

10. Find tan 180�. 10. __________________

11. Find the exact value of cos 330�. 11. __________________

12. Find the exact value of sin � for angle � in standard position 12. __________________if the point at (�1, 4) lies on its terminal side.

13. Suppose � is an angle in standard position whose 13. __________________terminal side lies in Quadrant II. If sin � � �11

23�, find

the value of sec �.

Chapter

5

Exercises 6–8

For Excercises 6–8, refer to the figure.

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 117


For Exercises 14 and 15, refer to the figure. The angle of elevationfrom the far side of the pool to the top of the waterfall is 68� andthe distance is 200 feet.14. Find the height of the waterfall 14. __________________



16. If 0� � x � 360�, solve sin x � ��23��. 16. __________________

17. Assuming an angle in Quadrant I, evaluate cos �tan�1 �152��. 17. __________________

18. Given the triangle at the right, 18. __________________find B to the nearest tenth of a degree if b � 12 and c � 18.

For Exercises 19 and 20, round answers to the nearest tenth.19. In �ABC, A � 47�, B � 58�, and b � 23. Find a. 19. __________________

20. If C � 37.2�, a � 17.9, and b � 22.3, find the area of �ABC. 20. __________________

21. Determine the number of possible solutions if A � 47�, 21. __________________a � 2, and b � 4.

22. Determine the greatest possible value for c if A � 15�, 22. __________________a � 8, and b � 13.

For Exercises 23–25, round answers to the nearest tenth.23. In �ABC, A � 67�, b � 10, and c � 5. Find a. 23. __________________

24. In �ABC, a � 8, b � 6, and c � 12. Find C. 24. __________________

25. If a � 18, b � 22, and c � 30, find the area of �ABC. 25. __________________

Bonus The terminal side of an angle � in standard Bonus: __________________position coincides with the line y � x in Quadrant I. Find tan � to the nearest thousandth.



5

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 118



NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear,concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond therequirements of the problem.

1. The point at (�3, ��3�) lies on the terminal side of an angle in standard position.

a. Give the degree measure of three angles that fit the description.

b. Tell how to find the cosine of such angles. Give the cosine of these angles.

c. Name angles in the first, second, and third quadrants that have the same reference angle as those above.

d. Write the coordinates of a point in Quadrant II. Find the values of the six trigonometric functions of an angle in standard position with this point on its terminal side.

2. A children’s play area is being built next to a circular fountain in the park. A fence will be erected around the play area for safety. A diagram of the area is shown below.

a. How long will the fence need to be in order to enclose the area?

b. The park commission is planning to enlarge the play area. Do you think it should be enlarged to the east or to the west? Why?

Chapter

5

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 119



2. If a �470� angle is in standard position, determine a 2. __________________coterminal angle that is between 0� and 360�. State the quadrant in which the terminal side lies.

For Exercises 3 and 4, use right triangle ABC to find each value.

3. Find the value of the cosine for �A. 3. __________________


5. If csc � � �3, find sin �. 5. __________________

6. Use the unit circle to find tan 180�. 6. __________________

7. Find the exact value of sin 300�. 7. __________________

8. Find the value of csc � for angle � in standard position if 8. __________________the point at (2, �1) lies on its terminal side.

For Exercises 9 and 10, use right triangle ABC to find each value to the nearest tenth.

9. Find b. 9. __________________

10. Find c. 10. __________________



5

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 120

1. Change 47.283� to degrees, minutes, and seconds. 1. __________________2. Write 122� 43′ 12″ as a decimal to the nearest thousandth 2. __________________

of a degree.3. Give the angle measure represented by 2.25 rotations 3. __________________

counterclockwise.4. Identify all coterminal angles between �360� and 360� 4. __________________

for the angle 480�.5. Find the measure of the reference angle for 323�. 5. __________________6. Find the value of the sine for �A. 6. __________________7. Find the value of the tangent for �A. 7. __________________8. Find the value of the secant for �A. 8. __________________

9. If cot � � ��23�, find tan �. 9. __________________

10. If sin � � 0.5, find csc �. 10. __________________

Use the unit circle to find each value. 1. __________________

1. sin (�90�) 2. csc 180� 2. __________________3. Find the exact value of cos 210�. 3. __________________4. Find the exact value of tan 135�. 4. __________________5. Find the value of sec � for angle � in standard position if 5. __________________

the point at (4, �5) lies on its terminal side.6. Suppose � is an angle in standard position whose terminal 6. __________________

side lies in Quadrant III. If tan � � �152�, find the value of sin �.

Refer to the figure. Find each value to the nearest tenth.

7. Find a. 7. __________________8. Find c. 8. __________________

A 100-foot cable is stretched from a stake in the ground to the topof a pole. The angle of elevation is 57°.

9. Find the height of the pole to the nearest tenth. 9. __________________10. Find the distance from the base of the pole to the 10. __________________

stake to the nearest tenth.


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

5

Chapter

5

Exercises 6–8

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 121

1. If 0� � x � 360�, solve: csc x � �2. 1. __________________

2. Assuming an angle in Quadrant I, evaluate tan �sec�1 �153��. 2. __________________

3. Given right triangle ABC, find B to the 3. __________________nearest tenth of a degree if b � 7 and c � 12.

Find each value. Round to the nearest tenth.4. In �ABC, A � 58� 21�, C � 97� 07�, and b � 23.8. Find a. 4. __________________

5. If B � 29.5�, C � 64.5�, and a � 18.8, find the area of �ABC. 5. __________________

1. Determine the number of possible solutions for �ABC 1. __________________if A � 28�, a � 4, and b � 11.

Find each value. Round to the nearest tenth.

2. For �ABC, determine the least possible value for B 2. __________________if A � 49�, a � 12, and b � 15.

3. In �ABC, B � 32�, a � 11, and c � 2.4. Find b. 3. __________________

4. In �ABC, a � 3.1, b � 5.4, and c � 4.7. Find C. 4. __________________

5. If a � 28.2, b � 36.5, and c � 40.1, find the area of �ABC. 5. __________________


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

5

Chapter

5

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 122



NAME _____________________________ DATE _______________ PERIOD ________


Multiple Choice1. The length of a diagonal of a square

is 6 units. Find the length of a side. DA 9�2� unitsB �2� unitsC 9 unitsD 3�2� unitsE None of these

2. If the area of �ABC is 30 square meters,what is the length of segment CD? DA 2 mB 3 m C 5 mD 8 mE 10 m

3. The midpoint of a diameter of a circle is(3, 4). If the coordinates of one endpointof the diameter are (�3, 6), what arethe coordinates of the other endpoint?A (9, 2) AB (9, 1) C (8, 2)D (3, 2)E (9, 14)

4. The endpoints of a diameter of a circleare (3, 2) and (11, 8). Find the area ofthe circle. CA 5 units2

B 25 units2

C 25� units2

D 10� units2

E 5� units2

5. What is the arithmetic mean of �13� and �14�?A �16� B �7

1� E

C �112� D �1

72�

E �274�

6. Which of the following products has thegreatest value less than 100? E

A 2 4 6B 2 4 9C 4 4 9D 3 3 9E 4 4 6

7. A diagonal of a rectangle is �1�5�inches. The length of the rectangle is�1�2� inches. Find the area of the rectangle. BA 3�2� in2

B 6 in2

C 9 in2

D 6�5� in2

E None of these

8. The diagonals of a rhombus are perpendicular and bisect each other.If the length of one side of a rhombus is 25 meters and the length of onediagonal is 14 meters, find the lengthof the other diagonal. DA 7 mB 12 mC 24 mD 48 mE 144 m

9. If c � �a1b�, what is the value of a when

c � 12�1 and b � 3? DA �36B ��3

16�

C �4D 4E None of these

10. How many times do the graphs of y � x2 and xy � 27 intersect? BA 0B 1C 2D 3E 4

Chapter

5

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 123


11. If �143x� is an integer, then the value of x

CANNOT be which of the following?A �112B �32C 0D 6E 8

12. Which of the following statementsmakes the expression a � b representa negative number?A b aB a bC b 0D a 0E b � a

13. A hiker travels 5 miles due north, then3 miles due west, and then 2 miles duenorth. How far is the hiker from hisbeginning point?A About 8.2 miB About 7.1 miC About 9.4 miD About 6 miE About 7.6 mi

14. A rectangular swimming pool is 100 meters long and 25 meters wide.Lucia swims from one corner of thepool to the opposite corner and back10 times. How far did she swim?A About 193.6 mB About 1030.8 mC About 2061.6 mD About 10,308.8 mE None of these

15. 4 � 3�2� is a root of which equation?A x2 � 8x � 2 � 0B x2 � 8x � 18 � 0C x2 � 16x � 2 � 0D x2 � 16x � 18 � 0E None of these

16. If ƒ(x) � x4 � 4x3 � 16x � 16 has azero of �2, with a multiplicity of 3,what is another zero of ƒ?A 3B 2C �1D 1E 8

17–18. Quantitative ComparisonA if the quantity of Column A is




Column A Column B17. The value of x in each figure

18. The length of the hypotenuse of aright triangle with the given leglengths

19. Grid-In A tent with a rectangularfloor has a diagonal length of 7 feetand a width of 5 feet. What is the areaof the floor to the nearest square foot?

20. Grid-In Julio wants to bury a waterpipe from one corner of his field to theopposite corner. How many feet ofpipe, to the nearest foot, does he needif the rectangular field is 200 feet by300 feet?



5

7 and 12 8 and 11

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 124



NAME _____________________________ DATE _______________ PERIOD ________

1. If ƒ(x) � x � 2 and g(x) � �21x� , find g( ƒ(x)). 1. __________________

2. Determine whether the graphs of 2x � y � 5 � 0 2. __________________and x � 2y � 6 are parallel, coinciding, perpendicular, or none of these.

3. Graph the inequality 2x � 4y � 8. 3.

4. Solve the following system of equations algebraically. 4. __________________5x � y � 162x � 3y � 3

5. Find BC if B � � � and C � � �. 5. __________________

6. Find the inverse of � �, if it exists. 6. __________________

7. Determine whether the graph of y � 2x4 � x2 � 3 is 7. __________________symmetric to the x-axis, the y-axis, both, or neither.

8. Describe the transformations relating the graph of 8. __________________y � ��x � 2� � 5 to its parent function, y � �x �.

9. Find the inverse of y � 2x � 6. State whether the inverse 9. __________________is a function.

10. Solve x2 � 6x � 6 � 0. 10. __________________

11. Solve �3

x� �� 2� � 7 � 4. 11. __________________

12. Find csc (�90°). 12. __________________

13. Given the right triangle ABC, 13. __________________find side c to the nearest tenth.

14. In �ABC, a � 8, b � 6, and c � 10. Find B to the 14. __________________nearest tenth.

�13

24

5�4

�38

�25

61

Chapter

5

107-125 A&E C05-0-02-834179 10/6/00 2:51 PM Page 125

Page 107

1. B

2. D

3. A

4. A

5. C

6. B

7. D

8. A

9. C

10. D

11. C

12. A

13. B

Page 108

14. B

15. A

16. C

17. B

18. D

19. B

20. D

21. A

22. C

23. C

24. D

25. D

Bonus: B

Page 109

1. D

2. A

3. D

4. C

5. D

6. B

7. D

8. B

9. B

10. A

11. C

12. D

13. C

Page 110

14. A

15. B

16. A

17. A

18. A

19. C

20. C

21. B

22. B

23. B

24. C

25. D

Bonus: A



126-132 A&E C05-0-02-83417 10/4/00 2:42 PM Page 126



Page 111

1. A

2. B

3. A

4. C

5. B

6. B

7. A

8. D

9. D

10. D

11. B

12. A

13. A

Page 112

14. C

15. A

16. B

17. A

18. B

19. B

20. C

21. C

22. B

23. D

24. D

25. C

Bonus: B

Page 113

1. 225° 38� 20.4�

2. 23.274°

3. �864°

4.�180° and 180°

5. 22°

6. �41�16��

7. �52�

46��

8. �151�

9. � �12

�

10. 1

11. ��3�

12. ��31�3��

13. ��153�

Page 114

14. 179 feet

15. 48 feet

16. 150° and 330°

17. �45�

18. 68.2°

19. 17.6

20. 103.7 units2

21. two

22. 3.9

23. 12.0

24. 22.2°

25. 136.8 units2

Bonus: 1.054

Form 1C Form 2A

126-132 A&E C05-0-02-83417 10/4/00 2:42 PM Page 127



Page 115

1. 124° 37� 48�

2. 48.538°

3. �450°

4. �90° and 270°

5. 50°

6. ��73�3��

7. �47�

8. ��43�3��

9. � �41�

10. 0

11. �2

12. ��44�1��

13. � �152�

Page 116

14. 186 feet

15. 135 feet

16. 210° and 330°

17. �1132�

18. 56.9°

19. 10.7

20. 164.9 units2

21. one

22. 9.9

23. 10.5

24. 129.8°

25. 154.7 units2

Bonus: �0.9487

Page 117

1. 25° 36� 00�

2. 75.500°

3. 540°

4. 135°

5. 55°

6. ��96�5��

7. �4�65

6�5��

8. �49�

9. � �31�

10. 0

11. ��23��

12. �4�17

1�7��

13. � �153�

Page 118

14. 185 feet

15. 75 feet

16. 240° and 300°

17. �153�

18. 41.8°

19. 19.8

20. 120.7 units2

21. none

22. 19.8

23. 9.3

24. 117.3

25. 196.7 units2

Bonus: 1.000

126-132 A&E C05-0-02-83417 10/4/00 2:42 PM Page 128




3 Superior • Shows thorough understanding of the concepts standard position, degree measure, quadrant, reference angle, and the six trigonometric functions of an angle.

• Computations are correct.• Uses appropriate strategies to solve problems.• Written explanations are exemplary.• Goes beyond requirements of some or all problems.

2 Satisfactory, • Shows understanding of the concepts standard position,with Minor degree measure, quadrant, reference angle, and the sixFlaws trigonometric functions of an angle.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts standard Satisfactory, position, degree measure, quadrant, reference angle, with Serious and the six trigonometric functions of an angle.Flaws • May not use appropriate strategies to solve problems.

• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts standard position, degree measure, quadrant, reference angle, and the six trigonometric functions of an angle.

• May not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are not satisfactory.• Does not satisfy requirements of problems.

126-132 A&E C05-0-02-83417 10/4/00 2:42 PM Page 129

Page 119

1a. 210°, 570°, 930°

1b. The cosine is �xr

� � �2�

�33�

�, or

��23��.

1c. first quadrant: 30°, second quadrant: 150°, third quadrant: 210°, fourth quadrant: 330°

1d. Sample answers:(�3, 4); sin A � �

54�,

cos A � � �35

�, tan A � � �43

�,

csc A � ��54

�, sec A � � �53

�,

cot A � � �43�

2a. The length of the missing sideto the east is given by

�s1in5

4ft0.°

� � �sin

x60°� , or 20.2 feet.

The length of the missing sidearound the fountain is given by �1

6� � 20�, or 10.5 feet.

The total length isapproximately 140.7 feet.

2b. Answers will vary but mightinclude issues such as thelength of fence required versusthe increase in park size, access and/orproximity to roads,maintenance and/orenhancement of the fountain, and so on.



126-132 A&E C05-0-02-83417 10/4/00 2:42 PM Page 130



1. 65° 46� 55.2�

2. 250°; III

3. �2�11

1�0��

4. �2�91�0��

5. � �31�

6. 0

7. ��23��

8. ��5�

9. 9.2

10. 11.0

Quiz APage 121

1. 47° 16� 59�

2. 122.7200°

3. 810°

4. 120° and �240°

5. 37°6. �3�

343�4��

7. �53�

8. ��53�4��

9. � �23�

10. 2

Quiz BPage 121

1. �1

2. undefined3. ��

23��

4. �15. ��

44�1��

6. ��153�

7. 6.6

8. 9.8

9. 83.9 feet

10. 54.5 feet

Quiz CPage 122

1. 210° and 330°

2. �152�

3. 35.7°

4. 48.8

5. 78.7 units2

Quiz DPage 122

1. none

2. 70.6°

3. 9.1

4. 60.1°

5. 498.0 units2


126-132 A&E C05-0-02-83417 10/4/00 2:42 PM Page 131

Page 123

1. D

2. D

3. A

4. C

5. E

6. E

7. B

8. D

9. D

10. B

Page 124

11. D

12. A

13. E

14. C

15. A

16. B

17. B

18. A

19. 24

20. 361

Page 125

1. �2x

1� 4�

2. perpendicular

3.

4. (3, �1)

5. � �

6.

7. y-axisreflected over x-axis; translated left 2 units

8. and down 5 units

9. y � �x �2

6� ; yes

10. 3 � �3�

11. �29

12. �1

13. 15.4

14. 36.9°

38�15

�3437



��25

�

�110�

�15

�

�130�

� �

126-132 A&E C05-0-02-83417 10/4/00 2:42 PM Page 132



NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right of each problem.1. Change 1400� to radian measure in terms of �. 1. ________

A. �709

�� B. �359

�� C. �1490�� D. None of these

2. Change �2397�� radians to degree measure. 2. ________

A. 5220� B. 141.1� C. 167.6� D. 66.6�

3. Determine the angular velocity if 0.75 revolutions are completed 3. ________in 0.05 seconds.A. 4.7 radians/s B. 47.1 radians/sC. 9.4 radians/s D. 94 radians/s

4. Determine the linear velocity of a point rotating at 25 revolutions 4. ________per minute at a distance of 2 feet from the center of the rotating object.A. 2.6 ft /s B. 314.2 ft /s C. 5.2 ft /s D. 78.5 ft /s

5. There are 20 rollers under a conveyor belt and each roller has a 5. ________radius of 15 inches. The rollers turn at a rate of 40 revolutions per minute. What is the linear velocity of the conveyor belt?A. 3769.9 ft/s B. 104.7 ft/s C. 5.2 ft/s D. 62.8 ft/s

6. Find the degree measure of the central angle associated with an arc 6. ________that is 16 inches long in a circle with a radius of 12 inches.A. 76.4� B. 270.0� C. 283.6� D. 43.0�

7. Find the area of a sector if the central angle measures 105� and the 7. ________radius of the circle is 4.2 meters.A. 7.7 m2 B. 16.2 m2 C. 32.3 m2 D. 926.1 m2

8. Write an equation of the sine function with amplitude 3, period �32��, 8. ________

and phase shift ��4�.

A. y � 3 sin ��32x� � ��4�� B. y � �3 sin ��43

x� � ��4��C. y � �3 sin ��32

x� � �38�� D. y � 3 sin ��43

x� � ��3��9. Write an equation of the tangent function with period �38

��, phase 9. ________shift ��5�, and vertical shift �2.A. y � tan ��83

x� � �81�5�� 2 B. y � tan ��83

x� � �81�5�� 2

C. y � tan ��136x� � �38

�0�� 2 D. y � tan ��83

x� � �34�0�� 2

10. State the amplitude, period, and phase shift of the function 10. ________y � �3 cos �3x � �32

��.A. 3; 2�; ��2� B. 3; �23

��; ��2� C. �3; 2�; ��2� D. 3; �23��; �32

��

11. State the period and phase shift of the function y � �4 tan ��12�x � �38��. 11. ________

A. 2�, ��34�� B. �, �38

�� C. 2�, �38�� D. �, ��38

��

12. What is the equation for the inverse of y � Cos x � 3? 12. ________A. y � Arccos (x � 3) B. y � Arccos x � 3C. y � Arccos x � 3 D. y � Arccos (x � 3)

Chapter

6

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13. Evaluate tan �Cos�1 ��23�� Tan�1 ��3

3��. 13. ________

A. ��33�� B. �3� C. 0 D. undefined

The paddle wheel of a boat measures 16 feet in diameter and is revolving ata rate of 20 rpm. The maximum depth of the paddle wheel under water is 1 foot. Suppose a point is located at the lowest point of the wheel at t � 0.14. Write a cosine function with phase shift 0 for the height of the 14. ________

initial point after t seconds.A. h � 8 cos ��23

��t� � 7 B. h � �8 cos 3t � 7

C. h � �8 cos ��23��t� � 7 D. h � 8 cos 3t � 7

15. Use your function to find the height of the initial point after 15. ________55 seconds.A. 7.5 ft B. 11 ft C. 10.4 ft D. 6.5 ft

16. Find the values of x for which the equation sin x � �1 is true. 16. ________A. 2�n B. ��2� � 2�n C. � � 2�n D. �32

�� 2�n

17. What is the equation of the graph shown below? 17. ________A. y � tan �2

x�

B. y � �cot 2x

C. y � �cot �2x�

D. y � tan 2x

18. What is the equation of the graph shown below? 18. ________A. y � �2 sin ��23

x� � ��B. y � 2 cos ��3

x� � ��2��C. y � �2 sin ��23

x� � ��2��D. y � 2 cos ��23

x� � ��19. What is the equation of the graph shown below? 19. ________

A. y � tan �4x � ��2��B. y � tan (4x � �)

C. y � cot �4x � ��2��D. y � cot (4x � �)

20. State the domain and range of the relation y � Arctan x. 20. ________A. D: {all real numbers}; R: ��2� � y � ��2��B. D: {all real numbers}; R: {0 � y � �}

C. D: ��2� � y � ��2��; R: {all real numbers}

D. D: {0 � y � �}; R: {all real numbers}

Bonus Evaluate cos �2� � Arctan �43��. Bonus: ________

A. �2245� B. ��2

75� C. �45� D. �5

3�



6

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NAME _____________________________ DATE _______________ PERIOD ________


1. Change �435� to radian measure in terms of �. 1. ________A. �51

�2� B. ��51

�2� C. �21

92�� D. ��21

92��


A. 355� B. 156� C. 162� D. 207.7�

3. Determine the angular velocity if 65.7 revolutions are completed 3. ________in 12 seconds.A. 5.5 radians/s B. 34.4 radians/sC. 17.2 radians/s D. 125.5 radians/s

4. Determine the linear velocity of a point rotating at an angular 4. ________velocity of 62� radians per minute at a distance of 5 centimeters from the center of the rotating object.A. 973.9 cm/min B. 310 cm/min C. 39.0 cm/min D. 1947.8 cm/min

5. There are three rollers under a conveyor belt, and each roller has 5. ________a radius of 8 centimeters. The rollers turn at a rate of 2 revolutions per second. What is the linear velocity of the conveyor belt?A. 0.50 m/s B. 50.26 m/s C. 100.53 m/s D. 1.005 m/s

6. Find the degree measure of the central angle associated with an 6. ________arc that is 21 centimeters long in a circle with a radius of 4 centimeters.A. 10.9� B. 59.2� C. 68.6� D. 300.8�

7. Find the area of a sector if the central angle measures 40� and 7. ________the radius of the circle is 12.5 centimeters.A. 54.5 cm2 B. 109.1 cm2 C. 8.7 cm2 D. 4.4 cm2

8. Write an equation of the cosine function with amplitude 2, 8. ________period �, and phase shift ��2�.A. y � �2 cos ��2

x� � ��4�� B. y � 2 cos ��2x� � ��

C. y � �2 cos �2x � ��2�� D. y � 2 cos �2x � ��9. Write an equation of the tangent function with period 3�, phase 9. ________

shift ��4�, and vertical shift 2.A. y � tan ��3

x� � �34�� 2 B. y � tan �3x � �1

�2�� 2

C. y � tan ��3x� � �1

�2�� 2 D. y � tan �3x � �34

�� 2

10. State the amplitude, period, and phase shift of the function 10. ________y � �13� sin �2x � ��3��.A. �13�; �; ��6� B. �13�; 4�; ��6� C. �13�; �; ��3� D. �13�; 4�; �3

��

11. State the period and phase shift of the function y � �12� cot �2x � ��4��. 11. ________A. ��2�; � �1

�6� B. ��2�; ��8� C. ��4�; ��4� D. ��2�; �4

��

Chapter

6

D

B

B

A

D

D

A

D

C

A

B

133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 135


12. What is the equation for the inverse of y � �12� Sin x? 12. ________A. y � Arcsin x B. y � Arcsin �12�x C. y � Arcsin 2xD. y � 2 Arcsin x

13. Evaluate cos �Tan�1 ��33�� Sin�1 �12��. 13. ________

A. ��23�� B. �3� C. 0 D. �2

1�

Kala is jumping rope, and the rope touches the ground every timeshe jumps. She jumps at the rate of 40 jumps per minute, and thedistance from the ground to the midpoint of the rope at itshighest point is 5 feet. At t � 0 the height of the midpoint is zero.14. Write a function with phase shift 0 for the height of the midpoint 14. ________

of the rope above the ground after t seconds.A. h � 2.5 cos (3�t) � 2.5 B. h � 2.5 sin (3�t) � 2.5C. h � �2.5 cos ��43

��t� � 2.5 D. h � �2.5 sin ��43��t� � 2.5

15. Use your function to find the height of the midpoint of her rope 15. ________after 32 seconds.A. 4.2 ft B. 2.5 ft C. 0.33 ft D. 3.75 ft

16. Find the values of x for which the equation cos x � �1 is true. 16. ________Let k represent an integer.A. 2�k B. ��2� � 2�k C. � � 2�k D. �32

�� 2�k

17. What is the equation of the graph 17. ________shown at the right?A. y � tan x B. y � cot xC. y � cot 2x D. y � tan 2x

18. What is the equation of the graph 18. ________shown at the right?A. y � 3 cos �23

x� B. y � 3 cos �34x�

C. y � �3 sin �34x� D. y � �3 sin �23

x�

19. What is the equation of the graph 19. ________shown at the right?A. y � tan 2x B. y � tan �2x � ��2��C. y � cot 2x D. y � cot �2x � ��2��

20. State the domain and range of the relation y � arcsin x. 20. ________A. D:{all reals}; R: ��2� � y � ��2�� B. D: {all reals}; R: {0 � y � �}

C. D: {�1 � x � 1}; R: {all reals} D. D: {�1 � x � 1}; R: ��2� � y � ��2��Bonus Evaluate cos �Arctan �34��. Bonus: ________

A. �275� B. �22

45� C. �45� D. �5

3�



6C

D

C

D

C

A

C

B

C

C

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NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right of each problem.1. Change �54� to radian measure in terms of �. 1. ________

A. �51�2� B. ��51

�2� C. ��31

�0� D. �1

30��


A. 190� B. 216� C. 67� D. 65.8�

3. Determine the angular velocity if 29 revolutions are completed 3. ________in 2 seconds.A. 14.5 radians/s B. 45.6 radians/sC. 91.1 radians/s D. 143.1 radians/s

4. Determine the linear velocity of a point rotating at an angular 4. ________velocity of 15 � radians per second at a distance of 12 feet from the center of the rotating object.A. 565.5 ft /s B. 56.5 ft /s C. 180 ft /s D. 3.9 ft /s

5. Each roller under a conveyor belt has a radius of 0.5 meters. The 5. ________rollers turn at a rate of 30 revolutions per minute. What is the linear velocity of the conveyor belt?A. 94.25 m/s B. 1.57 m/s C. 6.28 m/s D. 4.71 m/s

6. Find the degree measure of the central angle associated with an 6. ________arc that is 13.8 centimeters long in a circle with a radius of 6 centimeters.A. 2.3� B. 414� C. 131.8� D. 65.9�

7. Find the area of a sector if the central angle measures 30� and 7. ________the radius of the circle is 15 centimeters.A. 58.9 cm2 B. 117.8 cm2 C. 3.9 cm2 D. 7.9 cm2

8. Write an equation of the sine function with amplitude 5, period 3�, 8. ____________

and phase shift ��.A. y � 5 sin ��23

x� � �32�� B. y � 5 sin ��32

x� � �23��

C. y � 5 sin ��23x� � �� D. y � 5 sin ��23

x� � �23��

9. Write an equation of the tangent function with period ��4�, phase 9. ________shift �, and vertical shift 1.A. y � tan �4x � ��4�� 1 B. y � tan (4x � 4�) � 1

C. y � tan ��4x� � �� 1 D. y � tan (4x � 4�) � 1

10. State the amplitude, period, and phase shift of the function 10. ________y � �0.4 sin �10x � ��2��.A. 0.4; ��5�; ��2

�0� B. 0.4; ��5�; �2

�0� C. 0.4; 5�; �2

10� D. 0.4; ��5�; ��2

10�

11. State the period and phase shift of the function y � 3 tan �4x � ��3��. 11. ________

A. ��4�; ��3� B. 4�; �1�2� C. ��4�; ��1

�2� D. ��4�; �1

�2�

Chapter

6

133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 137


12. What is the equation for the inverse of y � Cos x � 1? 12. ________A. y � Arccos x B. y � Arccos (x � 1)C. y � 1 � Arccos x D. y � Arccos x � 1

13. Evaluate Cos�1 �tan ��4��. 13. ________

A. ��22�� B. � C. 0 D. �4

��

A tractor tire has a diameter of 6 feet and is revolving at a rate of 45 rpm. At t � 0, a certain point on the tread of the tire is at height 0.

14. Write a function with phase shift 0 for the height of the point 14. ________above the ground after t seconds.A. h � 3 cos ��32

��t� � 3 B. h � �3 cos ��32��t� � 3

C. h � 3 sin ��83��t� � 3 D. h � �3 sin ��83

��t� � 3

15. Use your function to find the height of the point after 1 minute. 15. ________A. 6 ft B. 3 ft C. 0 ft D. 1.5 ft

16. Find the values of x for which the equation cos x � 1 is true. 16. ________A. 2�n B. ��2� � 2�n C. � � 2�n D. �32

�� 2�n

17. What is the equation of the graph shown 17. ________at the right?A. y � tan x B. y � cot xC. y � cot 2x D. y � tan 2x

18. What is the equation of the graph 18. ________shown at the right?A. y � 2 cos �4

x� B. y � 2 cos 4xC. y � �2 sin �4

x� D. y � �2 sin 4x

19. What is the equation of the graph 19. ________shown at the right?A. y � tan ��2

x� � �� B. y � tan ��2x� � ��2��

C. y � tan ��4x� � �� D. y � tan ��4

x� � ��4��

20. State the domain and range of the relation y � arccos x. 20. ________A. D: {all real numbers}; R: ��2� � y � ��2��B. D: {all real numbers}; R: {0 � y � �}C. D: {�1 � x � 1}; R: {all real numbers}D. D: {�1 � x � 1}; R: ��2� � y � ��2��

Bonus Evaluate cot �Arctan �35��. Bonus: ________

A. �34� B. �43� C. �45� D. �35�



6

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NAME _____________________________ DATE _______________ PERIOD ________

1. Change �312� to radian measure in terms of �. 1. __________________

2. Change ��236

�� radians to degree measure. 2. __________________

3. Determine the angular velocity if 11.3 revolutions are 3. __________________completed in 3.9 seconds. Round to the nearest tenth.

4. Determine the linear velocity of a point rotating at 4. __________________15 revolutions per minute at a distance of 3.04 meters from the center of a rotating object. Round to the nearest tenth.

5. A gyroscope of radius 18 centimeters rotates 35 times 5. __________________per minute. Find the linear velocity of a point on the edge of the gyroscope.

6. An arc is 0.04 meters long and is intercepted by a central 6. __________________angle of ��8� radians. Find the diameter of the circle.

7. Find the area of sector if the central angle measures 225� 7. __________________and the radius of the circle is 11.04 meters.

8. Write an equation of the cosine function with amplitude �23�, 8. __________________period 1.8, phase shift �5.2, and vertical shift 3.9.

9. Write an equation of the cotangent function with period �32��, 9. __________________

phase shift �32��, and vertical shift ��43�.

10. State the amplitude, period, phase shift, and vertical shift 10. __________________for y � 2 � sin �3x � ��5��.

11. State the period, phase shift, and vertical shift for 11. __________________y � �2 � 3 tan (4x � �).

12. Write the equation for the inverse of y � 4 Arccot ��34x� � �23��. 12. __________________

13. Evaluate sin �Cos�1 ��22�� 4�. 13. __________________

Chapter

6

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The table shows the average monthly temperatures (°F) for Detroit, Michigan.

14. Write a sinusoidal function that models the 14. ___________________________monthly temperatures in Detroit, using t � 1 to represent January.

15. According to your model, what is the average 15. ___________________________temperature in October?

Graph each function.16. y � sin x for ��13

3�� x � ��83

�� 16.

17. y � cot x for ��32�� x � �2

�� 17.

18. y � ��52� cos (3x � �) 18.

19. y � csc �2x � ��2�� 3 19.

20. y � Arctan x 20.

Bonus Evaluate sin �2� � Arctan �152��. Bonus: __________________________



6

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.25.3� 27.1� 35.8� 48.2� 59.5� 69.1� 73.8� 72.1� 64.6� 53.4� 41.4� 30.2�

133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 140



NAME _____________________________ DATE _______________ PERIOD ________

1. Change 585� to radian measure in terms of �. 1. __________________

2. Change �113

�� radians to degree measure. 2. __________________

3. Determine the angular velocity if 12.5 revolutions are 3. __________________completed in 8 seconds. Round to the nearest tenth.

4. Determine the linear velocity of a point rotating at an 4. __________________angular velocity of 84� radians per minute at a distance of 2 meters from the center of the rotating object. Round to the nearest tenth.

5. The minute hand of a clock is 7 centimeters long. Find the 5. __________________linear velocity of the tip of the minute hand.

6. An arc is 21.4 centimeters long and is intercepted by a 6. __________________central angle of �38

�� radians. Find the diameter of the circle.

7. Find the area of a sector if the central angle measures 7. __________________�31

�0� radians and the radius of the circle is 52 centimeters.

8. Write an equation of the cosine function with amplitude 4, 8. __________________period 6, phase shift ��, and vertical shift �5.

9. Write an equation of the tangent function with period 2�, 9. __________________phase shift ��4�, and vertical shift �1.

10. State the amplitude, period, phase shift, and vertical shift 10. __________________for y � �4 cos ��2

x� � ��2�� 1.

11. State the period, phase shift, and vertical shift for 11. __________________y � cot ��4

x� � ��2�� 3.

12. Write the equation for the inverse of y � Arctan (x � 3). 12. __________________

13. Evaluate sin (2 Tan�1 �3�). 13. __________________

Chapter

6�1

43��

660°

9.8 radians/s

527.8 m/min

0.7 cm/min

about 36.3 cm

about 1274 cm2

y � �4 cos ��3x� � ��

32

�� 5

y � tan ��2x� � ��

8�� 1

4; 4�; �� ; 1

4�; �2�; 3

y � Tan x � 3

��23��

133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 141


The average monthly temperatures in degrees Fahrenheit for theMoline, Illinois, Quad City Airport are given below.

14. Write a sinusoidal function that models the monthly 14. __________________temperatures in Moline, using t � 1 to represent January.

15. According to your model, what is the average temperature 15. __________________in May?


16. y � cos x for ��94�� x � �4

�� 16.

17. y � tan x for 6� � x � 8� 17.

18. y � �6 sin ��23x�� 18.

19. y � csc �x � ��4�� 2 19.

20. y � Arccos x 20.

Bonus Evaluate sec �Arccot �35��. Bonus: __________________



6

Jan. Feb. March April May June21.0� 25.7� 36.9� 50.4� 61.2� 70.9�

July Aug. Sept. Oct. Nov. Dec

74.8� 72.9� 64.6� 53.2� 38.8� 26.2�

133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 142



NAME _____________________________ DATE _______________ PERIOD ________


2. Change �51�2� radians to degree measure. 2. __________________

3. Determine the angular velocity if 88 revolutions are 3. __________________completed in 5 seconds. Round to the nearest tenth.

4. Determine the linear velocity of a point rotating at an 4. __________________angular velocity of 7� radians per second at a distance of 10 feet from the center of the rotating object. Round to the nearest tenth.

5. The second hand of a clock is 10 inches long. Find the linear 5. __________________velocity of the tip of the second hand.

6. Find the length of the arc intercepted by a central angle of 6. __________________��8� radians on a circle of radius 8 inches.

7. Find the area of a sector if the central angle measures 7. __________________��3� radians and the radius of the circle is 9 meters.

8. Write an equation of the sine function with amplitude 2, 8. __________________period 5�, phase shift ��2�, and vertical shift 3.

9. Write an equation of the tangent function with period 3�, 9. __________________phase shift ��, and vertical shift 2.

10. State the amplitude, period, phase shift, and vertical shift 10. __________________for y � �32� sin �2x � ��4��.

11. State the period, phase shift, and vertical shift for 11. __________________y � tan (3x � �) � 2.

12. Write the equation for the inverse of y � Arccos (x � 5). 12. __________________

13. Evaluate tan ��12� Cos�1 0�. 13. __________________

Chapter

6

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The average monthly temperatures in degrees Fahrenheit for the Spokane International Airport, in Washington, are given below.

14. Write a sinusoidal function that models the 14. ____________________monthly temperatures in Spokane, using t � 1 to represent January.

15. According to your model, what is the average 15. ____________________temperature in March?

Graph each function.16. y � cos x for ��2� � x � �52

�� 16.

17. y � tan x for ��52�� x � �2

�� 17.

18. y � 3 sin ��23x�� 18.

19. y � sec (x � �) � 3 19.

20. y � Arcsin x 20.

Bonus Evaluate tan �Arccos �23��. Bonus: ____________________



6


133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 144



NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear,concise solution to each problem. Be sure to include all relevantdrawings and justify your answers. You may show your solution inmore than one way or investigate beyond the requirements of theproblem.1. a. Explain what is meant by a sine function with an amplitude of 3.

Draw a graph in your explanation.

b. Explain what is meant by a cosine function with a period of �.Draw a graph in your explanation.

c. Explain what is meant by a tangent function with a phase shiftof ��4�. Draw a graph in your explanation.

d. Choose an amplitude, a period, and a phase shift for a sinefunction. Write the equation for these attributes and graph it.

2. a. Write an equation to describe the motion of point A as the wheel shown at the right turns counterclockwise in place. The wheel completes a revolution every 20 seconds.

b. How would the equation change if the radius of the wheel were 5 inches?

c. How would the equation change if the wheel completed a revolution every 10 seconds?

d. How would the equation change if point A were at the top ofthe circle at t � 0?

3. Give an example of a rotating object. (Engineering and thesciences are good sources of examples.) What is the angularvelocity of the object? Choose a point on the object and find itslinear velocity.

Chapter

6

133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 145


1. Change �42� to radian measure in terms of �. 1. __________________

2. Change �41�5� radians to degree measure. 2. __________________

3. Given a central angle of 76.4�, find the length of the 3. __________________intercepted arc in a circle of radius 6 centimeters.Round to the nearest tenth.

4. Find the area of a sector if the central angle measures 4. __________________�71

�2� radians and the radius of the circle is 2.6 meters.

Round to the nearest tenth.

5. A belt runs a pulley that has a diameter of 12 centimeters. 5. __________________If the pulley rotates at 80 revolutions per minute, what is its angular velocity in radians per second and its linear velocity in centimeters per second?

6. Graph y � cos x for �72�� x � 5�. 6.

7. Determine whether the 7. __________________graph represents y � sin x,y � cos x, or neither. Explain.

State the amplitude and period for each function.8. y � �83� cos �65

�� 8. __________________

9. y � �2.3 sin �56�� 9. __________________

10. Write an equation of the sine function with amplitude 5 10. __________________and period �56

��.



6

133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 146


2. Change ��1�2� radians to degree measure. 2. __________________

3. Find the area of a sector if the central angle measures 66� 3. __________________and the radius of the circle is 12.1 yards. Round to the nearest tenth.

4. Determine the angular velocity if 57 revolutions are 4. __________________completed in 8 minutes. Round to the nearest tenth.

5. Determine the linear velocity of a point that rotates �51�8� 5. __________________

radians in 5 seconds and is a distance of 10 centimeters from the center of the rotating object.

Graph each function for the given interval.1. y � cos x, 3� � x � �12

3�� 1.

2. y � sin x, �8� � x � �5� 2.

State the amplitude and period for each function.3. y � ��32� cos 5� 3. __________________

4. y � 0.7 sin �32�� 4. __________________

5. Write an equation of the sine function with amplitude �85� 5. __________________and period �53

��.


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

6

Chapter

6

133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 147

1. State the phase shift and vertical shift for y � �3 cos ��3�� 2��. 1. __________________

2. Write an equation of a cosine function with amplitude 2.4, 2. __________________period 8.2, phase shift ��3�, and vertical shift 0.2.

3. Graph y � �12�x � sin x. 3.

The average monthly temperature in degrees Fahrenheit for the city of Wichita, Kansas, are given below.

4. Write a sinusoidal function that models the monthly 4. __________________temperatures, using t � 1 to represent January.

5. According to your model, what is the average monthly 5. __________________temperature in August?

1. Graph y � sec x for � � x � 4�. 1.

2. Graph y � tan �2x � ��2�� for 0 � x � 2�. 2.

3. Write an equation for a cotangent function with period ��3�, 3. __________________phase shift ��1

�2�, and vertical shift �4.

4. Write the equation for the inverse of y � Arcsin �2x�. 4. __________________

5. Evaluate tan �Sin�1 ��23�� Cos�1 ��2

3��. 5. __________________


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

6

Chapter

6


133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 148



NAME _____________________________ DATE _______________ PERIOD ________


Multiple Choice1. Find cos � if sin � � ��13� and

tan � 0. EA ��3

2�� B �23�

C ��13� D ��32��

E ��2�3

2��

2. Evaluate (2 sin �)(cos �) if � � 150�.A ��3� B �3� D

C ��23�� D ��2

3��

E None of these

3. If the product of (1 � 3), (3 � 7), and(7 � 13) is equal to two times the sumof 12 and x, then x � AA �36B 48C �60D �108E 12

4. From which of the following statements can you determine that m n? EI. 2m � n 12 II. m � n � 7

A Both I alone and II aloneB Neither I nor II, nor I and II

togetherC I alone, but not IID II alone, but not IE I and II together, but neither alone

5. Solve sin2 x � 1 � 0 for 0� � x � 360�.A 90� EB 180�C 270�D All of the aboveE Both A and C

6. At a point on the ground 27.6 metersfrom the foot of a flagpole, the angle ofelevation to the top of the pole is 60�.What is the height of the flagpole?A 15.9 m BB 47.8 mC 23.9 mD 13.8 mE None of these

7. �ABC is an isosceles triangle, and the coordinates of two vertices areA(�3, 2) and B(1, �2). What are thecoordinates of C? DA (4, 3)B (�1, 3)C (�3, �1)D (3, 4)E None of these

8. Determine the coordinates of Q, anendpoint of P�Q�, given that the otherendpoint is P(�2, 4) and the midpointis M(1, 5). CA (4, 14)B (0, 6)C (4, 6)D ��12�, �92��E (5, 6)

9. �(xx2

2

yy3

7

)2� � C

A �yx

2

2� B �xy

�

C �xy2� D �x

12�

E y

10. If kx2 � k2, for what values of k willthere be exactly two real values of x?A All values of k DB All values of k � 0C All values of k � 0D All values of k 0E None of these

Chapter

6

133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 149


11. If cot � � �185� and cos � � 0, evaluate

�1� ��1c�7o�s��.

A �157�

B �5�17

1�7��

C �1875�

D 5

E ��11�79�

�

12. In �ABC shown below, A � 30�,C � 60�, and AC � 10. Find BD.A �2

5�

B �5�

23�

�

C ��23��

D �21�

E ��33��

13. For what value of k will the line 3x � ky � 8 be perpendicular to theline 4x � 3y � 6?A 6B 4C 3D �3E �4

14. Which line is a perpendicular bisectorto A�B� with endpoints A(1, 3) andB(1, �2)?A x � 2y � 1B 2x � 2y � 1C y � 1D x � �2

1�

E None of these

15. For any integer k, which of the following represents three consecutive even integers?A 2k, 4k, 6kB k, k � 1, k � 2C k, k � 2, k � 4D 4k, 4k � 1, 4k � 2E 2k, 2k � 2, 2k � 4

16. If y varies directly as the square of x, what will be the effect on y ofdoubling x?A y will doubleB y will be half as largeC y will be 4 times as largeD y will decrease in sizeE None of these

17–18. Quantitative ComparisonA if the quantity of Column A is




Column A Column B17. k is a positive integer.

18. 4(x � y) � 24 and 3x � y � 6

19–20. Refer to the figure below.

19. Grid-In Find the value of a.

20. Grid-In Find the value of b � c.



6

(�1)2k (�1)2k � 1

x y

133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 150



NAME _____________________________ DATE _______________ PERIOD ________

1. State the domain of [ ƒ ° g](x) for ƒ(x) = �8x� and g(x) = x � 8. 1. __________________

2. State whether the linear programming problem 2. __________________represented by the graph below is infeasible, is unbounded,has an optimal solution, or has alternate optimal solutions for finding a minumum.

�ABC has vertices at A(-2, 3), B(1, 7), and C(4, -3). Find thecoordinates of the vertices of the triangle after each of thefollowing transformations.

3. dilation of scale factor 3 3. __________________

4. reflection over the y-axis 4. __________________

5. rotation of 270° counterclockwise about the origin 5. __________________

6. Find the value of k so that the remainder of 6. __________________(3x3 � 10x2 � kx � 6) � (x � 3) is 0.

7. Solve 3 2x � 4 � 20. 7. __________________

8. Find the area of �ABC if A � 24.4°, B � 56.3°, and 8. __________________c � 78.4 centimeters.

9. Find the area of �ABC if a � 15 inches, b � 19 inches, 9. __________________and c � 24 inches.

10. An oar floating on the water bobs up and down, covering 10. __________________a distance of 12 feet from its lowest point to its highest point. The oar moves from its lowest point to its highest point and back to its lowest point every 15 seconds.Write a cosine function with phase shift 0 for the height of the oar after t seconds.

Chapter

6x � 8

infeasible

(�6, 9), (3, 21),(12, �9)

(3, 2), (7, �1),(�3, �4)

(2, 3), (�1, 7),(�4, �3)

5

��43

� � x � �136�

1070 cm2

142.5 in2

h � �6 cos �125��t

133-151 A&E C06-0-02-834179 10/4/00 2:48 PM Page 151

Form 1BPage 133

1. A

2. B

3. D

4. C

5. C

6. A

7. B

8. D

9. B

10. B

11. A

12. D

Page 134

13. B

14. C

15. B

16. D

17. C

18. A

19. D

20. A

Bonus: D

Page 135

1. D

2. B

3. B

4. A

5. D

6. D

7. A

8. D

9. C

10. A

11. B

Page 136

12. C

13. D

14. C

15. D

16. C

17. A

18. C

19. B

20. C

Bonus: C


Chapter 6 Answer KeyForm 1A

152-158 A&E C06-0-02-83417 10/4/00 2:50 PM Page 152

Form 1C Form 2A



Page 137

1. C

2. B

3. C

4. A

5. B

6. C

7. A

8. D

9. B

10. A

11. D

Page 138

12. B

13. C

14. B

15. C

16. A

17. A

18. D

19. B

20. C

Bonus: D

Page 139

1. ��2165��

2. �690�

3.18.2 radians/s

4. 286.5 m/min

5.3958.4 cm/min

6. 0.2 m

7. 239.3 m2

8.

9. y � cot ��23x� � ��

34�

10. 1; �23��; �

1�5�; 2

11. �4�

�; ��4�

�; �2

12. y � �43

��23

� � Cot �4x��

13. 1

Page 140

14. Sample answer: y �

24.25 sin ��6�t� � �2

3�� 49.55

15. Sample answer:49.55�

16.

17.

18.

19.

20.

Bonus: 12�13

Sample answer:

y �

��23

� cos ��0�.x9� � 5.78� �

� 3.9

152-158 A&E C06-0-02-83417 10/4/00 2:50 PM Page 153

Form 2CForm 2B



Page 141

1. �134

��

2. 660�

3. 9.8 radians/s

4. 527.8 m/min

5. 0.7 cm/min

6. about 36.3 cm

7. about 1274 cm

y �8. �4 cos ��

�3x� � �

�3

2�� 5

y �

9. tan ��2x� � �

�8

�� 1

10. 4; 4� ; �� ; 1

11. 4� ; �2� ; 3

12. y � Tan x � 3

13. ��23��

Page 14214. Sample answer:

y � 26.9 sin ��6

�t � �23��

� 47.9

15. Sample answer:61.4�

16.

17.

18.

19.

20.

Bonus: ��33�4��

Page 142

1. �94��

2. 75�

3. 110.6 radians/s

4. 219.9 ft/s

5.

6. about 3.1 in.

7. about 42.4 m2

y � �2 sin8. ��2

5x� � �

�5

�� 3

9. y � tan ��3x� � �

�3

�� 2

10. �32

�; � ; ��2

�; 0

11. ��3

�; ��3

�; �2

12. y � Cos x � 5

13. 1

Page 14414. Sample answer:

y � 20.8 sin ��6

� t � �23��

� 47.9

15. Sample answer: 37.5�

16.

17.

18.

19.

20.

Bonus: ��25��

1.05 in./s or62.8 in./min

152-158 A&E C06-0-02-83417 10/4/00 2:50 PM Page 154




3 Superior • Shows thorough understanding of the concepts amplitude, period, and phase shift of a graph.

• Uses appropriate strategies to model motion of point on wheel.• Computations are correct.• Written explanations are exemplary.• Example and analysis of rotating object are appropriate and make sense.

• Graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.

2 Satisfactory, • Shows understanding of the concepts amplitude, period,with Minor and phase shift of a graph.Flaws • Uses appropriate strategies to model motion of point on wheel.

• Computations are mostly correct.• Written explanations are effective.• Example and analysis of rotating object are appropriate and make sense.

• Graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts amplitude, Satisfactory, period, and phase shift of a graph.with Serious • May not use appropriate strategies to model motion of point Flaws on wheel.

• Computations are mostly correct.• Written explanations are satisfactory.• Example and analysis of rotating object are mostly appropriate and sensible.

• Graphs are mostly accurate and appropriate.• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts amplitude, period, and phase shift of a graph.

• May not use appropriate strategies to model motion of point on wheel.

• Computations are incorrect.• Written explanations are not satisfactory.• Example and analysis of rotating object are not appropriate and sensible.

• Graphs are not accurate and appropriate.• Does not satisfy requirements of problems.

152-158 A&E C06-0-02-83417 10/4/00 2:50 PM Page 155

Page 145

1a. An amplitude of 3 for a sinefunction means that the yvalues of the function varybetween �3 and �3, as shownin the graph.

1b. A period for � for a cosinefunction means it takes � unitsalong the x-axis for thefunction to complete onecycle, as shown in the graph.

1c. A phase shift of � ��4

� for a tangent function means the graph has shifted ��

4� units

to the left, as shown in the graph.

1d. Sample answer: amplitude 2,period 4�, and phase shift ��

y � 2 sin ��12

�x � ��2

��

2a. y � 10 sin �1�0�t

�2x��

210�, x � �

1�0� rev/s

2b. The amplitude would changefrom 10 to 5. y � 5 sin �

1�0�t

2c. The period would change

from �1�0� to �

�5

� .

y � 10 sin ��5

�t�2x��

110�, x � �

�5

� rev/s

2d. Sample answer: The functionwould change from sine tocosine.y � 10 cos �

1�0�t

3. Earth is a rotating object. Itrotates once every 24 hours.Since 24 hours � 86,400seconds, the angular velocity is

� � ��t� � �

8624�00� � 0.000277 radian/s.

The radius of Earth is about 6400kilometers. The linear velocity ofa point at the equator is v � �r

t��

68460,04(020�)

� � 0.465 km/s.



152-158 A&E C06-0-02-83417 10/4/00 2:50 PM Page 156



1. ��73

�0�

2. 48�

3. 8.0 cm

4. 6.2 m2

�83�� radians/s;

5. 50.3 cm/s

6.

neither; the maximum value

7. is 2, not 1

8. �83

�; �53��

9. 2.3; �125��

10. y � � 5 sin �152��

Quiz APage 147

1. �359

��

2. �15�

3. 84.3 yd2

4. 44.8 radians/min

5. 1.75 cm/s

Quiz BPage 147

1.

2.

3. �32

� ; �25��

4. 0.7; �43��

5. y � � �85

� sin �65��

Quiz CPage 148

1. 6� ; 0y � � 2.4 cos ��4

�.�1� � �

1�2.

2

3��

2. � 0.2

3.

4.

Sample answer:5. 77.1�

Quiz DPage 148

1.

2.

3. y � cot �3x � ��4

�� 4

4. y � 2 Sin x

5. undefined


Sample answer: y � 24.55 sin ��

�6

�t � �23�� 55.85

152-158 A&E C06-0-02-83417 10/4/00 2:50 PM Page 157

Page 149

1. E

2. D

3. A

4. E

5. E

6. B

7. D

8. C

9. C

10. D

Page 150

11. A

12. B

13. B

14. E

15. E

16. C

17. A

18. C

19. 15

20. 65

Page 151

1. x � 8

2. infeasible

(�6, 9), (3, 21),3. (12, �9)

(2, 3), (�1, 7), 4. (�4, �3)

(3, 2), (7, �1), 5. (�3, �4)

6. 5

7. � �43

� x �136�

8. 1070 cm2

9. 142.5 in2

10. h � �6 cos �21

�5� t



152-158 A&E C06-0-02-83417 10/4/00 2:50 PM Page 158



NAME _____________________________ DATE _______________ PERIOD ________


1. Find an expression equivalent to �secs�in

ta�n ��. 1. ________

A. sec2 � B. cot � C. tan2 � D. cos2 �

2. If csc � � ��54� and 180� � � � 270�, find tan �. 2. ________

A. ��43� B. �34� C. �43� D. ��54�

3. Simplify �tan2 �ta

cns

2c2

�� 1�. 3. ________

A. csc2 � B. �1 C. tan2 � D. 1

4. Simplify �seccoxs

�x

1� � �seccoxs

�x

1�. 4. ________

A. 2 tan2 x B. 2 cos x C. 2 cos2 x � 1 D. 2 cot2 x

5. Find a numerical value of one trigonometric function of x if 5. ________�tcaont x

x� � �scoesc x

x� � �cs2c x�.

A. csc x � 1 B. sin x � ��12� C. csc x � �1 D. sin x � �21�

6. Use a sum or difference identity to find the exact value of sin 255�. 6. ________

A. ��2�4� �6�� B. �

�6� �4

�2�� C. �

�6� �4

�2�� D. �

�2� �4

�6��

7. Find the value of tan (� � �) if cos � � ��35�, sin � � �153�, 7. ________

90� � � � 180�, and 90� � � � 180�.

A. �6536� B. ��65

36� C. ��35

36� D. �5

363�

8. Which expression is equivalent to cos (� � �)? 8. ________A. �cos � B. cos � C. �sin � D. sin �

9. Which expression is not equivalent to cos 2�? 9. ________A. cos2 � � sin2 � B. 2 cos2 � � 1C. 1 � 2 sin2 � D. 2 sin � cos �

10. If cos � � 0.8 and 270� � � � 360�, find the exact value of sin 2�. 10. ________A. �0.96 B. �0.6 C. 0.96 D. 0.28

11. If csc � � ��53� and � has its terminal side in Quadrant III, find the 11. ________exact value of tan 2�.

A. �2245� B. �2

75� C. �27

4� D. ��275�

Chapter

7

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 159


12. Use a half-angle identity to find the exact value of cos 165�. 12. ________A. �12� �2� �� 3�� B. ��12� �2� �� 3��

C. �12� �2� �� 2�� D. ��12� �1� �� 3��

13. Solve 4 sin2 x � 4�2� cos x � 6 � 0 for all real values of x. 13. ________A. �34

�� 2�k, �54�� 2�k B. ��4� � 2�k, �74

�� 2�k

C. ��4� � 2�k, �54�� 2�k D. �34

�� 2�k, �74�� 2�k

14. Solve 2 cos2 x � 5 cos x � 2 � 0 for principal values of x. 14. ________A. 0� and 30� B. 30� C. 60� D. 60� and 300�

15. Solve 2 sin x � �3� � 0 for 0 � x � 2�. 15. ________A. �43

�� x � �53�� B. �23

�� x� �43��

C. �76�� x � �11

6�� D. �56

�� x � �76��

16. Write the equation 2x � 3y � 5 � 0 in normal form. 16. ________

A. �2�13

1�3��x � �3�13

1�3��y � �5�13

1�3�� 0 B. ��2�13

1�3��x � �3�13

1�3��y � �5�13

1�3�� 0

C. ��2�13

1�3��x � �3�13

1�3��y � �5�13

1�3�� 0 D. �2�13

1�3��x � �3�13

1�3��y � �5�13

1�3�� 0

17. Write the standard form of the equation of a line for which the 17. ________length of the normal is 6 and the normal makes an angle of 120�with the positive x-axis.A. x � �3� y � 12 � 0 B. x � �3� y � 12 � 0C. �3� x � y � 12 � 0 D. �3� x � y � 12 � 0

18. Find the distance between P(�4, 3) and the line with equation 18. ________2x � 5y � �7.

A. �14

2�92�9�

� B. 0 C. ��162�92�9�� D. �16

2�92�9��

19. Find the distance between the lines with equations 3x � y � 9 and 19. ________y � 3x � 4.

A. �54� B. �5�

21�0�� C. ��2

1�0�� D. �13�2

1�0��

20. Find an equation of the line that bisects the obtuse angles formed by 20. ________the lines with equations 3x � y � 1 and x � y � �2.A. (3�2� � �1�0�) x � (�1�0� � �2�) y � 2�1�0� � �2� � 0B. (3�2� � �1�0�) x � (�1�0� � �2�) y � 2�1�0� � �2� � 0C. (3�2� � �1�0�) x � (�1�0� � �2�) y � 2�1�0� � �2� � 0D. (3�2� � �1�0�) x � (�1�0� � �2�) y � 2�1�0� � �2� � 0

Bonus If 90� � � � 180� and cos � � ��45�, find sin 4�. Bonus: ________

A. ��4285� B. �42

85� C. �36

3265� D. ��36

3265�



7

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 160



NAME _____________________________ DATE _______________ PERIOD ________


1. Find an expression equivalent to sec � sin � cot � csc �. 1. ________A. tan � B. csc � C. sec � D. sin �

2. If sec � � ��54� and 180� � � � 270�, find tan �. 2. ________

A. ��35� B. ��45� C. �34� D. �53�

3. Simplify �tatna

2

n�2�

�

1�. 3. ________

A. csc2 � B. �1 C. tan2 � D. 1

4. Simplify �tsainn

xx� � �co

1s x�. 4. ________

A. 2 tan2 x B. 2 cos x C. 2 cos x � 1 D. 2 sec x

5. Find a numerical value of one trigonometric function of 5. ________x if sec x cot x � 4.

A. csc x � �14� B. sec x � 4 C. sec x � �14� D. csc x � 4


A. ��2�4� �6�� B. ��6� �

4�2�� C. ��6� �

4�2�� D. ��2� �

4�6��

7. Find the value of tan (� ��) if cos � � �45�, sin � � ��153�, 7. ________

270� � � � 360�, and 270� � � � 360�.

A. �1663� B. ��16

63� C. ��53

63� D. �53

63�

8. Which expression is equivalent to cos (� � �)? 8. ________A. �cos � B. cos � C. �sin � D. sin �

9. Which expression is not equivalent to cos 2�? 9. ________A. 2 cos2 � �1 B. 1 �2 sin2 � C. cos2 � � sin2 � D. cos2 � �sin2 �

10. If sin � � 0.6 and 90� � � � 180�, find the exact value of sin 2�. 10. ________A. �0.6 B. �0.96 C. 0.96 D. 0.28

11. If cos � � ��35� and � has its terminal side in Quadrant II, find 11. ________the exact value of tan 2�.A. �22

45� B. �2

75� C. �27

4� D. ��275�

12. Use a half-angle identity to find the exact value of cos 75�. 12. ________

A. �12��2� �� 3�� B. �12��2� �� 3�� C. �12��2� �� 2�� D. ��12��1� �� 3��

Chapter

7

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13. Solve csc x � 2 � 0 for 0 � x � 2�. 13. ________

A. ��6� and �56�� B. ��6� and �76

�� C. �43�� and �53

�� D. �76�� and �16

1��

14. Solve 2 cos2 x � cos x � 1 � 0 for principal values of x. 14. ________A. 0� and 120� B. 30� C. 60� D. 0� and 60�

15. Solve 4 sin2 x � 4 sin x � 1 � 0 for all real values of x. 15. ________A. ��6� � 2�k, �76

�� 2�k B. �76�� 2�k, �11

6�� 2�k

C. �56�� 2�k, �11

6�� 2�k D. ��6� � 2�k, �56

�� 2�k

16. Write the equation 3x�2y � 7 � 0 in normal form. 16. ________

A. �3�13

1�3��x � �2�13

1�3��y � �7�13

1�3�� 0 B. ��3�13

1�3��x � �2�13

1�3��y � �7�13

1�3�� 0

C. ��3�13

1�3��x � �2�13

1�3��y � �7�13

1�3�� 0 D. �3�13

1�3��x � �2�13

1�3��y � �7�13

1�3�� 0

17. Write the standard form of the equation of a line for which the 17. ________length of the normal is 3 and the normal makes an angle of 135�with the positive x-axis.A. �2�x ��2�y � 6 � 0 B. �2�x � �2�y � 6 � 0C. �2�x ��2�y � 6 � 0 D. �2�x � �2�y � 6 � 0

18. Find the distance between P(�2, 5) and the line with 18. ________equation x �3y � 4 � 0.

A. �171�01�0�

� B. 0 C. ��171�01�0�

� D. �131�01�0�

�

19. Find the distance between the lines with equations 19. ________5x � 12y � 12 and y � ��1

52� x � 3.

A. �2143� B. �41

83� C. ��41

83� D. �1

478�

20. Find an equation of the line that bisects the acute angles formed 20. ________by the lines with equations 2x � y � 5 � 0 and 3x � 2y � 6 � 0.A. (2�1�3� � 3�5�)x � (�1�3� � 2�5�)y � 5�1�3� � 6�5� � 0B. (�2�1�3� � 3�5�)x � (��1�3� � 2�5�)y � 5�1�3� � 6�5� � 0C. (�2�1�3� � 3�5�)x � (��1�3� � 2�5�)y � 5�1�3� � 6�5� � 0D. (�2�1�3� � 3�5�)x � (��1�3� � 2�5�)y � 5�1�3� � 6�5� � 0

Bonus If 90� � � � 180�, express cos � in terms of tan �. Bonus: ________

A. ��1� �� t1�a�n�2�� B. ��1� �� t

1�a�n�2�� C. �1� �� t�a�n�2�� D. ��1� �� t�a�n�2��



7D

A

D

B

A

D

A

B

A

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NAME _____________________________ DATE _______________ PERIOD ________


1. Find an expression equivalent to �csoins

��. 1. ________

A. tan � B. cot � C. sec � D. csc �

2. If sec � � �54� and 0� � � � 90�, find sin �. 2. ________

A. ��35� B. ��45� C. �34� D. �53�

3. Simplify �1 �

tanse

2c�

2 ��. 3. ________

A. csc2 � B. �1 C. tan2 � D. 1

4. Simplify �sec

12 x� � �

csc12 x�. 4. ________

A. 2 tan2 x B. 2 cos x C. 1 D. 2 cot2 x

5. Find a numerical value of one trigonometric function of x if 5. ________sin x cot x � �4

1�.

A. cos x � �41� B. sec x � �4

1� C. csc x � 4 D. cos x � 4


A. ��2�4��6�� B. �

�6� �

4�2�� C. �

�6� �

4�2�

� D. ��2� �

4�6�

�

7. Find the value of tan (� � �) if cos � � �35�, sin � � �153�, 0� � � � 90�, 7. ________

and 0� � � � 90�.

A. �6536� B. �61

36� C. �31

36� D. �35

36�

8. Which expression is equivalent to cos (� � 2�)? 8. ________A. �cos � B. sin � C. cos � D. �sin �

9. Which expression is equivalent to cos 2� for all values of �? 9. ________A. cos2 � � sin2 � B. cos2 � � 1C. 1 � sin2 � D. 2 sin � cos �

10. If cos � � 0.8 and 0� � � � 90�, find the exact value of sin 2�. 10. ________A. 9.6 B. 2.8 C. 0.96 D. 0.28

11. If sin � � �35� and � has its terminal side in Quadrant II, find the exact 11. ________value of tan 2�.

A. �2245� B. ��22

45� C. �27

4� D. ��274�

Chapter

7

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 163


12. Use a half-angle identity to find the exact value of sin 105�. 12. ________

A. ��12��2� �� 3�� B. �12��2� �� 3��C. �12��2� �� 2�� D. ��12��1� �� 3��

13. Solve 2 cos x � 1 � 0 for 0 � x � 2�. 13. ________A. ��6� and �56

�� B. ��3� and �53�� C. ��3� and �23

�� D. �76�� and �16

1��

14. Solve 2 sin2 x � sin x � 0 for principal values of x. 14. ________A. 60� and 120� B. 0� and 150� C. 0� and 30� D. 60�

15. Solve cos x tan x � sin2 x � 0 for all real values of x. 15. ________A. �k, ��2� � 2�k B. ��2� � �k, 2�k

C. ��2� � 2�k, �32�� 2�k D. �k, ��4� � 2�k

16. Write the equation 3x � 4y � 7 � 0 in normal form. 16. ________A. �35�x � �45�y � �7

5� � 0 B. ��35�x � �45�y � �7

5� � 0

C. ��35�x � �45�y � �75

� � 0 D. �35�x � �45�y � �75

� � 0

17. Write the standard form of the equation of a line for which the length 17. ________of the normal is 4 and the normal makes an angle of 45� with the positive x-axis.A. �2�x � �2�y � 8 � 0 B. 2x � 2y � 8 � 0C. �2�x � �2�y � 8 � 0 D. 2x � 2y � 8 � 0

18. Find the distance between P(�2, 1) and the line with equation 18. ________x � 2y � 4 � 0.

A. �4�

55�

� B. 0 C. � �4�

55�

� D. �45

�

19. Find the distance between the lines with equations 3x � 4y � 8 and 19. ________y � �34�x � 4.

A. �85� B. � �274� C. �27

4� D. �254�

20. Find an equation of the line that bisects the acute angles formed by 20. ________the lines with equations 4x � y � 3 � 0 and x � y � 2 � 0.A. (4�2� � �1�7�)x � (�2� � �1�7�)y � 3�2� � 2�1�7� � 0B. (�2� � �1�7�)x � (4�2� � �1�7�)y � 3�2� � 2�1�7� � 0C. (4�2� � �1�7�)x � (�2� � �1�7�)y � 3�2� � 2�1�7� � 0D. (�2� � �1�7�)x � (4�2� � �1�7�)y � 3�2� � 2�1�7� � 0

Bonus If 90� � � � 180�, express sin � in terms of cos �. Bonus: ________A. ��1� �� c�o�s2� �� B. ��1� �� c�o�s2� ��C. �1� �� c�o�s2� �� D. �1� �� c�o�s2� ��



7

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NAME _____________________________ DATE _______________ PERIOD ________

1. Simplify (sec � � tan �)(1 � sin �). 1. __________________

2. If tan � � ��34� and 90� � � � 180�, find sec �. 2. __________________

3. Simplify . 3. __________________

4. Simplify sin � � cos � tan �. 4. __________________

5. If �1 �

setcan

x

2 x� � sin2 x � �sec

12 x�, find the value of cos x. 5. __________________

6. Use a sum or difference identity to find the exact 6. __________________value of sin 285�.

7. Find the value of sin (� � �) if cos � � �1157�, cot � � �27

4�, 7. __________________________0� � � � 90�, and 0� � � � 90�.

8. Simplify cos ��2� � ��. 8. __________________

9. If sec � � 4, find the exact value of cos 2�. 9. __________________

10. If cos � � 0.6 and 270� � � � 360�, find the exact 10. __________________value of sin 2�.

11. If sin � � � �45� and � has its terminal side in Quadrant III, 11. __________________find the exact value of tan 2�.

sec2 ��tan � � cot2 � tan �

Chapter

7

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 165


12. Use a half-angle identity to find the exact value 12. __________________of cos 67.5�.

13. Solve 2 cos x � sin2 x � 2 � 0 for all real values of x. 13. __________________

14. Solve 2 cos2 x � �3� cos x for principal values of x. Express 14. __________________the solution(s) in degrees.

15. Solve 2 sin x � 1 � 0 for 0 � x � 2�. 15. __________________

16. Write the equation 3x � 2y � 4 � 0 in normal form. 16. __________________

17. Write the standard form of the equation of a line for which 17. __________________the length of the normal is 5 and the normal makes an angle of 120� with the positive x-axis.

18. Find the distance between P(3, �2) and the line with 18. __________________equation x � 2y � 3 � 0.

19. Find the distance between the lines with equations 19. __________________3x � y � 7 and y � �3x � 4.

20. Find an equation of the line that bisects the obtuse angles 20. __________________formed by the lines with equations 2x � y � 5 � 0 and 3x � 2y � 6 � 0.

Bonus If 180� � � � 270� and cos � � ��45�, find sin 4�. Bonus: __________________



7

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NAME _____________________________ DATE _______________ PERIOD ________

1. Simplify cos � tan2 � � cos �. 1. __________________

2. If cot � � ��34� and 90� � � � 180�, find sin �. 2. __________________

3. Simplify csc � � cot � cos �. 3. __________________

4. Simplify �11

��

csoins

2

2��

�. 4. __________________

5. If sin2 x sec x cot x � 3, find the value of csc x. 5. __________________

6. Use a sum or difference identity to find the exact value 6. __________________of cos 255�.

7. Find the value of sin (� � �) if tan � � �43�, cot � � �152�, 7. __________________

0� � � � 90�, and 0� � � � 90�.

8. Simplify sin ��2� � ��. 8. __________________

9. If � is an angle in the first quadrant and csc � � 3, find 9. __________________the exact value of cos 2�.

10. If sin � � �0.6 and 180� � � � 270�, find the exact 10. __________________value of sin 2�.

11. If cos � � �45� and � has its terminal side in Quadrant IV, 11. __________________find the exact value of tan 2�.

Chapter

7

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 167


12. Use a half-angle identity to find the exact value of 12. __________________cos 105�.

13. Solve tan x � �3� � 0 for 0 � x � 2�. 13. __________________

14. Solve 4 sin2 x � 1 � 0 for principal values of x. Express 14. __________________the solution(s) in degrees.

15. Solve cos4 x � 1 � 0 for all real values of x. 15. __________________


17. Write the standard form of the equation of a line for 17. __________________which the length of the normal is 7 and the normal makes an angle of 150� with the positive x-axis.

18. Find the distance between P(�1, 4) and the line with 18. __________________equation 4x � 2y � 3 � 0.

19. Find the distance between the lines with equations 19. __________________x � 2y � 3 and y � �12�x � 2.

20. Find an equation of the line that bisects the acute angles 20. __________________formed by the lines with equations 3x � y � 6 � 0 and 2x � y � 1 � 0.

Bonus Express �� in terms of sin �. Bonus: __________________tan2 ��sec2 � + cot2 � sec2 �



7

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NAME _____________________________ DATE _______________ PERIOD ________

1. Simplify �tsainn

��

�. 1. __________________

2. If cos � � ��45� and 90� � � � 180�, find cot �. 2. __________________

3. Simplify sec2 � � tan2 �. 3. __________________

4. Simplify �sinta

2

n�

2��

cos12 ��. 4. __________________

5. If tan x cos x � �21�, find the value of sin x. 5. __________________

6. Use a sum or difference identity to find the exact value 6. __________________of cos 15�.

7. Find the value of tan (� � �) if cos � � �153�, sin � � �35�, 7. __________________

0� � � � 90�, and 0� � � � 90�.

8. Simplify sin (� � �). 8. __________________

9. If � is an angle in the first quadrant and cos � � �12�, find 9. __________________the exact value of cos 2�.

10. If cos � � 0.6 and 0� � � � 90�, find the exact 10. __________________value of sin 2�.

11. If cos � � � �45� and � has its terminal side in Quadrant II, 11. __________________find the exact value of tan 2�.

Chapter

7

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 169


12. Use a half-angle identity to find the exact value of cos 22.5�. 12. __________________

13. Solve 2 sin x � 1 � 0 for 0 � x � 2�. 13. __________________

14. Solve tan x � �3� � 0 for principal values of x. Express 14. __________________the solution(s) in degrees.

15. Solve �scescc

xx� � 1 � 0 for all real values of x. 15. __________________


17. Write the standard form of the equation of a line for 17. __________________which the length of the normal is 9 and the normal makes an angle of 60� with the positive x-axis.

18. Find the distance between P(2, 3) and the line with 18. __________________equation 2x � 5y � 4 � 0.

19. Find the distance between the lines with equations 19. __________________2x � 2y � 5 and y � x � 1.

20. Find an equation of the line that bisects the acute angles 20. __________________formed by the lines with equations 3x � 4y � 5 � 0 and 5x � 12y � 3 � 0.

Bonus How are the lines that bisect the angles formed Bonus: __________________by the graphs of the equations 3x � y � 6 and x � 3y � 1 related to each other?



7

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NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answer. You may show your solution in more than one way or investigate beyond the requirements of the problem.

1. a. Verify that �1 �cos

si�n �

� � �1 �cos

si�n �� 0 is an identity.

b. Why is it usually easier to transform the more complicated side of the equation into the simpler side rather than the other way around?

c. Is the following method for verifying an identity correct? Why or why not? If not, write a correct verification.

sec A sin A � tan A

sec A sin A � �csoins A

A�

cos A sec A sin A � sin A

cos A �co1s A� sin A � sin A

sin A � sin A

2. a. Write the equation 2y � 3x � 6 in normal form. Then,find the length of the normal and the angle it makes with the positive x-axis. Explain how you determined the angle.

b. Find the distance from a point on the line in part a to the line with equation 6x � 4y � 16 � 0. Tell what the sign of the distance d means.

c. Will the sign of the distance from a point on the line with equation 6x � 4y � 16 � 0 to the line described in part abe the same as in part b? Why or why not?

d. When will the sign of the distance between two parallel lines be the same regardless of which line it is measured from?

Chapter

7

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 171


1. If csc A � 2, find the value of sin A. 1. __________________

2. If tan � � ��34� and 90� � � � 180�, find cos �. 2. __________________

3. Simplify csc x � cos x cot x. 3. __________________

4. Simplify �c1sc�

�

tatann2 �

��. 4. __________________

5. If tan x csc x � 3, find the value of cos x. 5. __________________

6. Use a sum or difference identity to find the exact value 6. __________________of sin 285�.

7. Find the value of tan (� � �) if csc � � �153�, tan � � �34�, 7. __________________

0� � � � 90�, and 0� � � � 90�.

8. If tan � � �34� and 180� � � � 270�, find the exact value 8. __________________of sin 2�.

9. If � is an angle in the first quadrant and csc � � 4, find the 9. __________________exact value of cos 2�.

10. Use a half-angle identity to find the exact value of sin 22.5�. 10. __________________



7

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 172

1. If sec � � 3, find the value of cos �. 1. __________________

2. If cot � � �43� and 180� � � � 270�, find csc �. 2. __________________

3. Simplify cot2 x sec2 x. 3. __________________

4. Simplify �11

�

�

ccoost2

2

�

��. 4. __________________

5. If sec � sin � � 2, find the value of cot �. 5. __________________

1. Use a sum or difference identity to find the exact value 1. __________________of cos 345 �.

2. Find the value of tan (� � �) if sin � � � �153�, cos � � �45�, 2. __________________

270� � � � 360�, and 270� � � � 360�.

3. If sec � � ��153� and 90� � � � 180�, find the exact 3. __________________

value of sin 2�.

4. If cos � � �45� and � has its terminal side in Quadrant IV, find 4. __________________the exact value of tan 2�.

5. Use a half-angle identity to find the exact value of sin 165�. 5. __________________


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

7

Chapter

7

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 173

1. Solve 2 sin x � 2 � 0 for 0 � x � 2�. 1. __________________

2. Solve 4 cos2 x � 3 � 0 for principal values of x. Express 2. __________________the solution(s) in degrees.

3. Solve tan x � 1 � 0 for all real values of x. 3. __________________

4. Write the equation x � 5y � 8 � 0 in normal form. 4. __________________

5. Write the standard form of the equation of a line for which 5. __________________the length of the normal is 3 and the normal makes an angle of 240� with the positive x-axis.

1. Find the distance between P(1, 4) and the line with 1. __________________equation x � 2y � 5 � 0.


3. Find the distance between the lines with equations 3. __________________2x � y � 5 and y � 2x � 3.

4. Find the distance between the lines with equations 4. __________________3x � 4y � 18 � 0 and y � ��34�x � 3.

5. Find an equation of the line that bisects the acute angles 5. __________________formed by the lines with equations x � 3y � 3 � 0 and x � 2y � 2 � 0.


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

7

Chapter

7

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 174



NAME _____________________________ DATE _______________ PERIOD ________


Multiple Choice1. In the figure below, the measure of

�A is 65�. If the measure of �C is�45� the measure of �A, what is the measure of �B? CA 50�B 52�C 63�D 65�E 68�

2. In the figure below, three lines intersect to form a triangle. Find thesum of the measures of the markedangles. CA 90�B 180�C 360�D 540�E It cannot be determined from the

information given.

3. If x � y � z, and x � y � 24 and x � 10,then z � AA �4 B 0C 4 D 8E 16

4. What are the roots of x2 � 169 � 0? EA 0, 169B 0, 13C 0, �13D 169, �169E 13, �13

5. If �ABC is equilateral, what is thevalue of x � ( y � z) � w?

A �60 AB 0C 20D 60E It cannot be determined from the

information given.

6. In the figure below, A�D� is parallel toB�C�. Find the value of x. E


information given.

7. Sin�1 ��23�� Cos�1 ��2

3�� DA 0B �6

��

C �3��

D �2��

E �

8. The lengths of the sides of a rectangleare 6 inches and 8 inches. Which of thefollowing can be used to find �, theangle that a diagonal makes with alonger side? CA sin � � �4

3�

B cos � � �43�

C tan � � �43�

D tan � � �34�

E cos � � �53�

9. Points A(�1, �2), B(2, 1), and C(4, �2)are vertices of parallelogram ABCD.What are the coordinates of D? BA (0, �4)B (1, �5)C (�1, �5)D (2, �5)E (2, �4)

Chapter

7

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 175

10. The vertices of a triangle are (2, 4),(7, 9), and (8, 2). Which of the followingbest describes this triangle?A scaleneB equilateralC rightD isoscelesE right, isosceles

11. In right �ABD, C�A� bisects �DAB.What is the value of x?A 20B 40C 70D 80E None of these

12. In the figure below, what is the valueof x in terms of y?A yB 2yC 180 � 2yD 180 � yE 360 � 2y

13. What is the greatest common factor of the terms in the expansion of 2(6x2y � 9xy3)(15a3x � 10ay2)?A 2B 6yC 10aD 30ayE None of these

14. If 5x � 4y � xy � 8 � 0 and x � 3 � 9,then 3 � y � A �19B �16C 8D 19E 22

15. Which of the following could be lengthsof the sides of a triangle?A 7, 8, 14B 8, 8, 16C 8, 9, 20D 9, 10, 100E 1, 50, 55

16. In the rectangle ABDC below, what isthe measure of � ACB?A 63�

B 53�

C 37�

D 45�





determined from the information given

Column A Column B

17. Side A�B� of triangle ABC is extendedbeyond B to point D.

18. Angles P, Q, and R are the angles of aright triangle.


19. Grid-In What is the value of x?

20. Grid-In What is the value of y?




7

The measure of �ABC

180� �m �P 90�

The measure of �DBC

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 176


Chapter 7 Cumulative Review (Chapters 1-7)

NAME _____________________________ DATE _______________ PERIOD ________

1. Find the standard form of the equation of the line that 1. __________________passes through (�1, 2) and has a slope of 3.

2. If A � � and B � � , find AB. 2. __________________

3. Given ƒ(x) � (x � 2)2 � 5, find ƒ�1(x). Then state whether 3. __________________ƒ�1(x) is a function.

4. If y varies inversely as the square of x and y � 18 when 4. __________________x � 3, find y when x � �9.

5. Write a polynomial equation of least degree with roots 3, 5. __________________2i, and �2i.

6. Use the Remainder Theorem to find the remainder when 6. __________________x2 � 5x � 2 is divided by x � 5. State whether the binomial is a factor of the polynomial.

7. Given the triangle at the 7. __________________right, find m� A to the nearest tenth of a degree if b � 12 and c � 16.

8. If a � 8, b � 11, and c � 13, find the area of �ABC to the 8. __________________nearest tenth.

9. State the amplitude, period, and phase shift for the graph 9. __________________of y � 3 sin(2x � 4).

10. Find the value of Cos�1�tan �4��. 10. __________________

11. Solve 4 sin2 x � 3 � 0 for principal values of x. 11. __________________Express the solution(s) in degrees.

12. Find the distance between P(2, 4) and the line with 12. __________________equation 2x � y � 5 � 0.

045

312

�14

�03

�2�2

Chapter

7

159-177 A&E C07-0-02-834179 10/4/00 2:52 PM Page 177

Form 1BPage 159

1. A

2. C

3. D

4. D

5. B

6. A

7. C

8. A

9. D

10. A

11. C

Page 160

12. B

13. B

14. C

15. A

16. D

17. A

18. D

19. C

20. C

Bonus: D

Page 161

1. B

2. C

3. A

4. D

5. D

6. C

7. B

8. A

9. C

10. B

11. C

12. B

Page 162

13. D

14. A

15. D

16. B

17. A

18. D

19. A

20. B

Bonus: A


Chapter 7 Answer KeyForm 1A

178-184 A&E C07-0-02-834179 10/4/00 3:21 PM Page 178

Form 1C Form 2A



Page 163

1. B

2. D

3. B

4. C

5. A

6. B

7. D

8. C

9. A

10. C

11. D

Page 164

12. B

13. B

14. C

15. A

16. D

17. C

18. B

19. D

20. A

Bonus: D

Page 165

1. cos �

2. ��45�

3. tan �

4. 2 sin �

5. 1

6. ��6�4� �2��

7. �48275

�

8. sin �

9. � �78

�

10. � �2245� or �0.96

11. � �274�

Page 166

12. �21��2� �� 2��

13. � � 2�k

14. 30� and 90�

15. �76�� x � �11

6��

16.

17. x � �3�y � 10 � 0

18. �4�5

5�� units

19. �3�10

1�0�� units

(2�1�3� � 3�5�)x �

(�1�3� � 2�5�)y �

20. 5�1�3� � 6�5� � 0

�632356� or

Bonus: 0.5376

�3�13

1�3��x � �2�13

1�3��y �

�4�13

1�3�� 0

178-184 A&E C07-0-02-834179 10/4/00 3:21 PM Page 179

Form 2CForm 2B



Page 167

1. sec �

2. �54�

3. sin �

4. cot2 �

5. �31�

6. ��2� �4

�6��

7. ��6156�

8. cos �

9. �97�

10. �225

4� or 0.96

11. � �274�

Page 168

12. �

13. ��3

�, �43��

14. �30�, 30�

15. �k

16.

17. �3�x � y � 14 � 0

18. �91�05��

19. ��55��

(3�5� � 2�1�0�)x �

(�5� � �1�0�)y �

20. 6�5� � �1�0� � 0

Bonus: sin2 �

Page 169

1. cos �

2. ��34�

3. 1

4. cos2 �

5. �21�

6. ��2� �4

�6��

7. ��1663�

8. sin �

9. ��21�

10. �225

4� or 0.96

11. � �274�

Page 170

12.

13. ��6

�, �56��

14. �60°

15. ��4

� � �k

16.

17. x � �3�y � 18 � 0

18. �7�29

2�9�� units

19. �3�4

2�� units

20. 7x � 56y � 25 � 0

They areperpendicularto each

Bonus: other.

�2� �� 2��2�2� �� 3��2

�2�29

2�9��x � �5�29

2�9��y �

�3�29

2�9�� 0

��31�3

1�3��x � �2�13

1�3��y �

�6�13

1�3�� 0

178-184 A&E C07-0-02-834179 10/4/00 3:21 PM Page 180




3 Superior • Shows thorough understanding of the concepts proof, identity, normal to a line, and distance from a point to a line.

• Uses appropriate strategies to prove identities and write equations in normal form.


2 Satisfactory, • Shows understanding of the concepts proof, identity, with Minor normal to a line, and distance from a point to a line.Flaws • Uses appropriate strategies to prove identities and

write equations in normal form.• Computations are mostly correct.• Written explanations are effective.• Graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts proof, Satisfactory, identity, normal to a line, and distance from a point to a line.with Serious • May not use appropriate strategies to prove identities and Flaws write equations in normal form.

• Computations are mostly correct.• Written explanations are satisfactory.• Graphs are mostly accurate and appropriate.• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts proof, identity, normal to a line, and distance from a point to a line.

• May not use appropriate strategies to prove identities and write equations in normal form.

• Computations are incorrect.• Written explanations are not satisfactory.• Graphs are not accurate and appropriate.• Does not satisfy requirements of problems.

178-184 A&E C07-0-02-834179 10/4/00 3:21 PM Page 181

Page 171

1a.�1

c�ossin�

�� 1 c

�ossin�

�� 0

� 0

�(c1o�s2

s�in

��c) c

oos2

s��

� � 0

�(1 � sin

0�) cos �� 0

0 � 0

1b. There are many ways to makea simple expressioncomplicated but few ways to simplify a complicatedexpression. Thus, it is easier to find the propersimplification.

1c. No, because in the third line ofthe attempted proof, theexpression has been treated asan equality by multiplying bothsides by cos A. A correctverifications follows.sec A sin A � tan A�co

1s A� sin A � tan A

�csoins

AA

� � tan Atan A � tan A

2a. ��

31�3��x � �

�21�3��y � �

�61�3�� 0 or

��3�13

1�3��x � �2�1 3

1�3��y � �6�13

1�3�� 0

Length of normal � ��

61�3��

or �6�13

1�3�� , tan � �

Since cos � � 0 and sin � � 0, � is in Quadrant II. The angle ofthe normal with the positive x-axis � 146�.

2b. The point at (0, 3) is on theline with equation 2y � 3x � 6.The distance from the point at(0, 3) to the line with equation6x � 4y � 16 � 0 is

d ��6(�0)

��3�6�

4(��3)

1��6�16� , or � �2�

131�3�� .

The negative sign indicatesthat the point and the originare on the same side of the line.

2c. No, because the origin and anypoint on the line with equation6x � 4y � 16 � 0 are onopposite sides of the line withequation 2y � 3x � 6.

2d. For parallel lines, d will havethe same sign only when theorigin is between the lines.

cos2 � � (1 � sin2 �)��(1 � sin �)cos �



�2�1 3

1�3��

��3�13

1�3��

� ��23�.

178-184 A&E C07-0-02-834179 10/4/00 3:21 PM Page 182



1. �21�

2. ��54�

3. sin x

4. cos �

5. �31�

6. ��6�4� �2��

7. �3536�

8. �2245�

9. �87�

10. �12

� �2� �� 2��

Quiz APage 173

1. �31�

2. � �35�

3. csc2 x

4. sin4 �

5. �21�

Quiz BPage 173

1. ��2� �4

�6��

2. � �3536�

3. ��112609

�

4. ��274�

5. �21��2� �� 3��

Quiz CPage 174

1. �32��

2. 30�, 150°

3. ��4�� k

4.

5. x � �3� y � 6 � 0

Quiz DPage 174

1. �2�5

5�� units

2. 0 units

3. �8�5

5�� units

4. 6 units(�5� � �1�0�)x �

(3�5� � 2�1�0�)y �

5. 3�5� � 2�1�0� � 0


��22�66��x � �5�

262�6��y �

�4�13

2�6�� 0

178-184 A&E C07-0-02-834179 10/4/00 3:21 PM Page 183

Page 175

1. C

2. C

3. A

4. E

5. A

6. E

7. D

8. C

9. B

Page 176

10. D

11. E

12. C

13. E

14. B

15. A

16. B

17. D

18. D

19. 70

20. 120

Page 177

1. 3x � y � 5 � 0

2. � �ƒ�1(x) �

3. 2 � �x� �� 5; no

4. 2

5. x3 � 3x2 � 4x � 12 � 0

6. �2; no

7. 41.4�

8. 43.8 units2

9. 13; � ; 2�

10. 0

11. �60�, 60�

12. �5�

532

85



178-184 A&E C07-0-02-834179 10/4/00 3:21 PM Page 184



NAME _____________________________ DATE _______________ PERIOD ________


1. The vector v� has a magnitude of 89.7 feet and a direction of 12° 48�. 1. ________Find the magnitude of its vertical component.A. 887.47 ft B. 19.87 ft C. 19.38 ft D. 87.58 ft

2. What is an expression for x� involving r�, s�, and t�? 2. ________A. �3r� � s� � t� B. �3r� � s� � t�C. 3r� � s� � t� D. 3r� � s� � t�

3. Find the ordered pair that represents the vector from 3. ________A(�4.3, �0.9) to B(�2.8, 0.2). Then find the magnitude of AB�.A. �1.5, 1.1�; 3.46 B. ��7.1, �0.7�; 7.13C. �1.5, 1.1�; 1.86 D. ��7.1, �1.1�; 7.18

4. Find the ordered triple that represents the vector from A(�1.4, 0.3, �7.2) 4. ________to B(0.4, �9.1, 8.2). Then find the magnitude of AB�.A. �1.8, �9.4, 15.4�; 18.13 B. ��1, �8.8, 1�; 8.91C. �1.8, �9.4, 15.4�; 12.33 D. ��1, �8.8, 1�; 8.80

5. Find an ordered pair to represent u� in u� � �34� w�� 2v� if w�� 23�, 4� 5. ________and v� � ��38�, �2�.A. ��14�, 7� B. ��54�, �1� C. ��14�, 4� D. ��54�, 7�

6. Find an ordered triple to represent x� in x� � �6 z� � �14�y� if y� � �2, 18, ��45�� 6. ________and z� � ��12�, �34�, ��16��.A. ��28

5�, 0, �45�� B. ��72�, 0, �45�� C. ��72�, 0, �65�� D. ��72�, �383�, �45��

7. Write MN� as the sum of unit vectors for M��34�, 5, �23�� and N�6, �9, �35��. 7. ________

A. �247� i� � 14 j� � �1

15� k� B. �24

1� i� � 14 j� � �115� k�

C. 9i� � �8290� j� � �94

3� k� D. �247� i� � 14 j� � �1

15� k�

8. Find the inner product of a� and bb� if a� � �4, �54�, ��13�� and 8. ________bb� � ��12�, �2, ��32��, and state whether the vectors are perpendicular.A. 5; no B. 5; yes C. 0; yes D. 0; no

9. Find the cross product of v� and w� if v� � ��13�, 4, ��38�� and w�� 6, ��45�, 4�. 9. ________

A. ��11507�, ��11

12�, ��31

556�� B. ��11

603�, �11

12�, ��31

556��

C. ��11507�, 3, ��31

556�� D. ��11

603�, ��11

12�, ��31

556��

10. Find the magnitude and direction of the 10. ________resultant vector for the diagram at the right.A. 8.2 N, 73° 35�B. 20 N, 18° 37�C. 6.5 N, 79° 7� D. 8.2 N, 83° 48�

11. A force F�1 of 35 newtons pulls at an angle of 15° north of due east. 11. ________A force F�2 of 75 newtons pulls at an angle of 55° west of due south.Find the magnitude and direction of the resultant force.A. 43.8 N, 54.1° west of due south B. 43.8 N, 39.1° west of due southC. 42.2 N, 54.1° west of due south D. 42.2 N, 27.4° west of due south

Chapter

8

185-203 A&E C08-0-02-834179 10/4/00 3:20 PM Page 185


Write a vector equation of the line that passes through point P and is parallel to a�. Then write parametric equations of the line.

12. P(�1, 3); a� � ��6, �1� 12. ________A. �x � 1, y � 3� � t��6, �1�; x � �1 � 6t, y � 3 � tB. �x � 1, y � 3� � t��6, �1�; x � 1 � 6t, y � �3 � tC. �x � 1, y � 3� � t��6, �1�; x � �1 � 6t, y � 3 � tD. �x � 1, y � 3� � t��6, �1�; x � 1 � 6t, y � �3 � t

13. P(0, 5); a� � �2, �9� 13. ________A. �x, y � 5� � t��2, 9�; x � �2t, y � 5 � 9tB. �x, y � 5� � t�2, �9�; x � 2t, y � 5 � 9tC. �x � 2, y �9� � t�0.5�; x � 2, y � �9 � 5tD. �x � 2, y � 9� � t�0, �5�; x � 2, y � �9 � 5t

14. Which graph represents a line whose parametric equations are 14. ________x � 2t � 4 and y � �t � 2?A. B. C. D.

15. Write parametric equations of �3x � �12�y � �23�. 15. ________

A. x � t; y � 6t � �43� B. x � t; y � 6t � �31�

C. x � t; y � 6t � �13� D. x � t; y � 6t � �34�

16. Write an equation in slope-intercept form of the line whose 16. ________parametric equations are x � ��12�t � �23� and y � t � �34�.

A. y � 2x � �172� B. y � 2x � �1

72� C. y � �2x ��1

72�D. y � �2x � �1

72�

Darius serves a volleyball with an initial velocity of 34 feet persecond 4 feet above the ground at an angle of 35°.17. What is the maximum height, reached after about 0.61 seconds? 17. ________

A. 2.14 ft B. 9.94 ft C. 5.94 ft D. 6.14 ft18. After how many seconds will the ball hit the ground if it landed 39 feet 18. ________

away and it is not to be returned?A. 1.2 B. 1.3 C. 1.4 D. 0.4

A triangular prism has vertices at A(2, �1, �1), B(2, 1, 4), C(2, 2, �1), D(�1, �1, �1), E(�1, 1, 4), and F(�1, 2, �1).19. Which image point has the coordinates (�3, 2, 1) after a translation 19. ________

using the vector ��5, 1, �3�?A. C� B. B� C. E� D. F�

20. What point represents a reflection of B over the yz-plane? 20. ________A. B�(�2, �1, 4) B. B�(�2, 1, 4)C. B�(�2, 2, �4) D. B�(�2, 1, �4)

Bonus Find the cross product of ��34� v� and �12� w� if v� � ��2, 12, �3� Bonus: ________and w� � ��7, 4, �6�.

A. ��425�, ��28

7�, ��527��B. ��62

3�, ��287�, ��52

7�� C. ��425�, �28

7�, ��527�� D. ��42

5�, ��287�, ��62

9��



8

185-203 A&E C08-0-02-834179 10/4/00 3:20 PM Page 186



NAME _____________________________ DATE _______________ PERIOD ________


1. The vector v� has a magnitude of 6.1 inches and a direction of 55°. Find 1. ________the magnitude of its vertical component.A. 5.00 in. B. 10.64 in. C. 7.45 in. D. 3.50 in.

2. What is an expression for x� involving r� and s�? 2. ________A. r� � 2s� B. �r� � 2s�C. r� � 2s� D. �r� � 2s�

3. Find the ordered pair that represents the vector from A(9, 2) to 3. ________B(�6, 3). Then find the magnitude of AB�.A. ��15, 1�; 15.03 B. �3, 5�; 5.83C. �15, �1�; 3.74 D. �3, 1�; 3.16

4. Find the ordered triple that represents the vector from A(�3, 5, 6) to 4. ________B(�6, 8, 6). Then find the magnitude of AB�.A. �3, �3, 0�; 4.24 B. ��9, 13, 12�; 19.85C. ��3, 3, 0�; 4.24 D. ��9, 3, 0�; 9.49

5. Find an ordered pair to represent u� in u� � 4w� � 2v� if w� � ��3, 4� 5. ________and v� � ��4, 0�.A. ��20, 16� B. ��4, 16� C. ��10, �8� D. ��22, 8�

6. Find an ordered triple to represent x� in x� � 3z� � 5y� if y� � �2, 11, �5� 6. ________and z� � ��2, 8, 6�.A. �4, 79, �7� B. ��16, �31, 43�C. ��2, 17, �1� D. �16, �7, �45�

7. Write MN� as the sum of unit vectors for M(�14, 8, 6) and N(7, 9, �2). 7. ________A. �7i� � j� � 8k� B. �7i� � j� � 8k�

C. 21i� � j� � 8k� D. 21i� � j� � 8k�

8. Find the inner product of a� and b� if a� � �4, �2, �2� and b� � ��7, �2, 4� 8. ________and state whether the vectors are perpendicular.A. 0; yes B. �32; yes C. �40; no D. �32; no

9. Find the cross product of v� and w� if v� � ��9, 4, �8� and w� � �6, �2, 4�. 9. ________A. ��54, �8, �32� B. �0, �12, �6�C. �32, 84, 42� D. ��6, �12, 0�

10. Find the magnitude and direction of the 10. ________resultant vector for the diagram at the right.A. 26.4 N; 51.8° B. 22.2 N; 58.8°C. 22.2 N; 38.8° D. 26.4 N; 31.8°

11. An 18-newton force acting at 56° and a 32-newton force acting at 124° 11. ________act concurrently on an object. What is the magnitude and direction of a third force that produces equilibrium on the object?A. 42.2 N; 100.7° B. 42.2 N; 280.7°C. 44.6 N; 36.5° D. 44.6 N; 216.5°

Chapter

8

185-203 A&E C08-0-02-834179 10/4/00 3:20 PM Page 187


Write a vector equation of the line that passes through point Pand is parallel to a�. Then write parametric equations of the line.12. P(�2, 5); a� � ��7, �6� 12. ________

A. �x � 2, y � 5� � t��7, �6�; x � �2 � 7t, y � 5 � 6tB. �x � 2, y � 5� � t��7, �6�; x � 2 � 7t, y � �5 � 6tC. �x � 2, y � 5� � t��7, �6�; x � 2 � 7t, y � �5 � 6tD. �x � 2, y � 5� � t��7, �6�; x � �2 � 7t, y � 5 � 6t

13. P(0, 3); a� � �1, �8� 13. ________A. �x, y � 3� � t��1, 8�; x � �t, y � 3 � 8tB. �x � 1, y � 8� � t(0, 3); x � 1, y � �8 � 3tC. �x, y � 3� � t�1, �8�; x � t, y � 3 � 8tD. �x � 1, y � 8� � t�0, �3�; x � 1, y � �8 � 3t

14. Which is the graph of parametric equations x � 4t � 5 and y � �4t � 5? 14. ________A. B. C. D.

15. Write parametric equations of x � 4y � 5. 15. ________A. x � t; y � �4t � �54� B. x � t; y � ��14�t � �54�

C. x � t; y � 4t � �54� D. x � t; y � �14�t � �54�

16. Write an equation in slope-intercept form of the line whose 16. ________parametric equations are x � �3t � 8 and y � �2t � 9.A. y � �23�x � �43

3� B. y � ��23�x � �433� C. y � ��23�x � �13

1� D. y � �23�x � �131�

Aaron kicked a soccer ball with an initial velocity of 39 feet persecond at an angle of 44° with the horizontal.17. After 0.9 second, how far has the ball traveled horizontally? 17. ________

A. 24.4 ft B. 12.3 ft C. 11.4 ft D. 25.2 ft

18. After 1.5 seconds, how far has the ball traveled vertically? 18. ________A. 6.1 ft B. 40.6 ft C. 4.6 ft D. 42.1 ft

A triangular prism has vertices at A(2, �1, 0), B(2, 1, 0), C(2, 0, 2), D(�1, �1, 0), E(�1, 1, 0), and F(�1, 0, 2).19. Which image point has the coordinates (�2, 1, 1) after a translation 19. ________

using the vector ��1, 2, 1�?A. C′ B. D′ C. E′ D. F′

20. What point represents a reflection of E over the xz-plane? 20. ________A. E′(1, �1, 0) B. E′(�1, �1, 0)C. E′(�1, 1, 0) D. E′(2, �1, 0)

Bonus Find 3v� � �2w� if v� � ��1, 5, 3� and w� � ��7, 5, �6�. Bonus: ________A. �270, 162, �180� B. �270, 90, 240�C. �270, �90, 240� D. �270, �162, �180�



8

D

C

D

B

A

D

C

B

B

A

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NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right of each problem.1. The vector v� has a magnitude of 5 inches and a direction of 32°. 1. ________

Find the magnitude of its vertical component.A. 4.24 in B. 2.65 in C. 2.79 in D. 31.88 in

2. What is an expression for x� involving r� and s� ? 2. ________A. �r� � s� B. r� � s�C. �r� � s� D. r� � s�

3. Find the ordered pair that represents the vector from 3. ________A(1, 2) to B(0, 3). Then find the magnitude of AB�.A. ��1, 1�; 1.41 B. �1, �1�; 2C. ��1, �1�; 1.41 D. �1, 1�; 2

4. Find the ordered triple that represents the vector from A(�4, 2, 1) to 4. ________B(�3, 0, 5). Then find the magnitude of AB�.A. ��7, �2, 4�; 8.31 B. ��1, �2, 4�; 4.58C. �1, �2, 4�; 4.58 D. ��7, 2, 6�; 9.43

5. Find an ordered pair to represent u� in u� � 2w� � v� if w� � ��2, 4� and 5. ________v� � �3, 1�.A. ��7, �7� B. ��1, �7� C. ��7, 7� D. ��1, 7�

6. Find an ordered triple to represent x� in x� � 3 y� � z� if y� � �2, �1, 5� 6. ________and z� � �1, �6, 6�.A. �7, 3, 9� B. �5, 3, 9� C. �5, 9, 9� D. �7, 3, 21�

7. Write MN� as the sum of unit vectors for M(�2, 3, 6) and N(1, 5, �2). 7. ________A. �i� � 2 j� � 8k� B. �i� � 2 j� � 4k�

C. 3i� � 2 j� � 4k� D. 3i� � 2 j� � 8k�

8. Find the inner product of a� and b� if a� � �3, 0, �1� and b� � �4, �2, 5� and 8. ________state whether the vectors are perpendicular.A. 7; no B. 0; yes C. 7; yes D. 0; no

9. Find the cross product of v� and w� if v� � ��1, 2, 4� and w� � ��3, �1, 5�. 9. ________A. �14, �7, �5� B. �14, 7, 7� C. �14, �7, 7� D. �6, �7, 7�

10. Find the magnitude and direction of the resultant 10. ________vector for the diagram at the right.A. 129.5 N, 46.5° B. 129.5 N, 11.5°C. 113.6 N, 13.1° D. 113.6 N, 48.1°

11. A 22-newton force acting at 48° and a 65-newton 11. ________force acting at 24° act concurrently on an object.What is the magnitude and direction of a third force that produces equilibrium on the object?A. 85.6 N; 30° B. 85.6 N; 6°C. 85.6 N; 210° D. 85.6 N; 186°

Chapter

8

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Write a vector equation of the line that passes through point P and is parallel to a� . Then write parametric equations of the line.12. P(�1, 3); a� � �2, �5� 12. ________

A. �x � 1, y � 3� � t�2, �5�; x � 1 � 2t, y � �3 � 2tB. �x � 1, y � 3� � t�2, �5�; x � �1 � 2t, y � 3 � 5tC. �x � 1, y � 3� � t�2, �5�; x � �1 � 2t, y � 3 � 5tD. �x � 1, y � 3� � t�2, �5�; x � �1 � 2t, y � �3 � 2t

13. P(1, �4); a� � �2, �5� 13. ________A. �x � 2, y � 5� � t�1, �4�; x � 2 � t, y � �5 � 4tB. �x � 2, y � 5� � t�1, �4�; x � �2 � t, y � 5 � 4tC. �x � 1, y � 4� � t�2, �5�; x � 1 � 2t, y � �4 � 5tD. �x � 1, y � 4� � t�2, �5�; x � �1 � 2t, y � 4 � 5t

14. Which graph represents a line whose parametric equations are 14. ________x � t � 2 and y � t � 2?A. B. C. D.

15. Write parametric equations of y � 2x � 3. 15. ________A. x � t; y � �12� t � 3 B. x � t; y � 2t � 3

C. x � t; y � �12� t � 3 D. x � t; y � 2t � 316. Write an equation in slope-intercept form of the line whose 16. ________

parametric equations are x � t � 4 and y � 2t � 1.A. y � 2x � 7 B. y � 2x � 9 C. y � 2x � 5 D. y � �12� x � 5

Jana hit a golf ball with an initial velocity of 102 feet per second at an angle of 67° with the horizontal.17. After 2 seconds, how far has the ball traveled horizontally? 17. ________

A. 27.9 ft B. 123.8 ft C. 79.7 ft D. 97.7 ft18. After 3 seconds, how far has the ball traveled vertically? 18. ________

A. 137.7 ft B. 119.6 ft C. 233.7 ft D. 52.6 ft

A triangular prism has vertices at A(2, 0, 0), B(2, 1, 3), C(2, 2, 0), D(0, 0, 0), E(0, 1, 3), and F(0, 2, 0).19. Which image point has the coordinates (1, 4, 3) after a translation 19. ________

using the vector �1, 2, 3�?A. C� B. D� C. E� D. F�

20. What point represents a reflection of B over the xy-plane? 20. ________A. B�(2, 1, �3) B. B�(�2, �1, 3)C. B�(�2, 1, �3) D. B�(2, �1, 3)

Bonus Find the cross product of v� and �2w� if v� � �2, 4, �1� and Bonus: ________w� � ��1, 2, �5�.

A. �44, 22, �16� B. �36, �22, �16� C. �36, 22, �16� D. �36, �22, 0�



8

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NAME _____________________________ DATE _______________ PERIOD ________

1. The vector v� has a magnitude of 11.4 meters and a direction 1. __________________of 248°. Find the magnitude of its vertical and horizontal components.

2. The vector u� has a magnitude of 89.6 inches. If v� � ��72� u�, 2. __________________what is the magnitude of v�?

Use a ruler and a protractor to determine the magnitude (in centimeters) and direction of each vector. Then find the magnitude and direction of each resultant.

3. �3 a� � �12� b� � �23� a� 3. __________________

4. �12� a� � �25� b� 4. __________________

5. Write the ordered pair that represents the vector from 5. __________________A(1.8, �3.8) to B(�0.1, 5.1). Then find the magnitude of AB�.

6. A force F�1 of 18.8 newtons pulls at an angle of 12° above 6. __________________due east. A force F�2 of 3.2 newtons pulls at an angle of 88° below due east. Find the magnitude and direction of the resultant force.

Find an ordered pair or ordered triple to represent u� in each equation if v� � �0, �1

2��, w� � �2, ��3

4��, r� � �1, ��1

4�, 2�, 7. __________________

and s� � �10, �6, �34

��. 8. __________________

7. u� � �v� � �13� w� 8. u� � �12� r� � 4s� 9. u� � ��23� s� � 3r� 9. __________________

10. Write the ordered triple that represents the vector from 10. __________________A(5.1, �0.8, 9) to B(�3.8, 7, �1.4). Then find the magnitude of AB�.

11. Write EF� as the sum of unit vectors for E(2.1, �2.6, 7) 11. __________________and F(�0.8, �7, 5).

Find each inner product and state whether the vectors are 12. __________________perpendicular. Write yes or no.

12. �8, �23�� 12�, �6� 13. ��2, 6, 8� � ��4, �2, ��12�� 13. __________________

Find each cross product. 14. __________________

14. �6, ��12�, 3� � �4, 2, ��13�� 15. ��14�, 7, �4� � ��5, �32�, 2� 15. __________________

Chapter

8

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16. Find the magnitude and direction of the 16. __________________resultant vector for the diagram at the right.

17. What force is required to push a 147-pound crate up 17. __________________a ramp that makes a 12° angle with the ground?

18. A 12.2-newton force acting at 12° and an 18.9-newton force 18. __________________acting at 75.8° act concurrently on an object. What is the magnitude and direction of a third force that produces equilibrium on the object?

19. Write a vector equation of the line that passes through 19. __________________point P ��32�, �5� and is parallel to a� � �2, 3�. Then write parametric equations of the line and graph it.

Write parametric equations for each equation. 20. __________________

20. y � ��34�x � 3 21. 2x � �13�y � 5 21. __________________

Write an equation in slope-intercept form of the line with the given parametric equations. 22. __________________

22. x � ��12�t � 6; y � 2t � 4 23. x � �2t � 5; y � 4t � �47� 23. _______________________________________________

24. Lisset throws a softball from a height of 4 meters, with 24. __________________an initial velocity of 20 meters per second at an angle of 45° with respect to the horizontal. When will the ball be a horizontal distance of 30 meters from Lisset?

25. A rectangular prism has vertices at A(1, �1, 3), B(1, 2, 3), 25. __________________C(1, 2, �1), D(1, �1, �1), E(�2, �1, 3), F(�2, 2, 3),G(�2, 2, �1), and H(�2, �1, �1). Find the vertices of the prism after a translation using the vector �1, �2, 1�and then a reflection over the xy-plane.

Bonus Write parametric equations for the line passing Bonus: __________________

through the point at ��23�, ��34�� and perpendicular

to the line with equation 4y � 8x � 3.



8

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NAME _____________________________ DATE _______________ PERIOD ________

1. The vector v� has a magnitude of 10 meters and a direction 1. __________________of 92°. Find the magnitude of its vertical and horizontal components.

2. The vector u� has a magnitude of 25.5 feet. If v� � ��13�u�, what 2. __________________is the magnitude of v�?


3. a� � 3b� 3. __________________

4. ��12�a� � b� 4. __________________

5. Write the ordered pair that represents the vector from 5. __________________A(0, �8) to B(�1, 7). Then find the magnitude of AB�.

6. A force F�1 of 27 newtons pulls at an angle of 23° above 6. __________________due east. A force F�2 of 33 newtons pulls at an angle of 55°below due west. Find the magnitude and direction of the resultant force. 7. __________________

Find an ordered pair or ordered triple to represent u� ineach equation if v� � �1, �6�, w� � �2, �5�, r� � �1, �1, 0�, and 8. __________________s� � �10, �6, 5�.

7. u� � v� � 3w� 8. u� � 3s� �2r� 9. u� � r� � �15�s� 9. __________________

10. Write the ordered triple that represents the vector from 10. __________________A(5, �8, 9) to B(�2, 2, 2). Then find the magnitude of AB�.

11. Write EF� as the sum of unit vectors for E(1, �2, 7) and 11. __________________F(�8, �7, 5).

Find each inner product and state whether the vectors are perpendicular. Write yes or no.12. �8, 2� � �0, �6� 12. __________________

13. �3, �7, 4� � ��4, �2, 1� 13. __________________

Find each cross product.14. �6, �4, 3� � �4, 2, �6� 14. __________________

15. ��2, 7, �4� � ��5, �6, 2� 15. __________________

Chapter

89.99 m, 0.35 m

8.5 ft

6.0 cm, 220°

2.3 cm, 43°

��1, 15�; 15.03

17.5 N; 70.21°below east

�7, �21�

�28, �16, 15�

�3, ��151�, 1�

��7, 10, �7�; 14.07

�9i� �5j� �2k�

�12; no

6; no

�18, 48, 28�

��10, 24, 47�

185-203 A&E C08-0-02-834179 10/4/00 3:20 PM Page 193

x � t, y � ��12

�t � �45�


16. Find the magnitude and direction of the resultant vector 16. __________________for the diagram below.

17. Anita is riding a toboggan down a hill. If Anita weighs 17. __________________120 pounds and the hill is inclined at an angle of 72°from level ground, what is the force that propels Anita down the hill?

18. A 15-newton force acting at 30° and a 25-newton force 18. __________________acting at 60° act concurrently on an object. What is the magnitude and direction of a third force that produces equilibrium on the object?

19. Write a vector equation of the line that passes through 19. __________________point P(1, 0) and is parallel to a� � ��3, �7�. Then write parametric equations of the line and graph it.

20. __________________Write parametric equations for each equation.20. y � �x �3 21. 2x � 4y � 5 21. __________________

Write an equation in slope-intercept form of the line 22. __________________with the given parametric equations.22. x � �t � 6; y � 2t � 4 23. x ��2t � 5; y � 4t � 2 23. __________________

24. Pablo kicks a football with an initial velocity of 30 feet 24. __________________per second at an angle of 58° with the horizontal. After 0.3 second, how far does the ball travel vertically?

25. A rectangular prism has vertices at A(2, 0, 2), B(2, 2, 2),C(2, 2, �2), D(2, 0, �2), E(0, 0, 2), F(0, 2, 2), G(0, 2, �2), and H(0, 0, �2). Find the vertices of the prism after a reflection over the xz-plane. 25. __________________

Bonus Write parametric equations for the line passing Bonus: __________________through (2, �2) and parallel to the line with equation 8x � 2y � �6.



8

A′(2, 0, 2), B′(2, �2, 2),

C′(2, �2, �2), D′(2, 0, �2),

E′(0, 0, 2), F′(0, �2, 2),

G′(0, �2, �2), H′(0, 0, �2)

�x � 1, y� �t��3, �7�;x � 1 � 3t,y � �7t

13.7 N; 65.4°

126.2 lb

38.72 N, 48.8°

x � t, y � �t � 3

y � �2x � 8

y � �2x � 8

about 6.19 ft

x � t; y � �4t � 6

185-203 A&E C08-0-02-834179 10/4/00 3:20 PM Page 194

1. The vector v� has a magnitude of 5 meters and a direction 1. __________________of 60°. Find the magnitude of its vertical and horizontal components.

2. The vector u� has a magnitude of 4 centimeters. 2. __________________If v� � ��3

1� u�, what is the magnitude of v� ?

Use a ruler and protractor to determine the magnitude (in centimeters) and direction of each vector. Then find the magnitude and direction of each resultant.

3. a� � b� 3. ____________

4. 2a� � b� 4. ____________

5. Write the ordered pair that represents the vector from 5. __________________A(3, �1) to B(�1, �2). Then find the magnitude of AB�.

6. A force F�1 of 25 newtons pulls at an angle of 20° above 6. __________________due east. A force F�2 of 35 newtons pulls at an angle of 60°above due east. Find the magnitude and direction of the resultant force.

Find an ordered pair or ordered triple to represent u� in each equation if v� � �2, �3�, w� � �1, 5�, r� � �1, �1, 1�, 7. __________________and s� � �0, �3, 2�.

8. __________________7. u� � �2v� � w� 8. u� � s� � r� 9. u� � 3s� � r�

9. __________________

10. Write the ordered triple that represents the vector from 10. __________________A(1, 3, 5) to B(�3, 0, 1). Then find the magnitude of AB�.

11. Write EF� as the sum of unit vectors for E(5, 1, �4) and 11. __________________F(9, 3, 1).

Find each inner product and state whether the vectors are perpendicular. Write yes or no. 12. __________________

12. �2, 0� � �0, �5� 13. �3, �4, �2� � ��2, �2, 1� 13. __________________


14. �2, �1, 3� � �1, 0, �5� 15. ��2, 2, �1� � �0, �2, 2� 15. __________________




8

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(–3, 2)

(–1, –4)

x

y

O


16. Find the magnitude and direction 16. __________________of the resultant vector for the diagram at the right.

17. Matt is pushing a grocery cart on a level floor with a force 17. __________________of 15 newtons. If Matt’s arms make an angle of 28° with the horizontal, what are the vertical and horizontal components of the force?

18. A 10-newton force acting at 45° and a 20-newton force 18. __________________acting at 130° act concurrently on an object. What is the magnitude and direction of a third force that produces equilibrium on the object?

19. Write a vector equation of the line that passes through 19. __________________point P(�3, 2) and is parallel to a� � ��2, 6�. Then write parametric equations of the line and graph it.

Write parametric equations for each equation.

20. y � 4x 21. y � 2x � 1 20. __________________

21. __________________

Write an equation in slope-intercept form of the line with the given parametric equations.

22. x � t; y � 2t 23. x � 2t; y � t � 5 22. __________________

23. __________________

24. Shannon kicks a soccer ball with an initial velocity of 24. __________________45 feet per second at an angle of 12° with the horizontal.After 0.5 second, what is the height of the ball?

25. A cube has vertices at A(2, 0, 0), B(2, 0, 2), C(2, 2, 2), 25. __________________D(2, 2, 0), E(0, 0, 0), F(0, 0, 2), G(0, 2, 2), and H(0, 2, 0).Find the vertices of the prism after a translation using the vector �1, �1, 2�.

Bonus Write parametric equations for the line passing Bonus: __________________through (0, 0) and parallel to 3y � 9x � 3.



8

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NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear,concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond therequirements of the problem.

1. Given the vectors below, complete the questions that follow.

c� � ��3, 1�, and d� � ��8, �11�

a. Show two ways to find a� � b�.

b. Find a� � b�. Explain each step.

c. Does a� � b� � b� � a�? Why or why not?

d. Does a� � b� � b� � a�? Defend your answer.

e. Tell how to find the sum c� � d�. Find the sum and its magnitude.

f. Find two vectors whose difference is �4, �1, 3�. Write the difference as the sum of unit vectors.

g. Find a vector perpendicular to �7, �3�. Explain how you know that the two vectors are perpendicular.

h. Find a� � b� if a� � �2, 1, 0� and b� � �1, 3, 0�. Graph the vectors and the cross product c� in three dimensions.

2. a. Find parametric equations for a line parallel to a� � �3, �1� and passing through (�2, 4).

b. Find another vector and point from which the parametric equations for the same line can be written.

3. A ball is thrown with an initial velocity of 56 feet per second at an angle of 30° with the ground.

a. If the ball is thrown from 8 feet above ground, when will it hit the ground?

b. How far will the ball travel horizontally before hitting the ground?

4. Find two pairs of perpendicular vectors. Then verify that they are perpendicular by calculating their dot products.

Chapter

8

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1. The vector v� has a magnitude of 12 inches and direction 1. __________________of 36°. Find the magnitude of its vertical and horizontal components.

2. The vector u� has a magnitude of 9.9 centimeters. If 2. __________________v� � �4u�, what is the magnitude of v�?


3. 2a� � 2b�

3. ____________

4. �3a� � b�

4. ____________

5. Write the ordered pair that represents the vector from 5. __________________A(4, �7) to B(0, �5). Then find the magnitude of AB�.

6. Write CD� as the sum of unit vectors for points C(4, �3) 6. __________________and D(1, �2).

7. Javier normally swims 3 miles per hour in still water. When 7. __________________he tries to swim directly toward shore at the beach, his course is altered by the incoming tide. If the current is 6 mph and makes an angle of 47 with the direct path to shore, what is Javier’s resultant speed?

Find an ordered pair to represent u� in each equation if 8. __________________v� � ��3, 8� and w� � �3, �4�. 9. __________________

8. u� � �5w� 9. u� � 2v� � 3w� 10. u� � 4w� � v� 10. __________________11. Write the ordered triple that represents the vector from 11. __________________

A(2, �2, 4) to B(6, 1, �8). Then find the magnitude of AB�.

12. Write EF� as a sum of unit vectors for E(1, �4, 3) and 12. __________________F(�4, �2, 3).

Find an ordered triple to represent u� in each equation if 13. __________________v� � �5, �2, 0� , w� � �3, �8, 1� , and x� � �0, �3, �4�. 14. __________________

13. u� � �v� � w� 14. u� � 3w� � 2x� 15. u� � x� � �12� v� 15. __________________

Find each inner product and state whether the vectors are perpendicular. Write yes or no. 16. __________________

16. �6, �4� � �2, 4� 17. �4, �3, 1� � �8, 12, 4� 17. __________________


18. �9, 1, 0� � ��3, 2, 5� 19. �6, �4, �2� � �1, 1, �3� 19. __________________

20. Find a vector that is perpendicular to both c� � �0, �3, 6� 20. __________________and d� � �4, 2, �5�.



8

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1. The vector v� has a magnitude of 13 millimeters and a 1. __________________direction of 84°. Find the magnitude of its vertical and horizontal components.

2. The vector a� has a magnitude of 6.3 meters. If b� � �2a� , 2. __________________what is the magnitude of b�?


3. 2a� � b� 3. __________________

4. �a� � 2b� 4. __________________

5. Write the ordered pair that represents the vector from 5. __________________A(1, �3) to B(�6, �8). Then find the magnitude of AB�.

6. Write CD� as a sum of unit vectors for C(7, �4) and D(�8, 1). 6. __________________

7. Two people are holding a box. One person exerts a force of 7. __________________140 pounds at an angle of 65.5 with the horizontal. The other person exerts a force of 115 pounds at an angle of 58.3 with the horizontal. Find the net weight of the box.

Find an ordered pair to represent u� in each equation if 8. __________________

v� � �6, �6� and w� � �3, �4�. 9. __________________

8. u� � �5w� 9. u� � 2v� � 3w� 10. u� � 4w� � v� 10. _____________

1. Write the ordered triple that represents the vector from 1. __________________A(3, 4, 10) to B(8, 4, �2). Then find the magnitude of AB�.

2. Write EF� as a sum of unit vectors for E(8, 2, �4) and 2. __________________F(5, �3, 0).

3. Find an ordered triple that represents 2v� � �31�w� � z� if 3. __________________

v� � �2, �1, 5�, w� � ��3, 4, �6�, and z� � �0, 3, �2�.

4. Find the inner product of a� and b� if a� � �7, �3, 8� and 4. __________________b� � �5, �2, �4�. Are a� and b� perpendicular?

5. Find the cross product of c� and d� if c� � �5, �5, 4� and 5. __________________d� � �2, 3, �6�. Verify that the resulting vector is perpendicular to c� and d�.


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

8

Chapter

8

185-203 A&E C08-0-02-834179 10/4/00 3:20 PM Page 199

(1, –3)

(–1,1)

x

y

O


NAME _____________________________ DATE _______________ PERIOD ________

Chapter

8

1. Find the magnitude and direction 1. __________________of the resultant vector forthe figure at the right.

2. Maggie is pulling on a tarp along level ground with a force of 2. __________________25 newtons. If the tarp makes an angle of 50 with the ground,what are the vertical and horizontal components of the force?

3. A 25-newton force acting at 75 and a 50-newton force acting at 3. __________________45° act concurrently on an object. What are the magnitude and direction of a third force that produces equilibrium on the object?

4. Write a vector equation of the line that passes through 4. __________________P(1, �3) and is parallel to q� � ��2, 4�. Then write parametric equations of the line and graph it.

Write parametric equations for each equation.5. 6x � y � 2 6. �2x � 5y � �4 5. __________________

6. __________________

Write an equation in slope-intercept form of the line with the given parametric equations. 7. __________________

7. x � 6t � 8 8. x � 3t � 10y � �t � 4 y � �4t � 2 8. __________________

While positioned 25 yards directly in front of the goalposts, Bill kicks the football withan initial velocity of 65 feet per second at an angle of 35� with the ground.

1. Write the position of the football as a pair of parametric 1. __________________equations. If the crossbar is 10 feet above the ground, does Bill’s team score?

2. What is the elapsed time from the moment the football is 2. __________________kicked to the time the ball hits the ground?

A rectangular prism has vertices at A(�1, �1, 1), B(�1, 1, 1), C(�1, 1, �2), D(�1, �1, �2), E(2, �1, 1), F(2, 1, 1), G(2, 1, �2), and H(2, �1, �2). Find the vertices of the prism after each transformation.

3. a translation using the vector �1, 2, �1� 3. __________________

4. a reflection over the yz-plane 4. __________________

5. the dimensions are increased by a factor of 3 5. __________________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

8

185-203 A&E C08-0-02-834179 10/4/00 3:20 PM Page 200



NAME _____________________________ DATE _______________ PERIOD ________


Multiple Choice1. If the area of a circle is 49�, what

is the circumference of the circle? DA 7B 7�C 14D 14�E 49

2. If all angles in the figure below areright angles, find the area of the shaded region. CA 12 units2

B 48 units2

C 144 units2

D 192 units2

E 240 units2

3. What is the equation of the perpendicular bisector of the segment from P(2, �1) to Q(3, 7)? AA 2x � 16y � 53B 2x � 16y � 53C 2x � 16y � 43D 2x � 16y � 43E None of these

4. If A(0, 0) and B(8, 4) are vertices of�ABC and �ABC is isosceles, whatare the coordinates of C? EA (5, �9)B (8, �3)C (5, 5)D (8, 0)E (1, 8)

5. The following are the dimensions offive rectangular solids. All have thesame volume EXCEPT AA 8 by 6 by 5B 4 by 15 by 2C �15� by 15 by 40

D �13� by 24 by 15

E �12� by 4 by 60

6. In �ABC, �A is a right angle. If BC � 25 and AB � 20, which is thearea of �ABC? DA 187.5 units2

B 250 units2

C 75�3�4� units2

D 150 units2

E 300 units2

7. If the measure of one angle in a parallelogram is 40°, what are the measures of the other three angles?A 60°, 100°, and 160° CB 40°, 280°, and 280°C 40°, 140°, and 140°D 40°, 150°, and 150°E None of these

8. Which of the following statements isNOT true for the diagram below? B

A m�6 � m�9B m�3 � m�6 � 90°C m�2 � m�6 � m�5 � 180°D m�8 � m�2 � m�3E m�4 � m�2 � m�9

9. If 2 y � 50 and y � 2x � 1, then which of the following statements is true? DA x � 13B 16.5 x 32.5C 2 x 2.5D 3 x 3.5E None of these

10. If x and y are real numbers and y2 � 6 � 2x, then which of the following statements is true? DA x � 6B x � 3C x � 6D x � 3E None of these

Chapter

8

185-203 A&E C08-0-02-834179 10/4/00 3:20 PM Page 201

A D

CB E


11. A circle is inscribed in a square asshown in the figure below. What is theratio of the area of the shaded region tothe area of the square?A �4

��

B �1 �4

��

C �4 �4

��

D ��4�

E �1 �4

��

12. Each angle in the figure below is aright angle. Find the perimeter of thefigure.A 11 unitsB 18 unitsC 22 unitsD 24 unitsE 28 units

13. Which number is �45� of �34� of 10?A 6 B 4C 3 D 1.5E 0.5

14. Evaluate 9[4�2(�2)4 � 3�2]�1.A 8 B �8

1�

C ��81� D �8

E None of these

15. A solid cube has 4-inch sides. Howmany straight cuts through the cubeare needed to produce 512 small cubesthat have half-inch sides?A 7 B 9C 16 D 21E None of these

16. A roll of wallpaper is 15 inches wideand can cover 39 square feet. How longis the roll?A 2.6 ft. B 21.7 ft.C 31.2 ft D 46.9 ft.E None of these





Column A Column B

17. Square X has sides of length x.Square Y has sides of length 2x.

18. ABCD is a rectangle.

19. Grid-In �BDE is contained in rectangle ABCD as shown below. Findthe area of �BDE in square units.

20. Grid-In The area of a rhombus is 28 square units. The length of one diagonal is 7 units. What is the lengthof the other diagonal in units?



8

A

D C

BE

9

3

4

4

7 Area of square XHalf the area of square Y

Area of �DBC Area of �AED

185-203 A&E C08-0-02-834179 10/4/00 3:20 PM Page 202



NAME _____________________________ DATE _______________ PERIOD ________

1. Find the zero of ƒ(x) � 12 � 4x. If no zero exists, write none. 1. __________________

2. Graph ƒ(x) � �x � 3. 2.

3. Triangle ABC has vertices A(�2, 3), B(2, 1), and C(0, �4). 3. __________________Find the image of the triangle after a reflection over the x-axis.

4. Find the inverse of � if it exists. If it does not 4. __________________

exist, write none.

5. Write the equation obtained when ƒ(x) � �x� is 5. __________________translated 3 units down and compressed horizontally by a factor of 0.5.

6. Solve �x � 3� 5. 6. __________________

7. Determine the rational roots of 2x3 � 3x2 �17x � 12 � 0. 7. __________________

8. Solve �x �1

1� � �2x� � 0. 8. __________________

9. Identify all angles that are coterminal with a 232 angle. 9. __________________Then find one positive angle and one negative angle coterminal with the given angle.

10. Find the area of �ABC if a � 4.2, A � 36, and B � 55°. 10. __________________

11. Find the amplitude and period of y � 3 cos �4x�. 11. __________________

12. Find the phase shift of y � 2 sin �x � ��6��. 12. __________________

13. If � is an angle in the second quadrant and cos � � � ��35��, 13. __________________

find tan 2�.

14. Write 2x � y � 5 in normal form. Then find the length 14. __________________of the normal and the angle it makes with the positive x-axis.

15. Write an equation in slope-intercept form of the line 15. __________________whose parametric equations are x � �3 � 7t and y � 4 � t.

�10

�35

Chapter

8

185-203 A&E C08-0-02-834179 10/4/00 3:20 PM Page 203

Page 185

1. B

2. A

3. C

4. A

5. D

6. B

7. A

8. C

9. A

10. D

11. B

Page 186

12. C

13. B

14. B

15. A

16. D

17. B

18. C

19. B

20. B

Bonus: A

Page 187

1. A

2. C

3. A

4. C

5. B

6. B

7. D

8. D

9. B

10. A

11. B

Page 188

12. D

13. C

14. D

15. B

16. A

17. D

18. C

19. B

20. B

Bonus: A



204-210 A&E C08-0-02-824179 10/4/00 3:18 PM Page 204


Page 189

1. B

2. C

3. A

4. C

5. C

6. B

7. D

8. A

9. C

10. A

11. C

Page 190

12. B

13. C

14. A

15. D

16. B

17. C

18. A

19. D

20. A

Bonus: B

Page 191

1. 10.57 m, 4.27 m

2. 313.6 in.

3. 4.6 cm, 60�

4. 0.5 cm, 289�

5. ��1.9, 8.9�; 9.10

18.5 N,6. 2.2� above east

7. ��23

�, ��34

��8. ��8

21�, ��19

83�, 4�

9. ��131�, �1

43�, �1

21��

��8.9, 7.8, �10.4�;10. 15.75

11. �2.9i� � 4.4 j� � 2k�

12. 0; yes

13. �8; no

14. �� 365�, 14, 14�

15. �20, �421�, �27

87��

Page 19216. 1 N, 313�

17. 30.6 lb

18. 26.6 N, 231.5�

�x � �32

�, y � 5� � t �2, 3�;x � �3

2� � 2t,

19. y � �5 � 3t

20. x � t, y � ��34

�t � 3

21. x � t, y � �6t � 15

22. y � �4x � 20

23. y � �2x � �343�

24.after 2.1 secondsA� (2, �3, �4), B� (2, 0, �4), C� (2, 0, 0), D� (2, �3, 0), E � (�1, �3, �4), F � (�1, 0, �4), G� (�1, 0, 0),

25. H� (�1, �3, 0)

x � t,Bonus: y � ��1

2�t � �

152�

Form 1C Form 2A


204-210 A&E C08-0-02-824179 10/4/00 3:18 PM Page 205



Page 193

1. 9.99 m, 0.35 m

2. 8.5 ft

3. 6.0 cm, 219�

4. 2.2 cm, 43�

5. ��1, 15�; 15.03

17.5 N; 70.2�6. below east

7. �7, �21�

8. �28, �16, 15�

9. �3, � �151�, 1�

10. ��7, 10, �7�; 14.07

11. �9i� � 5j� � 2k�

12. �12; no

13. 6; no

14. �18, 48, 28�

15. ��10, 24, 47�

Page 194

16. 13.7 N; 65.4�

17. 114.1 lb

18. 38.7 N, 228.8�

�x � 1, y� �t��3, �7�; x � 1 � 3t,

19. y � �7t

20. x � t, y � �t � 3

x � t,21. y � ��1

2�t � �

45�

22. y � �2x � 8

23. y � �2x � 8

24. about 6.19 ft

A�(2, 0, 2), B�(2, �2, 2),C�(2, �2, �2), D�(2, 0, �2), E�(0, 0, 2), F�(0, �2, 2),G�(0, �2, �2),

25. H�(0, 0, �2)

x � t,Bonus: y � �4t � 6

Page 195

1. 4.33 m, 2.5 m

2. �43

� cm

3. 3.9 cm, 49�

4. 3.8 cm, 71�

5. ��4, �1�; 4.12

56.5 N; 43.5�6. above east

7. ��3, 11�

8. ��1, �2, 1�

9. �1, �10, 7�

10. ��4, �3, �4�; 6.40

11. 4i� � 2 j� � 5k�

12. 0; yes

13. 0; yes

14. �5, 13, 1�15. �2, 4, 4�

Page 196

16. 5.0 N; 36.0�

17. 7.0 N, 13.2 N

18. 23.1 N, 284.5�

�x � 3, y � 2� �

t��2, 6�; x � �3 � 2t,19. y � 2 � 6t

20. x � t, y � 4t

21. x � t, y � 2t � 1

22. y � 2x

23. y � �21�x � 5

24. about 0.68 ftA�(3, �1, 2), B�(3, �1, 4), C�(3, 1, 4),D�(3, 1, 2), E �(1, �1, 2),F�(1, �1, 4), G�(1, 1, 4),

25. H�(1, 1, 2)

Bonus: x � t, y � 3t

204-210 A&E C08-0-02-824179 10/4/00 3:18 PM Page 206



3 Superior • Shows thorough understanding of the concepts vector addition, subtraction, cross multiplication, inner product, and parametric equations.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.

2 Satisfactory, • Shows understanding of the concepts vector addition, with Minor subtraction, cross multiplication, dot product, andFlaws parametric equations.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Diagrams and graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts vector Satisfactory, addition, subtraction, cross multiplication, dot product,with Serious and parametric equations.Flaws • May not use appropriate strategies to solve problems.

• Computations are mostly correct.• Written explanations are satisfactory.• Diagrams and graphs are mostly accurate and appropriate.• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts vector addition, subtraction, cross multiplication, dot product,and parametric equations.

• May not use appropriate strategies to solve problems. • Computations are incorrect.• Written explanations are not satisfactory.• Diagrams and graphs are not accurate or appropriate.• Does not satisfy requirements of problems.



204-210 A&E C08-0-02-824179 10/4/00 3:18 PM Page 207



Page 197

1a.

1b. a� � b� � a� � (� b�), as shown in thefigure below.

1c. Yes. They are the same diagonal of a parallelogram.

1d. No. a� � b� and b� � a� are shown inthe figures below.

1e. Add the first terms of each vectortogether, and then add the secondterms together. These termsrepresent the horizontal and verticalcomponents of the resultant vector,respectively.c� � d� � ��3 � �8, 1 � (�11)�, or��11, �10�The magnitude of c� � d� is �(��1�1�)2� �� (��1�0�)2�, or about 14.9.

1f. Sample answer: �1, 2, 3� ��3, 3, 0� � �4, �1, 3�; �4, �1, 3� �4i� � j� � 3k�

1g. Sample answer: �3, 7�; The vectorsare perpendicular because their dotproduct is zero.a1b1 � a2b2 � 7 � 3 � (�3)7 � 0

1h. � � � 0i� � 0j� � 5k�

2a. x � �2 � 3ty � 4 � t

2b. Sample answer: b� � �6, �2�, (1, 3)

3a.t(56)sin 30� � �1

2�(32)t2 �8 � 0

4t2 �7t � 2 � 0(4t � 1)(t � 2) � 0

t � 2The ball hits the ground after 2seconds.

3b. Distance: x � (2)(56) ��23��, or

about 97 feet

4. Sample answer: The vectors a� � �1, 0�and b� � �0, �1� are perpendicularbecause their inner product isa1b1 � a2b2 � 1(0) � 0(�1) or 0; a� � �5, 5�, and b� � �5, �5� areperpendicular because their innerproduct is a1b1 � a2b2 � 5(5) �

5(�5) � 25 � 25 � 0.

k�00

j�13

i�21


204-210 A&E C08-0-02-824179 10/4/00 3:18 PM Page 208



1. 7.05 in., 9.71 in.

2. 39.6 cm

3. 6.6 cm; 64�

4. 5.6 cm; 260�

5. ��4, 2�; 4.47

6. �3i� � j�

7. about 8.3 mph

8. ��15, 20�

9. �3, 4�

10. �15, �24�

11. �4, 3, �12�; 13

12. �5i� � 2j�

13. ��8, 10, �1�

14. �9, �30, �5�

15. ��52

�, �2, �4�

16. �4; no

17. 0; yes

18. �5, �45, 21�

19. �14, 16, 10�

Sample answer:20. �3, 24, 12�

Quiz APage 199

1. 12.93 mm, 1.36 mm

2. 12.6 m

3. 5.5 cm; 29�

4. 5.9 cm; 187�

5. ��7, �5�; 8.60

6. �15i� � 5j�

7. 225.25 lb

8. ��15, 20�

9. �21, �24�

10. �6, �10�

Quiz BPage 199

1. �5, 0, �12�; 13

2. �3i� � 5j� � 4k�

3. �3, ��131�, 10�

4. 9; no

�18, 38, 25�; 5. both inner products � 0

Quiz CPage 200

1. 11.4 N; 50.7�

2. 19.15 N, 16.07 N

3. 72.7 N, 234.9�

�x � 1, y � 3� � t��2, 4�;4. x � 1 � 2t, y � �3 � 4t

5. x � t, y � 6t � 2

6. x � t, y � �25

�t � �54�

7. y � ��16

�x � �136�

8. y � ��43

�x � �334�

Quiz DPage 200

x � 65t cos 35�,1. y � 65t sin 35� � 16t2; yes

2. about 2.33 s

A�(0, 1, 0), B�(0, 3, 0), C�(0, 3, �3), D�(0, 1, �3), E�(3, 1, 0), F�(3, 3, 0),

3. G�(3, 3, �3), H�(3, 1, �3)

A�(1, �1, 1), B�(1, 1, 1), C�(1, 1, �2), D�(1, �1, �2), E�(�2, �1, 1), F�(�2, 1, 1),

4. G�(�2, 1, �2), H�(�2, �1, �2)

A�(�3, �3, 3), B�(�3, 3, 3), C�(�3, 3, �6), D�(�3, �3, �6),E�(6, �3, 3), F�(6, 3, 3),

5. G�(6, 3, �6), H�(6, �3, �6)


204-210 A&E C08-0-02-824179 10/4/00 3:18 PM Page 209

Page 201

1. D

2. C

3. A

4. E

5. A

6. D

7. C

8. B

9. D

10. D

Page 202

11. C

12. C

13. A

14. E

15. D

16. C

17. B

18. C

19. 12

20. 8

Page 203

1. 3

2.

A�(�2, �3), B�(2, �1),3. C�(0,4)

4.

5. ƒ(x) � �2�x� � 3

6. {x x �2 or x � 8}

7. ��32

�, �1, 4

8. {x 0 x � �23

� or x � 1}

232� � 360k�, k is an integer; 9. Sample answers: 592�, �128�

10. 12.3 square units

11. 3, 8�

12. ��6

� units to the right

13. �4�5�

�2�5

5��x � ��55��y � �5� � 0;

14. �5�; 333�

15. y � ��17

�x � �275�



� ��15

�

��35

�

0

�1

204-210 A&E C08-0-02-824179 10/4/00 3:18 PM Page 210



NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at theright or each problem.

1. Find the polar coordinates that do not describe 1. ________the point in the given graph.A. (�4, 120�)B. (4, 300�)C. (4, �240�)D. (4, �60�)

2. Find the equation represented in the 2. ________given graph.A. � � ��3

��

B. r � �3��

C. � � 2D. r � �23

��

3. Find the distance between the points with polar coordinates ��2.5, ��6�� 3. ________and ��1.9, ��3��.A. 3.14 B. 2.91 C. 3.49 D. 1.65

4. Identify the graph of the polar equation r � 4 sin 2�. 4. ________A. B. C. D.

5. Find the equation whose graph is given. 5. ________A. r � 4 cos 2�B. r � 2 � 2 cos �C. r � 4 cos �D. r2 � 16 cos 2�

6. Find the polar coordinates of the point with rectangular 6. ________coordinates (�2, 2�3�).A. �4, ��3�� B. �4, �23

�� C. �4, �56�� D. �2, �23

��7. Find the rectangular coordinates of the point with polar coordinates 7. ________

�4, �54��.

A. (�2�2�, �2�2�) B. (2, 2�3�)C. (2�2�, 2�2�) D. (�2�3�, �2)

8. Write the rectangular equation x2 � y2 � 2x � 0 in polar form. 8. ________A. r � 2 sin � B. r2 � 2r sin � � 0C. r � cos 2� D. r � 2 cos �

9. Write the polar equation r2 � 2r sin � � 0 in rectangular form. 9. ________A. x � y � 2 � 0 B. x2 � y2 � 2x � 0C. x2 � y2 � 2y � 0 D. x � 2y

Chapter

9

211-229 A&E C09-0-02-834179 10/10/00 10:16 AM Page 211 (Black plate)


10. Identify the graph of the polar equation r � 2 csc (� � 60�). 10. ________A. B. C. D.

11. Write 2x � y � 5 in polar form. 11. ________A. ��5� � r cos (� � 27�) B. �5� � r cos (� � 27�)C. ��5� � r cos (� � 27�) D. �5� � r cos (� � 27�)

12. Simplify 2(3 � i14) � (5 � i23). 12. ________A. 1 � 3i B. 3 � i C. 1 � 2i D. 1 � i

13. Simplify (5 � 3i)2. 13. ________A. 16 � 30i B. 34 � 30i C. 16 � 30i D. 34 � 30i

14. Simplify �34��

25ii�. 14. ________

A. �421� � �24

31� i B. �24

21� � �24

31� i C. ��29� � �29

3� i D. �421� � �24

31� i

15. Express 5�3� � 5i in polar form. 15. ________A. 10�cos �11

6�� i sin �11

6�� B. 10�cos �11

6�� i sin �11

6��

C. 5�cos �116

�� i sin �116

�� D. 10�cos �53�� i sin �53

��16. Express 4�cos �34

�� i sin �34�� in rectangular form. 16. ________

A. ��2� � �2�i B. 2�2� � 2�2�iC. �2�2� � 2�2�i D. �2�2� � 2�2�i

For Exercises 17 and 18, let z1 � 8(cos �23�� i sin �2

3��) and

z2 � 0.5�cos ��3

� � i sin ��3

��.17. Write the rectangular form of z1z2. 17. ________

A. �4i B. 4 C. 4 � 4i D. �4

18. Write the rectangular form of �zz

1

2�. 18. ________

A. 8 � 8�3�i B. �8 � 8�3�i C. 16 � 16�3�i D. 8 � 8�3�i

19. Simplify (3�3� � 3i)�3 and express the result in rectangular form. 19. ________

A. �216i B. ��2116� i C. �2

116� i D. 216i

20. Which of the following is not a root of z3 � 1 � �3�i to the nearest 20. ________hundredth?A. �0.22 � 1.24i B. �0.97 � 0.81iC. 1.02 � 0.65i D. 1.18 � 0.43i

Bonus Find (cos � � i sin �)2. Bonus: ________A. cos 2� � i sin 2� B. cos2 � � i sin2 �C. cos2 � � i sin2 � D. cos 2� � i sin 2�



9




NAME _____________________________ DATE _______________ PERIOD ________


1. Find the polar coordinates that do not 1. ________describe the point in the given graph.A. (�3, 45°) B. (�3, �135°) C. (3, 225°) D. (�3, �315°)

2. Find the equation represented in 2. ________the given graph.A. r � 2 B. � � 2�C. � � 4 D. r � 4

3. Find the distance between the points with polar coordinates 3. ________(3, 120°) and (0.5, 49°).A. 2.88 B. 3.19 C. 3.49 D. 1.59

4. Identify the graph of the polar equation r � 2 � 2 sin �. 4. ________A. B. C. D.

5. Find the equation whose graph is given. 5. ________A. r � 4 sin �B. r � 2 � 2 sin �C. r � 4 sin 2�D. r2 � 16 sin 2�

6. Find the polar coordinates of the point with rectangular 6. ________coordinates (2, �2).

A. �2, ��4�� B. ��2�, �74�� C. �2�2�, ��4�� D. �2�2�, �74

��7. Find the rectangular coordinates of the point with polar 7. ________

coordinates ��2, �56��.

A. (��3�, �1) B. (�2�3�, 2) C. (�3�, �1) D. (2�3�, �2).

8. Write the rectangular equation y � x in polar form. 8. ________A. � � 45° B. r � tan � C. r � cos � D. � � 1

9. Write the polar equation r � 3 sin � in rectangular form. 9. ________A. y � 3x B. x2 � y2 � 3y � 0 C. x2 � y2 � 3x � 0 D. x � 3y

Chapter

9



10. Identify the graph of the polar equation r � 2 sec (� � 120°). 10. ________A. B. C. D.

11. Write 3x � 2y � 13 � 0 in polar form. 11. ________A. ��1�3� � r cos (� � 34°) B. �1�3� � r cos (� � 34°) C. ��1�3� � r cos (� � 34°) D. �1�3� � r cos (� � 34°)

12. Simplify (3 � i7) � 2(i6 � 5i). 12. ________A. 5 � 11i B. 5 � 9i C. 1 � 11i D. 5 � 9i

13. Simplify (5 � 3i)(2 � 4i). 13. ________A. �2 � 14i B. 22 � 14i C. 22 � 14i D. �2 � 14i


24ii�. 14. ________

A. �275� � �22

65�i B. �22

35� � �22

65�i C. �1 � �27

6�i D. �275� � �22

65�i

15. Express �2�2� � 2�2�i in polar form. 15. ________A. 4�cos �34

�� i sin �34�� B. 2�cos �34

�� i sin �34��

C. 4�cos �34�� i sin �34

�� D. 4�cos �74�� i sin �74

��16. Express 10�cos �76

�� i sin �76�� in rectangular form. 16. ________

A. �5�3� � 5i B. �5 � 5�3�i C. 5�3� � 5i D. �5�3� � 5i

For Exercises 17 and 18, let z1 � 12�cos �76�� i sin �7

6�� and

z2 � 3�cos ��6

� � i sin ��6

��. 17. Write the rectangular form of z1z2. 17. ________

A. �18 � 18�3�i B. �18 � 18�3�i C. 18 � 18�3�i D. 18 � 18�3�i


1

2�. 18. ________

A. 4 B. �4i C. �4 D. 4 � 4i

19. Simplify (1 � �3�i)5and express the result in rectangular form. 19. ________

A. 16 � 16�3�i B. 16�3� � 16i C. 16 � 16�3�i D. �16 � 16�3�i

20. Find �32�7�i�. 20. ________

A. �3�2

3�� 32�i B. �32� � �3�2

3��i C. ��3�2

3�� 32�i D. �3�2

3�� 32� i

Bonus Find (cos �� i sin �)2. Bonus: ________A. cos 2� � i sin 2� B. cos2 � � i sin2 �C. cos2 � � i sin2 � D. cos 2� � sin 2�



9




NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right ofeach problem.1. Find the polar coordinates that do not describe 1. ________

the point in the given graph.A. (�2, 30�)B. (�2, 210�)C. (2, 30�)D. (�2, �150�)

2. Find the equation represented in the given graph. 2. ________A. � � 3B. r � 3C. � � 2�D. r � 2

3. Find the distance between the points with polar 3. ________coordinates (2, 120�) and (1, 45�).A. 1.40 B. 2.98 C. 2.46 D. 1.99

4. Identify the graph of the polar equation r � 4 sin �. 4. ________A. B. C. D.

5. Find the equation whose graph is given. 5. ________A. r � 4 cos �B. r � 2 � 2 cos �C. r � 2 � 2 cos �D. r � 2 � 2 sin �

6. Find the polar coordinates of the point with rectangular 6. ________coordinates (�3�, 1).A. �2, ��3�� B. �2, ��6�� C. �2, ��4�� D. �1, ��6��

7. Find the rectangular coordinates of the point with polar 7. ________coordinates (3, 180�).A. (�3, 0) B. (0, 3) C. (3, 0) D. (0, �3)

8. Write the rectangular equation x � 3 in polar form. 8. ________A. r sin � � 3 B. r � 3C. � � 3 D. r cos � � 3

9. Write the polar equation r � 3 in rectangular form. 9. ________A. x2 � 9 � 0 B. x2 � y2 � 9y � 0C. x2 � y2 � 9 D. xy � 9

Chapter

9



10. Identify the graph of the polar equation r � 3 sec (� � 90�). 10. ________A. B. C. D.

11. Write 3x � 4y � 5 � 0 in polar form. 11. ________A. �1 � r cos (� � 53�) B. 1 � r cos (� � 53�)C. �1 � r cos (� � 53�) D. 1 � r cos (� � 53�)

12. Simplify 2(3 � 4i) � (5 � i15) 12. ________A. 10 � 7i B. 11 � 9i C. 12 � 8i D. 11 � 7i

13. Simplify (3 � i)(1 � i) 13. ________A. 2 � 2i B. 2 � 2i C. 4 � 2i D. 4 � 2i


ii�. 14. ________

A. �12� � �23�i B. �12� � �2

3�i C. �23� � �2

3�i D. 1 � 2i

15. Express 3�3� � 3i in polar form. 15. ________A. 3�cos ��6� � i sin ��6�� B. 6�cos ��6� � i sin ��6��C. 6�cos ��3� � i sin ��3�� D. 6�cos ��6� � i sin ��6��

16. Express 2�cos ��3� � i sin ��3�� in rectangular form. 16. ________

A. �1 � �3�i B. 1 � �3�i C. 1 � �3�i D. �3� � i

For Exercises 17 and 18, let z1 � 4(cos 135� � i sin 135�) andz2 � 2(cos 45� � i sin 45�).17. Write the rectangular form of z1z2. 17. ________

A. �8i B. �8 C. 8 � 8i D. 8


1

2�. 18. ________

A. 2i B. �2 C. �2i D. 2 � 2i

19. Simplify (�3� � i)4 and express the result in rectangular form. 19. ________A. 8 � 8�3�i B. 8 � 8�3�i C. 16 � 16�3�i D. �8 � 8�3�i

20. Find �3

i�. 20. ________

A. ��23�� 12� i B. ��2

3�� 12� i C. ��23�� 12� i D. �12� � ��2

3��i

Bonus If 2 � 2i � 2�2�(cos 45� � i sin 45�), find 2 � 2i. Bonus: ________

A. 2�2�(cos 45� � i sin 45�) B. 2�2�(cos 135� � i sin 135�)

C. 2�2�(cos 225� � i sin 225�) D. 2�2�(cos 315� � i sin 315�)



9




NAME _____________________________ DATE _______________ PERIOD ________

1. Write the polar coordinates of the point 1. __________________in the graph if r � 0 and 0� � � � 180�.

2. Graph the polar equation r � �3. 2.

3. Find the distance between the points with 3. __________________polar coordinates ��1.5, �34

�� and ��2, ��6��.

4. Graph the polar equation r � 4 sin 3�. 4.

5. Identify the classical curve represented by the equation 5. __________________r2 � 16 sin 2�.

6. Find the polar coordinates of the point with rectangular 6. __________________coordinates (�3, �3). Use 0 � � � 2� and r � 0.

7. Find the rectangular coordinates of the point with polar 7. __________________coordinates �6, �74

��.

8. Write the rectangular equation x � 2y � 5 � 0 in polar form. 8. __________________Round � to the nearest degree.

9. Write the polar equation r2 sin 2� � 8 in rectangular form. 9. __________________

Chapter

9



10. Graph the polar equation 1 � r cos (� � 15�). 10.

11. Write 3x � y � 10 in polar form. 11. __________________

12. Simplify 3(2i � i10) � 4(8 � i49). 12. __________________

13. Simplify (3 � 4i)(2 � 5i). 13. __________________


45ii�. 14. __________________

15. Express 2 � 2�3�i in polar form. 15. __________________

16. Express 8�cos �54�� i sin �54

�� in rectangular form. 16. __________________


3�� and

z2 � 2�cos ��6

� � i sin ��6

��.17. Write the rectangular form of z1z2. 17. __________________


1

2�. 18. __________________

19. Simplify (4 � 4i)�2 and express the result in 19. __________________rectangular form.

20. Solve the equation z3 � �2 � 2�3�i. 20. __________________

Bonus If 3 � �3�i � 2�3�(cos 30� � i sin 30�), find Bonus: __________________3 � �3�i.



9




NAME _____________________________ DATE _______________ PERIOD ________

1. Write the polar coordinates of the point in 1. __________________the graph if �90° � � � 0°.

2. Graph the polar equation � � �56��. 2.

3. Find the distance between the points with polar coordinates 3. __________________(2.5, 150°) and (1, 70°).

4. Graph the polar equation r � 2 � 2 sin �. 4.

5. Identify the classical curve represented by the equation 5. __________________r � 2 � 5 sin �.

6. Find the polar coordinates of the point with rectangular 6. __________________coordinates (1, ��3�). Use 0 � � � 2� and r � 0.


��.

8. Write the rectangular equation x2 � y2 � 4 in polar form. 8. __________________

9. Write the polar equation r2 � 8 in rectangular form. 9. __________________

Chapter

9



10. Graph the polar equation r � 2 csc (� � 60°). 10.

11. Write x � 2y � 5 � 0 in polar form. 11. __________________

12. Simplify 2(i21 � 7) � (5 � i3). 12. __________________

13. Simplify (3 � 2i)2. 13. __________________

14. Simplify �43 ��

52ii�. 14. __________________

15. Express �6 � 6i in polar form. 15. __________________

16. Express 4�cos ��6� � i sin ��6�� in rectangular form. 16. __________________


6�� and

z2 � 4�cos ��3

� � i sin ��3

��. 17. Write the rectangular form of z1z2. 17. __________________


1

2�. 18. __________________

19. Simplify (2�3� � 2i)3and express the result in 19. __________________

rectangular form.

20. Find �3

��6�4�i�. 20. __________________

Bonus Find �cos ��4� � i sin ��4��3. Express the result in Bonus: __________________

rectangular form.



9




NAME _____________________________ DATE _______________ PERIOD ________

1. Write the polar coordinates of the point 1. __________________in the graph if 0� � � � 180�.

2. Graph the polar equation � � ��3�. 2.

3. Find the distance between the points with polar 3. __________________coordinates (3.2, 120�) and (2, 45�).

4. Graph the polar equation r � 4 cos �. 4.

5. Identify the classical curve represented by the equation 5. __________________r � 4 sin 2�.

6. Find the polar coordinates of the point with rectangular 6. __________________coordinates (0, 1).

7. Find the rectangular coordinates of the point with 7. __________________polar coordinates �2, ��4��.

8. Write the rectangular equation y � 2 in polar form. 8. __________________

9. Write the polar equation r � 3 in rectangular form. 9. __________________

Chapter

9



10. Graph the polar equation r � 2 sec (� � 60�). 10.

11. Write x � y � 2 � 0 in polar form. 11. __________________

12. Simplify (3 � i17) � (2 � 3i). 12. __________________

13. Simplify (2 � 4i)(2 � 4i). 13. __________________


ii�. 14. __________________

15. Express 2�3� � 2i in polar form. 15. __________________

16. Express 6�2��cos �34�� i sin �34


For Exercises 17 and 18, let z1 � 12(cos 240� � i sin 240�)and z2 � 0.5(cos 30� � i sin 30�).17. Write the rectangular form of z1z2. 17. __________________


1

2�. 18. __________________

19. Simplify (2 � 2i)4 and express the result in rectangular 19. __________________form.

20. Find �3

��8�i�. 20. __________________

Bonus Find �cos ��4� � i sin ��4��2. Express the result in Bonus: __________________

rectangular form.



9




NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear,concise solution to each problem. Be sure to include all relevantdrawings and justify your answers. You may show your solution inmore than one way or investigate beyond the requirements of theproblem.1. a. Write the rectangular coordinates for a point in a plane.

b. Graph the point described in part a.

c. Find the polar coordinates for the point described in part a.Graph the point in the polar coordinate system.

d. Explain how the two graphs are related.

2. a. Write the polar coordinates for a point in a plane.

b. Graph the point described in part a.

c. Find the rectangular coordinates for the point described in part a. Graph the point in the rectangular coordinate system.

d. Explain how the two graphs are related.

3. a. Draw the graph of r � cos �.

b. Tell how the graph of r � 2 cos � differs from the graph in part a.

c. What type of classical curve is represented by r � cos 4�?

d. What type of classical curve is represented by r � 1 � cos �?

e. Write a polar equation for a classical curve. Graph the equationand name the type of curve.

4. a. Find two complex numbers a and b whose sum is 3 � 3i.

b. Express the complex numbers a and b in part a in polar form.Explain each step.

c. Find the product of a and b.

d. Show two ways to find (3 � 3i)4. Then find (3 � 3i)4.

e. Explain how to find (3 � 3i)�13

�

. Then find (3 � 3i)�13

�

.

Chapter

9



1. Write the polar coordinates of the point 1. __________________in the given graph if 0� � � � 180�.

2. Graph the polar equation r � 3. 2.

3. Find the distance between the points with 3. __________________polar coordinates (3, 150�) and (�2, 45�).

4. Graph the polar equation r � 2 � 2 cos �. 4.

5. Identify the type of classical curve represented by 5. __________________the graph of r � 3 cos 2�.

6. Find the polar coordinates of the point with rectangular 6. __________________coordinates (�3, �3). Use 0 � � � 2� and r � 0.

7. Find the rectangular coordinates of the point 7. __________________with polar coordinates (4, 150�).

8. Write the rectangular equation x2 � y2 � 5y in polar form. 8. __________________

9. Write the polar equation � � ��3� in rectangular form. 9. __________________

10. Graph the polar equation r � 2 sec (� � �) and state the 10. __________________rectangular form of the linear equation.



9



NAME _____________________________ DATE _______________ PERIOD ________

Chapter

9

1. Write the polar coordinates of the point at (3, 30�) if 1. __________________�180� � � � 0�.

Graph each polar equation.2. � � �56

�� 2.

3. r2 � 16 sin 2� 3.

4. Find the distance between the points with polar 4. __________________coordinates (�2, 210�) and (4, 60�).

1. Find the polar coordinates of the point with rectangular 1. __________________coordinates (4, �4�3�). Use 0 � � � 2� and r � 0.

2. Find the rectangular coordinates of the point with polar 2. __________________coordinates (6, 315�).

3. Write the rectangular equation x � 3y � 5 � 0 in 3. __________________polar form. Round � to the nearest degree.

4. Write the polar equation r � 5 in rectangular form. 4. __________________

5. Graph the polar equation r � 3 sec (� � 30�). 5.


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

9


Simplify.1. 2(3 � i11) � (4 � i) 1. __________________

2. (2 � 4i)(3 � 5i) 2. __________________

3. �45��

32ii� 3. ______________

4. Express 2�3� � 2i in polar form. 4. __________________

5. Express 8�cos �34�� i sin �34


Find each product, quotient, or power and express the result in rectangular form. Let z1 � 4(cos 120� � i sin 120�) and z2 � 0.5(cos 30� � i sin 30�).1. z1z2 1. ___________________________

2. �zz2

1� 2. __________________

3. z12 3. __________________

Find each power or root. Express the result in rectangular form.4. (�2� � �2�i)4 4. __________________

5. �5

��3�2�i� 5. __________________


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

9

Chapter

9




NAME _____________________________ DATE _______________ PERIOD ________


Multiple Choice1. A�C� is a diameter of the

circle at the right. Point Bis on the circle such that m�BAC � 2x°. Find m�BCA.A 6x°B ��12��180 � 2x��°C (2x � 90)°D [2(45 � x)]°E It cannot be determined from the

information given.

2. A chord with a length of 8 is 2 unitsfrom the center of a circle. Find thediameter.A �5�B 2�5�C 4�5�D 2�3�E 4�3�

3. 2 cos ��4� �

A 0 B �12�

C 1 D �2�E 2

4. �sin ��3��cos ��6�� cos ��3��sin ��6��

A ��12�

B �12�

C �34�

D 1 E �54�

5. Given that lines r and s intersect at P,m�1 � 3x°, and m�3 � m�1, findm�2.A x°B (180 � 3x)°C 6x°D (180 � 6x)°E 3x°

6. If � �� n , then I. Angles 3 and 5 are supplementary.II. m�7 � m�8 III. m�3 � m�7 � m�6 A I only B II only C III only D I and II only E I, II, and III

7. In the rectangle below, what is thearea of the shaded region? A 10wB 4x2

C 10w � 4xD 10w � x2

E 10w � 4x2

8. On a map, 1 inch represents 2 miles.A circle on the map has a circumference of 5� inches. What area does the circular region on themap represent? A 10� mi2

B 25� mi2

C 5� mi2

D 100� mi2

E 50� mi2

9. �101

15� � �101

16� �

A ��109

16� B �109

16�

C �110� D ��1

10�

E �101

16�

Chapter

9

xx

xx x

x

xx

10

w



10. Choose the expression that is notequivalent to the other three.A 4 � 2�5�B �12�(8 � �8�0�)C 6 � �8�0� � 2 � �2�0�D �2� (�8� � �1�0�)E They are all equivalent.

11. In the circle O below, if m�B � 20°,find m ACB�.A 40°B 140°C 220°D 320°E None of these

12. Given that A�B� is tangent to circle O atpoint A, O�A� is a radius, OA � 6, andOB � 8, find AB.A �7� B 2�7�C 4�7� D 5 E 10

13. If �(x � 33)(wy � 3)z�� 60, which of the

variables cannot be 3? A x B yC z D wE None of these

14. A function ƒ is described by ƒ(x) � 3x � 6 and a functions g isdescribed by g(x) � 12 � 6x. Which ofthe following statements is true? A g( ƒ(x)) � ƒ( g(x)) B g( ƒ(x)) � 2ƒ( g(x)) C g( ƒ(x)) � �2ƒ( g(x)) D g( ƒ(x)) � ƒ( g(x)) � 18 E None of these

15. � ABC is inscribed in a circle.m� A � 40°, and m�C � 80°. Which is the shortest chord? A A�B� B B�C�C C�A� D AC � BCE It cannot be determined from the

information given.

16. In circle O below, find mADC�.A 45°B 75°C 155°D 230°E 245°




determined from the informationgivenColumn A Column B

17.

18.

19–20. Use the diagram for Exercises 19 and 20.In the diagram, m � � n.

19. Grid-In m�2 is 60° less than twicem�3. Find m�1.

20. Grid-In m�10 � 3x � 30 and m�9 � x � 40. Find m�9.



9

c d

Length of BC� Length of BD�




NAME _____________________________ DATE _______________ PERIOD ________

1. Find [ƒ ° g](x) for ƒ(x) � �2 �1

x� and g(x) � 3x � 2. 1. __________________

2. Determine whether the graphs of 3x � 2y � 5 � 0 and 2. __________________y � �23�x � 4 are parallel, coinciding, perpendicular, or none of these.

3. Solve the system of equations. 5x � 3y � 11 3. __________________x � 2y � �16

4. Find the inverse of � �, if it exists. 4. __________________

5. Determine whether the function ƒ(x) = �x3� is odd, even, or 5. __________________neither.

6. Solve the equation 2x2 � 10x � 12 � 0. 6. __________________



9. State the amplitude, period, phase shift, and vertical 9. __________________shift for y � 5 � 3 sin (2� � �).

10. Find the value of Cos�1 �tan �34��. 10. __________________


csoins

2

2

��. 11. __________________

12. Write the ordered pair that represents the vector from 12. __________________M(�7, 4) to N(3, �1).

13. Find the cross product �6, 3, 2 3, 4, �1. 13. __________________

14. Find the distance between the points with polar 14. __________________coordinates (3, 150°) and (4, 70°).

15. Express 2�cos ��6� + i sin ��6�� in rectangular form. 15. __________________

�12

34

Chapter

9


Page 211

1. C

2. A

3. A

4. A

5. D

6. B

7. A

8. D

9. C

Page 212

10. C

11. B

12. B

13. C

14. D

15. A

16. D

17. D

18. A

19. B

20. C

Bonus: D

Page 213

1. B

2. D

3. A

4. B

5. C

6. D

7. C

8. A

9. B

Page 214

10. C

11. B

12. A

13. B

14. D

15. C

16. D

17. B

18. C

19. A

20. D

Bonus: A



230-236 A&E C09-0-02-834179 10/4/00 3:15 PM Page 230


Page 215

1. A

2. B

3. D

4. C

5. C

6. B

7. A

8. D

9. C

Page 216

10. A

11. B

12. D

13. C

14. A

15. D

16. B

17. B

18. A

19. D

20. C

Bonus: D

Page 217

1. (�2, 150�)

2.

3. 2.79

4.

5. lemniscate

6. �3�2�, �54��

7. (3�2�, �3�2�)

8. �5� � r cos (� � 117�)

9. xy � 4

Page 21810.

11. �1�0� � r cos (� � 18�)

12. �35 � 10i

13. 26 � 7i

14.��1249� � �

2293�i

15. 4�cos�53�� i sin�5

3��

16. �4�2� � 4�2� i

17. �24i

18.�3�3� � 3i

19. �312�i

1.22 � 1.02 i; �1.49 � 0.54 i;

20. 0.28 � 1.56i

2�3� (cos 330�

Bonus: � i sin 330� )

Form 1C Form 2A


230-236 A&E C09-0-02-834179 10/4/00 3:15 PM Page 231

Page 219

1. (3, �60�)

2.

3. 2.53

4.

5. limaçon

6. �2, �53��

7. ��1, �3��

8. r � � 2

9. x2 � y2 � 8

Page 220

10.

11.�5� � r cos (� � 63�)

12. 9 � i

13. 5 � 12i

14. �117� � �

137�i

6�2� �cos �34��

15. i sin �34��

16. 2�3� � 2i

17. �16�3� � 16i

18. 2i

19. �64i

20. 2�3� � 2i

Bonus: ��22��

22��i

Page 221

1. (2, 120� )

2.

3. 3.31

4.

5. rose

6. �1, ��2

��

7. (�2�, �2�)

r sin � � 2 or 8. r � 2csc �

9. x2 � y2 � 9

Page 222

10.

11. �2� � r cos (� � 45� )

12. 5 � 2i

13. 20

14. 1 � i

15. 4�cos ��6

� � i sin ��6

��

16. �6 � 6i

17. �6i

18. �12�3� � 12i

19. �64

20. �3� � i

Bonus: i



230-236 A&E C09-0-02-834179 10/4/00 3:15 PM Page 232




3 Superior • Shows thorough understanding of the concepts polar and rectangular coordinates, polar equations, and sum, product, and powers of complex numbers.

• Uses appropriate strategies to find complex numbers with known sum.


2 Satisfactory, • Shows understanding of the concepts polar and with Minor rectangular coordinates, polar equations, and sum,Flaws product, and powers of complex numbers.

• Uses appropriate strategies to find complex numbers with known sum.


1 Nearly • Shows understanding of most of the concepts polar and Satisfactory, rectangular coordinates, polar equations, and sum,with Serious product, and powers of complex numbers.Flaws • May not use appropriate strategies to solve problems.

• Computations are mostly correct.• Written explanations are satisfactory.• Diagrams and graphs are mostly accurate and appropriate.• Satisfies most requirements of problems.• Written explanations are satisfactory.• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts polar and rectangular coordinates, polar equations, and sum, product, and powers of complex numbers.

• May not use appropriate strategies to find complex numbers with known sum.

• Computations are incorrect.• Written explanations are not satisfactory.• Diagrams and graphs are not accurate or appropriate.• Does not satisfy requirements of problems.

230-236 A&E C09-0-02-834179 10/4/00 3:15 PM Page 233

Page 1971–2. Sample answers are given1a. (2, 2)1b.

1c. r � 2�2��

4�

�

1d. The two graphs locate the samepoint in different coordinatesystems. The graphs are related by the relationships x � r cos � andy � r sin �.

2a. �4, ��6

��2b.

2c. x � 4 cos ��6

��, or 2�3�y � 4 sin ��

�6

��, or �2

2d. The two graphs locate the samepoint in different coordinatesystems. The graphs are related bythe relationships x � r cos � and y � r sin �.

3a.

3b. The graph of r � 2 cos � is a circle ofradius 1 centered at (1, 0). Studentscan use the graph from part a in theirdescription in part b.

3c. The graph is an 8-petal rose.

3d. The graph is a cardioid passingthrough (2, 0) and (0, � ) andsymmetric about � � 0.

3e. Sample answer: r � 2 sin 2�; rose

4a– 4c. Sample answers are given.

4a. (1 � i) � (2 � 2i) � 3 � 3i

4b. r � �1�2�� (��1�)2� � �2�, � �Arctan ��1

1�, or ��4��

The polar form of 1 � i is �2� �cos ��

4�� i sin ��

4�� .

r � �2�2�� (��2�)2� � 2�2�, � �

Arctan ��22�, or ��

4��.

The polar form of 2 � 2i is

2�2� �cos ��4�� i sin ��

4�� .

� � 4i4d. (3 � 3i)4 � (3 � 3i)(3 � 3i)(3 � 3i)(3 � 3i)

or �324

� 324[cos (�� ) � i sin (�� )]� �324

� 1.56 � 0.42i



� 4�cos ��2�� i sin ��

2��

(3 � 3i)4 � �3�2��cos ��4�� i sin ��

4��

4

4e. (3 � 3i)�13�

� �3�2��cos ��4�� i sin ��

4��

�13�

� �6

1�8� �cos ��1�2�� i sin ��

1�2��

4c. (1 � i)(2 � 2i) � �2� �cos ��4�� i sin ��

4��

2�2� �cos ��4�� i sin ��

4��

230-236 A&E C09-0-02-834179 10/4/00 3:15 PM Page 234



1. (�3, 150�)

2.

3. 3.15

4.

5. rose

6. �3�2�, �54��

7. (�2�3�, 2)

8. r � 5 sin �9. �3�x � y � 0

10. x � �2

Quiz APage 225

1. (�3, �150� )

2.

3.

4. 2.48

Quiz BPage 225

1. �8, �53��

2. (3�2�, �3�2�)

3. ��21�0�� r cos (� � 72�)

4. x2 � y2 � 25

5.

Quiz CPage 226

1. 2 � 3i

2. 26 � 2i

3. �1249� � �2

239�i

4. 4�cos �116

�� i sin �11

6��

5. �4�2� � 4�2�i

Quiz DPage 226

1. ��3� � i

2. 8i

3. �8 � 8�3�i

4. �16

5. 1.90 � 0.62i


230-236 A&E C09-0-02-834179 10/4/00 3:15 PM Page 235

Page 227

1. D

2. C

3. D

4. B

5. D

6. E

7. E

8. B

9. B

Page 228

10. E

11. C

12. B

13. A

14. D

15. B

16. D

17. B

18. D

19. 80

20. 45

Page 229

1. �31x�

2. perpendicular

3. (-2, �7)

4. �110� � �

5. even

6. 2, 3

7. �1, �2, ��13

�, ��23

�

8. 40�

9. 3; �; ��2

�; 5

10. �

11. tan2 �

12. �10, �5�

13. ��11, 0, �33�

14. 4.56

15. �3� � i

13

2�4



230-236 A&E C09-0-02-834179 10/4/00 3:15 PM Page 236



NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right of each problem.Exercises 1–3 refer to the ellipse represented by 9x2 � 16y2 � 18x � 64y � 71 � 0.

1. Find the coordinates of the center. 1. ________A. (1, 2) B. (1, �2) C. (�1, 2) D. (�2, 1)

2. Find the coordinates of the foci. 2. ________A. (1 � �7�, �2) B. (1, �2 � �7�)C. (5, �2), (�3, �2) D. (1, 4), (1, �8)

3. Find the coordinates of the vertices. 3. ________A. (1, 2), (1, �6), (4, �2), (�2, �2) B. (4, 2), (�2, 2), (1, 1), (1, �5)C. (5, �2), (�3, �2), (1, 1), (1, �5) D. (5, �2), (�3, �2), (1, 2), (1, �6)

4. Write the standard form of the equation of the circle that passes 4. ________through the points at (4, 5), (�2, 3), and (�4, �3).A. (x � 5)2 � ( y � 4)2 � 49 B. (x � 3)2 � ( y � 2)2 � 50C. (x � 4)2 � ( y � 2)2 � 36 D. (x � 2)2 � ( y � 2)2 � 25

5. For 4x2 � 4xy � y2 � 4, find �, the angle of rotation about the origin, 5. ________to the nearest degree.A. 27° B. 63° C. 333° D. 307°

6. Find the rectangular equation of the curve whose parametric 6. ________equations are x � 5 cos 2t and y � �sin 2t, 0° � t � 180°.A. �x5

2� � y2 � 1 B. �x5

2� � y2 � 1

C. �2x52� � y2 � 1 D. �

(x �

52)2� � ( y � 2)2 � 1

7. Find the distance between points at (m � 4, n) and (m, n � 3). 7. ________A. 3.5 B. 5 C. 1 D. 7

8. Write the standard form of the equation of the circle that is tangent 8. ________to the line x � �3 and has its center at (2, �7).A. (x � 2)2 � ( y � 7)2 � 25 B. (x � 2)2 � ( y � 7)2 � 5C. (x � 2)2 � ( y � 7)2 � 16 D. (x � 2)2 � ( y � 7)2 � 25

9. Find the coordinates of the point(s) of intersection for the graphs of 9. ________x2 � 2y2 � 33 and x2 � y2 � 2x � 19.A. (5, 2), (�1, 4) B. (5, 4), (�1, 2)C. (5, �2), (�1, �4) D. Graphs do not intersect.

10. Identify the conic section represented by 10. ________9y2 � 4x2 � 108y � 24x � �144.A. parabola B. hyperbola C. ellipse D. circle

11. Write the equation of the conic section y2 � x2 � 5 after a rotation of 11. ________45° about the origin.A. x�y� � �2.5 B. x�y� � �5C. (y�)2 � (x�)2 � 2.5 D. (x�)2 � 2.5y�

12. Find parametric equations for the rectangular equation 12. ________(x � 2)2 � 4( y � 1).A. x � t, y � t2 � 2, �∞ t ∞B. x � t, y � �4

1�t2 � t � 2, �∞ t ∞

C. x � t, y � �41�t 2 � t � 2, �∞ t ∞

D. x � t, y � 4t2 � t � 2, �∞ t ∞

Chapter

10

237-255 A&E C10-02-834179 10/10/00 10:24 AM Page 237 (Black plate)


13. Which is the graph of 6x2 � 12x � 6y2 � 36y � 36? 13. ________A. B. C. D.

Exercises 14 and 15 refer to the hyperbola represented by �2x2 � y2 � 4x � 6y � �3.14. Write the equations of the asymptotes. 14. ________

A. y � 3 � �2(x � 1) B. y � 3 � ��12�(x � 1)

C. y � 3 � ��2�(x � 1) D. y � 3 � ��22��(x � 1)

15. Find the coordinates of the foci. 15. ________A. (1 � �2�, �3)B. (1 � �6�, �3)C. (1, �3 � �2�)D. (1, �3 � �6�)

16. Write the standard form of the equation of the hyperbola for which 16. ________the transverse axis is 4 units long and the coordinates of the foci are (1, �4 � �7�).

A. �(x �

31)2

� � �( y �

44)2

� � 1 B. �( y �

44)2

� � �(x �

31)2

� � 1

C. �( y �

34)2

� � �(x �

41)2

� � 1 D. �(x �

41)2

� � �( y �

34)2

� � 1

17. The graph at the right shows the solution set for 17. ________which system of inequalities?A. xy �6, 9(x � 1)2 � 4( y � 1)2 � 36B. xy � �6, 4(x � 1)2 � 9( y � 1)2 � 36C. xy �6, 4(x � 1)2 � 9( y � 1)2 � 36D. xy � �6, 9(x � 1)2 � 4( y � 1)2 � 36

18. Find the coordinates of the vertex and the equation of the axis of symmetry for the parabola represented 18. ________by x2 � 4x � 6y � 10 � 0.A. (�2, 1), y � 1 B. (1, �2), y � �2C. (�2, 1), x � �2 D. (1, �2), x � 1

19. Write the standard form of the equation of the parabola whose 19. ________directrix is x � �1 and whose focus is at (5, �2).A. ( y � 2)2 � 12(x � 2) B. y � 2 � 12(x � 2)2

C. x � 2 � �112� ( y � 2)2 D. x � 2 � �1

12� ( y � 2)2

20. Identify the graph of the equation 16x2 � 24xy � 9y2 � 30x � 40y � 0. 20. ________Then, find � to the nearest degree.A. hyperbola; �37° B. parabola; �37°C. parabola; 45° D. circle; �74°

Bonus Find the coordinates of the points of intersection of the Bonus: ________graphs of x2 � y2 � 3, xy � 2, and y � �2x � 5.

A. (�2, �1) B. (1, �2)C. (2, 1) D. Graphs do not intersect.



10




NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right of each problem.Exercises 1-3 refer to the ellipse represented by x2 � 25y2 � 6x � 100y � 84 � 0.

1. Find the coordinates of the center. 1. ________A. (2, 3) B. (3, 2) C. (�3, �2) D. (�2, �3)

2. Find the coordinates of the foci. 2. ________A. (3, 2 � 2�6�)B. (�2, 2), (8, 2) C. (3 � 2�6�, 2)D. (2 � 2�6�, 3)

3. Find the coordinates of the vertices. 3. ________A. (8, 2), (�2, 2), (3, 3), (3, 1) B. (8, 2), (�2, 2), (3, 7), (3 , �3)C. (4, 2), (2, 2), (3, 3), (3, 1) D. (4, 2), (2, 2), (3, 7), (3, �3)

4. Write 6x2 � 12x � 6y2 � 36y � 36 in standard form. 4. ________A. (x � 3)2 � (y � 1)2 � 16 B. (x � 1)2 � (y � 3)2 � 16C. (x � 1)2 � (y � 3)2 � 16 D. (x � 3)2 � (y � 1)2 � 16

5. For 2x2 � 3xy � y2 � 1, find �, the angle of rotation about the origin, 5. ________to the nearest degree.A. �9� B. 36� C. �36� D. 324�

6. Find the rectangular equation of the curve whose parametric equations 6. ________are x � 3 cos t and y � sin t, 0� � t � 360�.A. �x9

2� � y2 � 1 B. �x9

2� � y2 � 1 C. y2 � �x3

2� � 1 D. y2 � �x3

2� � 1

7. Find the distance between points at (�5, 2) and (7, �3). 7. ________A. �5� B. 13 C. �2�9� D. �1�1�9�

8. Write the standard form of the equation of the circle that is tangent to 8. ________the y-axis and has its center at (�3, 5).A. (x � 3)2 � ( y � 5)2 � 9 B. (x � 3)2 � ( y � 5)2 � 25C. (x � 3)2 � ( y � 5)2 � 3 D. (x � 3)2 � ( y � 5)2 � 9

9. Find the coordinates of the point(s) of intersection for the graphs of 9. ________x2 � y2 � 4 and y � 2x � 1.A. (1.3, 1.5) B. (1.3, 1.5), (�0.5, �1.9)C. (�1.3, �1.5), (�0.5, �1.9) D. Graphs do not intersect.

10. Identify the conic section represented by 3y2 � 3x2 � 12y � 18x � 42. 10. ________A. parabola B. hyperbola C. ellipse D. circle

11. Write the equation of the conic section y2 � x2 � 2 after a rotation of 11. ________45� about the origin.A. x′y′ � �1 B. x′y′ � �2 C. (y′)2 � (x′)2 � 2 D. (x′)2 � y′

12. Find parametric equations for the rectangular equation x2 � y2 � 25 � 0. 12. ________A. x � cos 5t, y � sin 5t, 0 � t � 2� B. x � cos t, y � 5 sin t, 0 � t � 2�C. x � 5 cos t, y � sin t, 0 � t � 2� D. x � 5 cos t, y � 5 sin t, 0 � t � 2�

Chapter

10



13. Which is the graph of x2 � (y � 2)2 � 16? 13. ________A. B. C. D.

Exercises 14 and 15 refer to the hyperbola represented by 36x2 � y2 � 4y � �32.

14. Write the equations of the asymptotes. 14. ________A. y � 1 � �6(x � 2) B. y � �6xC. y � 2 � �6(x � 1) D. y � 2 � �6x

15. Find the coordinates of the foci. 15. ________A. (0, 4 � �3�7�), (0, �8 � �3�7�) B. (0, 4), (0, �8)C. (0, 4 � �3�5�), (0, �8 � �3�5�) D. (0, �2 � �3�7�)

16. Write the standard form of the equation of the hyperbola for which 16. ________a � 2, the transverse axis is vertical, and the equations of the asymptotes are y � �2x.A. �x4

2� � y2 � 1 B. y2 � �x4

2� � 1 C. x2 � �

y4

2� � 1 D. �

y4

2� � x2 � 1

17. The graph at the right shows the solution set for 17. ________which system of inequalities?A. y � 1 �(x � 1)2, 4(x � 1)2 � 9( y � 3)2 � 36B. y � 1 �(x � 1)2, 4(x � 1)2 � 9( y � 3)2 36C. y � 1 � �(x � 1)2, 4(x � 1)2 � 9( y � 3)2 � 36D. y � 1 � �(x � 1)2, 4(x � 1)2 � 9( y � 3)2 36

18. Find the coordinates of the vertex and the equation of the axis of 18. ________symmetry for the parabola represented by x2 � 2x � 12y � 37 � 0.A. (�1, �3), x � �1 B. (�1, �6), x � �1C. (�1, �12), x � �5 D. (3, 2), y � �9

19. Write the standard form of the equation of the parabola whose 19. ________directrix is y � �4 and whose focus is at (2, 2).A. (y � 2)2 � 12(x � 2) B. y � 1 � 12(x � 2)2

C. (x � 2)2 � 12(y � 2) D. (x � 2)2 � 12(y � 1)

20. Identify the graph of the equation 4x2 � 5xy � 16y2 � 32 � 0. 20. ________A. circle B. ellipse C. parabola D. hyperbola

Bonus Find the coordinates of the points of intersection of the Bonus: ________graphs of x2 � y2 � 5, xy � �2, and y � �3x � 1.

A. (�2, �1) B. (1, �2) C. (2, 1) D. (�1, �2)



10




NAME _____________________________ DATE _______________ PERIOD ________


Exercises 1–3 refer to the ellipse represented by 4x2 � 9y2 � 18y � 27 � 0.

1. Find the coordinates of the foci. 1. ________A. (�1, 0) B. (0, �1) C. (1, 0) D. (0,1)

2. Find the coordinates of the foci. 2. ________A. (0, 1 � �5�) B. (�5�, 1), (��5�, 1)C. (�5�, 3), (��5�, 3) D. (1, 5), (1, �5)

3. Find the coordinates of the vertices. 3. ________A. (2, 1), (�2, 1) (0, 4), (0, �2) B. (3, 1), (�3, 1), (0, 3), (0, �1)C. (3, 1), (�3, 1), (0, 4), (0, �2) D. (2, 1), (�2, 1), (0, 3), (0, �1)

4. Write the standard form of the equation of the circle that is tangent 4. ________to the x-axis and has its center at (3, �2).A. (x � 3)2 � ( y � 2)2 � 4 B. (x � 3)2 � ( y � 2)2 � 4C. (x � 3)2 � ( y � 2)2 � 2 D. (x � 3)2 � ( y � 2)2 � 2

5. For 2x2 � xy � 2y2 � 1, find �, the angle of rotation about the origin, 5. ________to the nearest degree.A. 215° B. 150° C. 45° D. �30°

6. Find the rectangular equation of the curve whose parametric 6. ________equations are x � �cos t and y � sin t, 0° � t � 360°.A. y2 � x2 � 1 B. x2 � y2 � �1 C. x2 � y2 � 1 D. x2 � y2 � 1

7. Find the distance between points at (�1, 6) and (5, �2). 7. ________A. �1�4� B. 10 C. �3�4� D. 8

8. Write x2 � 4x � y2 � 2y � 4 in standard form. 8. ________A. (x � 2)2 � ( y � 1)2 � 9 B. (x � 2)2 � ( y � 1)2 � 9C. (x � 2)2 � ( y � 1)2 � 3 D. (x � 2)2 � ( y � 1)2 � 4

9. Find the coordinates of the point(s) of intersection for the graphs of 9. ________x2 � y2 � 20 and y � x � 2.A. (�2, �4), (4, 2) B. (�2, �4), (�4, �2)C. (2, 4), (�4, �2) D. (�2, �4), (4, 2)

10. Identify the conic section represented by x2 � y2 � 12y � 18x � 42. 10. ________A. parabola B. hyperbola C. ellipse D. circle

11. Write the equation of the conic section x2 � y2 � 16 after a rotation 11. ________of 45° about the origin.A. (x�)2 � ( y�)2 � 16 B. (x�)2 � ( y�)2 � 16C. (x�)2 � 2x�y� � ( y�)2 � 16 D. (x�)2 � 2x�y� � ( y�)2 � 16

12. Find parametric equations for the rectangular equation x2 � y2 � 16. 12. ________A. x � cos 4t, y � sin 4t, 0° � t � 360°B. x � cos 16t, y � sin 16t, 0° � t � 360°C. x � 4 cos t, y � 4 sin t, 0° � t � 360°D. x � 16 cos t, y � 16 sin t, 0° � t � 360°

Chapter

10



13. Which is the graph of (x � 3)2 � ( y � 2)2 � 9? 13. ________A. B. C. D.

Exercises 14 and 15 refer to the hyperbola represented by�16y 2 � 54x � 9x2 � 63.14. Write the equations of the asymptotes. 14. ________

A. y � 3 � � �43� x B. y � 3 � � �

34� x

C. y � � �43�(x � 3) D. y � � �

34�(x � 3)

15. Find the coordinates of the foci. 15. ________A. (5, 0), (�5, 0) B. (0, 5), (0, �5)C. (3, 5), (3, �5) D. (8, 0), (�2, 0)

16. Write the standard form of the equation of the hyperbola for 16. ________which a � 5, b � 6, the transverse axis is vertical, and the center is at the origin.

A. �2y5

2

� � �3x6

2

� � 1 B. �3x6

2

� � �2y5

2

� � 1 C. �2x5

2

� � �3y6

2

� � 1 D. �3y6

2

� � �2x5

2

� � 1

17. The graph at the right shows the solution set for 17. ________which system of inequalities?A. y2 � 4x2 � 1, x2 � y2 � 9B. y2 � 4x2 1, x2 � y2 � 9C. y2 � 4x2 � 1, x2 � y2 � 3D. y2 � 4x2 1, x2 � y2 � 3

18. Find the coordinates of the vertex and the equation of the axis of 18. ________symmetry for the parabola represented by y2 � 8x � 4y � 28 � 0.A. (3, 2), x � 3 B. (3, 2), y � 2C. (2, 3), y � 3 D. (2, 3), x � 2

19. Write the standard form of the equation of the parabola whose 19. ________directrix is x � �2 and whose focus is at (2, 0).A. ( y � 2)2 � 8(x � 2) B. ( y � 2)2 � 4(x � 2)C. y2 � 8x D. x2 � 8y

20. Write the equation for the translation of the graph of 20. ________y2 � 4x � 12 � 0 for T(�1, 1).A. ( y � 1)2 � 4(x � 2) B. ( y � 1)2 � 4(x � 4)C. ( y � 1)2 � 4(x � 2) D. ( y � 1)2 � 4(x � 4)

Bonus Find the coordinates of the points of intersection of the Bonus: ________graphs of x2 � y2 � 4, x2 � y2 � 4, and x � y � 2.

A. (2, 0) B. (0, 2) C. (0, �2) D. (�2, 0)



10




NAME _____________________________ DATE _______________ PERIOD ________

1. Find the distance between points at (m, n � 5) and 1. __________________(m � 3, n � 2).

2. Determine whether the quadrilateral ABCD with vertices 2. __________________

A��1, �12��, B��12�, ��12��, C�1, ��32��, and D��12�, �1� is a parallelogram.

3. Write the standard form of the equation of the circle that 3. __________________passes through the point at (2, �2) and has its center at (�2, 3).

4. Write the standard form of the equation of the circle that 4. __________________passes through the points at (1, 3), (7, 3), and (8, 2).

5. Write 2x2 � 10x � 2y2 � 18y � 1 � 0 in standard form. 5. __________________Then, graph the equation, labeling the center.

6. Write 3x2 � 2y2 � 24x � 4y � 26 � 0 in standard 6. __________________form. Then, graph the equation, labeling the center, foci, and vertices.

7. Find the equation of the ellipse that has its major axis 7. __________________parallel to the y-axis and its center at (4, �3), and that passes through points at (1, �3) and (4, 2).

8. Find the equation of the equilateral hyperbola that has 8. __________________its foci at (�2, �3 � 2�3�) and (�2, �3 � 2�3�), and whose conjugate axis is 6 units long.

9. Write �2x2 � 3y2 � 24x � 6y � 93 � 0 in standard 9. __________________form. Find the equations of the asymptotes of the graph. Then, graph the equation, labeling the center,foci, and vertices.

Chapter

10



10. Write x � y2 � 2y � 5 in standard form. Find the equations 10. __________________of the directrix and axis of symmetry. Then, graph the equation, labeling the focus, vertex, and directrix.

11. Find the equation of the parabola that passes through the 11. __________________point at �0, � �12��, has a vertical axis, and has a maximum

at (�2, 1).

12. Find the equation of the hyperbola that has eccentricity �95� 12. __________________and foci at (4, 6) and (4, �12).

13. Identify the conic section represented by xy � 3y � 4x � 0. 13. __________________

14. Find a rectangular equation for the curve whose 14. __________________parametric equations are x � ��12� cos 4t � 2,y � �2 sin 4t � 3, 0° � t � 90°.

15. Find parametric equations for the equation 15. __________________

�(x �

161)2

� � 2( y � 3)2 � 1.

Identify the graph of each equation. Write an equation of thetranslated or rotated graph in general form.16. x2 � 12y � �31 � 10x for T (3, �5) 16. ___________________________

17. x2 � xy � 2y2 � 2, � � �30° 17. __________________

18. Identify the graph of 4x2 � 7xy � 5y2 � �3. Then, find �, 18. __________________the angle of rotation about the origin, to the nearest degree.

19. Solve the system 4x2 � y2 � 32 � 4y and x2 � 7 � y 19. __________________algebraically. Round to the nearest tenth.

20. Graph the solutions for the system of inequalities. 20.9x2 � y2

x2 � 100 � y2

Bonus Find the coordinates of the point(s) of Bonus: __________________intersection of the graphs of 2x � 1 � y,x2 � 10 � y2, and y � 4x2 � �1.



10




NAME _____________________________ DATE _______________ PERIOD ________

1. Find the distance between points at (�2, �5) and (6, �1). 1. __________________

2. Determine whether the quadrilateral ABCD with vertices 2. __________________A(�1, 2), B(2, 0), C(�2, �2), and D(�5, 0) is a parallelogram.

3. Write the standard form of the equation of the circle that is 3. __________________tangent to x � �3 and has its center at (1, �3).

4. Write the standard form of the equation of the circle that 4. __________________passes through the points at (�6, 3), (�4, �1), and (�2, 5).

5. Write x2 � y2 � 6x � 14y � 42 � 0 in standard form. Then, 5. __________________graph the equation, labeling the center.

6. Write 4x2 � 9y2 � 24x � 18y � 9 � 0 in standard form. Then, 6. __________________graph the equation, labeling the center, foci, and vertices.

7. Find the equation of the ellipse that has its foci at (2, 1) and 7. __________________(2, �7) and b � 2.

8. Find the equation of the hyperbola that has its foci at (0, �4) 8. __________________and (10, �4), and whose conjugate axis is 6 units long.

9. Write 4x2 � y2 � 24x � 4y � 28 � 0 in standard form. Find 9. __________________the equations of the asymptotes of the graph. Then, graph the equation, labeling the center, foci, and vertices.

Chapter

10



10. Write �2x � y2 � 2y � 5 � 0 in standard form. Find the 10. __________________equations of the directrix and axis of symmetry. Then, graph the equation, labeling the focus, vertex and directrix.

11. Find the equation of the parabola that passes through 11. __________________the point at (�8, 15), has its vertex at (2, �5), and opens to the left.

12. Find the equation of the ellipse that has its center at the 12. __________________origin, eccentricity �2�

32��, and a vertical major axis of 6 units.

13. Identify the conic section represented by x2 � 3xy � y2 � 5. 13. __________________

14. Find a rectangular equation for the curve whose parametric 14. __________________equations are x � cos 3t, y � �2 sin 3t, 0� � t � 120�.

15. Find parametric equations for the equation �1x62� � �3

y6

2� � 1. 15. __________________

Identify the graph of each equation. Write an equation of thetranslated or rotated graph in general form.16. x2 � 2x � 2y � 9 � 0 for T(�2, 3) 16. __________________

17. 2x2 � 5y2 � 20 � 0, � � 30� 17. __________________

18. Identify the graph of 3x2 � 8xy � 3y2 � 3. Then, find �, the 18. __________________angle of rotation about the origin, to the nearest degree.

19. Solve the system 5x2 � 10 � 2y2 and 3y2 � 84 � 2x2 19. __________________algebraically. Round to the nearest tenth.

20. Graph the solutions for the system of inequalities. 20.(x � 2)2 � (y � 1)2 � 9�1x62� � y2 � 1

Bonus Find the coordinates of the point(s) of intersection Bonus: __________________of the graphs of x � y � 1 � 0, x2 � y2 � 5,y � �3x2 � 1.



10




NAME _____________________________ DATE _______________ PERIOD ________

1. Find the distance between points at (2, 1) and (5, �3). 1. __________________

2. Determine whether the quadrilateral ABCD with vertices 2. __________________A(5, 3), B(7, 3), C(5, 1), and D(2, 1) is a parallelogram.

3. Write the standard form of the equation of the circle that 3. __________________has its center at (�4, 3) and a radius of 5.

4. Write the standard form of the equation of the circle that 4. __________________passes through the points at (�2, 2), (2, 2), and (2, �2).

5. Write x2 � 8x � y2 � 4y � 16 � 0 in standard form. Then, 5. __________________graph the equation, labeling the center.

6. Write 9x2 � 54x � 4y2 � 16y � 61 � 0 in standard form. 6. __________________Then, graph the equation, labeling the center, foci, and vertices.

7. Find the equation of the ellipse that has its center at 7. __________________(�1, 3), a horizontal major axis of 6 units, and a minor axis of 2 units.

8. Find the equation of the hyperbola that has its center at 8. __________________(�2, 4), a � 3, b � 5, and a vertical transverse axis.

9. Write x2 � 4x � 4y2 � 8y � 16 � 0 in standard 9. __________________form. Find the equations of the asymptotes of the graph. Then, graph the equation, labeling the center, foci, and vertices.

Chapter

10



10. Write y2 � 4y � 4x � 12 � 0 in standard form. Find the 10. __________________equations of the directrix and axis of symmetry. Then,graph the equation, labeling the focus, vertex, and directrix.

11. Find the equation of the parabola that has its vertex at 11. __________________(5, �1), and the focus at (5, �2).

12. Find the eccentricity of the ellipse �(x �

42)2

� � �( y �

83)2

� � 1. 12. __________________

13. Identify the conic section represented by 13. __________________x2 � 2y2 � 16y � 42 � 0.

14. Find a rectangular equation for the curve whose parametric 14. __________________equations are x � 2 cos t, y � �3 sin t, 0° � t � 360°.

15. Find parametric equations for the equation x2 � y2 � 8. 15. __________________

Identify the graph of each equation. Write an equation of the translated or rotated graph in general form.

16. y2 � 8y � 8x � 32 � 0 for T(1, �4) 16. __________________

17. 5x2 � 4y2 � 20, � � 45° 17. __________________

18. Identify the graph of 4x2 � 9xy � 3y2 � 5. Then, find �, 18. __________________the angle of rotation about the origin, to the nearest degree.

19. Solve the system algebraically. Round to the nearest tenth. 19. __________________x2 � y2 � 92x2 � 3y2 � 18

20. Graph the solutions for the system of inequalities. 20.(x � 1)2 � 2( y � 3)2 16(x � 3)2 � �8( y � 2)

Bonus Find the coordinates of the point(s) of Bonus: __________________intersection of the graphs of ��2

3�x � y � 0,

x2 � y2 � 13, and y � 1 � �14�(x � 2)2.



10




NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include allrelevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond therequirements of the problem.

1. Consider the equation of a conic section written in the formAx2 � By2 � Cx � Dy � E � 0.a. Explain how you can tell if the equation is that of a circle.

Write an equation of a circle whose center is not the origin.Graph the equation.

b. Explain how you can tell if the equation is that of an ellipse.Write an equation of an ellipse whose center is not the origin.Graph the equation.

c. Explain how you can tell if the equation is that of a parabola.Write an equation of a parabola with its vertex at (�1, 2).Graph the equation.

d. Explain how you can tell if the equation is that of a hyperbola.Write an equation of a hyperbola with a vertical transverseaxis.

e. Identify the graph of 3x2 � xy � 2y2 � 3 � 0. Then find theangle of rotation � to the nearest degree.

2. a. Describe the graph of x2 � 4y2 � 0.

b. Graph the relation to verify your conjecture.

c. What conic section does this graph represent?

3. Give a real-world example of a conic section. Discuss how youknow the object is a conic section and analyze the conic section if possible.

Chapter

10



Find the distance between each pair of points with the given coordinates. Then,find the midpoint of the segment that has endpoints at the given coordinates.

1. __________________

1. (1, �4), (2, �9) 2. (s, �t), (6 � s, �5 � t) 2. __________________

3. Determine whether the quadrilateral ABCD with vertices 3. __________________A(�2, 2), B(1, 3), C(4, �1), and D(1, �2) is a parallelogram.

4. Write the standard form of x2 � 6x � y2 � 4y � 12 � 0. 4. __________________Then, graph the equation labeling the center.

5. Write the standard form of the equation of the circle that 5. __________________passes through the points (�2, 16), (�2, 0), and (�32, 0).Then, identify the center and radius.

For the equation of each ellipse, find the coordinates of the center, foci, and vertices. Then, graph the equation.

6. Write the standard form of 9x2 � y2 � 18x � 6y � 9 � 0. 6. __________________Then, find the coordinates of the center, the foci, and the vertices of the ellipse.

7. Graph the ellipse with equation �(x �25

5)2� � �

(y �

164)2

� � 1. 7.Label the center and vertices.

8. Write the equation of the 8. __________________hyperbola graphed at the right.

9. Find the coordinates of the center, the foci, and the vertices 9. __________________for the hyperbola whose equation is 4y2 � 9x2 � 48y �18x � 99 � 0. Then, find the equations of the asymptotes and graph the equation.



10


Find the distance between each pair of points with the given coordinates. Then, find the midpoint of the segment thathas endpoints at the given coordinates. 1. __________________

1. (�2, 4), (5, �3) 2. (a, b), (a � 4, b � 3) 2. __________________

3. Determine whether the quadrilateral ABCD with vertices 3. __________________A(�1, 1), B(3, 3), C(3, 0), D(�1, �1) is a parallelogram.

4. Write the standard form of x2 � 6x � y2 � 10y � 2 � 0. 4. __________________Then graph the equation, labeling the center.

5. Write the standard form of the equation of the circle that passes through the points (2, 10), (2, 0), and (�10, 0). 5. __________________Then, identify the center and radius.

1. Write the standard form of 16x2 � 4y2 � 96x � 8y � 84 � 0. 1. __________________Then, find the coordinates of the center, the foci, and the vertices of the ellipse.

2. Graph the ellipse with equation �(x �64

6)2� � �

(y1�00

1)2� � 1. 2.

Label the center and vertices.

3. Write the equation of the 3. __________________hyperbola shown in the graph at the right.

4. Find the coordinates of the center, the foci, and the vertices 4.for the hyperbola whose equation is 3x2 � 4y2 � 150x �16y � 109. Then, find the equations of the asymptotes and graph the equation.


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

10

Chapter

10


1. Write the standard form of x2 � 4x � 8y � 12 � 0. Identify 1. __________________the coordinates of the focus and vertex, and the equations of the directrix and axis of symmetry. Then, graph the equation.

2. Write the equation of the parabola that has a focus at 2. __________________(�2, 3) and whose directrix is given by the equation x � 4.

3. Identify the conic section represented by the equation 3. __________________4x2 � 25y2 � 16x � 50y � 59 � 0. Then, write the equation in standard form.

4. Find the rectangular equation of the curve whose parametric 4.equations are x � �3t2 and y � 2t, �2 � t � 2. Then graph the equation, using arrows to indicate the orientation.

5. Find parametric equations for the equation x2 � y2 � 100. 5. __________________

Identify the graph of each equation. Write an equation of the 1. __________________translated or rotated graph in general form.

1. 5x2 � 8y2 � 40 for T(�3, 5) 2. 4x2 � 18y2 � 36, � � 135° 2. __________________

3. Identify the graph of x2 � 3xy � 2y2 � 2x � y � 6 � 0. 3. __________________

4. Solve the system x2 � 25 � y2 and xy � �12 algebraically. 4. __________________

5. Graph the solutions for the system of inequalities. 5.x2 � 9y2 36x2 � 2y 4


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

10

Chapter

10




NAME _____________________________ DATE _______________ PERIOD ________


Multiple Choice1. Three points on a line are X, Y, and Z,

in that order. If XZ � YZ � 6, what is the ratio �X

YZZ�?

A 1 to 2B 1 to 3C 1 to 4D 1 to 5E It cannot be determined from the

information given.

2. A train traveling 90 miles per hour for 1 hour covers the same distance as atrain traveling 60 miles per hour forhow many hours?A �12� B �3

1�

C �23� D �32�

E 3

3. If xy � 48, then which of the followingCANNOT be true?A x � y 14 B x � y � 14C x � y � 14 D �x � y� 14E �x � y� � 14

4. If x � 0, then �(��

44xx)3

3� �

A �16B �1C 3D 1E 16

5. A framed picture is 5 feet by 8 feet,including the frame. If the frame is 8 inches wide, what is the ratio of thearea of the frame to the area of theframed picture, including the frame?A 7 to 18B 7 to 11C 11 to 18D 143 to 180E 37 to 180

6. In circle O, OA � 6 and O�A� ⊥ O�B�. Find the area of the shaded region.A 2� units2

B (� � 2) units2

C (6� � 9�3�) units2

D (9� � 18) units2

E (36� � 9�3�) units2

7. If 1 dozen pencils cost 1 dollar, howmany dollars will n pencils cost?A 12nB �1

n2�

C �1n2�

D �121n�


8. On a map drawn to scale, 0.125 inchrepresents 10 miles. What is the actualdistance between two cities that are 2.5 inches apart on the map?A. 12.5 miB 25 miC 200 miD 250 miE 1250 mi

9. In the diagram below, if � ��m, thenI. �3 and �4 are supplementary.

II. m�2 � m�10.III. m�6 � m�8 � m�9.

A I onlyB II onlyC III onlyD II and III onlyE I and III only

Chapter

10



10. P�C� and A�B� intersect at point Q.m�PQB � (2z � 80)�, m�BQC �(4x � 3y)�, m�CQA � 5w�, andm� AQP � 2z�. Find the value of w.A 20B 26C 75D 130E It cannot be determined from the

information given.11. Which point lies the greatest distance

from the origin?A (0, �9)B (�2, 9)C (�7, �6)D (8, 5)E (�5, 7)

12. The vertices of rectangle ABCD are thepoints A(0, 0), B(8, 0), C(8, k), and D(0, 5). What is the value of k?A 2B 3C 4D 5E 6

13. The measures of three exterior anglesof a quadrilateral are 37�, 58�, and 92�.What is the measure of the exteriorangle at the fourth vertex?A 7�B 173�C 83�D 49�E 81�

14. The diagonals of parallelogram A BCDintersect at point O. If AO � 2x � 1and AC � 5x � 5, then AO �A 5B 7C 15D 30E It cannot be determined from the

information given.

15. If Nancy earns d dollars in h hours,how many dollars will she earn in h � 25 hours?A �

25hd

� B d � �2

h5d�

C 26d D �h �

dh25�

E None of these

16. If 6n cans fill �n2� cartons, how manycans does it take to f ill 2 cartons?A 12 B 24nC 24 D 6n2

E �32� n2




determined from the information given

Column A Column B

17.

18.

19. Grid-In If 4 pounds of fertilizer cover1500 square feet of lawn, how manypounds of fertilizer are needed to cover2400 square feet?

20. Grid-In The distance between twocities is 750 miles. How many inchesapart will the cities be on a map with ascale of 1 inch � 250 miles?



10

Ratio of boys to girls in a class

with twice as manygirls as boys

Ratio of boys to girls in a class

with half as manyboys as girls

Ratio of �25� to �3

1�Ratio of �3

1� to �

15�




NAME _____________________________ DATE _______________ PERIOD ________

1. Write a linear function that has no zero. 1. __________________

2. Find BC if B � � � and C � � �. 2. __________________

3. Consider the system of inequalities 3x � 2y 10, 3. __________________x � 3y 9, x 0, and y 0. In a problem asking you to find the minimum value of ƒ(x, y) � x � 3y,state whether the situation is infeasible, hasalternate optimal solutions, or is unbounded.

4. Given ƒ(x) � �x �2

5�, find ƒ�1(x). Then, state whether 4. __________________ƒ�1(x) is a function.

5. If y varies directly as the square of x, inversely as w, 5. __________________inversely as the square of z, and y � 2 when x � 1,w � 4, and z � �2, find y when x � 3, w � �6, and z � �3.

6. Solve �xx��

13� � �3x� � �

x21�

23x

�. 6. __________________

7. Suppose � is an angle in standard position whose 7. __________________terminal side lies in Quadrant II. If cos � � ��1

73�,

find the value of csc �.

8. Solve � ABC if B � 47 , C � 68 , and b � 29.2. 8. __________________

9. Find the linear velocity of the tip of an airplane propeller 9. __________________that is 3 meters long and rotating 500 times per minute.Give the velocity to the nearest meter per second.

10. Solve tan � � cot � for 0 � � < 360 . 10. __________________

11. Jason is riding his sled down a hill. If the hill is inclined 11. __________________at an angle of 20 with the horizontal, find the force that propels Jason down the hill if he weighs 151 pounds.

12. Find (�3� � i)5. Express the result in rectangular form. 12. __________________

13. Write the equation in standard form 13. __________________of the ellipse graphed at the right.

0�4

�17

52

�21

36

Chapter

10


Page 237

1. B

2. A

3. C

4. B

5. C

6. C

7. B

8. A

9. C

10. C

11. A

12. B

Page 238

13. A

14. C

15. D

16. B

17. C

18. C

19. D

20. B

Bonus: C

Page 239

1. B

2. C

3. A

4. C

5. B

6. A

7. B

8. A

9. B

10. B

11. A

12. D

Page 240

13. B

14. D

15. D

16. D

17. A

18. A

19. D

20. B

Bonus: B



256-262A&EC10-0-02-834179 10/10/00 10:42 AM Page 256


Page 241

1. D

2. B

3. B

4. A

5. C

6. C

7. B

8. B

9. D

10. B

11. A

12. C

Page 242

13. C

14. D

15. D

16. A

17. A

18. B

19. C

20. B

Bonus: A

Page 243

1. �5�8�

2. no

(x � 2)2 �3. ( y � 3)2 � 41

(x � 4)2 �4. ( y � 1)2 � 25

5. �x � �52

��2� �y � �9

2��

2� 27

6.�(x �8

4)2

� � �( y

1�21)2

� � 1

7. �(x �

94)2

� � �( y �

253)2

� � 1

8.�( y �

33)2

� � �(x �

92)2

� � 1

�( y �

81)2

� � �(x �

126)2

� � 1asymptotes:

9. y � 1 � � ��36�� (x � 6)

Page 244

( y � 1)2 � x � 6;10. x � ��2

45�; y � 1

11. (x � 2)2 � ��83

�(y � 1)

12. �( y �

253)2

� � �(x �

564)2

� � 1

13. hyperbola

14. � �( y �

43)2

� � 1

Sample answer:x � 4 cos t � 1,

y � ��2

2�� sin t � 3,

15.0° � t � 360°

parabola;16. y2 � 4x � 12y � 70 � 0

ellipse; �5 � �3��x2 �

17. �2 � 2�3��xy � �7 � �3��y2 � 8 � 0

18. hyperbola, 19°

19. (�2.7, 0.5)

20.

Bonus: (1, 3)

(x � 2)2�

�14

�

Form 1C Form 2A


256-262A&EC10-0-02-834179 10/10/00 10:42 AM Page 257

Page 245

1. 4�5�

2. yes

(x � 1)2 �3. ( y � 3)2 � 16

(x � 3)2 �4. ( y � 2)2 � 10

(x � 3)2 �5. ( y � 7)2 � 100

6. �(x �

93)2

� � �( y �

41)2

� � 1

7. �(x �

42)2

� � �( y

2�

03)2

� � 1

8. �(x �

165)2

� � �( y �

94)2

� � 1

�(x �

13)2

� � �( y �

42)2


9. y � 2 � �2(x � 3)

Page 246( y � 1)2 � 2( x � 2);

10. x � �32

�; y � 1

11. ( y � 5)2 � �40(x � 2)

12. �x1

2� � �

y9

2� � 1

13. hyperbola

14. �x1

2� � �

y4

2� � 1

Sample answer:x � 4 cos t, y � 6 sin t,

15. 0� � t � 360�

parabola; 16. x2 � 2x � 2y � 3 � 0

ellipse;11(x�)2 � 6�3� x�y� �

17. 17 (y�)2 � 80 � 0

18. hyperbola, 27�

19. no solution

20.

Bonus: (1, �2)

Page 247

1. 5

2. no

(x � 4)2 �3. ( y � 3)2 � 25

4. x2 � y2 � 8

(x � 4)2 �5. ( y � 2)2 � 4

6. �(y �

92)2

� � �(x �

43)2

� � 1

7. �(x �

91)2

� � �( y �

13)2

� � 1

8. �( y �

94)2

� � �(x

2�

52)2

� � 1

�(x �

162)2

� � �( y �

41)2


9. y � 1 � � �12

�(x � 2)

Page 248

( y � 2)2 � 4(x � 2); 10. x � 1; y � �2

(x � 5)2 �11. �4( y � 1)

12. ��22��

13. hyperbola

14. �x42� � �

y9

2

� � 1

Sample answer:x � 2�2� cos t, y � 2�2� sin t,

15. 0° � t � 360�

parabola; 16. y2 � 8x � 16y � 72 � 0

ellipse; 9(x�)2 � 2x�y�

17. � 9 (y�)2 � 40 � 0

18. hyperbola, �26�

19. (3, 0), (�3, 0)

20.

Bonus: (2, 3)



256-262A&EC10-0-02-834179 10/10/00 10:42 AM Page 258





3 Superior • Shows thorough understanding of the concepts circle, ellipse,parabola, hyperbola, center, vertex, and angle of rotation.

• Uses appropriate strategies to identify equations of conic sections.

• Computations are correct.• Written explanations are exemplary.• Real-world example of conic section is appropriate and makes sense.


2 Satisfactory, • Shows understanding of the concepts circle, ellipse,with Minor parabola, hyperbola, center, vertex, and angle of rotation.Flaws • Uses appropriate strategies to identify equations of conic

sections.• Computations are mostly correct.• Written explanations are effective.• Real-world example of conic section is appropriate and makes sense.

• Graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts circle, ellipse,Satisfactory, parabola, hyperbola, center, vertex, and angle of rotation.with Serious • May not use appropriate strategies to identify equations Flaws of conic sections.

• Computations are mostly correct.• Written explanations are satisfactory.• Real-world example of conic section is mostly appropriate and sensible.


0 Unsatisfactory • Shows little or no understanding of the concepts circle,ellipse, parabola, hyperbola, center, vertex, and angle ofrotation.

• May not use appropriate strategies to identify equations of conic sections.

• Computations are incorrect.• Written explanations are not satisfactory.• Real-world example of conic section is not appropriate orsensible.

• Graphs are not accurate or appropriate.• Does not satisfy requirements of problems.

256-262A&EC10-0-02-834179 10/10/00 10:42 AM Page 259



Page 197

1a. The equation is a circle if A � B.Sample answer: (x � 1)2 � ( y � 2)2 � 4

1b. The equation is an ellipse if A Band A and B have the same sign.Sample answer: �(x �

42)2

� � �( y �

11)2

� � 1

1c. The equation is a parabola when Aor B is zero, but not both. Sample answer: y � 2 � 4(x � 1)2

1d. The equation is a hyperbola if A andB have opposite signs. Sample

answer: �y4

2� � �

x1

2� � 1

1e. The graph is an ellipse since (�1)2 � 4(3)(2) 0.

tan 2� � �3

��12

� � �1, 2� � �45�,

� � ��8

�, or �22.5�.

2a. The graph of x2 � 4y2 � 0 is two

lines of slope �12

� and slope ��12

� that intersect at the origin.x2 � 4y2 � 0

x2 � 4y2

|x| � 2|y|�|2x|� � |y|

�2x� � y or ��

2x� � y

2b.

2c. degenerate hyperbola

3. Sample answer: Most lamps withcircular shades shine a cone of light. When this light cone strikes anearby wall, the resulting shape is ahyperbola. The hyperbola is formedby the cone of light intersecting theplane of the wall.


256-262A&EC10-0-02-834179 10/10/00 10:42 AM Page 260


1. �2�6�, ��32

�, ��123��

2. �6�1�, �3 � s, ��52� 2t��

3. yes

4. (x � 3)2 � ( y � 2)2 � 25

(x � 17)2 � ( y � 8)2 � 5. 289; (�17, 8); 17

�(y�

93)2� � �

(x �1

1)2� � 1;

(�1, 3); (�1, 3 � 2�2�); 6. (�1, 6), (�1, 0), (0, 3), (�2, 3)

7.

8.�(x �16

3)2� � �

(y �9

2)2� � 1

(�1, 6); (�1, 6 � �1�3�); (�1, 9), (�1, 3);

9. y � 6 � � �32

� (x � 1)

Quiz APage 251

1. 7�2�, ��32

�, �12

��2. 5; �a � 2, �2b

2� 3��

3. no

4. (x � 3)2 � ( y � 5)2 � 36

(x � 4)2 � ( y � 5)2 � 61;5. (�4, 5); �6�1� � 7.8

Quiz BPage 251

�(y �

161)2

� � �(x �

43)2

� � 1;(3, �1); (3, �1 � 2�3�);

1. (3, 3), (3, �5), (5, �1), (1, �1)

2.

3. �( y �4

3)2� � �

(x �9

1)2� � 1

(3, �2); �3 ��2�9�, �2�;(5, �2), (1, �2);

4. y � 2 � ��52

�(x � 3)

Quiz CPage 252

1. (x � 2)2 � �8( y � 1), (2, �3);(2, �1); y � 1; x � 2

2. ( y � 3)2 � �12(x � 1)

3. ellipse; �(x �25

2)2� � �

( y �4

1)2� � 1

4. y2 � ��43

� x

Sample answer: x � 10 cos t, y � 10 sin t

5. 0 � t � 2�

Quiz DPage 252

hyperbola;1. (5x2 � 8y2 � 30x � 80y � 195 � 0

ellipse; 11(x�)2 �2. 14x�y� � 11(y�)2 � 36 � 0

3. hyperbola; (4, �3), (�4, 3)

4. (3, �4), (�3, 4)

5.



256-262A&EC10-0-02-834179 10/10/00 10:42 AM Page 261


Page 253

1. E

2. D

3. D

4. E

5. A

6. D

7. B

8. C

9. E

Page 254

10. B

11. D

12. D

13. B

14. C

15. B

16. C

17. A

18. C

19. 6.4

20. 3

Page 255

1. Sample answer: ƒ(x) � 3

2. � �3. Alternate optimal solutions

4. ƒ�1(x) � �2x

� � 5; yes

5. ��136�

6. �1

7. �13�60

3�0��

8. A � 65°, a � 36.2, c � 37.0

9. 79 m/s

10. 45°, 135°

11. 51.6 lb

12. �16�3� � 16i

13. �(x �

91)2

� � �( y �

42)2

� � 1

8�4

�171

1132


256-262A&EC10-0-02-834179 10/10/00 10:42 AM Page 262



NAME _____________________________ DATE _______________ PERIOD ________


1. Evaluate �9�12�

� 216�13�

��

12�

. 1. ________

A. ��13� B. �13� C. �3 D. 3

2. Simplify ��342

x

x�

4

2

y

y

4

��23�

. 2. ________

A. 2x�43�y B. 4x�

43�y2 C. 4x4y2 D. 2x4y2

3. Express �3

2�7�x�4y�6� using rational exponents. 3. ________

A. 3x�43�y2 B. 9x�

43�y2 C. 9x�

34�y D. 9x�

34�y2

4. Express (2x2)�13�(2x)�

12� using radicals. 4. ________

A. �6

3�2�x�5� B. �6

4�x�7� C. x�6

3�2�x� D. x�6

4�x�

5. Evaluate 9��

2�to the nearest thousandth. 5. ________

A. 14.137 B. 31.544 C. 497.521 D. 799.438

6. Choose the graph of y � 2�x. 6. ________A. B. C. D.

7. Choose the graph of y � 4x. 7. ________A. B. C. D.

8. In 2000, the bird population in a certain area is 10,000. The number of 8. ________birds increases exponentially at a rate of 9% per year. Predict the population in 2005.A. 15,137 B. 15,386 C. 15,489 D. 15,771

9. A scientist has 86 grams of a radioactive substance that decays at an 9. ________exponential rate. Assuming k � �0.4, how many grams of radioactive substance remain after 10 days?A. 21.5 g B. 15.8 g C. 3.7 g D. 1.6 g

10. Write 3�2 � �19� in logarithmic form. 10. ________

A. log3 (�2) � �19� B. log3 �19� � �2 C. log�2 �19� � 3 D. log

�2 3 � �19�

Chapter

11



11. Evaluate log9 �217�. 11. ________

A. �23� B. �32� C. ��23� D. ��32�

12. Solve log4 x � log4 (x � 2) � log4 15. 12. ________A. �3 only B. 5 only C. �3 or 5 D. �5 or 3

13. Choose the graph of y � log2 (x � 2). 13. ________A. B. C. D.

14. Find the value of log6 27.5 using the change of base formula. 14. ________A. 0.661 B. 1.439 C. 1.850 D. 2.232

15. Solve 5x � 3x�2 using common logarithms. 15. ________A. 2.732 B. 3.109 C. 4.117 D. 4.301

16. The pH of a water supply is 7.3. What is the concentration of hydrogen 16. ________ions in the tested water?A. 5.012 � 10�8 B.�0.863 C. 5.012 D. 1.995 � 107

17. Convert log5 47 to a natural logarithm and evaluate. 17. ________A. 0.770 B. 2.241 C. 2.392 D. 2.516

18. Solve e0.2x � 21.2 by using natural logarithms. 18. ________A. x � �1.898 B. x � 4.663 C. x � 8.234 D. x � 15.270

19. Banking Find the amount of time required for an investment to 19. ________double at a rate of 12.3% if the interest is compounded continuously.A. 5.635 years B. 6.241 years C. 7.770 years D. 8.325 years

20. Biology The table below shows the population of a given 20. ________bacteria colony.

Let x be the number of days and let y be the population in thousands.Linearize the data and find a regression equation for the linearized data.A. ln y � 0.0948x � 4.3321 B. ln y � 0.0798x � 4.5517C. ln y � 0.0722x � 4.7735 D. ln y � 0.0785x � 4.8203

Bonus Express �5

��x�6�� in exponential form. Assume x 0. Bonus: ________

A. x�35� B. x�

53� C. x�6

10� D. x�

45�



11

Time (days) 0 3 6 9 12

Population (thousands)

95 120 155 190 250




NAME _____________________________ DATE _______________ PERIOD ________


1. Evaluate �9�12

�

� 1�13

�

��1

2�

. 1. ________

A. ��12� B. �12� C. �2 D. 2

2. Simplify ��9xx�

3

1yy

3��

�32

�

. 2. ________

A. 6x2�y�3 B. 3x�43��y�3 C. 27x6�y�3 D. 27�x�3�y�3

3. Express �4

1�6�x�y�4� using rational exponents. 3. ________

A. 2x�14

��y� B. 4x�14

�y C. 2�x�y D. 4x4y

4. Express x�23

�y

�12

�using radicals. 4. ________

A. �3

x�2y� B. �6

x�2y� C. �6

x�2y�3� D. �6

x�4y�3�

5. Evaluate 3� to the nearest thousandth. 5. ________A. 9.425 B. 27.001 C. 31.026 D. 31.544

6. Choose the graph of y � ��13��x. 6. ________

A. B. C. D.

7. Choose the graph of y 4x. 7. ________A. B. C. D.

8. In 2000, the deer population in a certain area was 800. The number 8. ________of deer increases exponentially at a rate of 7% per year. Predict thepopulation in 2009.A. 1408 B. 1434 C. 1471 D. 1492

9. Find the balance in an account at the end of 8 years if $6000 is 9. ________invested at an interest rate of 12% compounded continuously.A. $15,670.18 B. $15,490.38 C. $14,855.78 D. $14,560.22

10. Write 2�3 � �18� in logarithmic form. 10. ________

A. log�3 �18� � 2 B. log

�3 2 � �18� C. log2 �18� � �3 D. log2 (�3) � �18�

Chapter

11



11. Evaluate log9 �811�. 11. ________

A. ��12� B. �12� C. �2 D. 2

12. Solve log4 x2 � log4 5 � log4 125. 12. ________A. �5 or 5 B. 5 only C. 25 only D. �25 or 25

13. Choose the graph of y � log2 (x � 1). 13. ________A. B. C. D.


15. Solve 4x�2 � 3 using common logarithms. 15. ________A. 2.023 B. 2.247 C. 2.541 D. 2.792

16. If the concentration of hydrogen ions in a sample of water is 16. ________5.31 � 10�8 moles per liter, what is the pH of the water?A. 8.0 B. 7.3 C. 8.7 D. 5.3


18. Solve e3x 48 by using natural logarithms. 18. ________A. x 1.290 B. x 1.337 C. x 1.452 D. x 1.619

19. Banking How much time would it take for an investment to 19. ________double at a rate of 10.2% if interest is compounded continuously?A. 6.011 years B. 6.241 years C. 6.558 years D. 6.796 years

20. Population The table below shows the population of a given 20. ________urban area.

Let x be the number of years since 1900 and let y be the population in thousands. Linearize the data and find a regression equation for the linearized data.A. ln y � 0.0395x � 3.9039 B. ln y � 0.0327x � 3.8166C. ln y � 0.0412x � 4.0077 D. ln y � 0.0365x � 4.2311

Bonus Express �4��x��6�� in exponential form. Assume x 0. Bonus: ________

A. x�32

�B. x

�34

�

C. x�214�

D. x�23

�



11

Year 1900 1920 1940 1960 1980

Population 50 106 250 520 1170(thousands)




NAME _____________________________ DATE _______________ PERIOD ________


1. Evaluate �16�12�

��1

2�. 1. ________

A. ��12� B. �12� C. �2 D. 2

2. Simplify ��25xxy

3y3

��32�

. 2. ________

A. 5x2�y�3 B. 125x�92�y�

92� C. 25�x�3�y�3 D. 125�x�3�y�3

3. Express �4

1�6�x� using rational exponents. 3. ________

A. 2x�14� B. 4x�

14� C. 2x D. 4x4

4. Express x�23� using radicals. 4. ________

A. �3

x�2� B. �6

x� C. �3

x� D. �x�3�

5. Evaluate 3�2� to the nearest thousandth. 5. ________A. 4.278 B. 4.578 C. 4.729 D. 4.927

6. Choose the graph of y � 2x. 6. ________A. B. C. D.

7. Choose the graph of y 3x. 7. ________A. B. C. D.

8. In 1998, the wolf population in a certain area was 1200. The number of 8. ________wolves increases exponentially at a rate of 3% per year. Predict the population in 2011.A. 1598 B. 1645 C. 1722 D. 1762

9. Find the balance in an account at the end of 14 years if $5000 is 9. ________invested at an interest rate of 9% that is compounded continuously.A. $16,998.14 B. $17,234.72 C. $17,627.11 D. $17,891.23

10. Write 43 � 64 in logarithmic form. 10. ________A. log3 4 � 64 B. log4 64 � 3 C. log3 64 � 4 D. log64 3 � 4

Chapter

11



11. Evaluate log4 �116�. 11. ________

A. ��12� B. �12� C. �2 D. 2

12. Solve log4 x � log4 5 � log4 60. 12. ________A. 3 B. 12 C. 120 D. 300

13. Choose the graph of y � log2 x. 13. ________A. B. C. D.


15. Solve 5x � 32 using common logarithms. 15. ________A. 2.023 B. 2.153 C. 2.241 D. 2.392

16. Evaluate log �453�. 16. ________

A. 2.505 B. 1.107 C. 0.380 D. 2.549


18. Solve e2x 37 by using natural logarithms. 18. ________A. x 1.805 B. x 1.822 C. x 1.931 D. x 1.955

19. Banking What is the amount of time required for an investment 19. ________to double at a rate of 8.2% if the interest is compounded continuously?A. 8.275 years B. 8.453 years C. 8.613 years D. 8.772 years

20. Biology The table below shows the population for a given 20. ________ant colony.

Let x be the number of days and let y be the population in thousands.Write a regression equation for the exponential model of the data.A. y � 39.4033(1.0611)x B. y � 39.2666(1.0723)x

C. y � 39.2701(1.0522)x D. y � 39.2741(1.0604)x

Bonus Express �4 �3�x�� in exponential form. Assume x 0. Bonus: ________

A. x�32� B. x�

34� C. x�1

12� D. x�3

4�



11

Time (days) 0 5 10 15 20

Population(thousands)

40 50 73 96 125




NAME _____________________________ DATE _______________ PERIOD ________

1.Evaluate ��5

(��

2�34�23�)4�

�. 1. __________________

2. Simplify (8x6 � 32y5)�12�. 2. __________________

3. Express 5�4

8�1�x�3y�8� using rational exponents. 3. __________________

4. Express (3x)�15� (3x2)�

13� using radicals. 4. __________________

5. Evaluate 8��7

� to the nearest thousandth. 5. __________________

6. Sketch the graph of y � 3�x. 6.

7. Sketch the graph of y � ��14��x. 7.

8. Suppose $1750 is put into an account that pays an annual 8. __________________rate of 6.25% compounded weekly. How much will be in the account after 36 months?

9. A scientist has 37 grams of a radioactive substance that 9. __________________decays exponentially. Assuming k � �0.3, how many grams of radioactive substance remain after 9 days? Round your answer to the nearest hundredth.

10. Write ��16��4

� 1296 in logarithmic form. 10. __________________

Chapter

11



11. Evaluate log16 �18�. 11. __________________

12. Solve log4 x � log4 (x � 2) � log4 35. 12. __________________

13. Sketch the graph of y log3 (x � 2). 13.

For Exercises 14-18, round your answers to the nearest thousandth.

14. Find the value of log5 87.2 using the change of base formula. 14. __________________

15. Solve 4x�3 � 7x using common logarithms. 15. __________________

16. The pH of a sample of seawater is approximately 8.1. What 16. __________________is the concentration of hydrogen ions in the seawater?

17. Convert log7 324 to a natural logarithm and evaluate. 17. __________________

18. Solve e�0.5x � 41.6 by using natural logarithms. 18. __________________

19. Banking What interest rate is required for an 19. __________________investment with continuously compounded interest to double in 8 years?

20. Biology The table below shows the population for a given 20. __________________bacteria colony.

Let x represent the number of days and let y represent apopulation in thousands. Linearize the data and find aregression equation for the linearized data.

Bonus Express �4

��x1��2�� in exponential form. Assume x 0. Bonus: __________________



11

Time (days) 0 4 8 12 16

Population (thousands)

87 112 135 173 224




NAME _____________________________ DATE _______________ PERIOD ________

1. Evaluate (�4

8�1�)3. 1. __________________

2. Simplify ��x2

�

54��3

2�. 2. __________________

3. Express �5

3�2�x�3y�10� using rational exponents. 3. __________________

4. Express 4x�13

�y

�23

�using radicals. 4. __________________

5. Evaluate 5� to the nearest thousandth. 5. __________________

6. Sketch the graph of y � 3x. 6.

7. Sketch the graph of y � ��12��x. 7.

8. A 1991 report estimated that there were 640 salmon 8. __________________in a certain river. If the population is decreasing exponentially at a rate of 4.3% per year, what is the expected population in 2002?

9. Find the balance in an account at the end of 12 years 9. __________________if $4000 is invested at an interest rate of 9% that is compounded continuously.

10. Write 16�34

�� 8 in logarithmic form. 10. __________________

Chapter

11



11. Evaluate log4 �614�. 11. __________________

12. Solve log2 (x � 6) � log2 3 � log2 30. 12. __________________


For Exercises 14-18, round your answers to the nearestthousandth.14. Evaluate log �2

93�. 14. __________________


16. Solve 5x�2 � 7 using common logarithms. 16. __________________


18. Solve e4x < 98.6 by using natural logarithms. 18. __________________

19. Banking Find the amount of time in years required for 19. __________________an investment to double at a rate of 6.2% if the interest is compounded continuously.

20. Biology The table below shows the population of mold 20. __________________spores on a given Petri dish.

Let x represent the number of days and let y represent the populations in thousands. Linearize the data and find a regression equation for the linearized data.

Bonus Express 2 logb a � logb c as a single log. Bonus: __________________Assume b > 0.



11

Time (days) 0 2 4 6 8

Population (thousands) 45 51 63 74 81




NAME _____________________________ DATE _______________ PERIOD ________

1. Evaluate 8�23�

� 4�12�. 1. __________________

2. Simplify ��27

yy4

��23�

. 2. __________________

3. Express �3

1�2�5�x�5� using rational exponents. 3. __________________

4. Express x�25� using radicals. 4. __________________

5. Evaluate 5�3� to the nearest thousandth. 5. __________________



8. In 1990, the elk population in a certain area was 750. The 8. __________________number of elk increases exponentially at a rate of 6% per year. Predict the elk population in 2004.

9. Find the balance in an account at the end of 8 years if 9. __________________$7000 is invested at an interest rate of 12% compounded continuously.

10. Write 52 � 25 in logarithmic form. 10. __________________

Chapter

11



11. Evaluate log3 �19�. 11. __________________

12. Solve log2 x � log2 3 � log2 12. 12. __________________

13. Sketch the graph of y log3 x. 13.

For Exercises 14–18, round your answers to the nearest thousandth.

14. Evaluate log 3(6)2. 14. __________________


16. Solve 3x � 47 using common logarithms. 16. __________________


18. Solve e3x � 89 by using natural logarithms. 18. __________________

19. Banking Find the amount of time in years required for 19. __________________an investment to double at a rate of 9.5% if the interest is compounded continuously.

20. Biology The table below shows the population for a 20. __________________given bacteria colony.

Let x represent the number of days and let y represent the population in thousands. Write a regression equation for the exponential model of the data.

Bonus Express 2 logb a � 3 logb c as a single log. Bonus: __________________Assume b 0.



11

Time (days) 0 4 8 12 16





NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include allrelevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond therequirements of the problem.

1. a. Graph y � 3x.

b. Compare the graphs of y � 5x and y � 3x. Do the graphs intersect? If so, where? Graph y � 5x.

c. Compare the graphs of y � ��13��x

and y � 3x. Do the graphs

intersect? If so, where? Graph y � ��13��x.

d. Compare the graphs of y � 3x and y � log3 x. Do the graphs intersect? If so, where? Graph y � log3 x.

e. Compare the graphs of y � log5 x and y � log3 x. Do the graphs intersect? If so, where? Graph y � log5 x.

f. Tell how the graphs of y � log2 x and y � log8 x are related.Justify your answer.

2. Write a word problem for the equation below. Then solve for x and explain what the answer means.

178 � 9 � 2x

3. Solve the equation log2 (x � 3) � 3 � log2 (x � 2). Explain each step.

4. Solve the equation e2x � 3ex � 2 � 0. Explain each step.

5. Before calculators and computers were easily accessible, scientistsand engineers used slide rules. Using the properties of logarithms,they performed mathematical operations, including finding aproduct. To see the principle involved, pick two positive numbersthat are less than 10 and that each have two places after the deci-mal point. Calculate their product using only the properties of thelogarithm and the exponential function, without calculating theproduct directly. When you have found the product, check youranswer by calculating the product directly.

Chapter

11



For Exercises 1-3, evaluate each expression.

1. �16�12�

� 64�31

��31

�

1. ____________________________________________________________________

2. 2. __________________

3. �1�5� � �6�0� 3. __________________

4. Express �38�x�2y�6� using rational exponents. 4. __________________

5. Evaluate 7�

to the nearest thousandth. 5. __________________

6. Sketch the graph of y � 4�x. 6.

7. The number of seniors at Freedmont High School was 7. __________________241 in 1993. If the number of seniors increases exponentially at a rate of 1.7% per year, how many seniors will be in the class of 2005?

8. Jasmine invests $1500 in an account that earns an interest 8. __________________rate of 11% compounded continuously. Will she have enough money in 4 years to put a $2500 down payment on a new car? Explain.

9. A city’s population can be modeled by the equation 9. __________________y � 29,760e�0.021t, where t is the number of years since 1986. Find the projected population in 2012.

10. Evaluate log4 �614�. 10. __________________

11. Solve log3 x � log3 (x � 6) � log3 16. 11. __________________

12. Sketch the graph of y � log2 (x � 1). 12.

�8�31

�

�8

Chapter 11 Mid-Chapter Test (Lessons 11-1 and 11-4)


11


Evaluate each expression.

1. �81�12�

� 42��

12�

1. __________________

2. 64�13�

� 64��13� 2. __________________

3. Express 16�17� using radicals. 3. __________________

Graph each exponential function or inequality.4. y � 2x�1 4.

5. y � 3x. 5.

1. Finance Find the balance in an account at the end of 1. __________________12 years if $6500 is invested at an interest rate of 8% compounded continuously.

2. Write 3�4 � �811� in logarithmic form. 2. __________________

3. Evaluate log�5� 125. 3. __________________

4. Solve log5 72 � log5 x � 3 log5 2. 4. __________________



NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

11

Chapter

11


For Exercises 1-5, round your answers to the nearest thousandth.


2. Solve 5x�2 � 87 using common logarithms. 2. __________________

3. Given that log 4 � 0.6021, evaluate log 40,000. 3. __________________


5. Evaluate ln �0.145�. 5. __________________

Find the amount of time required for an investment to double at the given rate if interest is compounded continuously.

1. 9.5% 1. __________________

2. 5.0% 2. __________________

3. Population The table shows the population for a given 3. __________________urban area.

Let x be the number of years since 1900 and let y be thepopulation in thousands. Linearize the data and find a regression equation for the linearized data.


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

11

Chapter

11

Year 1900 1910 1920 1930 1940





NAME _____________________________ DATE _______________ PERIOD ________


Multiple Choice1. Bobbi scored 75, 80, and 85 on three

tests. What must she score on herfourth test to keep her average testscore the same?A 75 B 80C 85 D 90E None of these

2. The average of 3, 4, x, and 12 is 7.What is the value of x?A 5 B 6C 7 D 8E 9

3. What is in terms of (x � y)?

A (x � y)2

B (x � y)�21

�

C (x � y)�31

�

D (x � y)E None of these

4. For some positive x, if x2 � xy � 3(x � y),then what is the value of x?A 3B �3C Both A and BD Neither A and BE It cannot be determined from the

information given.

5. The average of a set of four numbers is22. If one of the numbers is removed,the average of the remaining numbersis 21. What is the value of the numberthat was removed?A 1 B 2C 22 D 25E It cannot be determined from the

information given.

6. What is the average age of a group of15 students if 9 students are 15, 3 are16, and 3 are 17?A 15.4 years oldB 15.5 years oldC 15.6 years oldD 15.7 years oldE 15.8 years old

7. cot �43��

A ��33��

B �3�C ��3�D ��3

3��

E None of these

8. �tan � �2

cot ��

A sin �B cos 2�C cos �D 2 sin �E 2 sin � cos �

9. A basket contains 12 marbles, somegreen and some blue. Which of the following is not a possible ratio ofgreen marbles to blue marbles?A 1:1B 1:2C 1:3D 1:4E 1:5

10. The ratio of two integers is 5:4, andtheir sum is equal to 54. How muchlarger than the smaller number is thebigger number?A 45B 30C 24D 12E 6

�(x� �� y�)3��(x� �� y�)�

Chapter

11



11. A rectangular solid is cut diagonally asshown below. What is the surface areaof the wedge?A 60 units2

B 56 units2

C 54 units2

D 44 units2

E 36 units2

12. Square ABCD is divided into 4 equalsquares. If the perimeter of eachsmaller square is four, what is the areaof the larger square?A 2 units2

B 4 units2

C 8 units2

D 16 units2

E None of these

13. What is the average price of a dozen rolls if �13� of the customers buy the larger rolls for $3.00 per dozen and �23� of the customers buy the smaller rolls for $2.25 per dozen? A $2.60B $2.50C $2.45D $2.40E $2.70

14. What is the average speed of Jane’s car if Jane drives for 30 minutes at 50 miles per hour, then at 65 miles perhour for 2 hours, then at 45 miles perhour for 20 minutes, and at 30 milesper hour for 10 minutes?A 68.5 mphB 58.3 mphC 56.4 mphD 52.6 mphE None of these

15. If �ABC has two sides that are eachone unit long, which of the followingcannot be the length of the third side?

A ��22��

B 1C �2�D �3�E 2�2�

16. In the figure slown, two equilateral triangles have a common vertex.Find p � q.


information given.





Column A Column B

17.

18. The values of x and y are positive.

19. Grid-In For what positive value of ydoes �9

y� � �2y

5�?

20. Grid-In If xy � 270 and x � y � 20 � (x � y), what is x?



11

Average speed of atrain that travels

150 miles in 3 hours

Average speed of atrain that travels 50 miles in �13� hour

Average of x and y

Average of x2 and y2




NAME _____________________________ DATE _______________ PERIOD ________

1. If ƒ(x) � 2x2 � 1, find ƒ(4). 1. __________________

2. Solve the system algebraically. 2. __________________5x � y � 212x � 3y � 5

3. Describe the end behavior for y � x4 � 3x. 3. __________________

4. Solve the equation �x� �� 1�5� � 7 � 12. 4. __________________

5. Find sin (�180�). 5. __________________

6. State the amplitude, period, and phase shift for the graph 6. __________________of y � 4 sin (2x � 6�).

7. Find the polar coordinates of the point with rectangular 7. __________________coordinates (1, �3�).

8. Write the equation of the circle with center (0, 2) and 8. __________________radius 3 units.


10. Sketch the graph of y � log3 x. 10.

Chapter

11


Page 263

1. B

2. C

3. A

4. C

5. B

6. A

7. C

8. B

9. D

10. B

Page 264

11. D

12. B

13. A

14. C

15. D

16. A

17. C

18. D

19. A

20. B

Bonus: A

Page 265

1. B

2. C

3. A

4. D

5. D

6. B

7. A

8. C

9. A

10. C

Page 266

11. C

12. D

13. B

14. B

15. D

16. B

17. C

18. A

19. D

20. A

Bonus: B



282-288 A&E C11-0-02-834179 10/10/00 10:32 AM Page 282


Page 267

1. B

2. D

3. A

4. A

5. C

6. B

7. C

8. D

9. C

10. B

Page 268

11. C

12. D

13. A

14. D

15. B

16. B

17. A

18. A

19. B

20. D

Bonus: C

Page 269

1. �9

2. 16�x�3y�25

�

3. 15x�34�

y2

4. �15

6�5�6�1�x�13�

5. 102.858

6.

7.

8. $2110.67

9. 2.49 g

10. log�16�

1296 � �4

Page 270

11. ��43�

12. 5

13.

14. 2.776

15. �7.432

16. 7.943 � 10�9

17. 2.971

18. x � �7.456

19. 8.66%

ln y �20. 0.0582x � 4.4657

Bonus: x�23�

Form 1C Form 2A


282-288 A&E C11-0-02-834179 10/10/00 10:32 AM Page 283

Page 271

1. 27

2. 125x6

3. 2x�35�

y2

4. 4�3x�y�2�

5. 156.993

6.

7.

8. 395

9. $11,778.72

10. log16 8 � �43�

Page 272

11. �3

12. 4

13.

14. 0.051

15. 4.120

16. �0.791

17. 3.138

18. x � 1.148

19. 11.180 years

In y � 0.0774x �20. 3.8065

Bonus: logb �ac2�

Page 273

1. 8

2. 9y2

3. 5x�35�

4. �5

x�2�

5. 16.242

6.

7.

8. 1696

9. $18,281.88

10. log5 25 � 2

Page 274

11. �2

12. 4

13.

14. 2.033

15. 3.182

16. 3.505

17. 3.712

18. x � 1.496

19. 7.296 years

20. y � 32.0703(1.0645)x

Bonus: logb a2c3



282-288 A&E C11-0-02-834179 10/10/00 10:32 AM Page 284




3 Superior • Shows thorough understanding of the concepts exponential and logarithmic functions and their graphs.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Word problem concerng exponential equation is appropriate and makes sense.


2 Satisfactory, • Shows understanding of the concepts exponential andwith Minor logarithmic functions and their graphs.Flaws • Uses appropriate strategies to solve problems.

• Computations are mostly correct.• Written explanations are effective.• Word problem concerng exponential equation is appropriate and makes sense.

• Graphs are accurate and appropriate.• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts Satisfactory, exponential and logarithmic functions and their graphs.with Serious • May not use appropriate strategies to solve problems.Flaws • Computations are mostly correct.

• Written explanations are satisfactory.• Word problem concerning exponential equation is mostlyappropriate and sensible.


0 Unsatisfactory • Shows little or no understanding of the concepts exponential and logarithmic functions and their graphs.

• May not use appropriate strategies to solve problems. • Computations are incorrect.• Written explanations are not satisfactory.• Word problem concerning exponential equation is not appropriate or sensible.

• Graphs are not accurate or appropriate.• Does not satisfy requirements of problems.

282-288 A&E C11-0-02-834179 10/10/00 10:32 AM Page 285



Page 2751a.

1b. For x � 0, 5x � 3x. For x � 0, 5x � 3x. For x � 0, 5x � 3x. They intersect at x � 0.

1c. The graph of y � ��13

��xis the

reflection of y � 3x about the y-axis. They intersect at x � 0.

1d. The graph of y � log3 x is thereflection of y � 3x over the line y � x. The graphs do not intersect.

1e. For x � 1, log5 x � log3 x. For x � 1, log5 x � log3 x. For x � 1, log5 x � log3 x. They intersect at x � 1.

1f. The graph of y � log8 x is the graph ofy � log2 x compressed vertically by afactor of �1

3�. By the Change of Base

Formula, log8 x � �lloo

gg

2

2

8x

�, or �13

� log2 x.

2. Sample answer: A colony of ninebacteria doubles every minute. When will the population of the colonybe 178?

178 � 9 � 2x

log 178 � log 9 � x log 2

x ��log 17

lo8g�2

log 9�, or about 4.3

The colony will number 178 in about 4.3 minutes.

3. log2 (x � 3) � log2 (x � 2) � 3log2 [(x � 3)(x � 2)] � 3

(x � 3)(x � 2) � 23

x2 � x � 6 � 8

� ��1 �2�5�7��

The solution x � ��1 �2�5�7�� is

extraneous because it makes bothlogarithms undefined.

4. e2x � 3ex � 2 � 0(ex � 1)(ex � 2) � 0ex � 1 � 0 or ex � 2 � 0

ex � 1 ex � 2x � In 1, or 0 x � In 2

5. Sample answer: Let the two positivenumbers be 5.36 and 8.45.In (5.36 � 8.45) � ln 5.36 � ln 8.45

� 1.6790 � 2.1342� 3.8132

The product is5.36 � 8.45 � e3.8132

� 45.295The actual product is 45.292. Thedifference is due to rounding.


Product PropertyDefinition of logarithmMultiply.

Factor. e2x � ex � ex

Definition of logarithm

x ��1 �2

1� �� 5�6��

282-288 A&E C11-0-02-834179 10/10/00 10:32 AM Page 286



1. 2

2. ��41�

3. 30

4. 2x�23�

y2

5. 451.808

6.

7. 295

No; she will have only 8. $2329.06.

9. 17,239

10. �3

11. 8

12.

Quiz APage 277

1. �51�

2. �145�

3. �71�6�

4.

5.

Quiz BPage 277

1. $16,976.03

2. log3 �811� ��4

3. 6

4. 9

5.

Quiz CPage 278

1. 2.289

2. 0.775

3. 4.6021

4. 2.806

5. 0.799

Quiz DPage 278

1. 7.296 years

2. 13.863 years

3. ln y = 0.0677x + 3.3983


282-288 A&E C11-0-02-834179 10/10/00 10:32 AM Page 287

Page 279

1. B

2. E

3. D

4. E

5. D

6. C

7. A

8. E

9. D

10. E

Page 280

11. A

12. B

13. B

14. B

15. E

16. A

17. B

18. D

19. 15

20. 27

Page 281

1. 31

2. (4, �1)

As x→, y→3. As x→�, y→

4. 40

5. 0

6. 4; �; 3�

7. �2, ��3

��

8. x2 � ( y � 2)2 � 9

9. log5 �1125� � �3

10.



282-288 A&E C11-0-02-834179 10/10/00 10:32 AM Page 288



NAME _____________________________ DATE _______________ PERIOD ________


1. Find the 15th term in the arithmetic sequence 14, 10.5, 7, . . . . 1. ________A. �21 B. �63 C. 63 D. �35

2. Find the sum of the first 36 terms in the arithmetic series 2. ________�0.2 � 0.3 � 0.8 � . . . .A. 318.6 B. 332.2 C. 307.8 D. 315

3. Find the sixth term in the geometric sequence �3� y3, �3y5, 3�3� y7, . . . . 3. ________A. �27y13 B. 9�3� y13 C. 27y13 D. �9�3� y13

4. Find the sum of the first f ive terms in the geometric series 4. ________��32� � 1 � �23� � . . . .

A. �5554� B. ��55

54� C. �52

57� D. ��52

57�

5. Find three geometric means between ��2� and �4�2�. 5. ________A. 2, �2�2�, 4 B. �2, 2�2�, �4C. 2, 2�2�, 4 D. A or C

6. Find limn→∞ �1 � �

(�n1)n

� �. 6. ________

A. 1 B. 0 C. �1 D. does not exist

7. Find the sum of �2�7� � �9� � �3� � . . . . 7. ________

A. �12�(9 � 9 �3�) B. 9 � 9�3� C. �21�(9 � 9 �3�) D. does not exist

8. Write 3.1�2�3� as a fraction. 8. ________

A. �13034303� B. �13

03430� C. �10

3430� D. �1

3034303�

9. Which of the following series is convergent? 9. ________A. �3� � 3 � 3�3� � . . . B. 6�2� � 12 � 12�2� � . . .C. 6�2� � 6 � 3�2� � . . . D. 6�2� � 12 � 12�2� � . . .

10. Which of the following series is divergent? 10. ________

A. 1 � 3��14�� 9��14��2

� 27��14��3

� . . . B. 1 � 3��15�� 9��15��2

� 27��15��3

� . . .

C. 1 � 3��17�� 9��17��2

� 27��17��3

� . . . D. 1 � 3��12�� 9��12��2

� 27��12��3

� . . .

11. Write �3

k�0��2

1��k

in expanded form and then find the sum. 11. ________

A. ��12� � �14� � �18�; ��78� B. 1 � �12� � �14� � �18�; �18�

C. ��12� � �14� � �18�; �83� D. 1 � �12� � �14� � �18�; �58�

Chapter

12



12. Express the series �27 � 9 � 3 � 1 � . . . using sigma notation. 12. ________

A. �∞

k�0�3k B. �

3

k�0�27��13��

k

C. �∞

k�0�27��13��

kD. �

∞

k�027��13��

k

13. The expression 81p4 � 108p3r3 � 54p2r6 � 12pr9 � r12 is the 13. ________expansion of which binomial?A. ( p � 3r3)4 B. (3p � r3)4 C. (3p3 � r)4 D. (3p � 3r3)4

14. Find the fifth term in the expansion of (3x2 � �y�)6. 14. ________A. 135x4y2 B. 45x4y2 C. �135x4y2 D. �45x4y2

15. Use the first f ive terms of the trigonometric series to find the value 15. ________of sin �1

�2� to four decimal places.

A. 0.2618 B. 0.2588 C. 0.7071 D. 0.2648

16. Find ln (�91.48). 16. ________A. 4.5161 B. i� � 4.5161 C. i� � 4.5161 D. �4.5161

17. Write 3 � �3�i in exponential form. 17. ________A. 9ei�

116�� B. 9ei�

53�� C. 2�3�ei�

116�� D. 2�3�ei�

53��

18. Find the first three iterates of the function ƒ(z) � �z � i for 18. ________z0 � 2 � 3i.A. 2 � 2i, 2 � 3i, 2 � 2i B. �2 � 2i, 2 � 3i, �2 � 2iC. 2 � 3i, �2 � 2i, 2 � 3i D. �2 � 2i, 2 � 3i, �2 � 2i

19. Find the first three iterates of the function ƒ(z) � z2 � c for 19. ________c � 1 � 2i and z0 � 1 � i.A. �1 � 4i, �15 � 8i, 219 � 194iB. �1 � 4i, �16 � 6i, 220 � 192iC. �1 � 4i, �16 � 6i, 221 � 194iD. �1 � 4i, �16 � 6i, 219 � 194i

20. Suppose in a proof of the summation formula 20. ________1 � 5 � 25 � . . . � 5n�1 � �14�(5n � 1) by mathematical induction,you show the formula valid for n � 1 and assume that it is valid for n � k. What is the next equation in the induction step of this proof ?A. 1 � 5 � 25 � . . . � 5k�1 � 5k�1�1 � �14� (5k � 1) � �14� (5k�1 � 1)

B. 1 � 5 � 25 � . . . � 5k � 5k�1 � �14� (5k � 1) � 5k�1�1

C. 1 � 5 � 25 � . . . � 5k�1 � 5k�1�1 � �14� (5k�1 � 1) � 5k�1�1

D. 1 � 5 � 25 � . . . � 5k�1 � 5k�1�1 � �14� (5k � 1) � 5k�1�1

Bonus Solve �6

n�0(3n � 2x) � 7 for x. Bonus: ________

A. �121� B. 8 C. 4 D. �13

9�



12




NAME _____________________________ DATE _______________ PERIOD ________


1. Find the 27th term in the arithmetic sequence �8, 1, 10, . . . . 1. ________A. 174 B. 242 C. 235 D. 226

2. Find the sum of the first 20 terms in the arithmetic series 2. ________14 � 3 � 8 � . . . .A. �195 B. �1810 C. 195 D. 1810

3. Find the sixth term in the geometric sequence 11, �44, 176, . . . . 3. ________A. 11,264 B. �11,264 C. 45,056 D. �45,056

4. Find the sum of the first five terms in the geometric series 4. ________2 � �43� � �89� � . . . .

A. �8515� B. �12

37� C. �18

110� D. �28

715�

5. Find three geometric means between ��23� and �54. 5. ________A. 2, 6, 18 B. �2, 6, �18 C. 2, �6, 18 D. A or C

6. Find lim �5n43n�

3 �7n

72n�

2

3�. 6. ________

A. �54� B. 0 C. �45� D. does not exist

7. Find the sum of �151� � �35

35� � �6

9095� � . . . . 7. ________

A. �17201� B. ��17

201� C. �27

2� D. does not exist

8. Write 0.1�2�3� as a fraction. 8. ________A. �43

13� B. �3

43133� C. �3

4313� D. �34

313�

9. Which of the following series is convergent? 9. ________A. 7.5 � 1.5 � 0.3 � . . . B. 1.2 � 3.6 � 10.8 � . . .C. 1.2 � 3.6 � 10.8 � . . . D. �2.5 � 2.5 � 2.5 � . . .

10. Which of the following series is divergent? 10. ________A. �3

12� � �6

12� � �9

12� � . . . B. �23

2� � �26

4� � �29

6� � . . .

C. �13�12� � �23

�23� � �33

�34� � . . . D. �0.

352� � �0.

654� � �0.

956� � . . .

11. Write �5��23��kin expanded form and then find the sum. 11. ________

A. 5��23��2� ��23��2

� ��23��2; �29

8� B. ��5 3� 2��2

� ��5 3� 2��3

� ��5 3� 2��4

; �15

8,7100

�

C. 5��23��1� 5��23��2

� 5��23��3; �12

970� D. 5��23��2

� 5��23��3� 5��23��4

; �38810�

Chapter

12

n→∞

k�2

4



12. Express the series 0.7 � 0.007 � 0.00007 � . . . using sigma notation. 12. ________

A. �0.7(10)k � 1 B. �7(10)1 � 2k C. �7(10)1 � k D. �0.7(10)�k

13. The expression 243c5 � 810c4d � 1080c3d2 � 720c2d3 � 240cd4 � 32d5 13. ________is the expansion of which binomial?A. (3c � d)5 B. (c � 2d)5 C. (2c � 3d)5 D. (3c � 2d)5

14. Find the third term in the expansion of (3x � y)6. 14. ________A. 1215x4y2 B. 1215x2y4 C. �1215x2y4 D. �1215x4y2

15. Use the first five terms of the exponential series 15. ________ex � 1 � x � �x2

2

!� ��x33

!� � �x44

!� � . . . to approximate e3.9.

A. 39.40 B. 24.01 C. 32.03 D. 90.11

16. Find ln (�102). 16. ________A. 4.6250 B. i� � 4.6250 C. i� � 4.6250 D. �4.6250

17. Write 15�3� � 15i in exponential form. 17. ________

A. 30ei�11

6�� B. 30ei�

56�� C. 30ei�

76�� D. 15ei�

116��

18. Find the first three iterates of the function ƒ(z) � �2z for z0 � 1 � 3i. 18. ________A. 2 � 6i, 4 � 12i, 8 � 24i B. �2 � 6i, 4 � 12i, �8 � 24iC. �2 � 6i, 4 � 12i, 8 � 24i D. 2 � 6i, �4 � 12i, 8 � 24i

19. Find the first three iterates of the function ƒ(z) � z2 � c for c � i 19. ________and z0 � 1.A. 1 � i, 2 � 3i, �5 � 13i B. 1 � i, �3i, �9 � iC. 1 � i, �3i, 9 � i D. 1 � i, �2i, �4 � i

20. Suppose in a proof of the summation formula 7 � 9 � 11 � . . . � 20. ________2n � 5 � n(n � 6) by mathematical induction, you show the formula valid for n � 1 and assume that it is valid for n � k. What is the next equation in the induction step of this proof ?A. 7 � 9 � 11 � . . . � 2k � 5 � 2(k � 1) � 5 � k(k � 6) � (k � 1)(k � 1 � 6)B. 7 � 9 � 11 � . . . � 2(k � 1) � 5 � k(k � 6)C. 7 � 9 � 11 � . . . � 2k � 5 � k(k � 6)D. 7 � 9 � 11 � . . . � 2k � 5 � 2(k � 1) � 5 � k(k � 6) � 2(k � 1) � 5

Bonus If a1, a2, a3, . . ., an is an arithmetic sequence, where an � 0, Bonus: ________then �a

11�, �a

12�, �a

13�, . . ., �a

1n� is a harmonic sequence. Find one

harmonic mean between �12� and �18�.

A. �14� B. �15� C. �16� D. �156�



12

∞

k�1

∞

k�1

∞

k�1

∞

k�1




NAME _____________________________ DATE _______________ PERIOD ________


1. Find the 21st term in the arithmetic sequence 9, 3, �3, . . . . 1. ________A. �111 B. �129 C. �117 D. �126

2. Find the sum of the first 20 terms in the arithmetic series 2. ________�6 � 12 � 18 � . . . .A. �2520 B. �1266 C. �1140 D. �1260

3. Find the 10th term in the geometric sequence �2, 6, �18, . . . . 3. ________A. 118,098 B. �118,098 C. 39,366 D. �39,366

4. Find the sum of the first eight terms in the geometric series 4. ________�4 � 8 � 16 � . . . .A. �342 B. �1020 C. �340 D. 340

5. Find one geometric mean between 2 and 32. 5. ________A. �16 B. 8 C. 12 D. 4

6. Find limn→∞

�n3

n�2

5� . 6. ________

A. �23� B. �5 C. �85� D. does not exist

7. Find the sum of 16 � 4 � 1 � . . . . 7. ________

A. 64 B. �654� C. 20 D. does not exist

8. Write 0.8� as a fraction. 8. ________

A. �98989� B. �9

8� C. �989� D. �8

9�

9. Which of the following series is convergent? 9. ________A. 8 � 8.8 � 9.68 � . . . B. 8 � 6 � 4 � . . .C. 8 � 2.4 � 0.72 � . . . D. 8 � 8 � 8 � . . .

10. Which of the following series is divergent? 10. ________

A. 1 � �21

2� � �

21

4� � �

21

6� � . . . B. 1 � �

31

2� � �

31

4� � �

31

6� � . . .

C. 1 � �21

2� � �

21

4� � �

21

6� � . . . D. 1 � ��32��

2� ��32��

4� ��32��

6� . . .

11. Write �4

k�13k�1 in expanded form and then find the sum. 11. ________

A. 1 � 3 � 9 � 27; 40 B. 1 � �13� � �19� � �217�; �42

07�

C. 3 � 9 � 27 � 81; 120 D. 0 � 2 � 8 � 26; 36

Chapter

12



12. Express the series 5 � 9 � 13 � . . . � 101 using sigma notation. 12. ________

A. �∞

k�1(4k � 1) B. �

25

k�1(4k � 1) C. �

25

k�1(4k � 1) D. �

24

k�1(4k � 1)

13. The expression 32x5 � 80x4 � 80x3 � 40x2 � 10x � 1 is the 13. ________expansion of which binomial?A. (2x � 1)5 B. (x � 2)5 C. (2x � 2)5 D. (2x � 1)5

14. Find the fourth term in the expansion of (3x � y)7. 14. ________A. 105x4y3 B. 420x4y3 C. 1701x4y3 D. 2835x4y3

15. Use the first f ive terms of the exponential series 15. ________ex � 1 � x � �x2

2

!� � �x33

!� � �x44

!� � . . . to approximate e5.

A. 65.375 B. 148.41 C. 48.41 D. 76.25

16. Find ln (�21). 16. ________A. 3.0445 B. i� � 3.0445 C. i� � 3.0445 D. �3.0445

17. Write 1 � i in exponential form. 17. ________A. �2� e i�

�4� B. �2�ei�

74�� C. ei�

74�� D. ei�

�4�

18. Find the first three iterates of the function ƒ(z) � z � i for z0 � 1. 18. ________A. 1, 1 � i, 1 � 2i B. 1 � i, 2 � 2i, 3 � 3iC. 1 � i, 1 � 2i, 1 � 3i D. 1 � i, 1 � i, 1 � i

19. Find the first three iterates of the function ƒ(z) � z2 � c for c � i 19. ________and z0 � i.A. �1 � i, �3i, �9 � i B. �1 � i, 3i, �9C. 1 � i, �3i, 9 � i D. �1 � i, 2i, �4 � i

20. Suppose in a proof of the summation formula 20. ________1 � 5 � 9 � . . . � 4n � 3 � n(2n � 1) by mathematical induction,you show the formula valid for n � 1 and assume that it is valid for n � k. What is the next equation in the induction step of this proof ?A. 1 � 5 � 9 � . . . � 4k � 3 � 4(k � 1) � 3 � k(2k � 1) � 4(k � 1) � 3B. 1 � 5 � 9 � . . . � 4k � 3 � k(2k � 1) � 4(k � 1) � 3C. 1 � 5 � 9 � . . . � 4k � 3 � k(2k � 1)D. 1 � 5 � 9 � . . . � 4k � 3 � 4(k � 1) � 3 � k(2k � 1) � (k � 1)[2(k � 1) � 1]

Bonus If a1, a2, a3, . . . , an is an arithmetic sequence, where an � 0, Bonus: ________

then �a1

1�, �a

12�, �a

13�, . . . , �a

1n� is a harmonic sequence. Find one

harmonic mean between 2 and 3.

A. �25� B. �52� C. �152� D. �15

2�



12




NAME _____________________________ DATE _______________ PERIOD ________

1. Find d for the arithmetic sequence in which a1 � 14 and 1. __________________a28 � 32.

2. Find the 15th term in the arithmetic sequence 2. __________________11�45�, 10 �25�, 9, 7�35�, . . . .

3. Find the sum of the first 27 terms in the arithmetic series 3. __________________35.5 � 34.3 � 33.1 � 31.9 � . . . .

4. Find the ninth term in the geometric sequence 25, 10, 4, . . . . 4. __________________

5. Find the sum of the first eight terms in the geometric series 5. __________________�15� � 2 � 20 � . . . .

6. Form a sequence that has three geometric means between 6. __________________6 and 54.

7. Find limn→∞

�9n

133n

�

4

5�

n25n

�

2

4� or state that the limit does not exist. 7. __________________

8. Find the sum of the series 6�2� � 6 � 3�2� � 3 � . . . or 8. __________________state that the sum does not exist.

9. Write 0.06�4� as a fraction. 9. __________________

Determine whether each series is convergent or divergent.

10. �2�2�

1� 13� � �

2�2�1

� 23� � �2�2�

1� 33� � . . . 10. __________________

11. �211� � �22

2� � �23

3� � . . . 11. __________________

Chapter

12





12

12. Write �7

k�227��13��

k�2in expanded form and then find 12. __________________

the sum.

13. Express the series �31�09� � �31

�211� � �31

�413� � . . . � �32

�423� 13. __________________

using sigma notation.

14. Use the Binomial Theorem to expand (1 � �3�)5. 14. __________________

15. Find the fifth term in the expansion of (3x3 � 2y2)5. 15. __________________

16. Use the first f ive terms of the exponential series 16. __________________to approximate e2.7.

17. Find ln (�12.7) to four decimal places. 17. __________________

18. Find the first three iterates of the function ƒ(z) � 3z � 1 18. __________________for z0 � 2 � i.

19. Find the first three iterates of the function ƒ(z) � z2 � c 19. __________________for c � 1 � i and z0 � 2i.

20. Use mathematical induction to prove that 20. __________________1 � 5 � 25 � . . . � 5n�1 � �4

1� (5n � 1). Write your proof on a separate piece of paper.

Bonus If ƒ(z) � z2 � z � c is iterated with an initial Bonus: __________________value of 3 � 4i and z1 � 4 � 11i, f ind c.




NAME _____________________________ DATE _______________ PERIOD ________

1. Find d for the arithmetic sequence in which a1 � 6 1. __________________and a13 � �42.

2. Find the 40th term in the arithmetic sequence 2. __________________7, 4.4, 1.8, �0.8, . . . .

3. Find the sum of the first 30 terms in the arithmetic series 3. __________________10 � 6 � 2 � 2 � . . . .

4. Find the ninth term in the geometric sequence �217�, �19�, �13�, . . . . 4. __________________

5. Find the sum of the first eight terms in the geometric series 5. __________________64 � 32 � 16 � 8 � . . . .

6. Form a sequence that has three geometric means between 6. __________________�4 and �324.

7. Find lim �2nn�3


8. Find the sum of the series 12 � 8 � �136� � . . . or state 8. __________________

that the sum does not exist.



10. �41� � �17� 2� � �1 �

120

� 3� � . . . 10. __________________

11. �311� � �32

2� � �33

3� � . . . 11. __________________

Chapter

12

n→�



12. Write �(k � 1)(k � 2) in expanded form and then find the 12. __________________sum.

13. Express the series �1 2� 0� � �2 3

� 1� � �3 4� 2� � . . . � �10

11� 9� using 13. __________________

sigma notation.

14. Use the Binomial Theorem to expand (2p � 3q)4. 14. __________________

15. Find the fifth term in the expansion of (4x � 2y)7. 15. __________________

16. Use the first five terms of the cosine series 16. __________________cos x � 1 � �x2

2

!� � �x44

!� � �x66

!� � �x88

!� � . . . to approximate

the value of cos �4�� to four decimal places.

17. Find ln (�13.4) to four decimal places. 17. __________________

18. Find the first three iterates of the function ƒ(z) � 0.5z 18. __________________for z0 � 4 � 2i.

19. Find the first three iterates of the function ƒ(z) � z2 � c 19. __________________for c � 2i and z0 � 1.

20. Use mathematical induction to prove that 7 � 9 � 11 � . . . � 20. __________________(2n � 5) � n(n � 6). Write your proof on a separate piece of paper.

Bonus Find the sum of the coefficients of the expansion Bonus: __________________of (x � y)7.



123

k�0




NAME _____________________________ DATE _______________ PERIOD ________

1. Find d for the arithmetic sequence in which a1 � 5 and 1. __________________a12 � 38.

2. Find the 31st term in the arithmetic sequence 2. __________________9.3, 9, 8.7, 8.4, . . . .

3. Find the sum of the first 23 terms in the arithmetic 3. __________________series 6 � 11 � 16 � 21 � . . . .

4. Find the fifth term in the geometric sequence 4. __________________�10, �40, �160, . . . .

5. Find the sum of the first 10 terms in the geometric 5. __________________series 3 � 6 � 12 � 24 � . . . .

6. Form a sequence that has two geometric means 6. __________________between 9 and �13�.

7. Find limn→∞

�n2

2n�

2


8. Find the sum of the series �112� � �12� � 3 � . . . or state 8. __________________

that the sum does not exist.



10. �21

0� � �

21

2� � �

21

4� � . . . 10. __________________

11. �211� � �22

2� � �23

3� � . . . 11. __________________

Chapter

12



12. Write �7

k�43k in expanded form and then find the sum. 12. __________________

13. Express the series �12� 2� � �2 4

� 3� � �36� 4� � . . . � �81

�69� using 13. __________________

sigma notation.

14. Use the Binomial Theorem to expand (2p � 1)4. 14. __________________

15. Find the fourth term in the expansion of (2x � 3y)4. 15. __________________

16. Use the first f ive terms of the sine series sin x � 16. __________________x � �x3

3

!� � �x55

!� � �x77

!� � �x99

!� � . . . to f ind the value of

sin ��5� to four decimal places.

17. Find ln (�58) to four decimal places. 17. __________________

18. Find the first three iterates of the function ƒ(z) � 2z for 18. __________________z0 � 1 � 4i.

19. Find the first three iterates of the function ƒ(z) � z2 � c 19. __________________for c � i and z0 � 1.

20. Use mathematical induction to prove that 1 � 5 � 9 � . . . � 4n � 3 � n(2n � 1). Write your proof 20. __________________on a separate piece of paper.

Bonus Find the sum of the coefficients of the expansion Bonus: __________________of (x � y)5.



12




NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear, concisesolution to each problem. Be sure to include all relevant drawingsand justify your answers. You may show your solution in more thanone way or investigate beyond the requirements of the problem.

1. a. Write a word problem that involves an arithmetic sequence.Write the sequence and solve the problem. Tell what the answerrepresents.

b. Find the common difference and write the nth term of thearithmetic sequence in part a.

c. Find the sum of the first 12 terms of the arithmetic sequence inpart a. Explain in your own words why the formula for the sum ofthe first n terms of an arithmetic series works.

d. Does the related arithmetic series converge? Why or why not?

2. a. Write a word problem that involves a geometric sequence. Writethe sequence and solve the problem. Tell what the answerrepresents.

b. Find the common ratio and write the nth term of the geometricsequence in part a.

c. Find the sum of the first 11 terms of the sequence in part a.

d. Describe in your own words a test to determine whether a geometric series converges. Does the geometric series in part a converge?

3. a. Explain in your own words how to use mathematical induction toprove that a statement is true for all positive integers.

b. Use mathematical induction to prove that the sum of the first n terms of a geometric series is given by the formula

Sn � �a1

1�

�

ar1r

n

�, where r � 1.

4. Find the fourth term in the expansion of ��

y2

x��

�yx��

6.

Chapter

12



1. Find the 20th term in the arithmetic sequence 1. __________________15, 21, 27, . . . .

2. Find the sum of the first 25 terms in the arithmetic 2. __________________series 11 � 14 � 17 � 20 � . . . .

3. Find the 12th term in the geometric sequence 3. __________________2�4, 2�3, 2�2, . . . .

4. Find the sum of the first 10 terms in the geometric 4. __________________series 2 � 6 � 18 � 54 � . . . .

5. Write a sequence that has two geometric means 5. __________________between 64 and �8.

6. Find limn→∞

�2n

n2

2

�

�

31n

� or state that the limit 6. __________________

does not exist.

7. Find the sum of the series �18� � �14� � �12� � . . . or state 7. __________________that the sum does not exist.



9. 5 � �15�

2

2� � �1 �52

3

� 3� � �1 � 25�

4

3 � 4� � . . . 9. __________________

10. �222� � �23

3� � �24

4� � �25

5� � . . . 10. __________________



12


1. Find the 11th term in the arithmetic sequence 1. __________________�3� � �5�, 0, ��3� � �5�, . . . .

2. Find n for the sequence for which an � 19, a1 � �13, 2. __________________and d � 2.

3. Find the sum of the first 17 terms in the arithmetic 3. __________________series 4.5 � 4.7 � 4.9 � . . . .

4. Find the fifth term in the geometric sequence for which 4. __________________a3 � �5� and r � 3.

5. Find the sum of the first six terms in the geometric 5. __________________series 1 � 1.5 � 2.25 � . . . .

6. Write a sequence that has one geometric mean 6. __________________between �13� and �2

57�.

Find each limit, or state that the limit does not exist. 1. __________________1. lim

n→∞�2n

32n�

4

5� 2. lim

n→∞�(2n �

21n)(

2

n � 2)� 3. lim

n→∞�nn

2�

�

14

�2. __________________

3. __________________

Find the sum of each series, or state that the sum 4. __________________does not exist.

4. �12� � �14� � �18� � �116� � . . . 5. ��35� � 1 � �53� � . . . 5. __________________

6. Write the repeating decimal 0.4�5� as a fraction. 6. __________________

Determine whether each series is convergent or divergent.7. 0.002 � 0.02 � 0.2 � . . . 7. __________________

8. �58� � �59� � �150� � �1

51� � . . . 8. __________________

9. �11� 2� � �2

1� 3� � �3

1� 4� � �4

1� 5� � . . . 9. __________________

10. �12� 2� � �2

3� 3� � �3

4� 4� � . . . 10. __________________


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

12

Chapter

12


1. Write �4

n�2�2n�1 � �12�� in expanded form and then find the sum. 1. __________________

2. Express the series �1861� � �2

87� � �49� � . . . using sigma notation. 2. __________________

3. Express the series 1 � 2 � 3 � 4 � 5 � 6 � . . . � 199 � 200 3. __________________using sigma notation.

4. Use the Binomial Theorem to expand (3a � d)4. 4. __________________

5. Use the first five terms of the exponential series 5. __________________ex � 1 � x � �x2

2

!� � �x33

!� � �x44

!� � . . . to approximate e4.1

to the nearest hundredth.

6. Use the first five terms of the trigonometric series 6. __________________sin x � x � �x3

3

!� � �x55

!� � �x77

!� � �x99

!� � . . . to approximate sin �

�3� to four decimal places.

1. Find the first four iterates of the function ƒ(x) � �110� x � 1 1. __________________

for x0 � 1.

2. Find the first three iterates of the function ƒ(z) � 2z � i 2. __________________for z0 � 3 � i.

3. Find the first three iterates of the function ƒ(z) � z2 � c 3. __________________for c � �1 � 2i and z0 � i.

4. Use mathematical induction to prove that 4. __________________1 � 3 � 5 � . . . � (2n � 1) � n2. Write your proof on a separate piece of paper.

5. Use mathematical induction to prove that 5. __________________5 � 11 � 17 � . . . � (6n � 1) � n(3n � 2). Write your proof on a separate piece of paper.


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

12

Chapter

12




NAME _____________________________ DATE _______________ PERIOD ________


Multiple Choice1. In a basket of 80 apples, exactly 4 are

rotten. What percent of the apples arenot rotten?A 4%B 5%C 20%D 95%E 96%

2. Which grade had the largest percentincrease in the number of studentsfrom 1999 to 2000?

A 8B 9C 10D 11E 12

3. Find the length of a chord of a circle ifthe chord is 6 units from the center andthe length of the radius is 10 units.A 4B 8C 16D 2�3�4�E 4�3�4�

4. A chord of length 16 is 4 units fromthe center of a circle. Find the diameter.A 2�5�B 4�5�C 8�5�D 4�3�E 8�3�

5. If x � y � 4 and 2x � y � 5, then x � 2y �A 1B 2C 4D 5E 6

6. If �3kx15

�k

36� � 1 and x � 4, then k �

A 2B 3C 4D 8E 12

7. If 12% of a class of 25 students do nothave pets, how many students in theclass do have pets?A 3B 12C 13D 20E 22

8. In a senior class there are 400 boysand 500 girls. If 60% of the boys and50% of the girls live within 1 mile ofschool, what percent of the seniors donot live within 1 mile of school?A about 45.6%B about 54.4%C about 55.5%D about 44.4%E about 61.1%

9. In �ABC below, if BC BA, which ofthe following is true?A x yB y zC y xD y � xE z x

Chapter

12

Grade 8 9 10 11 121999 60 55 65 62 602000 80 62 72 72 70



10. Which is the measure of each angle ofa regular polygon with r sides?A �1r�(360°)B (r � 2)180°C 360°D �1r�(r � 2)180°E 60°

11. A tractor is originally priced at $7000.The price is reduced by 20% and thenraised by 5%. What is the net reduction in price?A $5950 B $5880C $1400 D $1120E $1050

12. The original price of a camera provided a profit of 30% above thedealer’s cost. The dealer sets a newprice of $195, a 25% increase abovethe original price. What is the dealer’s cost?A $243.75 B $202.80C $156.00 D $120.00E None of these

13. Segments of the lines x � 4, x � 9,y � �5, and y � 4 form a rectangle.What is the area of this rectangle insquare units?A 6 B 10C 18 D 20E 45

14. In the figure below, the coordinates ofA are (4, 0) and of C are (15, 0). Findthe area of �ABC if the equation ofAB�� is 2x � y � 8.A 100 units2

B 121 units2

C 132 units2

D 144 units2

E 169 units2

15. In 1998, Bob earned $2800. In 1999,his earnings increased by 15%. In2000, his earnings decreased by 15%from his earnings in 1999. What werehis earnings in 2000?A $2800.00 B $2380.98C $2381.15 D $2737.00E None of these

16. Londa is paid a 15% commission onall sales, plus $8.50 per hour. Oneweek, her sales were $6821.29. Howmany hours did she work to earn$1371.69?A 68.3 B 41.0C 120.4 D 28.2E None of these





Column A Column B

17. x > 0

18. One day, 90% of the girls and 80% ofthe boys were present in class.

19. Grid-In How many dollars must beinvested at a simple-interest rate of7.2% to earn $1440 in interest in 5 years?

20. Grid-In Seventy-two is 150% ofwhat number?



12

0.75xx plus an increase

of 75% of x

Number ofboys absent

Number ofgirls absent




NAME _____________________________ DATE _______________ PERIOD ________

1. Solve the system by using a matrix equation. 1. __________________2x � 3y � 11y � 12 � x

2. Determine whether ƒ(x) � �xx2 �

�35x� is continuous at x � 3. 2. __________________

Justify your response using the continuity test.

3. Determine the binomial factors of 2x3 � x2 � 13x � 6. 3. __________________

4. Write an equation of the sine function with amplitude 1, 4. __________________period �23

��, phase shift �1�5�, and vertical shift 2.

5. Find the distance between the parallel lines 2x � 5y � 10 5. __________________and 2x � 5y � �5.

6. A 300-newton force and a 500-newton force act on the 6. __________________same object. The angle between the forces measures 95°.Find the magnitude and direction of the resultant force.

7. Find the product 4�cos �23�� i sin �23

�� 3�cos �56�� i sin �56

��. 7. __________________Then write the result in rectangular form.

8. Write the equation of the ellipse 6x2 � 9y2 � 54 after a 8. __________________rotation of 45° about the origin.

9. If $1500 is invested in an account bearing 8.5% interest 9. __________________compounded continuously, find the balance of the accountafter 18 months.

10. Express the series �33� � �65� � �97� � �192� � . . . � �32

01� using 10. __________________

sigma notation.

Chapter

12


Page 289

1. D

2. C

3. A

4. B

5. A

6. A

7. A

8. B

9. C

10. D

11. D

Page 290

12. C

13. B

14. A

15. B

16. C

17. C

18. B

19. D

20. D

Bonus: C

Page 291

1. D

2. B

3. B

4. C

5. C

6. C

7. A

8. C

9. A

10. B

11. D

Page 292

12. B

13. D

14. A

15. C

16. B

17. A

18. B

19. B

20. D

Bonus: B



308-314 A&E C12-0-02-834179 10/10/00 10:35 AM Page 308


Page 293

1. A

2. D

3. C

4. D

5. B

6. D

7. B

8. B

9. C

10. D

11. A

Page 294

12. B

13. A

14. D

15. A

16. C

17. B

18. C

19. A

20. A

Bonus: C

Page 295

1. �23

�

2. �7�54�

3. 537.3

4. �15

2,56625

� or 0.16384

5. 2,222,222.2

6, 6�3�, 18, 18�3�, 54 or 6, �6�3�, 18,

6. �18�3�, 54

7. does not exist

8. 12�2� � 12

9. �43925

�

10. convergent

11. divergent

Page 29627 � 9 � 3 � 1 �

�13

� � �19

�; �1892� or

12. 20�92�

13. �12

k�5 �3(2

2k

k� 1)�

1 � 5�3� � 30 �14. 30�3� � 45 � 9�3�

15. 240x3y8

16. 12.84

17. i� � 2.5416

7 � 3i, 22 � 9i,18. 67 � 27i

�3 � i, 9 � 5i,19. 57 � 89i

20. See students’ work.

Bonus: 8 � 17i

Form 1C Form 2A


308-314 A&E C12-0-02-834179 10/10/00 10:35 AM Page 309

Page 297

1. �4

2. �94.4

3. �1440

4. 243

5. 42.5

�4, �12, �36,�108, �324 or �4,

6. 12, �36,108, �324

7. 0

8. 36

9. �9101�

10. convergent

11. divergent

Page 298

12. 2 � 6 � 12 � 20; 40

13.

14.

15. 35,840x3y4

16. 0.7071

17. i� � 2.5953

18.

19.


Bonus: 128

Page 299

1. 3

2. 0.3

3. 1403

4. �2560

5. �1023

6. 9, 3, 1, �13

�

7. 2

8. does not exist

9. �9593�

10. convergent

11. divergent

Page 300

12. 12 � 15 � 18 � 21; 66

13. �8

k�1�k(k

2�k

1)�

14.

15. �216xy3

16. 0.5878

17. i� + 4.0604

18.

19.

20.See students’ work.

Bonus: 32



Sample answer;

�10

n�1�n

n(n

��

11)

�

16p4 � 96p3q �

216p2q2 � 216pq3 �81q4

2 � i, 1 � 0.5i, 0.5 � 0.25i

1 � 2i, �3 � 6i,�27 � 34i

16p4 � 32p3 �24p2 � 8p � 1

2 � 8i, 4 � 16i, 8 � 32i

1 � i, 3i, �9 � i

308-314 A&E C12-0-02-834179 10/10/00 10:35 AM Page 310




3 Superior • Shows thorough understanding of the concepts arithmetic and geometric sequences and series,common differences and ratios of terms, the binomial theorem, and mathematical induction.

• Uses appropriate strategies to solve problems and prove a formula by mathematical induction.

• Computations are correct.• Written explanations are exemplary.• Word problems concerning arithmetic and geometric sequences are appropriate and make sense.

• Goes beyond requirements of some or all problems.

2 Satisfactory, • Shows understanding of the concepts arithmetic and with Minor geometric sequences and series, common differences Flaws and ratios of terms, the binomial theorem, and

mathematical induction.• Uses appropriate strategies to solve problems and prove a formula by mathematical induction.

• Computations are mostly correct.• Written explanations are effective.• Word problems concerning arithmetic and geometric sequences are appropriate and make sense.

• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts arithmetic Satisfactory, and geometric sequences and series, common differences with Serious and ratios of terms, the binomial theorem, and Flaws mathematical induction.

• May not use appropriate strategies to solve problems or prove a formula by mathematical induction.

• Computations are mostly correct.• Written explanations are satisfactory.• Word problems concerning arithmetic and geometric sequences are appropriate and sensible.


0 Unsatisfactory • Shows little or no understanding of the concepts arithmetic and geometric sequences and series, common differences and ratios of terms, the binomial theorem, and mathematical induction.

• May not use appropriate strategies to solve problems or prove a formula by mathematical induction.

• Computations are incorrect.• Written explanations are not satisfactory.• Word problems concerning arithmetic and geometric sequences are not appropriate or sensible.


308-314 A&E C12-0-02-834179 10/10/00 10:35 AM Page 311



Page 3011a. Sample answer: Mr. Ling opened a savings

account by depositing $50. He plans todeposit $25 more per month into theaccount. What is his total deposit afterthree months? The sequence is 50 �(n � 1)25, and $100 is his total deposit afterthree months.

1b. Sample answer: The common difference is$25. The nth term is $50 � (n � 1)$25.

1c. Sample answer:

S12 � �122�(50 � 325) � 2250

S12 � 50 � 75 � 100 � 125 � 150 � 175 �

200 � 225 � 250 � 275 � 300 � 325

S12 � (50 � 325) � (75 � 300) �(100 � 275) � (125 � 250) �(150 � 225) � (175 � 200)

Since the sums in parentheses are all equal.

S12 � 6(50 � 325), or �122� (50 � 325), or

�2n� (a1 � an)

1d. No; arithmetic series have no limits; it isdivergent.

2a. Sample answer: Mimi has $60 to spend onvacation. If she spends half of her moneyeach day, how much will she have left afterthe third day?

$60 � ��12

��3

� $7.50

After the third day, she has $7.50.

2b. Sample answer: The common ratio is �12

�.The nth term is 60��1

2��

n�1.

2c. Sample answer:

S11 � � 120

2d. If r � limn→∞

�aan�

n

1� � 1, the series

converges; r � �12

�; the series converges.

3a. Prove that the statement is true for n � 1.Then prove that if the statement is true forn, then it is true for n � 1.

3b. Here Sn is defined as

a1 � a1r � a1r2 � . . . � a1r

n�1 � �a1

1�

�

ar1r

n

�

Step 1: Verify that the formula is valid for n � 1.

Since S1 � a1 and S1 � �a1

1�

�

ar1r

1

�

� �a1

1(1

�

�

rr)

�

� a1,

the formula is valid for n � 1.

Step 2: Assume that the formula is for n � k and derive a formula for n � k � 1.

Sk ⇒ a1 � a1r � a1r2 � . . . � a1r

k�1 �

�a1

1�

�

ar1r

k

�

Sk�1 ⇒ a1 � a1r � a1r2 � . . . � a1r

k�1 �

a1rk�1 � 1

� �a1

1�

�

ar1r

k

� � a1rk�1 � 1

� �a1

1�

�

ar1r

k

� � a1rk

�

�

Apply the original formula for n � k � 1.

Sk�1 ⇒�a1 �

1a�

1rr

( k�1)

�

The formula gives the same result as adding the (k � 1) term directly. thus, if the formula is valid for n � k, it is also validfor n � k � 1. Since the formula is valid forn � 2, it is valid for n � 3, it is also valid forn � 4, and so on indefinitely. Thus, theformula is valid for all integral values of n.

4. From the binomial expansion, the fourth term

of ��y2x��

�yx�

��6is

�36!3!!

��y2x��3��

�yx�

��3� �20 � �

y1

3� or ��2y03�.

a1 � a1rk�1

��1 � r

a1 � a1rk � a1r

k � a1rk�1

��1 � r

60 � 60��12

��11

��1 � �

12�


308-314 A&E C12-0-02-834179 10/10/00 10:35 AM Page 312



1. 129

2. 1175

3. 27 or 128

4. �29,524

5. 64, �32, 16, �8

6. �21�

7. does not exist

8. �6939� or �

171�

9. convergent

10. divergent

Quiz APage 303

1. �9�3� � 9�5�

2. 17

3. 103.7

4. 9�5�

5. 20.78125

6. �13

�, � ��95��, �

257�, . . .

Quiz BPage 303

1. does not exist

2. 1

3. 0

4. �31�

5. does not exist

6. �4959� or �

151�

7. divergent

8. divergent

9. convergent

10. divergent

Quiz CPage 304

1. 2�12

� � 4�12

� � 8�12

�; 15�12

�

2. �

n�1�1861� � ��

23��

n�1

3. �100

k=1(2k � 1)(2k)

81a4 � 108a3d �4. 54a2d2 � 12ad3 � d4

5. 36.77

6. 0.8660

Quiz DPage 304

1. 1.1, 1.11, 1.111, 1.1111

2. �2 � 2 i, �1 � 6i, �36 �14i

3. �1 � 2i, �4 � 2i, 11 � 18i




Sample answer:

308-314 A&E C12-0-02-834179 10/10/00 10:35 AM Page 313

Page 305

1. D

2. A

3. C

4. C

5. D

6. E

7. E

8. A

9. C

Page 306

10. D

11. D

12. D

13. E

14. B

15. D

16. B

17. A

18. D

19. 4000

20. 48

Page 307

1. (�5, 7)

No, ƒ(x) is2. undefined when x � 3.

3. (x � 2) (2x � 1) (x � 3)

4. y � �sin �3x � ��5

�� 2

5. �152�92�9��

6. 560.2 N, 32.2

7. 12�cos �96�� i sin �96

��; �12i

15(x�)2 � 30 x�y� �8. 15( y�)2 � 108

9. $1704

10. �10

n�1�(�

21n)n

�

�1

13n

�



308-314 A&E C12-0-02-834179 10/10/00 10:35 AM Page 314



NAME _____________________________ DATE _______________ PERIOD ________


1. A school has different course offerings: 4 in math, 6 in English, 5 in science,1. ________and 3 in social studies. How many different 4-course student schedules are possible if a student must have one course from each subject area?A. 4 B. 24 C. 120 D. 360

2. How many different ways can the letters in the word social be arranged 2. ________if the letter c must be directly followed by the letter i?A. 120 B. 720 C. 24 D. 256

3. How many sets of 5 books can be chosen from a set of 8? 3. ________A. 32,768 B. 120 C. 56 D. 40,320

4. How many different starting teams consisting of 1 center, 2 forwards, 4. ________and 2 guards can be chosen from a basketball squad consisting of 3 centers, 6 forwards and 7 guards?A. 945 B. 120 C. 126 D. 5292

5. Find the possible number of license plates consisting of 2 letters followed 5. ________by 4 digits if digits can be repeated but letters cannot.A. 3,276,000 B. 3,407,040 C. 6,500,000 D. 6,760,000

6. How many ways can the letters in the word bookkeeper be arranged? 6. ________A. 3,628,800 B. 151,200 C. 302,400 D. 362,880

7. How many ways can 10 different chairs be arranged in a circle? 7. ________A. 362,880 B. 120 C. 3,628,800 D. 10,000,000,000

8. Find the number of ways that 7 people can be seated at a circular table 8. ________if 1 seat has a microphone in front of it.A. 720 B. 5040 C. 46,656 D. 823,543

For Exercises 9 and 10, consider a class with 10 sophomores, 8 juniors, and 6 seniors. Two students are selected at random.

9. What is the probability of selecting 1 junior and 1 senior? 9. ________A. �1

12� B. �2

43� C. �1

738� D. �1

138�

10. Find the odds of selecting 2 students who are not seniors. 10. ________A. �54

11� B. �59

12� C. �55

1� D. �34�

11. The probability of getting 2 heads and 1 tail when three coins are 11. ________tossed is 3 in 8. Find the odds of not getting 2 heads and 1 tail.A. �35� B. �58� C. �53� D. �38�

Chapter

13



12. The odds of rolling a sum of 5 when two number cubes are rolled are �18�. 12. ________What is the probability of rolling a sum of 5 when two number cubes are rolled?A. �49� B. �19� C. �1

58� D. �14�

13. One red and one green number cube are tossed. What is the probability 13. ________that the red number cube shows an even number and the green number cube shows a number greater than 2?A. �12� B. �16� C. �13� D. �23�

14. If two cards are drawn at random from a standard deck of cards with 14. ________no replacement, find the probability that both cards are queens.A. �6

376� B. �1

169� C. �5

12� D. �2

121�

15. A basket contains 3 red, 4 yellow, and 5 green balls. If one ball is taken 15. ________at random, what is the probability that it is yellow or green?A. �19� B. �3

56� C. �2

32� D. �34�

16. A company survey shows that 50% of employees drive to work, 30% of 16. ________employees have children, and 20% of employees drive to work and have children. What is the probability that an employee drives to work or has children?A. 1 B. �35� C. �45� D. �2

30�

17. Three number cubes are tossed. Find the probability of exactly 17. ________two number cubes showing 6 if the first number cube shows 6.A. �1

58� B. �3

56� C. �1

508� D. �2

516�

18. In a certain health club, half the members are women, one-third of 18. ________the members use free weights, and one-fifth of the members are women who use free weights. A female member is elected treasurer.What is the probability that she uses free weights?A. �14� B. �18� C. �13

90� D. �25�

19. Four coins are tossed. What is the probability that at least 2 of the 19. ________4 coins show heads?A. �11

16� B. �1

56� C. �2

156� D. �12

14�

20. Nine out of every 10 students have a calculator. Expressed as a decimal 20. ________to the nearest hundredth, what is the probability that exactly 7 out of 8 students in a given class have a calculator?A. 0.05 B. 0.72 C. 0.38 D. 0.79

Bonus Two number cubes are thrown twice. What is the Bonus: ________probability of getting a sum that is a prime number less than 6 on both throws?

A. �376� B. �3

2254� C. �1

42996� D. �1

1261�



13




NAME _____________________________ DATE _______________ PERIOD ________


1. Given 3 choices of sandwiches, 4 choices of chips, and 2 choices of 1. ________cookies, how many different sack lunches can be prepared containing one choice of each item?A. 12 B. 24 C. 84 D. 288

2. How many ways can the letters in the word capitol be arranged if the 2. ________first letter must be p?A. 120 B. 720 C. 5040 D. 40,320

3. How many 4-letter codes can be formed from the letters in the word 3. ________capture if letters cannot be repeated?A. 28 B. 840 C. 2401 D. 5040

4. A class consisting of 10 boys and 12 girls must select 2 boys and 2 girls 4. ________to serve on a committee. How many variations of the committee can there be?A. 2970 B. 120 C. 4800 D. 7315

5. On a long city block, 4-digit house numbers must begin with the 5. ________digit 4 and end with either 0 or 1. How many different variations of house numbers are possible?A. 1000 B. 144 C. 180 D. 200

6. How many ways can 3 identical green candles and 7 identical blue 6. ________candles be arranged in a row in any variation?A. 21 B. 3,628,800 C. 30,240 D. 120

7. How many ways can 8 keys be arranged on a key ring with no chain? 7. ________A. 2520 B. 40,320 C. 720 D. 64

8. How many ways can 9 numbers be arranged on a small rotating wheel 8. ________relative to a fixed point?A. 81 B. 5040 C. 40,320 D. 362,880

For Exercises 9 and 10, consider a basket that contains 15 slips ofpaper numbered from 1 to 15. Two slips of paper are drawn at random.

9. What is the probability of drawing 2 even numbers? 9. ________A. �2

4295� B. �14� C. �15� D. �1

75�

10. What are the odds of drawing an even number less than 7 and an 10. ________odd number greater than 10?A. �3

25� B. �3

32� C. �25� D. �3

353�

11.The probability of rolling a sum of 4 when two number cubes are 11. ________tossed is 1 in 12. What are the odds of rolling a sum of 4 when two number cubes are tossed?A. �1

11� B. �11

23� C. �1

12� D. �1

121�

Chapter

13



12. The odds that it will rain in the city of Houston tomorrow are �14�. 12. ________What is the probability of rain in Houston tomorrow?A. �45� B. �34� C. �15� D. �14�

13. Two number cubes, 1 red and 1 green, are tossed. What is the 13. ________probability that the red number cube shows a number less than 3 and the green number cube shows a 6?A. �12� B. �16� C. �13� D. �1

18�

14. If two cards are drawn at random from a standard deck of cards 14. ________with no replacement, find the probability that both cards are hearts.A. �14� B. �1

16� C. �1

17� D. �5

12�

15. A bucket contains 4 red, 2 yellow, and 3 green balls. If one ball is 15. ________taken at random, what is the probability that it is red or green?A. �23� B. �79� C. �2

47� D. �8

71�

16. A school survey shows that 10% of students are in band, 12% of 16. ________students are in athletics, and 6% of students are in both band andathletics.What is the probability that a student is in band or athletics?A. �6

10� B. �15

10� C. �2

45� D. �2

75�

17. Three coins are tossed. Find the probability that exactly 2 coins 17. ________show heads if the first coin shows heads.A. �12� B. �38� C. �14� D. �18�

18. Given the integers 1 through 33, what is the probability that one 18. ________of these integers is divisible by 4 if it is a multiple of 6?A. �14� B. �3

53� C. �58� D. �25�

19. A survey shows that 20% of all cars are white. What is the probability 19. ________that exactly 3 of the next 4 cars to pass will be white?A. �14� B. �6

1265� C. �6

425� D. �1

125�

20. Eight out of every 10 houses have a garage. Express as a decimal to 20. ________the nearest hundredth the probability that exactly 9 out of 12 houses on a given block have a garage.A. 0.24 B. 0.42 C. 0.56 D. 0.88

Bonus Two number cubes are thrown twice. What is the Bonus: ________probability of getting a sum less than 4 on both throws?

A. �316� B. �1

144� C. �1

12� D. �16�



13




NAME _____________________________ DATE _______________ PERIOD ________


1. Susan must wear one of 5 blouses and one of 4 skirts. How many 1. ________different possible outfits consisting of 1 blouse and 1 skirt does she have?A. 20 B. 24 C. 120 D. 9

2. How many ways can the letters in the word country be arranged? 2. ________A. 120 B. 720 C. 5040 D. 40,320

3. How many 3-letter codes can be formed from the letters in the word 3. ________picture if letters cannot be repeated?A. 21 B. 210 C. 343 D. 5040

4. A class consisting of 24 people must select 3 people among them to 4. ________serve on a committee. How many different variations are there?A. 72 B. 2024 C. 12,144 D. 13,824

5. How many 5-digit ZIP codes are possible if the first number cannot be 0? 5. ________A. 10,000 B. 5040 C. 90,000 D. 30,240

6. How many ways can the letters in the word stereo be arranged? 6. ________A. 120 B. 46,656 C. 720 D. 360

7. Given 10 different stones, how many ways can all of the stones be 7. ________arranged in a circle?A. 5040 B. 40,320 C. 362,880 D. 3,628,800

8. Find the number of possible arrangements for 6 chairs around a 8. ________circular table with 1 chair nearest the door.A. 120 B. 5040 C. 46,656 D. 720

For Exercises 9 and 10, consider a bucket that contains 4 redmarbles and 5 blue marbles. Two marbles are drawn at random.

9. What is the probability of drawing 2 red marbles? 9. ________A. �16� B. �18

61� C. �12

65� D. �5

4�

10. What are the odds of drawing 1 red and 1 blue marble? 10. ________A. �1

58� B. �59� C. �45� D. �4

5�

11. The probability of getting a jack when a card is drawn from a 11. ________standard deck of cards is 1 in 13. What are the odds of getting a jack when a card is drawn?A. �1

11� B. �11

34� C. �1

12� D. �11

23�

Chapter

13



12. The odds that Lee will attend a movie this weekend are �35�. What is the 12. ________probability that Lee will attend a movie this weekend?A. �38� B. �25� C. �58� D. �2

3�

13. Using a standard deck of playing cards, find the probability of selecting 13. ________a king and then selecting a heart once the king has been returned to the deck.A. �2

421� B. �5

12� C. �15

72� D. �1

16�

14. Two ribbons are selected at random from a container holding 5 purple 14. ________and 6 white ribbons. Find the probability that both ribbons are white.A. �1

31� B. �1

3261� C. �1

61� D. �1

21�

15. A number cube is tossed. What is the probability that the number 15. ________cube shows a 1 or a number greater than 4?A. �23� B. �13� C. �1

18� D. �2

1�

16. A school survey shows that 40% of students like rock music, 16. ________20% of students like rap, and 10% of students like both rock and rap.What is the probability that a student likes either rock or rap music?A. �12� B. �35� C. �1

70� D. �5

2�

17. Two coins are tossed. Find the probability that both coins turn up 17. ________heads if the first coin turns up heads.A. �13� B. �34� C. �14� D. �2

1�

18. Given the integers 1 through 14, what is the probability that one of 18. ________these integers is divisible by 3 if it is less than 10?A. �14� B. �13� C. �1

34� D. �1

94�

19. Four coins are tossed. What is the probability of getting 3 heads and 19. ________1 tail?A. �1

56� B. �14� C. �34� D. �8

3�

20. Two out of every 10 houses in a neighborhood have a front porch. 20. ________Expressed as a decimal to the nearest hundredth, what is the probability that exactly 2 out of 6 houses on a given block have a front porch?A. 0.02 B. 0.33 C. 0.25 D. 0.20

Bonus How many ways can 5 cups and 5 glasses be arranged on a Bonus: ________shelf if all of the glasses must be kept together?A. 3,628,800 B. 120 C. 86,400 D. 15,625



13




NAME _____________________________ DATE _______________ PERIOD ________

1. A bakery’s dessert list consists of 3 kinds of cakes, 9 kinds 1. __________________of pies, and 10 kinds of brownies. How many combinations of three desserts will Jana have if she buys one of each kind?

2. How many ways can the letters in the word laughter be 2. __________________arranged if the g must be followed by the letter h?

3. How many 4-digit codes can be formed from the digits 3. __________________1, 2, 3, 4, 5, 6, and 7 if digits cannot be repeated?

4. How many different committees of 5 members can be 4. __________________chosen from a club with 25 members?

5. A test has 4 multiple-choice questions, and each 5. __________________question has 4 answer choices. The multiple-choice questions are followed by 3 true-false questions. How many ways can a student answer the questions if no answers can be left blank?

6. How many ways can the letters in the word entertain be 6. __________________arranged?

7. How many ways can 9 people arrange themselves in a circle 7. __________________around a campfire?

8. Find the number of ways that 7 people can sit around a 8. __________________circular table with one seat near a window.

For Exercises 9 and 10, consider a bag that contains 5 red, 3 blue,and 4 yellow marbles.

9. If five marbles are drawn at random, what is the probability 9. __________________that there will be 2 red, 1 blue, and 2 yellow marbles?

10. If three marbles are drawn at random, what are the odds of 10. __________________selecting 3 red marbles?

11. The probability of getting a red queen when a card is drawn 11. __________________from a standard deck of cards is 1 in 26. What are the odds of not getting a red queen when a card is drawn?

12. The odds that Sandra will attend a sporting event each week 12. __________________are �25�. What is the probability that Sandra will attend sporting events 2 weeks in a row?

Chapter

13



13. A coin collection consists of 4 quarters, 5 dimes, and 13. __________________7 nickels. One coin is selected and replaced. A second coin is selected. What is the probability that 1 dime and 1 nickel are selected?

14. If two cards are drawn from a standard deck of playing cards 14. __________________without replacement, find the probability of selecting an ace and a face card.

15. From a collection of 5 blue and 4 red ink pens, three are 15. __________________selected at random. What is the probability that at least two are red?

16. In a lakeside community, 50% of the residents own a boat, 16. __________________80% of the residents own fishing equipment, and 40% of the residents own both fishing equipment and a boat.Find the probability that a resident owns a boat or fishing equipment.

17. Two number cubes are tossed. Find the probability that the 17. __________________sum of the number cubes is an even number, given that the first number cube shows a 3.

18. At a certain gym, half the members are men, one-fourth 18. __________________of the members swim, and one-sixth of the members are men who swim. What is the probability that amale member who enters the gym is also a swimmer?

19. A survey shows that 60% of all students at one school 19. __________________complete their homework. Find the probability that at least 3 of 4 students who enter a class have completed their homework.

20. Six coins are tossed. What is the probability that at least 20. __________________4 coins turn up tails?

Bonus Different configurations of flags in a row represent Bonus: __________________different signals. How many different signals can be sent using 2 red, 4 blue, and 3 green signal flags?



13




NAME _____________________________ DATE _______________ PERIOD ________

1. Stephanie has 3 sweaters, 7 blouses, and 6 pairs of slacks 1. __________________in her closet. If she chooses one of each, how many different outfits could she have?

2. How many ways can a family of 7 be arranged for a 2. __________________photo if the mother is seated in the middle?

3. A club has 12 members. How many ways can a president, 3. __________________a secretary and a treasurer be chosen from among the members?

4. How many color schemes for a backdrop consisting of four 4. __________________colors are possible if there are 10 colors from which to choose?

5. Find the number of possible 7-digit local phone numbers 5. __________________if the first digit cannot be 0 or 1.

6. How many ways can the letters in the word photograph 6. __________________be arranged?

7. How many ways can 11 people arrange themselves in a 7. __________________circle around a flagpole?

8. Find the number of ways that 8 people can sit around a 8. __________________circular table with 7 blue chairs and 1 green chair.

For Exercises 9 and 10, consider a box containing 6 red, 4 blue,and 3 yellow blocks.

9. If three blocks are drawn at random, what is the 9. __________________probability that 2 blocks are red and 1 block is blue?

10. If two blocks are drawn at random, what are the odds 10. __________________of drawing 2 blue blocks?

11. The probability of getting all heads when four coins are 11. __________________tossed is1 in 16. What are the odds of getting all tails when four coins are tossed?

12. The odds that Jonathan will attend a concert each month 12. __________________are �29�. What is the probability that Jonathan will attend concerts 2 months in a row?

Chapter

13



13. A book rack contains 5 novels and 7 dictionaries. One book 13. __________________is selected and replaced. A second book is selected. What is the probability that 2 novels are selected?

14. If two cards are drawn from a standard deck of playing 14. __________________cards without replacement, find the probability of selecting a heart and a diamond.

15. One pen is randomly selected from a collection of 5 blue, 15. __________________3 black, and 4 red ink pens. What is the probability that the pen is red or black?

16. In a certain community, 70% of residents own a VCR, 16. __________________60% of residents own a stereo, and 50% of residents own both a VCR anda stereo. Find the probability that a resident owns a VCR or a stereo.

17. Two number cubes are tossed. Find the probability that 17. __________________the sum of the number cubes is less than 6, given that the first number cube shows a 3.

18. Given the integers 1 through 100, what is the probability 18. __________________that one of these integers is divisible by 4 if it ends in a 0?

19. A survey shows that 80% of all students wear jeans on 19. __________________Friday. If 4 students enter a class, what is the probability that exactly 3 of them are wearing jeans?

20. Four out of every 10 college students own a bike. What 20. __________________is the probability that exactly 4 out of 5 students in a project group own bikes?

Bonus Two number cubes are rolled. What is the Bonus: __________________probability that at least one number cube shows an 8?



13




NAME _____________________________ DATE _______________ PERIOD ________

1. A student has 12 pencils and 10 pens. How many ways can 1. __________________the student choose one pencil and one pen?

2. How many ways can 8 different videos be arranged in a row 2. __________________for a display?

3. How many ways can first place, second place, and third place 3. __________________be chosen in a contest in which there are 11 entries?

4. In an algebra class of 20 students, how many different ways 4. __________________can a subgroup of 6 students be chosen for a group project?

5. Find the number of possible variations of 4-digit street 5. __________________addresses if the first digit cannot be a 0.

6. How many ways can the letters in the word fishing be 6. __________________arranged?

7. How many ways can 12 people arrange themselves around 7. __________________a circular trampoline?

8. Find the number of ways that 6 people can arrange 8. __________________themselves around the circular base of a flagpole at the end of a sidewalk.

For Exercises 9 and 10, consider a bag containing 5 red, 3 blue, and 6 yellow marbles.

9. If two marbles are drawn at random, what is the probability 9. __________________of getting 1 red marble and 1 blue marble?

10. If two marbles are drawn at random, what are the odds of 10. __________________selecting 2 red marbles?

11. The probability of getting all tails when three coins are 11. __________________tossed is 1 in 8. What are the odds of getting all tails when three coins are tossed?

12. The odds that Marilyn will go to see a movie each week 12. __________________

are �15�. What is the probability that Marilyn will go to see a movie next week?

Chapter

13



13. A book rack contains 8 cookbooks and 3 novels. One book is 13. __________________selected at random and replaced. A second book is selected at random. What is the probability that 1 cookbook and 1 novel are selected?

14. Two cards are drawn from a standard deck of playing cards 14. __________________without replacement. Find the probability of selecting a jack and a king.

15. From a collection of 2 blue, 6 yellow, and 4 red crayons, 15. __________________one is selected at random. What is the probability that the crayon is red or blue?

16. In a certain community, 60% of residents own a microwave 16. __________________oven, 40% of residents own a computer, and 30% of residents own both a microwave oven and a computer.Find the probability that a resident owns a microwave oven or a computer.

17. Two number cubes are tossed. Find the probability that 17. __________________the sum of the number cubes is greater than 5 given that the first number cube shows a 3.

18. Given the integers 1 through 50, what is the probability 18. __________________that one of these integers is a multiple of 3 if it ends in a 5?

19. A survey shows that 80% of all students on one campus 19. __________________carry backpacks. If 4 students enter a class, find the probability that exactly 2 of them are carrying backpacks.

20. Eight out of every 10 working adults own a car. Expressed 20. __________________as a decimal to the nearest hundredth, what is the probability that exactly 6 out of 8 working adults in an office own cars?

Bonus Eight points lie on a circle. How many different Bonus: __________________inscribed pentagons can be drawn using the points as vertices?



13




NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear,concise solution to each problem. Be sure to include all relevantdrawings and justify your answers. You may show your solution inmore than one way or investigate beyond the requirements of theproblem.1. Men’s socks are to be displayed along an aisle of a department

store.a. If there are 3 styles, 5 colors, and 3 sizes of socks, how many

different arrangements are possible?

b. Not all the possible arrangements make sense; that is, somemay confuse a customer who is trying to locate a particularpair of socks. Describe a poor arrangement.

c. Sketch an example of a good arrangement of the socks.Explain why this is a better arrangement than the onedescribed in part b.

2. Seven different dress styles are to be arranged on a circular rack.a. How many different arrangements are possible?

b. How does the number of arrangements on a circular rack differfrom the number of arrangements on a straight rack? Explainthe reason for the differences.

3. A store has 7 different fashion scarves for sale.a. If the manager wants to display a combination of four of these

scarves by the checkout, what is the number of possiblecombinations?

b. Explain the difference between a combination and apermutation.

4. For the season, Chad’s free-throw percentage is 70%.a. If shooting consecutive free throws are independent events,

what is the probability that Chad will make two consecutiveshots?

b. Describe a situation in which two free throws would not beindependent events. What factors might affect the shots?

5. Eight distinct points are randomly located on a circle.a. How many different triangles can be formed by using the

points as vertices? Justify your answer.

b. How many different quadrilaterals can be formed? Justify youranswer.

Chapter

13



1. A lunch line offers 3 choices of salad, 2 choices of meat, 1. __________________4 choices of vegetable, and 2 choices of dessert. How many menu combinations are possible that include one of each course?

2. How many ways can the letters in the word decimal be 2. __________________arranged?

3. Find the number of possible arrangements of 9 different 3. __________________videos in a display window using exactly 4 at a time.

4. If 5 blocks are drawn at random from a box containing 4. __________________7 blue and 5 green blocks, how many ways can 3 blue and 2 green blocks be chosen?

5. How many ways can the letters in the word attitude be 5. __________________arranged?

6. Find the number of ways 6 keys can be arranged on a key 6. __________________ring with no chain.

7. How many ways can 10 people be seated around a circular 7. __________________conference table if there is a laptop computer on the table in front of one of the seats?

8. Two number cubes are tossed. Find the probability that the 8. __________________sum of the number cubes is 6.

9. If two marbles are selected at random from a bag containing 9. __________________6 red and 4 blue marbles, find the odds that both marbles are red.

10. The odds of all three coins showing heads when three 10. __________________coins are tossed are 1 to 7. What is the probability of tossing 3 heads when three coins are tossed?



13


1. A toolbox contains 12 wrenches, 8 screwdrivers, and 1. __________________5 pairs of pliers. How many ways can a mechanic choose 3 tools, if he needs one of each?

2. How many ways can a mother, father, and six children be 2. __________________arranged in a row for a photograph?

3. How many ways can 3 blue, 4 red, and 2 yellow notebooks 3. __________________be arranged in a row in any variation?

4. For dinner you have chicken, mashed potatoes, and corn. 4. __________________Are eating chicken first and eating mashed potatoes second dependent or independent events?

5. How many ways can 8 pins be arranged on a circular hatband? 5. __________________

1. What is the probability that a given month of the year begins 1. __________________with the letter J?

2. Two number cubes are tossed. What are the odds that they 2. __________________show a sum greater than 9?

3. From a box containing 12 slips of paper numbered 1 to 12, 3. __________________2 slips are drawn. Find the probability that the numbers on both slips are divisible by 3.

4. The probability of getting a sum of 7 when two number cubes 4. __________________are tossed is 1 in 6. What are the odds of getting a sum of 7 when two number cubes are tossed?

5. The odds of a student selected at random being a band member 5. __________________are 3 to 10. What is the probability that a student selected at random is in the band?

Chapter 13, Quiz B (Lesson 13-3)

NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

13

Chapter

13


1. Two number cubes, one red and one blue, are tossed. What is 1. __________________the probability that the red number cube shows a 5 and the blue number cube shows an even number?

2. Are selecting a king and selecting a black card from a standard 2. __________________deck of cards mutually exclusive or mutually inclusive events?What is the probability of selecting a king or a black card?

3. A basket contains 8 red, 3 blue, and 5 green balls. If one ball is 3. __________________taken at random, what is the probability that it is blue or green?

4. A class survey shows 60% of the students like rock music, 40% 4. __________________of the students are juniors, and 30% of the students are both rock music fans and juniors. Find the probability that a student likes rock music or is a junior.

5. If two number cubes are tossed, what is the probability of 5. __________________getting a sum that is less than 6, given that one number cube shows a 3?

For Exercises 1 and 2, consider that 5 coins are tossed.1. What is the probability of getting exactly 4 heads? 1. __________________

2. Find the probability of getting at least 2 tails. 2. __________________

For Exercises 3 and 4, consider survey results that show 25% of all cars in a community have tinted windows.3. Find the probability that exactly 3 of the next 5 cars to 3. __________________

pass will have tinted windows.

4. Find the probability that none of the next four cars will 4. __________________have tinted windows.

5. Three out of every 5 classrooms have a computer. Find the 5. __________________probability, expressed as a decimal to the nearest hundredth,that exactly 8 out of 10 classrooms in a given school have a computer.


NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

13

Chapter

13




NAME _____________________________ DATE _______________ PERIOD ________


Multiple Choice1. Determine the number of ways that

5 students can be chosen for a teamfrom a class of 30.A 1293B 142,506C 3,542,292D 17,100,720E None of these

2. If a number cube is rolled, what is theprobability that the cube will stop withan even number facing up?A �2

1�

B �31�

C �23�

D �32�

E �61�

3. What is the length of a line segmentjoining two points whose coordinatesare (�2, �7) and (6, 8)?A 4B 5C 7�12�

D 8�12�

E 17

4. In the figure below, �AOB and �PCBare isosceles right triangles with equal areas. What are the coordinatesof point P?A (6, 0)B (6, 12)C (12, 0)D (0, 12)E (12, 6)

5. C and D are distinct points on A�B�and C is the midpoint of A�B�. What isthe probability that D is the midpointof A�B�?A 0B �2

1�

C �23�

D 1E It cannot be determined from the

information given.

6. P is a point on the bisector of �ABC.What is the probability that P isequidistant from the sides of the angle?A 0B �2

1�

C �14�

D 1E It cannot be determined from the

information given.

7. If P � �h(a

2� b)�, what is the average of a

and b when P � 30 and h � 5?A 2B 6C 15D �32

5�

E 150

8. If a � 7b, then what is the average of aand b?

A 2b B 3bC 3�12�b D 4b

E 8b9. A 7-hour clock is shown below. If at

noon today the pointer is at 0, wherewill the pointer be at noon tomorrow?

A 2B 3C 4D 5E 6

Chapter

13



10. P � 2 � �190�, Q � 2 � 0.099, R � 2 � 9

Which list below shows P, Q, and R inorder from greatest to least?A P, Q, R B Q, P, RC R, P, Q D P, R, QE R, Q, P

11. One number cube is rolled. What isthe probability that when the cubestops rolling the number on top is aneven number or a number less than 4?A �3

2� B �21�

C �65� D �3

1�

E 1

12. Among a group of 6 people, how manycommittees of 3 people can be formedif 2 of the 6 people cannot be on thesame committee?A 12 B 9C 10 D 60E 16

13. An acute angle can have a measure of:I. 89.999° II. 0.0001° III. 90.0001°A I onlyB II onlyC III onlyD I and II onlyE I and III only

14. In circle O, A�B� is a chord, O�A� and O�B� are radii, m�AOB � 120°, andAB � 12. Find the distance from thechord to the center of the circle.A 2�3� B 4�3�C 3 D 6E It cannot be determined from the

information given.

15. In a detective game, there are 6 suspects, 6 weapons, and 9 rooms.What is the probability that the crimewas committed by the housekeeper inthe library with a candlestick holder?A �1

108� B �21

16�

C �3124� D �5

14�

E None of these

16. Two disks are selected at random froma box containing 10 disks numberedfrom 1 to 10. What is the probabilitythat one disk has an even number andthe other has an odd number if thefirst disk is not replaced before thesecond disk is selected?A �12� B �1

58�

C �59� D �53�

E �52�





Column A Column B

17.

18.

19. Grid-In José has 6 pennies,5 nickels, and 4 dimes in his pocket.What is the probability that a coin he draws at random is a penny?

20. Grid-In A 3-person committee is tobe chosen from a group of 6 males and4 females. What is the probabilitythat the committee will consist of 2 males and 1 female?



13

The number of waysto select 2 males or

2 females from agroup of 6 males

and 4 females

The number of waysto select 1 male and

1 female from agroup of 6 males

and 4 females

C(50, 0) C(30, 30)




NAME _____________________________ DATE _______________ PERIOD ________

1. Write the standard form of the equation of the line with an 1. __________________x-intercept of 2 and a y-intercept of 3.

2. Solve �3�t�� 7� � 7 � 0. 2. __________________

3. Suppose � is an angle in standard position whose terminal 3. __________________side lies in Quadrant IV. If cos � � �45�, what is the value of tan �?

4. Write the equation y � �x � 4 in parametric form. 4. __________________


��.

6. Write the equation of the parabola whose focus is at (�2, 6) 6. __________________and whose directrix has the equation y � 2.

7. Evaluate log9 27. 7. __________________

8. Find the sum of the series 18 � 12 � 8 � . . ., or state that 8. __________________the sum does not exist.

9. A sample of 3 fuses from a box of 100 fuses is to be inspected. 9. __________________How many ways can the sample be chosen?

10. Two number cubes are tossed. What is the probability that 10. __________________they show a sum of either 2 or 11?

Chapter

13


Page 315

1. D

2. A

3. C

4. A

5. C

6. B

7. A

8. B

9. B

10. A

11. C

Page 316

12. B

13. C

14. D

15. D

16. B

17. A

18. C

19. A

20. C

Bonus: C

Page 317

1. B

2. B

3. B

4. A

5. D

6. D

7. A

8. D

9. C

10. B

11. A

Page 318

12. C

13. D

14. C

15. B

16. C

17. A

18. D

19. B

20. A

Bonus: B



334-340 A&E C13-0-02-834179 10/10/00 10:43 AM Page 334


Page 319

1. A

2. C

3. B

4. B

5. C

6. D

7. C

8. D

9. A

10. D

11. C

Page 320

12. A

13. B

14. A

15. D

16. A

17. D

18. B

19. B

20. C

Bonus: C

Page 321

1. 270

2. 5040

3. 840

4. 53,130

5. 2048

6. 45,360

7. 40,320

8. 5040

9. �252�

10. �211�

11. �215�

12. �449�

Page 322

13. �23556

�

14. �22

81

�

15. �4127�

16. �190�

17. �21�

18. �31�

19. �622957�

20. �3121�

Bonus: 1260

Form 1C Form 2A


334-340 A&E C13-0-02-834179 10/10/00 10:43 AM Page 335

Page 323

1. 126

2. 720

3. 1320

4. 210

5. 8,000,000

6. 453,600

7. 3,628,800

8. 40,320

9. �13403

�

10. �112�

11. �115�

12. �12

41

�

Page 324

13. �12454

�

14. �21034

�

15. �172�

16. �54�

17. �31�

18. �21�

19. �265265

�

20. �64285

�

Bonus: 0

Page 325

1. 120

2. 40,320

3. 990

4. 38,760

5. 9000

6. 2520

7. 39,916,800

8. 720

9. �9115�

10. �1801�

11. �17�

12. �61�

Page 326

13. �12241

�

14. �66

43

�

15. �21�

16. �170�

17. �32�

18. �52�

19. �69265

�

20. 0.29

Bonus: 56



334-340 A&E C13-0-02-834179 10/10/00 10:43 AM Page 336




3 Superior • Shows thorough understanding of the concepts permutation, combination, probability, and independent events.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Sketch is detailed and sensible.• Goes beyond requirements of some or all problems.

2 Satisfactory, • Shows understanding of the concepts permutation,with Minor combination, probability, and independent events.Flaws • Uses appropriate strategies to solve problems.

• Computations are mostly correct.• Written explanations are effective.• Sketch is detailed and sensible.• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts Satisfactory, permutation, combination, probability, and independent with Serious events.Flaws • May not use appropriate strategies to solve problems.

• Computations are mostly correct.• Written explanations are satisfactory.• Sketch is detailed and sensible.• Satisfies all requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts polar permutation, combination, probability, and independent events.

• May not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are not satisfactory.• Sketch is not detailed does not make sense.• Does not satisfy requirements of problems.

334-340 A&E C13-0-02-834179 10/10/00 10:43 AM Page 337



Page 327

1a. There are 3 � 5 � 3, or 45,possible arrangements.

1b. A poor arrangement is any thatis random, such as anarrangement that is notlogically ordered by size, color,or style.

1c.

This arrangement has apattern that allows thecustomer to locate a particularpair of socks easily.

2a. �77!� � 720 arrangements

2b. On a circular rack, eacharrangement has six othersjust like it, the result of rotatingthe arrangement. Thus, thereare only one-seventh as manyarrangements on a circularrack as on a straight rack.

3a. �47! 3!!

� � 35 combinations

3b. Order is not considered in acombination.

4a. The probability of Chad makingboth free throws if the shotsare independent events is 0.7 � 0.7 � 0.49, or 49%.

4b. If missing the first free throwmakes Chad lose confidence inhis ability to make the second,then the events are notindependent. Likewise, makingthe first shot may boost hisconfidence and increase hischances of making the secondshot. Fatigue and crowd noiseare two other factors thatmight affect his shots.

5a. Three of the eight points arechosen as vertices for eachtriangle. Order is notconsidered in choosing thevertices, so we will use theformula for the number ofcombinations of 8 objectstaken 3 at a time.

C(8, 3) � �5!

83!!

�

�

� 8 � 7

� 56

56 different triangles can beformed.

5b. Four of the points are chosenas vertices for eachquadrilateral.

C(8, 4) � �4!

84!!

�

�

� 7 � 5 � 2

� 70

Seventy different quadrilateralscan be formed.

8 � 7 � 6 � 5 � 4 � 3 � 2 � 1��4 � 3 � 2 � 1 � 4 � 3 � 2 � 1

8 � 7 � 6 � 5 � 4 � 3 � 2 � 1��5 � 4 � 3 � 2 � 1 � 3 � 2 � 1


334-340 A&E C13-0-02-834179 10/10/00 10:43 AM Page 338



1. 48

2. 5040

3. 3024

4. 350

5. 6720

6. 60

7. 3,628,800

8. �356�

9. �21�

10. �81�

Quiz APage 329

1. 480

2. 40,320

3. 1260

4. dependent

5. 5040

Quiz BPage 329

1. �41�

2. �51�

3. �111�

4. �51�

5. �133�

Quiz CPage 330

1. �112�

2. mutually inclusive: �173�

3. �21�

4. �170�

5. �141�

Quiz DPage 330

1. �352�

2. �1163�

3. �54152

�

4. �28516

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5. 0.12


334-340 A&E C13-0-02-834179 10/10/00 10:43 AM Page 339

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6. D

7. B

8. D

9. B

Page 332

10. B

11. C

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13. D

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15. C

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18. C

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Page 333

1. 3x � 2y � 6 � 0

2. t � 14

3. ��43�

4. x � t; y � �t � 4

5. (�2�2�, �2�2�)

6. (x � 2)2 � 8(y � 4)

7. �32

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8. 54

9. 161,700

10. �112�



334-340 A&E C13-0-02-834179 10/10/00 10:43 AM Page 340



NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at theright of each problem.

The playing times of 20 songs on a top-hits radio station arerecorded in the chart below. Use the chart for Exercises 1-6.

1. In a stem-and-leaf plot of this data where 33�2 represents 3:32, 1. ________which of the following stems has the fewest number of leaves?A. 25 B. 30 C. 45 D. 51

2. In a frequency distribution of this data, how many data values are in 2. ________the class 3:00–3:30?A. 11 B. 5 C. 6 D. 7

3. In a histogram of this data, which bar would have the greatest height? 3. ________A. 3:00–3:15 B. 3:15–3:30 C. 3:30–3:45 D. 3:45–4:00

4. What is the mean of the data? 4. ________A. 3:29 B. 3:48 C. 3:67 D. 3:40

5. What is the median of the data? 5. ________A. 3:55 B. 3:33 C. 3:20 D. 3:07

6. What is the mode of the data? 6. ________A. 3:07 B. 3:33 C. 3:40 D. None of these

7. Find the value of x so that the mean of �5x, �32� x, x � 9, �x� is �1. 7. ________

A. 2 B. �12� C. ��12� D. �2

8. Find the mean of the data stem leaf 8. ________represented by the stem-and-leaf 1 0 0 1 4 6 8 8plot at the right. 2 2 2 5 6 7 7A. 2.2 B. 2.5 3 0 0 1 5C. 2.6 D. 2.7 1|0 � 1.0

For Exercises 9 and 10, use the frequency distribution below.

9. Estimate the mean of the data. 9. ________A. 20.07 B. 19.57 C. 20.25 D. 18.64

10. Estimate the median of the data. 10. ________A. 19 B. 18.75 C. 20 D. 18.25

Chapter

14

Playing Time (minutes:seconds)

3:32 3:24 2:54 3:07 4:52 3:45 2:39 3:09 3:34 3:26

3:35 4:17 4:03 3:52 5:10 4:59 3:07 4:00 3:07 2:56

Amount Frequency

0–8 18

8–16 15

16–24 16

Amount Frequency

24–32 11

32–40 9

40–48 6



For Exercises 11–13, use the data in the table below.

11. Find the mean deviation of the temperatures. 11. ________A. 11.84 B. 12.36 C. 10.47 D. 12.15

12. Find the standard deviation of the temperatures. 12. ________A. 11.84 B. 12.36 C. 10.47 D. 12.15

13. What values are used to create a box-and-whisker plot for the data? 13. ________A. 50.7, 54.5, 69, 79, 84 B. 50.7, 57.2, 69.35, 80.15, 84C. 50.7, 54.5, 68.7, 79, 84 D. 50.7, 54.5, 69.35, 79, 84

For Exercises 14-16, a set of 750 values has a normal distribution with a mean of 12.5 and a standard deviation of 0.36.14. What percent of the data is between 12.25 and 12.75? 14. ________

A. 38.3% B. 51.6% C. 20.1% D. 45.1%

15. Find the interval about the mean within which 45% of the data lie. 15. ________A. 12.28–12.72 B. 12.32–12.68 C. 12.15–12.85 D. 12.49–12.51

16. Find the probability that a value selected at random from this 16. ________data is between 11.67 and 13.33.A. 98.4% B. 57.6% C. 97.9% D. 83.0%

In a random sample of 30 tires of the same type, it is found that the average life span of a tire is 36,200 miles with a standard deviation of 3800 miles.17. Find the standard error of the mean. 17. ________

A. 126.67 B. 693.78 C. 9.53 D. 587.24

18. Find the interval about the sample mean that has a 1% level of 18. ________confidence.A. 26,396–46,004 B. 35,506–36,894C. 33,425–38,975 D. 34,410–37,990

19. Find the interval about the sample mean such that the probability 19. ________is 0.75 that the mean number lies within the interval.A. 35,402–36,998 B. 31,830–40,570C. 35,436–36,964 D. 35,367–37,033

20. Find the probability that the mean of the population will be less 20. ________than 560 miles from the mean of the sample.A. 80.7% B. 57.6% C. 1.3% D. 8.1%

Bonus Find the probability that the true mean is between Bonus: ________35,000 and 36,000.

A. 67.5% B. 33.75% C. 57.35% D. 83.8%



14

Average Monthly Temperatures in New Braunfels, Texas (Fahrenheit degrees)

Jan. Feb. Mar. April May June July Aug. Sept. Oct. Nov. Dec.

50.7 54.5 61.7 68.7 75.0 81.3 84.0 84.0 79.0 70.0 59.9 52.7Source: WorldClimate




NAME _____________________________ DATE _______________ PERIOD ________


The playing times for 20 movies are recorded in the chart below.Use the chart for Exercises 1–6.

1. In a stem-and-leaf plot of this data where 10 2 represents 102, 1. ________which stem has the greatest number of leaves?A. 9 B. 10 C. 11 D. 12

2. In a frequency distribution of this data, how many data values 2. ________are in the class 130–140?A. 0 B. 1 C. 2 D. 3

3. In a histogram of this data, which bar would have the greatest height? 3. ________A. 105–110 B. 110–115 C. 115–120 D. 120–125

4. What is the mean of the data? 4. ________A. 112.85 B. 112 C. 110.91 D. 124

5. What is the median of the data? 5. ________A. 115 B. 113 C. 112 D. 111

6. What is the mode of the data? 6. ________A. 102 B. 108 C. 124 D. All of the above

7. Find the value of x so that the mean of {x, x � 2, 2x � 1, 1.4x} is 6. 7. ________

A. 3�89� B. 5 C. 1�23� D. None of these

8. Find the mean of the data stem leaf 8. ________represented by the stem-and-leaf 7 6 7 9plot at the right. 8 0 0 1 3 3 3 5 5 6 6 8A. 85.5 B. 82.45 9 0 1 1 1 2 5 6 9C. 86.23 D. 86.38 7|6 � 76


9. Estimate the mean of the data. 9. ________A. 24.75 B. 24.25 C. 25.25 D. 23.25

10. Estimate the median of the data. 10. ________A. 22.07 B. 23.15 C. 28.15 D. 27.93

Chapter

14

Playing Time of Movies (minutes)

102 128 123 132 104 95 109 121 108 124

92 140 117 102 124 115 113 89 111 108

Amount Frequency

15–20 21

20–25 46

25–30 13

Amount Frequency

30–35 10

35–40 7

40–45 3



For Exercises 11–13, use the data in the table below.

11. Find the mean deviation of the data. 11. ________A. 0.5 B. 0.4 C. 0.3 D. 0.2

12. Find the standard deviation of the data. 12. ________A. 0.36 B. 0.43 C. 0.41 D. 0.38

13. What values are used to create a box-and-whisker plot for the data? 13. ________A. 0.6, 0.9, 1.3, 1.3, 1.8 B. 0.6, 0.9, 1.15, 1.4, 1.8C. 0.6, 0.9, 1.2, 1.5, 1.8 D. 0.6, 0.9, 1.15, 1.5, 1.8

For Exercises 14–16, a set of 300 values has a normal distribtuionwith a mean of 50 and a standard deviation of 5.14. What percent of the data is between 45 and 55? 14. ________

A. 38.3% B. 50% C. 68.3% D. 95.5%

15. Find the interval about the mean within which 90% of the data lie. 15. ________A. 40–60 B. 42.5–57.5 C. 38.75–61.25 D. 41.75–58.25

16. Find the probability that a value selected at random from this 16. ________data is between 49.5 and 50.5.A. 8% B. 9.2% C. 9.8% D. 7.66%

In a random sample of 700 refreshment-dispensing machines, it isfound that an average of 8.1 ounces is dispensed with a standarddeviation of 0.75 ounce.17. Find the standard error of the mean. 17. ________

A. 0.0107 B. 0.0011 C. 0.2833 D. 0.0283

18. Find the interval about the sample mean that has a 1% level 18. ________of confidence.A. 8.045–8.155 B. 8.027–8.173C. 6.165–10.035 D. 8.053–8.147

19. Find the interval about the sample mean such that the probability 19. ________is 0.90 that the mean number lies within the interval.A. 8.045–8.155 B. 8.027–8.173C. 6.165–10.035 D. 8.053–8.147

20. Find the probability that the mean of the population will be less 20. ________than 0.085 ounce from the mean of the sample.A. 30% B. 99.7% C. 49.9% D. 25%

Bonus Find the probability that the true mean is between Bonus: ________8.157 and 8.185.

A. 2.1% B. 4.45% C. 0.4% D. 0.2%



14

Numbers of Cars Rented Each Month in the U.S. (thousands)


0.7 0.6 0.9 1.2 1.3 1.5 1.8 1.7 1.1 0.9 0.9 1.2




NAME _____________________________ DATE _______________ PERIOD ________


The number of cars sold by 20 salespeople in one week are recorded in the chart below. Use the chart for Exercises 1-6.

1. In a frequency distribution of this data, how many data values are 1. ________in the class 4-6?A. 3 B. 7 C. 4 D. 5

2. In a histogram of this data, which bar would have the greatest height? 2. ________A. 4–6 B. 6–8 C. 8–10 D. 10–12

3. In a histogram of this data, which bar would have the least height? 3. ________A. 2–4 B. 4–6 C. 6–8 D. 8–10

4. What is the mean of the data? 4. ________A. 6.5 B. 6.8 C. 7 D. 7.2

5. What is the median of the data? 5. ________A. 6.8 B. 6 C. 7 D. 7.5

6. What is the mode of the data? 6. ________A. 5 B. 8 C. 6 D. 7

7. Find the value of x so that the mean of {x, 2x, 3x, 4x} is 5. 7. ________A. 2 B. �2 C. �12� D. 2.5

8. Find the mean of the data stem leaf 8. ________represented by the stem-and-leaf 18 5 9plot at the right. 19 1 3 3 5 6 8 8A. 194 B. 193.8 20 0C. 193 D. 194.2 18|5 � 185


9. Estimate the mean of the data. 9. ________A. $29.34 B. $32.21 C. $34.34 D. $59.62

10. Estimate the median of the data. 10. ________A. $25 B. $29.62 C. $45 D. $39.62

Chapter

14

Number of Cars

10 7 6 9 7 3 5 6 8 4

8 2 7 5 7 9 11 5 7 10

Amount Frequency

$0–$10 15

$10–$20 21

$20–$30 26

Amount Frequency

$30–$40 28

$40–$50 20

$50–$60 12



For Exercises 11-13, use the data in the table below.

11. Find the mean deviation of the data. 11. ________A. 3.20 B. 3.89 C. 3.13 D. 2.79

12. Find the standard deviation of the data. 12. ________A. 3.20 B. 3.89 C. 3.13 D. 2.79

13. What values are used to create a box-and-whisker plot for the data? 13. ________A. 0, 0.6, 3, 5.9, 9 B. 0, 0.9, 3.2, 6.3, 9C. 0, 0.75, 3.2, 6.25, 9 D. 0, 0.6, 3.2, 5.9, 9

For Exercises 14-16, a set of data has a normal distribution with a mean of 120 and a standard deviation of 10.14. What percent of the data is between 110 and 130? 14. ________

A. 38.3% B. 50% C. 68.3% D. 95.5%

15. Find the interval about the mean within which 90% of the data lie. 15. ________A. 94.2–145.8 B. 103.5–136.5C. 113.68–126.32 D. 100.4–139.6

16. Find the probability that a value selected at random from this data 16. ________is between 100 and 140.A. 99.9% B. 90% C. 99% D. 95.5%

In a random sample of 1000 exams, the average score was 500 points with a standard deviation of 80 points.17. Find the standard error of the mean. 17. ________

A. 15.81 B. 2.53 C. 3.58 D. 8

18. Find the interval about the sample mean that has a 1% level of 18. ________confidence.A. 499–501 B. 479–520 C. 495–505 D. 493–507

19. Find the interval about the sample mean such that the probability 19. ________is 0.90 that the mean number lies within the interval.A. 499–501 B. 495–505 C. 496–504 D. 368–632

20. Find the probability that the mean score of the population will be 20. ________less than five points from the mean score of the sample.A. 95.5% B. 38.3% C. 98.8% D. 62.5%

Bonus Find the probability that the true mean is between Bonus: ________495 and 500.

A. 95% B. 47.75% C. 95.5% D. 68.3%



14

Average Rainfall in China Flat, California (inches)


8.8 6.6 5.9 3.0 2.5 0.9 0.1 0.0 0.6 3.4 5.9 9.0Source: WorldClimate




NAME _____________________________ DATE _______________ PERIOD ________

Fifty randomly–selected people going to a science fiction moviewere asked their age. The results are recorded in the chart below.Use the chart for Exercises 1-6.

1. List the stems that would be used in a stem-and-leaf plot of 1. __________________the data.

2. Find the range of the data. 2. __________________

3. Make a histogram of the data. 3.

4. Find the mean of the data. 4. __________________

5. Find the median of the data. 5. __________________

6. Find the mode of the data. 6. __________________

7. Find the value of x so that the mean of 7. __________________{2x, �13�x, �52�x � 3, x � 2} is 8.

8. Find the mean of the data below. 8. __________________stem leaf

51 2 5 6 7 952 0 2 3 5 8 8 9 953 1 1 4

51|2 � 5120


9. Estimate the mean of the data. 9. __________________

10. Estimate the median of the data. 10. __________________

Chapter

14

Ages of Science-Fiction Moviegoers (years)

17 42 21 78 16 21 31 29 29 16

49 19 81 16 69 69 18 31 22 14

21 75 42 78 18 41 22 16 18 80

42 42 42 16 16 21 19 18 44 18

22 14 49 17 16 18 18 18 17 23

Class Frequency

600–615 2

615–630 8

630–645 16

645–660 3

660–675 5

Class Frequency

675–690 2

690–705 38

705–720 24

720–735 7

735–750 3



The table below gives the save percentages for some goalies at a certain point in a recent NHL season. Use the table for Exercises 11–13.

11. Find the mean deviation of the percentages. 11. __________________

12. Find the standard deviation of the percentages. 12. __________________

13. Make a box-and-whisker plot of the percentages. 13.

For Exercises 14 –16, a set of 1000 values has a normaldistribution with a mean of 400 and a standard deviation of 30.

14. What percent of the data is between 385 and 415? 14. __________________

15. Find the interval about the mean within which 60% of 15. __________________the data lie.

16. Find the probability that a value selected at random from 16. __________________this data is greater than 350.

A random sample of 225 homes showed an average of 5.2 clocks in each home. The standard deviation was 0.8.

17. Find the standard error of the mean. 17. __________________

18. Find the interval about the sample mean that has a 18. __________________1% level of confidence.

19. Find the interval about the sample mean such that the 19. __________________probability is 0.75 that the mean number lies within the interval.

20. Find the probability that the mean of the population will 20. __________________be less than 0.10 from the mean of the sample.

Bonus Find the probability that the true mean is Bonus: __________________between 5 and 5.1.



14

Save Percentages

0.940 0.937 0.930 0.929 0.926 0.925 0.925 0.923 0.923 0.923

0.921 0.920 0.919 0.918 0.916 0.914 0.911 0.911 0.910 0.910

0.909 0.907 0.907 0.904 0.903 0.903 0.902 0.900 0.898 0.895




NAME _____________________________ DATE _______________ PERIOD ________

Fifty students recorded the number of hours that the televisionwas on in their homes during one week. The results are given inthe chart below. Use the chart for Exercises 1–6.

1. List the leaves for stem 1 in a stem-and-leaf plot of the data. 1. __________________

2. List the class marks for the intervals 0–20, 20–40, 40–60, 2. __________________60–80, and 80–100 in a frequency distribution of the data.





7. Find the value of x so that the mean of 7. __________________{3x � 3, x � 5, �3x, 2x � 7} is 9.


0 4 7 8 9 91 0 1 2 3 3 4 5 6 6 8 92 1 2 5 8

0|4 � 4




Chapter

14

Weekly Television Hours (to the nearest hour)

54 28 9 15 3 54 35 32 0 34

72 57 62 33 58 23 57 53 24 27

36 63 34 58 53 13 12 75 66 57

18 53 53 46 77 26 32 42 43 88

44 71 22 57 45 73 44 11 45 34

Class Frequency

325–375 8

375–425 10

425–475 30

475–525 20

525–575 12

Class Frequency

575�625 10

625�675 6

675�725 2

725�775 1

775�825 1



The table below gives the percent pay raise for 13 employees. Usethe table for Exercises 11–13.

11. Find the mean deviation of the percentages. 11. __________________

12. Find the standard deviation of the percentages. 12. __________________

13. Make a box-and-whisker plot of the percentages. 13.

For Exercises 14-16, a set of data has a normal distribution with a mean of 120 and a standard deviation of 10.14. What percent of the data is between 100 and 140? 14. __________________


16. Find the probability that a value selected at random 16. __________________from this data is between 105 and 135.

In a random sample of 256 people, it was found that each personate fast food an average of 2.6 times per week with a standarddeviation of 0.4.17. Find the standard error of the mean. 17. __________________

18. Find the interval about the sample mean that has a 1% 18. __________________level of confidence.


20. Find the probability that the mean of the population 20. __________________will be less than 0.1 from the mean of the sample.

Bonus Find the probability that the true mean is Bonus: __________________between 2.55 and 2.575.


14 Chapter 14 Test, Form 2B (continued)

Percent Pay Raise

3.2% 4.4% 4.1% 3.8% 1.5% 2.4% 3.3% 1.7% 9.2% 4.5% 4.2% 5.1% 4.6%




NAME _____________________________ DATE _______________ PERIOD ________

The speeds of 50 cars in a 70 mile-per-hour zone are recorded in the chart below. Use the chart for Exercises 1–6.

1. List the stems that would be used in a stem-and-leaf plot 1. __________________of the data.

2. List the class marks for the intervals 50–60, 60–70, 70–80, 2. __________________and 80–90 in a frequency distribution of the data.





7. Find the value of x so that the 7. __________________mean of {x, 3x, 2x � 1, 2x � 5} is 15.


6 4 5 5 5 6 7 8 8 8 8 97 0 0 0 1 1 1 2 2 2 2 3 78 0 2

6|4 � 6.4




Chapter

14

Speed (miles per hour)

66 69 70 67 74 88 71 65 68 73

69 72 67 69 68 68 68 58 69 67

72 65 72 59 63 73 72 65 66 67

58 63 73 66 77 66 62 52 63 81

64 71 72 67 65 73 64 71 65 64

Class Frequency

50-60 7

60-70 5

70-80 4

80-90 1

90-100 3

Class Frequency

0-10 3

10-20 6

20-30 8

30-40 7

40-50 6


The table below gives the number of traffic tickets issued per day over a 20-day period by a police officer. Use the table for Exercises 11–13.

11. Find the mean deviation of the data. 11. __________________

12. Find the standard deviation of the data. 12. __________________

13. Make a box-and-whisker plot of the data. 13.

For Exercises 14–16, a set of data has a normal distribution with amean of 8 and a standard deviation of 1.4.



16. Find the probability that a value selected at random from 16. __________________this data is between 7.3 and 8.7.

In a random sample of 100 band students, it was found that eachstudent practiced an average of 10.5 hours per week with astandard deviation of 1.4 hours.




20. Find the probability that the mean of the population will 20. __________________be less than 15 minutes from the mean of the sample.

Bonus Find the probability that the true mean is Bonus: __________________between 10.2 and 10.6 hours.




14

Number of Traffic Tickets Issued

7 12 10 8 7 12 15 10 10 7

14 6 10 12 9 8 8 2 9 12




NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear,concise solution to each problem. Be sure to include allrelevant drawings and justify your answers. You may showyour solution in more than one way or investigate beyondthe requirements of the problem.

1. The table and graph below show the distribution of grades for an English test.

a. Are the grades on the test normally distributed? Why or why not?

b. Which measure of central tendency (mean, median, or mode) will be the greatest? Why?

c. Which measure of central tendency will be the least? Why?

2. The combined test scores for all of the advanced mathematics classes in a school arenormally distributed. The mean score is 85, and the standard deviation is 10. Thereare 200 students in the classes.

a. Those who had scores above 100 were given a grade of A. How many studentsreceived an A? Explain your reasoning.

b. What are the mode and median for the set of scores? How do you know?

c. Those who had scores between 80 and 90 were given a grade of C. How manystudents received a C? Explain your reasoning.

d. If the teacher changes the range for the grade of C to scores from 75 to 85, willthere be an increase or decrease in the number of C grades? Explain yourreasoning.

3. Consider the statement “Given two sets of data, the mean of the combination of thetwo sets equals the mean of the means.” Provide an example that disproves thestatement. In what situation is the statement true?

Chapter

14

English ScoresClass Class FrequencyLimits Mark

62.5–67.5 65 1

67.5–72.5 70 2

72.5–77.5 75 3

77.5–82.5 80 5

82.5–87.5 85 7

87.5–92.5 90 9

92.5–97.5 95 7

97.5–102.5 100 3



The amount of money 13 families spent on food in one year is recorded in the table below. Use the table for Exercises 1–7.

1. Organize the data into a frequency distribution. 1. __________________

2. List the stems that would be used in a stem-and-leaf 2. __________________plot of the data.




6. What is the mean deviation for the data? 6. __________________

7. What is the standard deviation for the data? 7. __________________

The weights of baseball players on the Chicago White Sox 1999 roster are recorded in the table below. Use the table for Exercises 8–10.

8. Estimate the mean weight of the players. 8. __________________

9. Estimate the median weight of the players. 9. __________________

10. Estimate the standard deviation of the weights. 10. __________________



14

Food Expenses (thousands of dollars)

4.5 5.2 6.5 2.9 2.7 4.6 3.9 6.0 4.7 4.2 5.2 4.6 7.2

Weight (in pounds) Frequency

160–180 6

180–200 13

200–220 8

220–240 3

240–260 2Source: Yahoo! Sports


Use the data in the table below for each exercise.

1. Find the range and determine an appropriate class interval. 1. __________________

2. Find the class marks. 2. __________________

3. Draw a histogram of the data. 3.

4. Draw a stem-and-leaf plot of the data. 4.

5. Find the mean, median, and mode of the data. 5. __________________

Use the data in the table below for Exercises 1-14.

1. Find the mean deviation of the heights. 1. __________________

2. Find the interquartile range for the data. 2. __________________

3. Find the semi-interquartile range for the data. 3. __________________

4. What is the standard deviation of the heights? 4. __________________



NAME _____________________________ DATE _______________ PERIOD ________


NAME _____________________________ DATE _______________ PERIOD ________


Chapter

14

Chapter

14

28 16 37 31 21 26 35 29 12

24 28 34 32 26 19 25 35 31

28 26 19 22 30 24 19 29 33

Heights of Women Basketball Players on the Houston Comets 1999 Roster (inches)

71 70 67 74 77 75 66 72 74 76 73 79Source: WNBA

stem leaf


A set of 900 values has a normal distribution with a mean of 150 and a standard deviation of 8.


2. How many values are within one standard deviation of 2. __________________the mean?

3. How many values fall in the interval between one and two 3. __________________standard deviations of the mean?

4. Find the probability that a value selected at random from 4. __________________the data will be greater than 162.

5. Find the interval about the mean that includes 90% 5. __________________of the data.

In a sample of 100 adults, the average time each adult kepta car was 6.2 years. The standard deviation was 1.1 years.




4. Find the probability that the mean of the population will 4. __________________be less than 0.1 year from the mean of the sample.

5. Find the probability that the mean of the population will 5. __________________be less than 0.5 year from the mean of the sample.


NAME _____________________________ DATE _______________ PERIOD ________

Chapter 14, Quiz C (Lesson 14-4)

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

14

Chapter

14




NAME _____________________________ DATE _______________ PERIOD ________


Multiple Choice1. Use the graph below to determine how

many more items were sold in Januarythan in May.A 5250B 3500C 1750D 4250E 1000

2. In the graph below, which of thefollowing could be the percent changefrom the number of widgets built in1985 to the number built in 1990?

A 150% decreaseB 67% decreaseC 60% decreaseD 67% increaseE 300% increase

3. If 5 more than x is 2 less than y, whatis y in terms of x?A x � 3 B y � 7C y � 3 D x � 7E x � 7

4. If 5x � 3y � 23 and x and y arepositive integers, then y can equalwhich of the following?A 3 B 4C 5 D 6E 7

5. Of the 50 students in a class, exactly 30 are women. What percent of thestudents are men?A 20% B 30%C 40% D 50%E 60%

6. Five is what percent of 2?A 2.5% B 25%C 40% D 250%E 400%

7.

Based on the table above, what is thecost of a 30-minute call from City B toCity C?A $4.90B $5.05C $7.30D $7.55E It cannot be determined from the

information given.

8.

Based on the table above, what is themean age of the students?A 14.6 B 15.0C 15.8 D 16.5E 16.8

9. Point A is on line m. If two points, Band C, are each placed to the right ofpoint A so that AB � 2AC, what will be the value of �B

ACC�?

A 1 B 2

C �21� D �4

1�


Chapter

14

Long Distance Rates

From FirstEach

City A MinuteAdditional

Minute

to City B $0.55 $0.15

to City C $0.25 $0.05

Attendance

Age 14 15 16 17Number of 5 6 10 10Students



10. In � ABC below, which of the followingcould be a value of y?A 2 B 4C 6 D 8E 12

11. Mosin’s monthly expenses are $1375per month and are distributed asshown in the pie chart below. Howmuch does he spend for “other”expenses each month?A $206.25B $137.50C $68.75D $13.75E $10

12. If Mosin’s monthly expenses are $1225per month and are distributed asshown in the pie chart above, howmuch more does he spend on rent thanon food?A $68.75 B $72.50C $7.25 D $61.25E $612.50

13. Which of the following ratios is equalto the ratio of �14� to 4?A �13� to 3 B 4 to �4

1�

C �18� to 2 D �12� to 2E None of these

14. A car traveling 60 miles per hour for30 minutes covers the same distance as a car traveling 20 miles per hour forhow many hours?A �3

2� B 1

C 1�21� D 3

E �31�

15. Use the chart below to determine thenumber of boxes of cabbage sold by afarmer in 1998.A 1000 B 4000C 8000D 10,000E 40,000

16. Use the chart above to determine howmany more boxes of cabbage were soldin 1999 than in 2000.A 250 B 500C 5000 D 1000E 2500





Column A Column B17.

18. Set X � {1, 2, 2, 3, 3, 3, 4, 4, 4, 4}

Mode of Set X Median of Set X

19–20. In a class of 250 students, 4 arerunning for the position of classpresident. Every student in the class voted exactly once. The voteswere distributed as shown below.Candidate Number of Votes

D.J. 75Belinda 20Darius 45

Lou Ann x

19. Grid-In How many votes did Lou Annget?

20. Grid-In What was the median number of votes per candidate?



14

Mean of 101,202, and 303

Mean of 101, 202,303, and 10




NAME _____________________________ DATE _______________ PERIOD ________

1. Write the ordered triple that represents CD� for C(5, 0, �1) 1. __________________and D(3, �2, 6).

2. Write an equation of the cosine function with amplitude 6, 2. __________________phase shift 0, vertical shift 0, and period 12.

3. Write the polynomial equation of least degree with roots 3. __________________7i and �7i.

4. Find the area of �A BC to the nearest tenth if c � 11.4, 4. __________________B � 31.6� and C � 120.3�.

5. Find �3

��8�i�. 5. __________________

6. Write in general form the equation of a parabola whose 6. __________________focus is at (5, �1) and whose directrix is x � 3.

7. Write log5 �1125� � �3 in exponential form. 7. __________________

8. Find the first three iterates of the function 8. __________________ƒ(z) � z2 � z � i if z0 � i.

9. The serial number for a product is formed from the 9. __________________digits 1, 2, and 3 and the letters A and B. No letters or numbers are repeated. What is the probability that the serial number ends in 2A given that it ends in a letter?

10. A set of data has a normal distribution with a mean 10. __________________of 16 and a standard deviation of 0.3. What percent of the data is in the interval 15.2�16?

Chapter

14


Page 341

1. D

2. C

3. A

4. D

5. B

6. A

7. D

8. A

9. B

10. D

Page 342

11. C

12. A

13. B

14. B

15. A

16. C

17. B

18. D

19. D

20. B

Bonus: B

Page 343

1. B

2. B

3. D

4. A

5. C

6. D

7. B

8. C

9. A

10. B

Page 344

11. C

12. A

13. B

14. C

15. D

16. A

17. D

18. B

19. D

20. B

Bonus: A



360-366 A&E C14-0-02-834179 10/10/00 10:48 AM Page 360


Page 345

1. C

2. B

3. A

4. B

5. C

6. D

7. A

8. B

9. A

10. B

Page 346

11. D

12. C

13. C

14. C

15. B

16. D

17. B

18. D

19. C

20. A

Bonus: B

Page 347

1. 1, 2, 3, 4, 6, 7, 8

2. 67

3.

4. 31.78

5. 21

6. 18

7. 5�3151�

8. 5236.875

9. 68510. 697

Page 348

11. 0.0097

12. 0.0114

13.

14. 38.3%

15. 376–424

16. 95.55%

17. 0.0533

18. 5.06–5.34

19. 5.14–5.26

20. 94.3%

Bonus: 2.8%

Form 1C Form 2A


Sample answer:

360-366 A&E C14-0-02-834179 10/10/00 10:48 AM Page 361

Page 349

1. 1, 2, 3, 5, 82. 10, 30, 50, 70, 90

3. Sample answer:

4. 42.425. 446. 53 and 577. 15

8. 14.5

9. 497.5

10. 480

Page 350

11. 1.25%

12. 1.85%

13.

14. 95.5%

15. 107-133

16. 86.6%

17. 0.025

18. 2.5355–2.6645

19. 2.5675–2.6325

20. 99.99%

Bonus: 13.6%

Page 351

1. 5, 6, 7, 8

2. 55, 65, 75, 85

3.

4. 67.94

5. 67.5

6. 65, 67, 72

7. 8

8. 7.024

9. 44

10. 41.67

Page 352

11. 2.3

12. 2.9

13.

14. 51.6%

15. 5.69–10.31

16. 38.3%

17. 0.14

18. 10.23 –10.77

19. 10.27–10.73

20. 92.9%

Bonus: 74%



Sample answer:

360-366 A&E C14-0-02-834179 10/10/00 10:48 AM Page 362




3 Superior • Shows thorough understanding of the concepts histogram, normal distribution, mean, median, mode, central tendency, and standard deviation.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Goes beyond requirements of problems.

2 Satisfactory, • Shows understanding of the concepts histogram, with Minor normal distribution, mean, median, mode, central, Flaws tendency, and standard deviation.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Satisfies all requirements of problems.

1 Nearly • Shows understanding of most of the concepts Satisfactory, histogram, normal distribution, mean, median, mode,with Serious central tendency, and standard deviation.Flaws • May not use appropriate strategies to solve problems.

• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the conceptshistogram, normal distribution, mean, median, mode,central tendency, and standard deviation.

• May not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are not satisfactory.• Does not satisfy requirements of problems.

360-366 A&E C14-0-02-834179 10/10/00 10:48 AM Page 363



Page 353

1a. No; it is not a symmetric bell curve.

1b. Mode; it is not affected by extremescores.

1c. Mean; it is most affected by extremescores.

2a. �10100-85� � 1.5 standard deviations

Of the scores, 0.866 lie within 1.5standard deviation of the mean.

Thus, �1.00-20.866�, or 0.067, lie 1.5

standard deviations above themean. 0.067 � 200 � 13.4, so about13 students received an A.

2b. They are both 85 because for anormally distributed set of data the mean, median, and mode are equal.

2c. Of the grades, 0.383 are within 0.5 standard deviation of the mean.0.383 � 200 � 76.6, so about 77students received a C.

2d. A decrease; there are not as manyscores in the 75 to 80 range as thereare in the 85 to 90 range.

3. Counterexample: Let the data sets be{1} and {100, 100, 100, 100}. The meanof the first data set is 1, and the meanof the second data set is 100, so the mean of the means is �1 �

2100� � 50.5.

The combination of the two data setsis {1, 100, 100, 100, 100}, which has amean of 80.2. Thus, the statement isnot true in this case. The statement istrue, however, if the two data setshave the same size, n. Let Z� be themean of the combination of two datasets of the same size, and let Xi and Yibe the values in the two sets. Then

Z� �

� �12

� � ��

21� � � �

� �12

� ( X� � Y� ) � �X� �2

Y��

So, when the data sets are the samesize, the mean of the combinationdata set equals the mean of the means.

Y1 � Y2 � . . . � Yn��nX1 � X2 � . . . � Xn��n

X1 � X2 � . . . � Xn � Y1 � Y2 � . . . � Yn��n

X1 � X2 � . . . � Xn � Y1 � Y2 � . . .�Yn��

2n


360-366 A&E C14-0-02-834179 10/10/00 10:48 AM Page 364

stem leaf1 2 6 9 9 92 1 2 4 4 5 6 6 6 8 8 8 9 93 0 1 1 2 3 4 5 5 7



1.

2. 2, 3, 4, 5, 6, 7

3.

4. 4.78

5. 4.6 and 5.2

6. 0.95

7. 1.23

8. 198.75 lb

9. 195.4 lb

10. 21.8

Quiz APage 355

1. 25; Sample answer: 5

2.

3.

4.

5. 26.6; 28; 19, 26, and 28

Quiz BPage 355

1. about 3.03

2. 5 in.

3. 2.5 in.

4. about 3.72

5.

Quiz CPage 356

1. 95.5%

2. 615

3. 245

4. 6.7%

5. 136.8-163.2

Quiz DPage 356

1. 0.11

2. 5.984-6.416

3. 5.916-6.484

4. 63.2%

5. 100%


Sample answer:

Costs Frequency2–4 34–6 76–8 3

Sample answer:

1|2 � 12

Sample answer: 12.5,17.5, 22.5, 27.5, 32.5, 37.5

360-366 A&E C14-0-02-834179 10/10/00 10:48 AM Page 365

Page 357

1. C

2. B

3. E

4. D

5. C

6. D

7. E

8. C

9. A

Page 358

10. D

11. B

12. D

13. C

14. C

15. D

16. E

17. A

18. A

19. 110

20. 60

Page 359

1. ��2, �2, 7�

2. y � � 6 cos ��6t�

3. x2 � 49 � 0

4. 18.6 units2

5. 2i

6. y2 � 2y � 4x � 17 � 0

7. 5�3 � �1125�

8. �1, �i, �1 � 2i

9. �81�

10. 49.6%



360-366 A&E C14-0-02-834179 10/10/00 10:48 AM Page 360



NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right of each problem.For Exercises 1 and 2, use the graph of y � ƒ(x) to find each value.

1. limx→1

ƒ(x) 1. ________

A. 1 B. 2C. 3 D. 0

2. limx→3

ƒ(x) 2. ________

A. 1 B. 2C. 3 D. 0

Evaluate each limit.

3. limx→3

�xx2

��

39� 3. ________

A. 0 B. 1 C. 6 D. 9

4. limx→4

��xx��

�

42

� 4. ________

A. 0 B. 1 C. �14� D. �21�

5. limx→1

�xx3

��

11� 5. ________

A. 2 B. 3 C. 1 D. 0

6. Find the derivative of ƒ(x) � x3 � x. 6. ________A. 3x2 � x B. 3x � 1 C. 3x2 D. 3x2 � 1

7. Find the derivative of ƒ(x) � (4x � 5)2. 7. ________A. 4x � 5 B. 8x � 10 C. 32x D. 32x � 40

8. Find the slope of the tangent line to the graph of the equation 8. ________y � (2x2 � 3)(x � 1) at x � 2.A. 11 B. 8 C. 19 D. 16

9. Find the antiderivative of ƒ(x) � x3(x � 2)2. 9. ________A. �16�x6 � �45�x5 � x4 � C B. �16�x6 � �45�x5 � �14�x4 � C

C. x6 � x5 � x4 � C D. x5 � 4x4 � 4x3 � C

10. The distance s(t) between an object and its starting point is given by 10. ________the antiderivative of the velocity function v(t). Find the distance between the object and its starting point after 15 seconds if v(t) � 0.2t2 � 2t � 10 meters per second.A. 85 m B. 825 m C. 450 m D. 600 m

11. Find the area of the shaded region 11. ________in the graph.A. �12

5� B. 9

C. �134� D. 7

Chapter

15



Find the area between each curve and the x-axis for the given interval.

12. y � x4 � 5 from x � 0 to x � 5 12. ________A. 620 B. 630 C. 640 D. 650

13. y � 3x2 � 5x � 1 from x � 1 to x � 4 13. ________A. 97.5 B. 60 C. 82.5 D. 95

14. y � 4x � x3 from x � 0 to x � 2 14. ________A. 0 B. 4 C. 8 D. 16

Evaluate each indefinite integral.

15. �(x3 � 2x) dx 15. ________

A. x4 �2x2 � C B. x4 � x2 � C

C. �14�x4 � x2 � C D. �14�x4 � x2 � C

16. �(x2 � 1)2 dx 16. ________

A. x5 � 2x3 � x � C B. �15�x5 � �23�x3 � x � C

C. �15�x5 � �23�x3 � x � C D. �15�x5 � �23�x3 � 1 � C

17. Find the area of the shaded region 17. ________in the given graph.A. 16 B. �63

4�

C. �332� D. 8

Evaluate each definite integral.

18. �3

0(3x2 � x3) dx 18. ________

A. �247� B. 9 C. �24

43� D. 6

19. �4

2(x3 � 2x2) dx 19. ________

A. 22 B. �638� C. 32 D. ��23

92�

20. �2

1x2(5x2 � 4) dx 20. ________

A. �635� B. �63

4� C. 63 D. 21

Bonus Find the area of the region bounded by the Bonus: ________curve y � �10

x03

0� and the x-axis from x � 0 to x � 1.

A. 0.00015 B. 0.0002 C. 0.00025 D. 0.00033



15




NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right ofeach problem.For Exercises 1 and 2, use the graph of y � ƒ(x) to find each value.

1. limx→�1

ƒ(x) 1. ________A. �1 B. 1C. 2 D. 0

2. lim ƒ(x) 2. ________

A. �1 B. 1C. 2 D. 0

Evaluate each limit.3. lim (2x2 � 5x) 3. ________

A. 27 B. 41 C. 21 D. 33

4. lim �xx2 �

�146� 4. ________

A. 1 B. 0 C. 8 D. 4

5. lim ��xx��

11� 5. ________

A. 0 B. 1 C. �12� D. �41�

6. Find the derivative of ƒ(x) � 3x2 � x. 6. ________A. 6x � 1 B. 3x � 1 C. 6x D. 2x � 1

7. Find the derivative of ƒ(x) � x2(x � 2). 7. ________A. 3x2 � 4 B. 3x2 � 4x C. 2x D. 3x � 2

8. Find the slope of the tangent line to the graph of the equation 8. ________y � (2x � 3)(x � 2) at x � 3.A. 11 B. 12 C. 13 D. 15

9. Find the antiderivative of ƒ(x) � 5x4 � 3x2 � 3. 9. ________A. �15�x5 � �13�x3 � x � C B. �15�x5 � �13�x3 � 3x � CC. x5 � x3 � 3x2 � C D. x5 � x3 � 3x � C

10. The distance s(t) between an object and its starting point is given 10. ________by the antiderivative of the velocity function v(t). Find the distance between the object and its starting point after 12 seconds if v(t) � 0.5t2 � 2 meters per second.A. 74 m B. 12 m C. 312 m D. 144 m

11. Find the area of the shaded region 11. ________in the graph.A. �83� B. �13

4�

C. �136� D. �23

8�

Chapter

15

x→3

x→4

x→1

x→1



Find the area between each curve and the x-axis for the giveninterval.12. y � x2 � 4 from x � 0 to x � 3 12. ________

A. 13 B. 15 C. 21 D. 39

13. y � 4x2 � 2x from x � 1 to x � 4 13. ________A. 99 B. 101 C. 103 D. 105

14. y � x3 � x from x � 1 to x � 3 14. ________A. 8 B. 16 C. 24 D. 32


15. �(x3 � 4x) dx 15. ________

A. x4 � 4x2 � C B. �14�x4 � x2 � C

C. �14�x4 � 2x2 � C D. �14�x4 � 2x2 � C

16. �x(x � 2) dx 16. ________

A. x3 � x2 � C B. �13�x3 � x2 � C

C. �13�x3 � x2 � C D. �13�x3 � �12�x2 � C

17. Find the area of the shaded region 17. ________in the graph.A. 5 B. �13

6�

C. 6 D. �332�


18. �2

0(x2 � 4x) dx 18. ________

A. �332� B. �12

38� C. �13

6� D. �634�

19. �6

2(x2 � 2x) dx 19. ________

A. �1132� B. 32 C. 64 D. �23

24�

20. �3

1x(4x2 � 3) dx 20. ________

A. 202.5 B. 135 C. 95 D. 68

Bonus The acceleration a(t) of a moving object is given by Bonus: ________the derivative of the velocity function v(t). Find the acceleration of the object at time t � 3 seconds if v(t) � 16t2 � 10t meters per second.

A. 53 m/s2 B. 96 m/s2 C. 106 m/s2 D. 116 m/s2



15




NAME _____________________________ DATE _______________ PERIOD ________


For Exercises 1 and 2, use the graph of y � f(x) to find each value.

1. limx→0

ƒ(x) 1. ________

A. 0 B. 1C. 2 D. �1

2. limx→2

ƒ(x) 2. ________

A. 0 B. 1C. 2 D. �1


3. limx→4

(3x � 2) 3. ________

A. 12 B. 14 C. 16 D. 20

4. limx→�2

(� x2 � 5x) 4. ________

A. 6 B. 14 C. �14 D. �9

5. limx→1

�xx2

��

11� 5. ________

A. 4 B. 1 C. 0 D. 2

6. Find the derivative of ƒ(x) � 3x2 � 2. 6. ________A. 6x � 2 B. 6x � 1 C. 6x D. 3x

7. Find the derivative of ƒ(x) � 8x. 7. ________A. 4x2 B. 8x C. 0 D. 8

8. Find the slope of the tangent line to the graph of the equation 8. ________y � 2x3 � 3 at x � 2.A. 16 B. 24 C. 19 D. 27

9. Find the antiderivative of ƒ(x) � 3x � 5. 9. ________A. �32� x2 � 5x � C B. �32� x2 � 5x � C

C. 3x2 � 5x � C D. 3

10. The distance s(t) between an object and its starting point is 10. ________given by the antiderivative of the velocity function v(t). Find the distance between the object and its starting point after 10 seconds if v(t) � 3t � 12 meters per second.A. 162 m B. 270 m C. 312 m D. 420 m

11. Find the area of the shaded region 11. ________in the graph.A. �13� B. 6

C. 21 D. �38�

Chapter

15




12. y � x2 � 1 from x � 0 to x � 3 12. ________A. 12 B. 6 C. 30 D. 18

13. y � 2x � 4 from x � 2 to x � 4 13. ________A. 4 B. 12 C. 16 D. 20

14. y � 8 � x3 from x � 0 to x � 2 14. ________A. 8 B. 10 C. 12 D. 16


15. �x6 dx 15. ________

A. 6x5 � C B. 6x � C C. �17� x7 � C D. x7 � C

16. �(4x2 � 3x) dx 16. ________

A. 4x3 � 3x2 � C B. �43� x3 � �32� x2 � C

C. �43�x3 � 3x2 � C D. 8x � 3 � C

17. Find the area of the shaded region 17. ________in the graph.A. �83� B. 2

C. �43� D. �25�


18. �1

0(3 � x2) dx. 18. ________

A. 2 B. �83� C. 3 D. �134�

19. �4

12x3 dx. 19. ________

A. �2525� B. 128 C. �25

45� D. 64

20. �5

2(4x3 � 5x) dx 20. ________

A. 140.625 B. 278.25 C. 340 D. 556.5

Bonus Find the area of the region bounded by the curve y � �10x04

0� Bonus: ________and the x-axis from x � 0 to x � 2.

A. 0.0008 B. 0.0016 C. 0.0032 D. 0.0064



15




NAME _____________________________ DATE _______________ PERIOD ________


1. limx→�1

ƒ(x) and ƒ(�1) 1. __________________

2. limx→2

ƒ(x) and ƒ(2) 2. __________________


3. limx→3

�x2 �

x2�x

3� 15� 3. __________________

4. limx→16

��x �

x� �

164

� 4. __________________

5. limx→0

�sinx3x� 5. __________________

6. Find the derivative of ƒ(x) � 2x3 � 5. 6. __________________

7. Find the derivative of ƒ(x) � (3x � 4)2. 7. __________________

8. Find the slope of the tangent line to the graph of the 8. __________________equation y � x(5x2 � 3x � 4) at x � �2.

9. Find the antiderivative of ƒ(x) � x2(x � 1)2. 9. __________________

10. The distance s(t) between an object and its starting point 10. __________________is given by the antiderivative of the velocity function v(t).Find the distance between the object and its starting point after 10 seconds if v(t) � 0.3t2 � 4t � 8 meters per second.

11. Find the area of the shaded 11. __________________region in the graph.

Chapter

15




12. y � 4x3 � 2 from x � 0 to x � 4 12. __________________

13. y � 5x2 � 3x � 2 from x � 2 to x � 5 13. __________________

14. y � x3 � 9x from x � �3 to x � 0 14. __________________


15. �(x5 � 3x2) dx 15. __________________

16. �(3x2 � 1)2 dx 16. __________________

17. Find the area of the shaded region 17. __________________in the graph.


18. �4

1(x4 � x) dx 18. __________________

19. ��2

�4(x2 � 3x) dx 19. __________________

20. �3

0x3 (2x2 � 3x � 8) dx 20. __________________

Bonus Find the area of the region bounded by the Bonus: __________________curve y � �10

x,0

4

00� and the x-axis from x � 0 to x � 1.



15




NAME _____________________________ DATE _______________ PERIOD ________

For Exercises 1 and 2, use the graph of y � f(x) to find each value.1. lim ƒ(x) and ƒ(�1) 1. __________________

2. lim ƒ(x) and ƒ(1) 2. __________________

Evaluate each limit.3. lim (3x2 � 2x) 3. __________________

4. lim �xx2 �

�255� 4. __________________

5. lim ��xx��

93� 5. __________________

6. Find the derivative of ƒ(x) � 4x2 � 2x. 6. __________________

7. Find the derivative of ƒ(x) � (2x � 5)(3x � 4). 7. __________________

8. Find the slope of the tangent line to the graph of the 8. __________________equation y � x2(x � 3) at x � �1.

9. Find the antiderivative of ƒ(x) � 2x3 � 2x � 4. 9. __________________

10. The distance s(t) between an object and its starting 10. __________________point is given by the antiderivative of the velocity function v(t). Find the distance between the object and its starting point after 15 seconds if v(t) � 0.4t2 � 6 meters per second.


Chapter

15

x→�1

x→1

x→4

x→5

x→9



NAME _____________________________ DATE _______________ PERIOD ________ChapterChapter

15Find the area between each curve and the x-axis for the giveninterval.12. y � 6x2 � 5 from x � 0 to x � 5 12. __________________

13. y � 2x2 � x � 1 from x � 3 to x � 6 13. __________________

14. y � 16x � x3 from x � 0 to x � 4 14. __________________


15. �(x4 � 6x) dx 15. __________________

16. �x2(3x � 1) dx 16. __________________



18. �4

0x(x � 3) dx 18. __________________

19. �5

2(x3 � 2x) dx 19. __________________

20. ��1

�3x2(2x � 7) dx 20. __________________

Bonus The acceleration a(t) of a moving object is Bonus: __________________given by the derivative of the velocity function v(t). Find the acceleration of the object at time t � 5 seconds if v(t) � 12t2 � 8t meters per second.

Chapter 15 Test, Form 2B (continued)Chapter

15




NAME _____________________________ DATE _______________ PERIOD ________


1. limx→0

ƒ(x) and ƒ(0) 1. __________________

2. limx→2

ƒ(x) and ƒ(2) 2. __________________


3. limx→3

(2x � 5) 3. __________________

4. limx→�1

(x2 � 5x) 4. __________________

5. limx→2

�xx2

��

24� 5. ______________

6. Find the derivative of ƒ(x) � 5x2 � 1 6. __________________

7. Find the derivative of ƒ(x) � 9x � 7. 7. __________________

8. Find the slope of the tangent line to the graph of the 8. __________________equation y � 2x2 � 4x � 3 at x � 4.

9. Find the antiderivative of ƒ(x) � 5x � 3. 9. __________________

10. The distance s(t) between an object and its starting point 10. __________________is given by the antiderivative of the velocity function v(t).Find the distance between the object and its starting point after 10 seconds if v(t) � 0.3t2 � 8 meters per second.


Chapter

15





12. y � 2x3 � 3 from x � 0 to x � 3 12. __________________

13. y � 4x � 6 from x � 1 to x � 5 13. __________________

14. y � x2 from x � �2 to x � 2 14. __________________


15. �3x5 dx 15. __________________

16. �(2x2 � 5x) dx 16. __________________



18. �4

23x5 dx 18. __________________

19. �5

2(x2 � 3) dx 19. __________________

20. �3

0(3x2 � 7x) dx 20. __________________

Bonus Find the area between the curve y � 4x � x3 Bonus: __________________and the x-axis from x � 1 to x � 2.

Chapter 15 Test, Form 2C (continued)Chapter

15





Instructions: Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include allrelevant drawings and justify your answers. You may show yoursolution in more than one way or investigate beyond therequirements of the problem.

1. Let ƒ(x) � �xx2

2��

56xx�.

a. Make a table of values for ƒ(x). Then draw the graph of thefunction.

b. Using the graph in part a, explain how to find the limit of ƒ(x)as x approaches 3. Then explain how to find the limit as xapproaches 0.

c. Find the limit of the numerator, x2 � 5x, as x approaches 3.Justify your answer.

d. Find the limit of the denominator, x2 � 6x, as x approaches 3.Justify your answer.

e. Find the limit as x approaches 3 of �xx2

2��

56xx� algebraically.

f. Find the limit as x approaches 0 of �xx2

2��

56xx� algebraically.

2. Find two different functions, ƒ and g, such that limx→0

ƒ(x) � limx→0

g(x).

3. The speed of an object is given by s � 4t � t2, where t is in seconds.a. Graph the function.

b. Use two different methods to find the area or approximate areabounded by the curve and the t-axis from t � 0 to t � 4. Whichmethod is more accurate? Why?

c. The area of a rectangle is given by a � l � w. In this case, l � s(speed) and w � t (time). What do you think the area under thecurve represents? Why?

Chapter

15





1. limx→�1

ƒ(x) and ƒ(�1) 1. __________________

2. limx→2

ƒ(x) and ƒ(2) 2. __________________


3. limx→3

(4x3 � 5x) 3. __________________

4. limx→4

�x2 �

x7�x

4� 12� 4. _______

5. limx→25

��x �

x� �

255

� 5. ____________


7. Find the derivative of ƒ(x) � (3x � 1)(5x � 6). 7. __________________

8. Find the slope of the tangent line to the graph of the 8. __________________equation y � x2(x � 2) at x � �2.


10. The distance s(t) between an object and its starting point 10. __________________is given by the antiderivative of the velocity function v(t).Find the distance between the object and its starting point after 12 seconds if v(t) � 5t � 3 meters per second.

Chapter 15 Mid-Chapter Test (Lessons 15-1 and 15-2)Chapter

15



NAME _____________________________ DATE _______________ PERIOD ________

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

15

Chapter

15

Chapter 15, Quiz A (Lesson 15-1)


1. limx→�2

ƒ(x) and ƒ(�2) 1. __________________

2. limx→1

ƒ(x) and ƒ(1) 2. __________________


3. limx→3

(�5x2 � 2x � 4) 3. __________________

4. limx→�4

�xx2 �

�146� 4. ____________

5. limx→36

��x �

x� �

366

� 5. ____________


2. Find the derivative of ƒ(x) � 4x2(x � 5). 2. __________________

3. Find the slope of the tangent line to the graph of the 3. __________________equation y � (x � 5)(2x � 3) at x � 1.


5. The distance s(t) between an object and its starting point 5. __________________is given by the antiderivative of the velocity function v(t).Find the distance between the object and its starting point after 12 seconds if v(t) � 0.2t2 � 15 meters per second.



NAME _____________________________ DATE _______________ PERIOD ________

Chapter 15, Quiz C (Lesson 15-3)

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

15

Chapter

15

1. Use a limit to find the area of the 1. __________________shaded region in the graph.

Use limits to find the area between each curve and the x-axis for the given interval.

2. y � 4x3 from x � 0 to x � 3 2. __________________

3. y � 5x2 from x � 2 to x � 5 3. __________________

Use limits to evaluate each integral. 4. __________________

4. �5

0(x2 � 4) dx. 5. �2

0x(x � 5) dx. 5. __________________


1. � (2x3 � 4x) dx 1. __________________

2. � x2(5x � 2) dx 2. __________________


4. Find the area between the curve y � x � x3 4. __________________and the x-axis from x � 0 to x � 1.

5. Evaluate �4

2(x5 � 3x) dx. 5. __________________





15 After working each problem, record thecorrect answer on the answer sheetprovided or use your own paper.

Multiple Choice1. How many lines are determined by

10 points, with no 3 points beingcollinear?A 20 B 45C 81 D 100E 900

2. If a � b is defined to be �aa�b

b�, then (3 � 3) � 3 �A 3 B 1C �32� D �4

9�

E �29�

3. Which of the following statements istrue in the figure below?A m�1 � m�BB m�A � m�BC �A

ADB� � �AA

EC�

D �AB

DC� � �A

DEB�

E m�A � m�1

4. If x, y, and z are in the ratio 3:1:2 inthe figure below, find z.A 15B 30C 45D 60E 90

5. How many even numbers with one,two, or three digits can be formed fromthe digits 6, 7, and 8 if no digit isrepeated?A 6 B 8C 9 D 10E None of these

6. A sports arena has 4 west gates and 6 east gates. How many ways can youenter the arena through a west gateand later leave through an east gate?A 10B 20C 18D 24E None of these

7. The area of each circle inside thesquare is 16� units2. What is the areaof the shaded region?A (256 � 64�) units2

B (256 � 16�) units2

C (64� � 256) units2

D (64 � 64�) units2

E (64� � 64) units2

8. What is the maximum possible area ofa rectangle with a width of x units anda length of 14 � x units?A 7 units2

B 14 units2

C 49 units2

D 196 units2

E None of these

9. Which of the following is a root of2x2 � 15x � 18 � 0?A �6B �1C �2

1�

D 3E 6

10. If � �23�, what is the value of

x � 4?A �1B �5C 4�5

1�

D 5E 9

4��6 � �x �

55�

Note: Figure is NOTdrawn to scale.




11. If {x} is defined to be x � 2, find thevalue of {{3} � {5}}.A 8B 10C 12D 14E None of these

12. If <<a, b, c>> is defined to be , then<<3, 9, 3>> � A <<1, 3, 1>>B <<3, 1, 3>>C <<12, 9, 12>>D <<9, 3, 9>>E <<6, 18, 6>>

13. Find the arithmetic mean of3.4x, �7.3x, and 2.1(x � 3).A 6.3 � 0.6xB 1.575 � 0.45xC �2.1 � 0.6xD �2.025xE �2.7x

14. The arithmetic mean of two numbers is 2. If twice the smaller number isadded to the larger number, the sumis �4. Find both numbers.A �8 and 12B �6 and 10C �12 and 16D �4 and 8E None of these

15. If //x// is defined to be x(x � 1)(x � 2), then �//

//82////� �

A 4B 10C 20D 30E 36

16. (p, q) � (m, n) is defined to be (pm � qn, pn � qm). If (a, b) � (x, y) �(a, b) for all (a, b), then (x, y) � A (0, 0)B (1, 0)C (0, 1)D (1, 1)E None of these

17–18. Quantitative ComparisonA if the quantity in Column A is greaterB if the quantity in Column B is greaterC if the two quantities are equalD if the relationship cannot be


Column A Column B

17. For all positive integers x,x � x if x is even x � x � 1 if x is odd

18. For all real numbers x and y,x * y � xy � x � y


19. Grid-In What is the greatest numberof triangles that can be formed usingpoints on the circle above as vertices ofthe triangle?

20. Grid-In What is the greatest numberof line segments that can be formedusing points on the circle above asendpoints?

�ab��c

61 � 65 62 � 64

5 * x (�5) * x

Chapter

15 SAT and ACT Practice (continued)




Chapter 15 Cumulative Review (Chapters 1–15)Chapter

151. Solve the following system of equations algebraically. 1. __________________

x � y � 63x � 5y � 26

2. If y varies directly as the cube of x and y � 16 when 2. __________________x � 2, find y when x � �3.

3. In � ABC, a � 9, b � 6, and c � 10. Find m�B to 3. __________________the nearest tenth.

4. Find the value of Sin�1 �tan ��4��. 4. __________________

5. Given csc � � 4 and 0 � � � ��2�, find the exact 5. __________________value of cos 2�.

6. Evaluate 8�32

�

. 6. _______________________________________________

7. Find the 17th term in the arithmetic 7. __________________sequence �2, 4, 10, . . . .

8. How many ways can the letters in the 8. __________________word temperate be arranged?

9. Evaluate lim ��3xx2

��

32x��. 9. __________________

10. Find the area of the shaded region in 10. __________________the given graph.

x→4


Page 367

1. A

2. B

3. C

4. C

5. B

6. D

7. D

8. C

9. A

10. D

11. A

Page 368

12. D

13. A

14. B

15. D

16. C

17. D

18. A

19. B

20. A

Bonus: C

Page 369

1. A

2. B

3. D

4. C

5. C

6. A

7. B

8. C

9. D

10. C

11. D

Page 370

12. C

13. A

14. C

15. D

16. B

17. B

18. A

19. A

20. D

Bonus: C



386-392 A&E C15-0-02-83417 10/10/00 10:51 AM Page 386


Page 371

1. B

2. C

3. B

4. A

5. D

6. C

7. D

8. B

9. A

10. B

11. D

Page 372

12. A

13. D

14. C

15. C

16. B

17. A

18. B

19. A

20. D

Bonus: D

Page 373

1. 2; undefined

2. 0; 2

3. 8

4. �81�

5. 3

6. 6x2

7. 18x � 24

8. 76

9.

10. 380 m

11. �136�

Page 374

12. 264

13. 169.5

14. �841�

15. �16

�x6 � x3 � C

16. �59�x5 � 2x3 � x � C

17. 4

18. �119071�

19. �1310�

20. 259.2

Bonus: 0.00002

Form 1C Form 2A


�15

�x5 � �12

�x4 � �13

�x3 � C

386-392 A&E C15-0-02-83417 10/10/00 10:51 AM Page 387

Page 375

1. �2; undefined

2. 2; 1

3. 40

4. 10

5. �61�

6. 8x � 2

7. 12x � 23

8. 9

9. �21�x4 � x2 � 4x � C

10. 540 m

11. �332�

Page 376

12. 275

13. 142.5

14. 64

15. �51�x5 � 3x2 � C

16. �43�x4 � �1

3�x3 � C

17. 10

18. �1336�

19. �5425�

20. �632�

Bonus: 128 m/s2

Page 377

1. 1; undefined

2. 2; 2

3. 1

4. 6

5. 4

6. 10x

7. 9

8. 20

9. �25�x2 � 3x � C

10. 180 m

11. 4

Page 378

12. 49.5

13. 72

14. �332�

15. �12

�x6 � C

16. �32�x3 � �5

2�x2 � C

17. �134�

18. 152

19. 1348

20. 58.5

Bonus: 2.25



386-392 A&E C15-0-02-83417 10/10/00 10:51 AM Page 388




3 Superior • Shows thorough understanding of the concepts limit, integration, and finding areas of regions.

• Uses appropriate strategies to solve problems and find limits.• Computations are correct.• Written explanations are exemplary.• Graphs are accurate and appropriate.• Goes beyond requirements of problems.

2 Satisfactory, • Shows understanding of the concepts limit, integration, with Minor and finding areas of regions.Flaws • Uses appropriate strategies to solve problems and find limits.


1 Nearly • Shows understanding of most of the concepts limit, Satisfactory, integration, and finding areas of regions.with Serious • May not use appropriate strategies to solve problems

and find limits. Flaws • Computations are mostly correct.

• Written explanations are satisfactory.• Graphs are mostly accurate and appropriate.• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the conceptshistogram, normal distribution, mean, median, mode,central tendency, and standard deviation.

• May not use appropriate strategies to solve problems and find limits.

• Computations are incorrect.• Written explanations are not satisfactory.• Graphs are not accurate and appropriate.• Does not satisfy requirements of problems.

386-392 A&E C15-0-02-83417 10/10/00 10:51 AM Page 389



Page 3791a.

1b. The graph indicates that ƒ(x) iscontinuous at x � 3. Therefore, limx→3

ƒ(x) � ƒ(3), or � �83

�. As x

approaches 0, the y-coordinatesapproach a value ofapproximately �1, so lim

x→0ƒ(x)

is about �1.

Draw a vertical line through thex-axis at x � 3. The intersection ofthis line and the curve is the limit asx→3, a value of approximately �3.For the limit as x→0, follow thesame procedure as describedabove, except at x � 0. The limit asx→0 is approximately �1.

1c. limx→3

(x2 � 5x) � limx→3

x2 � 5 � limx→3

x,

or 24

1d. limx→3

(x2 � 6x) � limx→3

x2 � 6 � limx→3

x,

or �9

1e. limx→3

�xx

2

2

��

65

xx� � , or ��

38�

3f. limx→0

�xx

2

2

�

�

65

xx� � lim

x→0�xx((xx

��

65))

�

� limx→0

�xx

��

65�

� �l

l

i

i

mx→

mx→

0

0

(

(

x

x

�

�

6

5

)

)�

� ��56� or ��

65�

2. Sample answer: ƒ(x) � x2 and g(x) � xlimx→0

ƒ(x) � 0 � limx→0

g(x)

3a.

3b. Method 1: Integrate

�4

0(4t � t2) dt � 2t2 � �1

3�t3 �4

0

� 32 � �634�

� �332� units2

Method 2: Estimate using rectangles1 � s(1) � 1 � s(2) � 1 � s(3) � 1 � s(4) �3 � 4 � 3 � 0 or 10 units2

Method 1 is more accurate becauseintegrating yields the exact areaunder the curve. Method 2 yields anapproximation.

3c. The area under the curve representsthe distance between the object andits starting point after 4 secondsbecause d � st.

limx→3

(x2 � 5x)��

limx→3

(x2 � 6x)


x ƒ(x)

�2 ��38

�

1 ��47

�

0 undefined

1 ��65

�

2 ��74

�

3 ��83

�

4 ��92

�

5 �10

6 undefined

7 12

386-392 A&E C15-0-02-83417 10/10/00 10:51 AM Page 390



1. 1; 3

2. 2; undefined

3. 93

4. 1

5. �110�

6. 12x � 5

7. 30x � 13

8. 4

9. �45

�x5 � �32

�x2 � 5x � C

10. 396 m

Quiz APage 381

1. 2; 1

2. 3; undefined

3. �47

4. �8

5. �112�

Quiz BPage 381

1. 14x � 3

2. 12x2 � 40x

3. �3

4. �23

�x3 � 3x2 � x � C

5. 295.2 m

Quiz CPage 382

1. �143�

2. 84

3. 202.5

4. �1385�

5. �338�

Quiz DPage 382

1. �21�x4 � 2x2 � C

2. �45�x4 � �2

3�x3 � C

3. 12

4. �14

�

5. 654


386-392 A&E C15-0-02-83417 10/10/00 10:51 AM Page 391

Page 383

1. B

2. B

3. A

4. D

5. D

6. D

7. A

8. C

9. A

10. E

Page 384

11. D

12. C

13. C

14. A

15. D

16. B

17. A

18. D

19. 20

20. 15

Page 385

1. (2, 4)

2. �54

3. 36.3�

4. ��2

�

5. �87�

6. 4

7. 94

8. 30, 240

9. 40

10. 4



386-392 A&E C15-0-02-83417 10/10/00 10:51 AM Page 392

Given that x is an integer, state therelation representing each equation as aset of ordered pairs. Then, state whetherthe relation is a function. Write yes or no.

1. y � 3x �1 and �1 � x � 32. y � �2 � x� and �2 � x � 3

Find [ƒ ° g](x) and [g ° ƒ](x) for each ƒ(x)and g(x).

3. ƒ(x) � 3x � 1g(x) � x � 3

4. ƒ(x) � 4x2

g(x) � �x3

5. ƒ(x) � x2 � 25g(x) � 2x � 4

Find the zero of each function.6. ƒ(x) � 4x � 107. ƒ(x) � 15x8. ƒ(x) � 0.75x � 3

Write the slope-intercept form of theequation of the line through the pointswith the given coordinates.

9. (4, �4), (6, �10)10. (1, 2), (5, 4)

Write the standard form of the equationof each line described below.11. parallel to y � 3x � 1

passes through (�1, 4)12. perpendicular to 2x � 3y � 6

x-intercept: 2

The table below shows the number of T-shirts sold per day during the firstweek of a senior-class fund-raiser.

13. Use the ordered pairs (2, 21) and (4, 43) to write the equation of a best-fit line.

14. Predict the number of shirts sold onthe eighth day of the fund-raiser.Explain whether you think theprediction is reliable.

Graph each function.15. ƒ(x) � �x � 2�16. ƒ(x) � �2x� � 1

17. ƒ(x) � �Graph each inequality.18. x � 3y � 1219. y � ��23�x � 5

Solve each system of equations.20. y � �4x

x � y � 521. x � y � 12

2x � y � �422. 7x � z � 13

y � 3z � 1811x � y � 27

Use matrices A, B, C, and D to find eachsum, difference, or product.

A � � � B � � �

23. A � B 24. 2A � B25. CD 26. AB � CD

Use matrices A, B, and E above to findthe following.27. Evaluate the determinant of matrix A.28. Evaluate the determinant of matrix E.29. Find the inverse of matrix B.

Solve each system of inequalities bygraphing. Name the coordinates of the vertices of each polygonal convexset. Then, find the maximum andminimum values for the function ƒ(x, y) � 2y � 2x � 3.30. x � 0 31. x � 2

y � 0 y � �32y � x � 1 y � 5 � x

y � 2x � 8

67

�45

2�3

63

x � 2 if x � �12x if �1 � x � 1�x if x � 2


Unit 1 Review, Chapters 1–4

NAME _____________________________ DATE _______________ PERIOD ________

Day Number of Shirts Sold1 122 213 324 435 56

D � � � E � � �5�2�1

1�4

2

�3�1

3

0�3�1

26

�5

UNIT1

C � � ��11

2�8

3�5

393-398 U01-0-02-834179 10/10/00 10:52 AM Page 393 (Black plate)

Determine whether each function is aneven function, an odd function, orneither.

32. y � �3x3

33. y � 2x4 � 534. y � x3 � 3x2 � 6x � 8

Use the graph of ƒ(x) � x3 to sketch agraph for each function. Then, describethe transformations that have takenplace in the related graphs.

35. y � �ƒ(x) 36. y � ƒ(x � 2)

Graph each inequality.

37. y � �x � 3�38. y �

3x � 4

Find the inverse of each function.Sketch the function and its inverse. Isthe inverse a function? Write yes or no.

39. y � �12� x � 540. y � (x � 1)3 � 2

Determine whether each graph hasinfinite discontinuity, jump discontinuity,point discontinuity, or is continuous.Then, graph each function.

41. y � �xx2

��

11�

42. y � �Find the critical points for the functionsgraphed in Exercises 43 and 44. Then,determine whether each point is amaximum, a minimum, or a point ofinflection.43.

44.

Determine any horizontal, vertical, orslant asymptotes or point discontinuityin the graph of each function. Then,graph each function.

45. y � �(2x � 1

x)(x � 2)�

46. y � �xx2

��

39�


47. x2 � 8x � 16 � 048. 4x2 � 4x � 10 � 0

49. �x �4

2� � �x �4

3� � 6

50. 2x � �2 �1

x� �12�

51. 9 � �x � 1 � 1

52. �x � 8 � �x � 35 � �3

Use the Remainder Theorem to find theremainder for each division.

53. (x2 � x � 4) (x � 6)54. (2x3 � 3x � 1) (x � 2)

Find the number of possible positivereal zeros and the number of possiblenegative real zeros. Determine all of therational zeros.

55. ƒ(x) � 3x2 � x � 256. ƒ(x) � x4 � x3 � 2x2 � 3x � 1

Approximate the real zeros of eachfunction to the nearest tenth.

57. ƒ(x) � x2 � 2x � 558. ƒ(x) � x3 � 4x2 � x � 2

x � 1 if x � 0x � 3 if x � 0


Unit 1 Review, Chapters 1–4 (continued)

NAME _____________________________ DATE _______________ PERIOD ________UNIT1


UNIT1


Unit 1 Test, Chapters 1–4

NAME _____________________________ DATE _______________ PERIOD ________

1. Find the maximum and minimum values of ƒ(x, y) � 3x � y 1. __________________for the polygonal convex set determined by x � 1, y � 0, and x � 0.5y � 2.

2. Write the polynomial equation of least degree that has the 2. __________________roots �3i, 3i, i, and �i.

3. Divide 4x3 � 3x2 � 2x � 75 by x � 3 by using synthetic 3. __________________division.

4. Solve the system of equations by graphing. 4.3x � 5y � �8x � 2y � 1

5. Complete the graph so that it is the graph of an 5.even function.

6. Solve the system of equations. 6. __________________x � y � z � 2x � 2y � 2z � 33x � 2y � 4z � 5

7. Decompose the expression �4n

127�

n2�

3n23

� 6� into partial fractions. 7. __________________

8. Is the graph of �x92� � �2

y5

2� � 1 symmetric with respect to the 8. __________________

x-axis, the y-axis, neither axis, or both axes?

9. Without graphing, describe the end behavior of the graph 9. __________________of ƒ(x) � �5x2 � 3x � 1.

10. How many solutions does a consistent and dependent 10. __________________system of linear equations have?

11. Solve 3x2 � 7x � 6 � 0. 11. __________________

12. Solve 3y2 � 4y � 2 � 0. 12. __________________



Unit 1 Test, Chapters 1–4 (continued)

NAME _____________________________ DATE _______________ PERIOD ________

13. If ƒ(x) � �4x2 and g(x) � �2x�, find [ g ° ƒ](x). 13. __________________

14. Are ƒ(x) � �21�x � 5 and g(x) � 2x � 5 inverses of each other? 14. __________________

15. Find the inverse of y � �1x02�. Then, state whether the 15. __________________

inverse is a function.

16. Determine if the expression 4m5 � 6m8 � m � 3 is a 16. __________________polynomial in one variable. If so, state the degree.

17. Describe how the graph of y � �x � 2� is related to 17. __________________its parent graph.

18. Write the slope-intercept form of the equation of the line 18. __________________that passes through the point (�5, 4) and has a slope of �1.

19. Determine whether the figure with vertices at 19. __________________(1, 2), (3, 1), (4, 3), and (2, 4) is a parallelogram.

20. A plane flies with a ground speed of 160 miles per hour 20. __________________if there is no wind. It travels 350 miles with a head wind in the same time it takes to go 450 miles with a tail wind.Find the speed of the wind.

21. Solve the system of equations algebraically. 21. __________________�13� x � �13� y � 12x � 2y � 9

22. Find the value of by using expansion by minors. 22. __________________

23. Solve the system of equations by using augmented matrices. 23. __________________y � 3x � 10x � 12 � 4y

24. Approximate the greatest real zero of the function 24. __________________g(x) � x3 � 3x � 1 to the nearest tenth.

25. Graph ƒ(x) � �x �1

1�. 25.

�242

30

�1

514

UNIT1




NAME _____________________________ DATE _______________ PERIOD ________

26. Write the slope-intercept form 26. __________________of the equation 6x � y � 9 � 0.Then, graph the equation.

27. Write the standard form of the equation of the line 27. __________________that passes through (�3, 7) and is perpendicular to the line with equation y � 3x � 5.

28. Use the Remainder Theorem to find the remainder of 28. __________________(x3 � 5x2 � 7x � 3) (x � 2). State whether the binomial is a factor of the polynomial.

29. Solve x � �2x � 1 � 7. 29. __________________

30. Determine the value of w so that the line whose equation 30. __________________is 5x � 2y � �w passes through the point at (�1, 3).

31. Determine the slant asymptote for ƒ(x) � �x2 � 5

xx � 3�. 31. __________________

32. Find the value of . 32. __________________

33. State the domain and range of {(�5, 2), (4, 3), (�2, 0), (�5, 1)}. 33. __________________Then, state whether the relation is a function.

34. Determine whether the function ƒ(x) � �x � 1� is odd, 34. __________________even, or neither.

35. Find the least integral upper bound of the zeros of the 35. __________________function ƒ(x) � x3 � x2 � 1.

36. Solve �2 � 3x� � 4. 36. __________________

37. If A � � � and B � � �, f ind AB. 37. __________________

38. Name all the values of x that are not in the domain of 38. __________________ƒ(x) � �2x

��

x5

2�.

39. Given that x is an integer between �2 and 2, state the 39. __________________relation represented by the equation y � 2 � �x � by listing a set of ordered pairs. Then, state whether the relation is a function. Write yes or no.

�25

01

34

21

5�2

37

UNIT1



40. Determine whether the system of 40. __________________inequalities graphed at the right is infeasible, has alternate optimal solutions, or is unbounded for the function ƒ(x, y) � 2x � y.

41. Solve 1 � ( y � 3)(2y � 2). 41. __________________

42. Determine whether the function y � ��x32� has infinite 42. __________________

discontinuity, jump discontinuity, or point discontinuity,or is continuous.

43. Find the slope of the line passing through the points 43. __________________at (a, a � 3) and (4a, a � 5).

44. Together, two printers can print 7500 lines if the first 44. __________________printer prints for 2 minutes and the second prints for 1 minute. If the first printer prints for 1 minute and the second printer prints for 2 minutes, they can print 9000 lines together. Find the number of lines per minute that each printer prints.

45. A box for shipping roofing nails must have a volume of 45. __________________84 cubic feet. If the box must be 3 feet wide and its height must be 3 feet less than its length, what should the dimensions of the box be?

46. Solve the system of equations. 46. __________________�3x � 2y � 3z � �12x � 5y � 3z � �64x � 3y � 3z � 22

47. Solve 4x2 � 12x � 7 � 0 by completing the square. 47. __________________

48. Find the critical point of the function y � �2(x � 1)2 � 3. 48. __________________Then, determine whether the point represents a maximum,a minimum, or a point of inflection.

49. Solve � � � � � � � �. 49. __________________

50. Write the standard form of the equation of the line 50. __________________that passes through (5, �2) and is parallel to the line with equation 3x � 2y � 4 � 0.

�51

xy

�31

1�1





Unit 1 Review

1. {(�1, �2), (0, 1), (1, 4), (2, 7), (3, 10)}; yes

2. {(�2, 4), (�1, 3), (0, 2), (1, 1), (2, 0), (3, 1)}; yes

3. 3x � 10; 3x � 4

4. 4x6; �64x6

5. 4x2 �16x � 9; 2x2 � 54

6. �25� 7. 0 8. �4

9. y � �3x � 8

10. y � �12

�x � �23�

11. 3x � y � 7 = 0

12. 3x � 2y � 6 = 0

13. y � 11x � 114. 87; No, because as the sale

continues, fewer studentswill be left to buy T-shirts.The number of shirts sold will have to decrease eventually.

15.

16.

17.

18.

19.

20. (1, �4) 21. ��83

�, �238��

22. (3, �6, 8) 23. � �

24. � � 25. � �

26. � � 27. �24

28. 19 29.

30. (0, 0), (1, 0), �0, �12

��; �2, �5

31. (2, �3), (5.5, �3), (3, 2), (2, 3); �1, �20

32. odd 33. even 34. neither

35. reflected over x-axis

36. translated 2 units right

37.

38.

39. y � 2(x � 5); yes

40. y � �3

x� �� 2� � 1; yes

4520

9�90

�523

23�63

�2�13

161

84

28

Unit 1 Answer Key

��578� �

239��

558� �

229��

399-400 U01-0-02-834179 10/10/00 10:53 AM Page 399

41. point discontinuity

42. jump discontinuity

43. max.: (�1, 1); min.: (1, 1)

44. pt. of inflection: (1, 0)

45. x � ��12

�, x � �2, y � 0

46. slant asymptote: y � x � 3point discontinuity: x � �3

47. 4 48. �1��2

1�1�� 49. �225�

50. 0 � x � 2 or x � �49�

51. no real solution

52. �8 � x � 1

53. 34 54. 11

55. 1; 1; �1, �32�

56. 3 or 1; 1; none

57. �1.4, 3.4

58. �1, �3.6, 0.6

Unit 1 Test

1. max.: 6; min.: 3

2. x4 � 10x2 � 9 � 0

3. 4x2 � 9x � 25

4. (�1, 1)

5.

6. (5, 1, 2)

7. �n �

56

� � �4n

3� 1�

8. both axes

9. as x → ∞, ƒ(x) → �∞; as x → �∞, ƒ(x) → �∞

10. infinitely many

11. ��23

�, 3

12. ��2 �3

�1�0�� y � ��2 �3

�1�0��

13. ��21x2� 14. no

15. no; y ��1�0�x� 16. yes; 8

17. translated 2 units to the right

18. y � �x � 1 19. yes

20. 20 mph 21. no solution

22. 64 23. (4, 2) 24. 1.5

25.

26. y � �6x � 9

27. x � 3y � 18 � 0

28. 5; no 29. 12 30. 11

31. y � x � 5 32. �41

33. D � {�5, �2, 4}; R � {0, 1, 2, 3}; no

34. neither 35. 1

36. ��23

� � x � 2

37. � � 38. �5

39. {(�1, 1), (0, 2), (1, 1)}; yes

40. infeasible 41. ��2 �23�2��

42. infinite discontinuity

43. ��38a� 44. 2000; 3500

45. 3 ft 7 ft 4 ft

46. (4, �1, 3) 47. �12

�, ��27�

48. max.: (1, �3) 49. (1, 2)

50. 3x � 2y � 11 � 0

1118

34


Unit 1 Answer Key (continued)

399-400 U01-0-02-834179 10/10/00 10:53 AM Page 400

Find the value of the given trigonometricfunction for angle � in standard positionif a point with the given coordinates lieson its terminal side.

1. cos �; (2, 3) 2. tan �; (10, 2)3. sin �; (�4, 1) 4. sec �; (1, 0)

Solve each problem. Round to thenearest tenth.

5. If A � 25° and a � 12.1, find b.6. If a � 3 and B � 59° 2’, find c.7. If c � 24 and B � 63°, find a.

Evaluate each expression.

8. cos �Arccos �14��9. cot �Cos�1 �23��

10. cos (Sin�1 0) � sin (Tan�1 0)

Determine the number of possiblesolutions for each triangle. If a solutionexists, solve the triangle. Round to thenearest tenth.11. A � 46°, a � 86, c � 20012. a � 19; b � 20, A � 65°13. A � 73°; B � 65°, b � 38

Find the area of each triangle. Round tothe nearest tenth.14. a � 5, b � 9, c � 615. a � 22, A � 63°, B � 17°

Change each radian measure to degreemeasure.

16. ��2� 17. �34��

18. �72�� 19. ��1

72��

Solve.20. Given a central angle of 60°, find the

length of its intercepted arc in acircle of radius 6 inches. Round to thenearest tenth.

Find each value by referring to the graphof the sine or the cosine function.

21. sin � 22. cos �2��

23. sin �72�� 24. cos (�6�)

State the amplitude and period for eachfunction.25. y � 2 cos 3x26. y � �5 tan 5x27. y � 4 cot ��2

x� � �2��


28. y � �12� cos 2x

29. y � 3 tan �2x � �2��

30. y � x � 2 sin 3x

Write the equation for the inverse ofeach relation. Then graph the relationand its inverse.31. y � arccos x 32. y � cot x

Use the given information to determineeach trigonometric value.

33. sec � � �43�, 0° � � � 90°; cos �

34. cos � � �13�, 0° � � � 90°; sin �

35. sin � � �13�, 0° � � � 90°; cot �

Verify that each equation is an identity.36. tan x � tan x cot2 x � sec x csc x37. sin (180° � �) � tan � cos �

Use sum or difference identities to findthe exact value of each trigonometricfunction.38. sin 105° 39. cos 135°40. tan 15° 41. sin (�210°)


Unit 2 Review, Chapters 5-8




If x is an angle in the first quadrant andsin x � �2

5�, find each value.

42. cos 2x 43. sin �2x�

44. tan �2x� 45. sin 2x

Solve each equation for 0° � x � 180°.46. sin2 x � sin x � 047. cos 2x � 4 cos x � 348. 5 cos x � 1 � 3 cos 2x

Write each equation in normal form.Then find the length of the normal andthe angle that it makes with the positive x-axis.49. 2x � 3y � 2 � 050. 5x � �2y � 851. y � 3x � 7

Find the distance between the pointwith the given coordinates and the linewith the given equation. 52. (2, 5); 2x � 2y � 3 � 053. (�2, 2); �x � 4y � �654. (1, �3); 4x � y � 1 � 0

Use vectors a� and b� for Exercises 55-56.

55. Use a ruler and a protractor to determine the magnitude (in centimeters) and direction of theresultant a� � b�.

56. Find the magnitude of the verticaland horizontal components of a�.

Find an ordered pair to represent a� ineach equation if b � �1, �3� and c� � �2, �2�.57. a� � b� � c� 58. a� � b� � c�59. a� � 3b� � 2c� 60. a� � �3b� � c�

Find an ordered triple to represent u� ineach equation if v� � �3, 1, �1� and w� � ��5, 2, 3� . Then write u� as the sumof unit vectors.61. u� � 2v� � w� 62. u� � v� � 2w�63. u� � 3v� � 3w� 64. u� � 4v� � 2w�

Find each inner product or crossproduct.65. �4, �2� � ��2, 3�66. �3, �4, 1� � �4, �2, 2�67. �5, �2, 5� � ��1, 0, �3�

Write a vector equation of the line thatpasses through point P and is parallel to v�. Then write parametric equations ofthe line.68. P(0, 5), v� � ��1, 5�69. P(4, �3), v� � ��2, �2�

Unit 2 Review, Chapters 5-8




UNIT2 Unit 2 Test, Chapters 5-8

NAME _____________________________ DATE _______________ PERIOD ________

1. True or false: sin (�85°) � �sin 85°. 1. __________________

2. Find the area of �ABC if a � 12, b � 15, and c � 23. Round 2. __________________to the nearest square unit.

3. Write the equation 5x � y � 2 � 0 in normal form. 3. __________________

4. Graph the function y � 2 cos �� 3��. 4.

5. Given a central angle of 60°, find the length of its 5. __________________intercepted arc in a circle of radius 15 inches. Round to the nearest tenth.

6. A vector has a magnitude of 18.3 centimeters and a direction 6. __________________of 38°. Find the magnitude of its vertical and horizontal components to the nearest tenth.

7. Write parametric equations of y � 5x � 2. 7. __________________

8. Find the value of Sin�1 �sin �56��. 8. __________________

9. Use the Law of Sines to solve �ABC when a � 1.43, 9. __________________b � 4.21, and A � 30.4°. If no solution exists, write none.

10. Use the sum or difference identity to find the exact value 10. __________________of tan 105°.

11. Find the distance between P(7, �4) and the line with 11. __________________equation x � 3y � 5 � 0. Round to the nearest tenth.

12. Find the inner product of the vectors �2, 5� and �4, �2�. 12. __________________Then state whether the vectors are perpendicular.Write yes or no.



Unit 2 Test, Chapters 5-8 (continued)

NAME _____________________________ DATE _______________ PERIOD ________

13. Find the value of sin � for angle � in standard position if a 13. __________________point with coordinates (�3, 2) lies on its terminal side.

14. Solve sin � � �1 for all real values of �. 14. __________________

15. A car’s f lywheel has a timing mark on its outer edge. 15.The height of the timing mark on the rotating flywheel is given by y � 3.55 sin �x � �

�4��. Graph one full cycle

of this function.

16. Find the ordered pair that represents �3 w� if w� � �6, �4�. 16. __________________

17. Write XY� as the sum of unit vectors for X(8, 2, �9) and 17. __________________Y(�12, �1, 10).

18. In the triangle at the right, b � 6.2 18. __________________and c � 8.2. Find � to the nearest tenth.

19. If 0° � � � 90° and tan � � ��23�� , f ind cos �. 19. __________________

20. Solve sin2 x � sin x � 2 � 0 for 0° x � 360°. 20. __________________

21. If �849° is in standard position, determine a coterminal 21. __________________angle that is between 0° and 360°. State the quadrant in which the terminal side lies.

22. Verify that �tansexccxsc x� � 1 is an identity. Write your 22. __________________

answer on a separate piece of paper.

UNIT2




NAME _____________________________ DATE _______________ PERIOD ________

23. Find the cross product of the vectors �2, �1, 4� and �6, �2, 1�. 23. __________________Is the resulting vector perpendicular to the given vectors?

24. A triangular shelf is to be placed in a curio cabinet whose 24. __________________sides meet at an angle of 105°. If the edges of the shelf along the sides measure 56 centimeters and 65 centimeters, how long is the outside edge of the shelf ? Round to the nearest tenth.

25. If sin � � �35� and � is a second quadrant angle, find tan 2�. 25. __________________

26. Graph the function y � sin x on 26.the interval ��2

�� x �2��.

27. Change �79�� radians to degree measure. 27. __________________

28. Nathaniel pulls a sled along level ground with a force of 28. __________________30 newtons on the rope attached to the sled. If the rope makes an angle of 20° with the ground when it is pulled taut, find the horizontal and vertical components of the force. Round to the nearest tenth.

29. State the amplitude, period, and phase shift of the 29. __________________function y � �2 sin (4� � 2�).

30. If � and � are two angles in Quadrant II such that 30. __________________

tan � � ��12� and tan � � ��23�, find cos (� � � ).

31. A surveyor sets a stake and then walks 150 feet north, 31. __________________where she sets a second stake. She then walks 300 feet east and sets a third stake. How far from the first stake is the third stake? Round to the nearest tenth.

UNIT2



32. Find the value of Tan�1 ��13��. 32. __________________

33. Use the Law of Cosines to solve � ABC with A � 126.3°, 33. __________________b � 45, and c � 62.5. Round to the nearest tenth.

34. Write an equation in slope-intercept form of the line with 34. __________________parametric equations x � 2 � 3t and y � 4 � t.

35. Verify that cos (90° � A) � �sin A is an identity. 35. __________________

36. Write the equation for the inverse of the function 36. __________________y � Cos x. Then graph the function and its inverse.

37. Find sin (Sin�1 �14�). 37. __________________

38. Find the area of a sector if the central angle measures 38. __________________

�56�� radians and the radius of the circle is 8 centimeters.

Round to the nearest tenth.


40. A golf ball is hit with an initial velocity of 135 feet per 40. __________________second at an angle of 22° above the horizontal. Will the ball clear a 25-foot-wide sand trap whose nearest edge is 300 feet from the golfer?





Unit 2 Answer KeyUnit 2 Review

1. �2�13

1�3�� 2. �51�

3. ��11�77�� 4. 1 5. 25.9

6. 5.8 7. 10.9

8. �14

� 9. �2�5

5�� 10. 1

11. no solution

12. two; B � 72.6�, c � 14.1, and C � 42.4�, orB � 107.4�, c � 2.8, and C � 7.6�

13. one; a � 40.1,c � 28.1, C � 42�

14. 14.1 15. 78.2

16. 90� 17. 135�

18. 630� 19. �105�

20. 6.3 in. 21. 0

22. 0 23. �1

24. 1 25. 2, �23��

26. none, ��5

�

27. none, 2�

28.

29.

30.

31.

32.

33. �34

� 34. �2�3

2�� 35. 2�2�

36. tan x � tan x cot2 x� sec x csc x

tan x (1 � cot2 x) � sec x csc x

��csoinsxx

��sin1

2 x��

� sec x csc x

��co1s x��sin

1x

�� sec x csc x

sec x csc x � sec x csc x

37. sin (180� � �) � tan � cos �

sin 180� cos � � cos 180� sin �� tan � cos �

0(cos �) � (�1) sin �� tan � cos �

sin � � tan � cos �

sin � ��ccoo

ss

��

�� tan � cos �

�csoins

��

� � cos � � tan � cos �

tan � cos � � tan � cos �

38. ��2� �4

�6�� 39. ��22��

40. 2 � �3� 41. �21�

42. �1275� 43. ��5� ��1

��0�2��1��44. �5 �

2�2�1�� 45. �4�

252�1��

46. 0�, 90�, 180� 47. 0�

48. 120�

49. �2�13

1�3��x � �3�13

1�3��y �

�2�13

1�3�� 0; �2�13

1�3�� ; 56�

50. �5�29

2�9��x � �2�29

2�9��y �

�8�29

2�9�� 0; �8�29

2�9��; 22�

407-408 U02-0-02-834179 10/10/00 10:56 AM Page 407



51. �3�10

1�0��x � ��11�00��y � �7�

101�0�� 0;

�7�10

1�0��; 342�

52. �3�42�� 53. �16��

171�7��

54. �6��17

1�7�� 55. 5.7 cm, 89�

56. 2.6 cm, 1.5 cm

57. �3, �5� 58. ��1, �1�

59. �7, �13� 60. ��1, 7�

61. �1, 4, 1�; u� � i� � 4j� � k�

62. �13, �3, �7�;u� � 13i� � 3j� � 7k�

63. ��6, 9, 6�; u� � �6i� � 9j� � 6k�

64. �22, 0, �10�; u� � 22i� � 10k�

65. �14 66. 22

67. �6, 10, �2�

68. �x, y � 5�� t ��1, 5�;x � �t, y � 5 � 5t

69. �x � 4, y � 3� �t ��2, �2�; x � 4 � 2t, y � �3 � 2t

Unit 2 Test

1. true 2. 81 units2

3. �5�26

2�6��x � ��22�66��y � ��

12�36�� 0

4.

5. 15.7 in.

6. v: 11.3 cm; h: 14.4 cm

7. x � t, y � 5t � 2

8. ��6

� 9. none

10. �2 � �3� 11. 7.6

12. �2; no 13. �2�13

1�3��

14. �32�� 2�k

15.

16. ��18, 12�

17. �20i� � 3j� � 19k�

18. 40.9� 19. �2�7

7��

20. 270� 21. 231�; III

22. �tansxec

csxc x� � 1

��csoinsxx

�� sin1

x��

� 1�co

1s x�

�co

1s x�

�co

1s x�

23. �7, 22, 2�; yes

24. 96.2 cm 25. ��274�

26.

27. 140�

28. 28.2 N; 10.3 N

29. 2, ��2

�, ��2

� 30. �4�65

6�5��

31. 335.4 ft 32. ��6

�

33. a � 96.2, B � 22.1�, C � 31.6�

34. y � �13

�x � �130�

35. cos (90� � A) � �sin Acos 90� cos A � sin 90� sin A

� �sin A 0 � cos A � 1 � sin A � �sin A

�sin A � �sin A

36. y � Arccos x

37. �14

� 38. 83.8 cm2

39. 40� 40. yes

� 1

1 � 1

407-408 U02-0-02-834179 10/10/00 10:56 AM Page 408


UNIT3

Graph the point that has the given polarcoordinates. Then, name three otherpairs of polar coordinates for eachpoint.

1. A(2, 60°) 2. B(�4, 45°)

3. C�1.5, ��6�� 4. D��2, ��23��

Graph each polar equation. Identify thetype of curve each represents.

5. r � �5�

6. � � 60°

7. r � 3 cos �

8. r � 2 � 2 sin �

Find the polar coordinates of each pointwith the given rectangular coordinates.Use 0 � � � 2� and r � 0.

9. (�2, �2) 10. (2, 2)

11. (2, �3) 12. (�3, 1)

Write each equation in rectangular form.

13. 2 � r cos �� 2��14. 4 � r cos �� 3��

Simplify.

15. i45

16. (3 � 2i) � (3 � 2i)

17. i4(3 � 3i)

18. (�i � 5)(i � 5)

19. �22

��

3ii�

Express each complex number in polarform.

20. �3i 21. 3 � 3i

22. �1 � 3i 23. 4 � 5i

Find each product. Express the result inrectangular form.

24. 2�cos ��2� � i sin ��2�� 4�cos ��2� � i sin ��2��25. 1.5(cos 3.1 � i sin 3.1) �

2(cos 0.5 � i sin 0.5)

Solve.

26. Find (1 � i)7 using De Moivre’sTheorem. Express the result inrectangular form.

27. Solve the equation x5 � 1 � 0 for allroots.

Find the distance between each pair ofpoints with the given coordinates. Then,find the coordinates of the midpoint ofthe segment that has endpoints at thegiven coordinates.

28. (�4, 6), (11, 2) 29. (5, 0), (3, �2)

30. (1, 9), (�4, �3)

For the equation of each circle, identifythe center and radius. Then graph theequation.

31. 4x2 � 4y2 � 49

32. x2 � 10x � y2 � 8y � 20

For the equation of each ellipse, find thecoordinates of the center, foci, andvertices. Then, graph the equation.

33. (x � 1)2 � 2(y � 3)2 � 25

34. 4(x � 2)2 � 25(y � 2)2 � 100

For the equation of each hyperbola, findthe coordinates of the center, foci, andvertices, and the equations of theasymptotes. Then, graph the equation.

35. 4x2 � y2 � 27

36. �( y �

43)2

� � �(x �

91)2� � 1


NAME _____________________________ DATE _______________ PERIOD ________



For the equation of each parabola, findthe coordinates of the focus and vertex,and the equations of the directrix andaxis of symmetry. Then graph theequation.37. (x � 2)2 � 2( y � 4)38. y2 � 2y � 5x � 18 � 0

Graph each equation and identify theconic section it represents.39. 12y � 3x � 2x2 � 1 � 040. 4x2 � 25y2 � 8x � 150y � 321 � 041. x2 � 4x � y2 � 12y � 4 � 0

Find the rectangular equation of thecurve whose parametric equations aregiven. Then graph the equation usingarrows to indicate orientation.42. x � 2t, y � 3t2, �2 � t � 243. x � sin t, y � 3 cos t, 0 � t � 2�

Write an equation in general form ofeach translated or rotated graph.44. x � 3( y � 2)2 � 1 for T(1, �5)

45. x2 � �1y6

2

� � 1, � � �3�

�

Graph each system of equations orinequalities. Then solve the system of equations.46. x2 � 2x � 2y � 2 � 0

(x � 4)2 � �8y

47. x2 � ( y � 1)2 � 14x2 � 9( y � 3)2 36

Simplify each expression.48. �1�6�x�2y�7�

49. �3

5�4�a�4b�3c�8�

50. �32c3d5��51�

51. (3x)2(3x2)�2

Graph each exponential function.52. y � 2�x

53. y � 2x�2

A city’s population can be modeled bythe equation y � 17,492e�0.027t, where tis the number of years since 1996.54. What was the city’s population in

1996?55. What is the projected population in

2007?

Solve each equation.56. logx 36 � 257. log2 (2x) � log2 27

58. log5 x � �13� log5 64 � 2 log5 3

Find the value of each logarithm usingthe change of base formula.59. log6 43160. log0.5 7861. log7 0.325

Use natural logarithms to solve eachequation.62. 2.3x � 23.463. x � log4 1664. 5x�2 � 2x

Solve each equation by graphing. Roundsolutions to the nearest hundredth.65. 46 � ex

66. 18 � e4k

67. 519 � 3e0.035t

Unit 3 Review, Chapters 9–11 (continued)




UNIT3 Unit 3 Test, Chapters 9–11

NAME _____________________________ DATE _______________ PERIOD ________

1. Evaluate log6 �6�. 1. __________________

2. Identify the conic section represented by 2. __________________3x2 � 4xy � 2y2 � 3y � 0.

3. Find the coordinates of the vertex and the equation of the 3. __________________axis of symmetry for the parabola with equation 2x2 � 2x � y � �3.

4. Write the rectangular equation x2 � y2 � 6 in polar form. 4. __________________

5. Simplify �31

��

2ii�. 5. __________________

6. Graph the point with polar coordinates ��2, �32��. 6.

7. Solve 6 � e0.2t. Round your answer to the nearest hundredth. 7. __________________

8. Evaluate (�2�8�9�)�3. 8. __________________

9. Find the rectangular equation of the curve whose 9. __________________parametric equations are x � �2 sin t and y � cos t,where 0 � t � 2�.

10. Find the balance after 15 years of a $2500 investment 10. __________________earning 5.5% interest compounded continuously.

11. Find the product 2(cos 10° � i sin 10°) � 4(cos 20° � i sin 20°). 11. __________________Then, express the result in rectangular form.

12. Identify the classical curve that the graph of r � 1 � sin � 12. __________________represents.

13. Write the standard form of the equation of the circle that 13. __________________passes through (0, 4) and has its center at (�3, �1).




NAME _____________________________ DATE _______________ PERIOD ________

14. Graph the polar equation 14.2 � r cos (� � 180°).

15. Solve 6 � 151�x by using logarithms. Round your answer 15. __________________to the nearest thousandth.

16. Find the principal root (�64)�61�. Express the result in 16. __________________

the form a � bi.

17. Find the distance between points at (6, �3) and (�1, 4). 17. __________________

18. Write the equation of the ellipse with foci at (0, ��3�) 18. __________________and (0, �3�) and for which 2a � 4.

19. Find the future value to the nearest dollar of $2700 19. __________________invested at 8% for 5 years in an account that compounds interest quarterly.

20. Use a calculator to find ln 36.9 to the nearest ten thousandth. 20. __________________

21. Graph the exponential function y � ��14��x. 21.

22. For the ellipse with equation 22.5x2 � 64y2 � 30x � 128y � 211 � 0,find the coordinates of the center, foci,and vertices. Then, graph the equation.

UNIT3




NAME _____________________________ DATE _______________ PERIOD ________

23. Find the rectangular coordinates of the point whose polar 23. __________________coordinates are (20, 140°). Round to the nearest hundredth.

24. Write the standard form of the equation of the circle that 24. __________________is tangent to x � �2 and has its center at (2, �4).

25. Evaluate (1 � i)12 by using De Moivre’s Theorem. 25. __________________Express the result in rectangular form.

26. Express �3

8�a�3y�5� using rational exponents. 26. __________________

27. Simplify (�1 � 2i) � (4 � 6i). 27. __________________

28. Graph the system of equations. Then solve. 28.x2 � y2 � 10 xy � 3

29. Express 8i in polar form. 29. __________________

30. Convert log7 0.59 to a natural logarithm and evaluate to 30. __________________the nearest ten thousandth.

31. Identify the graph of the equation 4x2 � 25y2 � 100. 31. __________________Then write the equation of the translated graph for T(5, �2) in general form.

32. Write the polar equation 3 � r cos (� � 315°) in 32. __________________rectangular form.

33. True or false: The graph of the polar equation 33. __________________r2 � 3 sin 2� is a limaçon.

34. Write 3x � 6y � �14 in polar form. Round � to the 34. __________________nearest degree.

35. Express x�23�( y5z)�

13� using radicals. 35. __________________

36. Write the equation log343 7 � �13� in exponential form. 36. __________________

UNIT3



37. Write the standard form of the equation of the circle that 37. __________________passes through the points at (0, 8), (8, 0), and (16, 8). Then identify the center and radius of the circle.

38. Use a calculator to find antiln (�0.049) to the nearest 38. __________________ten thousandth.

39. Find the equation of the hyperbola whose vertices are 39. __________________at (�1, �5) and (�1, 1) with a focus at (�1, �7).

40. Find the quotient 3�cos �512�� i sin �

512�� 6�cos �1

�2� � i sin �1

�2��. 40. __________________

The express the quotient in rectangular form.

41. Find the coordinates of the focus and the equation of the 41. __________________directrix of the parabola with equation y2 � 8y � 8x � 24 � 0.

42. Solve 92 x�3 � 4. Round your answer to the nearest 42. __________________hundredth.

43. Express 2(cos 300° � i sin 300°) in rectangular form. 43. __________________

44. Find the equation of the ellipse whose semi-major axis 44. __________________has length 6 and whose foci are at (3, �2��1�1�).

45. Graph the system of inequalities. 45.

�(x �

93)2� � �

( y �

42)2

� � 1(x � 3)2 � ( y � 2)2 � 4

46. Write the polar equation � � 45° in rectangular form. 46. __________________

47. What interest rate is required for an investment with 47. __________________continuously compounded interest to double in 6 years?

48. Write the equation 26 � 64 in logarithmic form. 48. __________________

49. Simplify (3 � 2i)(2 � 5i). 49. __________________

50. Find the coordinates of the center, the foci, and the vertices, 50. __________________and the equations of the asymptotes of the graph of the equation �(x �

21)2� � �

y8

2

� � 1.





Unit 3 ReviewGraph for Exercises 1 and 2

1. Sample answer: (2, �300�), (�2, �120�), (�2, 240�)

2. Sample answer: (�4, �315�),

(4, �135�), (4, 225�)

Graph for Exercises 3 and 4

3. Sample answer: �1.5, ��116��,

��1.5, ��56��, ��1.5, �7

6��

4. Sample answer: ��2, �43��, �2, �

�3

��,�2, ��5

3��

5. circle

6. line

7. circle

8. cardioid

9. �2�2�, �54��

10. �2�2�, ��4

��

11. (3.61, 5.30)

12. (3.16, 2.82)

13. y � 2

14. x � �3�y � 8 � 0

15. i 16. 6

17. 3 � 3i 18. 26

19. �113� � �

183�i

20. 3�cos �32�� i sin �3

2��

21. 3�2��cos ��4

� � i sin ��4

��

22.�1�0�(cos 1.89 � i sin 1.89)

23. �4�1�[cos (5.39) �i sin (5.39)]

24. �8 25. �2.69 � 1.33i

26. 8 � 8i

27. 1, 0.31 � 0.95i, �0.81 � 0.59i,�0.81 � 0.59i, 0.31 � 0.95i

28. �2�4�1�; (3.5, 4)

29. 2�2�; (4, �1)

30. 13; (�1.5, 3)

31. (0, 0); �27�

32. (�5, �4); �6�1�

Unit 3 Answer Key

415-418 U03-0-02-834179 10/10/00 10:58 AM Page 415

33.

center: (1, 3); foci:

�1 � �5�2

2��, 3�; vertices:

(6, 3), (�4, 3), �1, 3 � �5�2

2��

34.

center: (�2, 2); foci: (�2 � �2�1�, 2); vertices: (�7, 2), (3, 2), (�2, 4), (�2, 0)

35.

center: (0, 0); foci:

��3�2

1�5��, 0�; vertices:

��3�2

3��, 0�; asymptotes: y � �2x

36.

center: (�1, �3); foci: (�1, �3 � �1�3�); vertices: (�1, �1), (�1, �5); asymptotes: y � 3 � � �2

3�(x � 1)

37.

vertex: (2, 4); focus: (2, 4.5); directrix: y � 3.5, axis of symmetry: x � 2

38.

vertex: (3.4, �1); focus: (4.65, �1); directrix: x � 2.15; axis of symmetry: y � �1

39. parabola

40. hyperbola

41. circle

42. y � �34x2�



415-418 U03-0-02-834179 10/10/00 10:58 AM Page 416


43. x2 � �y9

2� � 1

44. 3y2 � x � 42y � 149 � 0

45. 13( x�)2 � 34�3�x�y� �47( y�)2 � 64 � 0

46. (2.58, �0.25), (0.62, �1.43)

47.

48. 4|x|y3 �y�

49. 3abc2 �32�a�c�2�

50. d�5

9�c�3� 51. �x12�

52.

53.

54. 17,492 55. 12,997

56. 6 57. �227� 58. 36

59. 3.3856 60. �6.2854

61. �0.5776 62. 3.79

63. 2 64. 3.51

65. 3.83

66. 0.72

67. 147.24

Unit 3 Test

1. �21�

2. ellipse

3. ��12

�, �52

��; x � ��21�

4. r � ��6� 5. �113� � �

153�i

6.

7. 8.96 8. �49

113�

9. x2 � 4y2 � 4

10. $5704.70

11. 4�3� � 4i

12. cardioid

13. (x � 3)2 � ( y � 1)2 � 34

14.


415-418 U03-0-02-834179 10/10/00 10:58 AM Page 417

15. 0.338 16. �3� � i

17. 7�2�

18. �x1

2� � �

y4

2� � 1

19. $4012

20. 3.6082

21.

22. center: (�3, �1);foci: (�3 � �5�9�, �1); vertices: (5, �1), (�11, �1), (�3, �1 ��5�)

23. (�15.32, 12.86)

24.(x � 2)2 � ( y � 4)2 � 16

25. �64 26. 2ay

27. 3 � 8i

28. (3, 1), (�3, �1), (1, 3), (�1, �3)

29. 8 �cos ��2

� � i sin ��2

��

30. �0.2711

31. hyperbola;

4x2 � 25y2 � 40x � 100y �

100 � 0

32. �2�x � �2�y � 6 � 0

33. false

34.�141�5

5�� r cos (� � 117�)

35. y�3

x�2y�2z�

36. 343�13

�

� 7

37. (x � 8)2 � ( y � 8)2 � 64;(8, 8); 8

38. 0.9522

39. �( y �

92)2

� � �(x �

161)2

� � 1

40. �14

� � ��4

3��i

41. (3, 4); x � �1

42. x � 1.82

43. 1 � �3� i

44. �( y

3�62)2

� � �(x �

253)2

� � 1

45.

46. y � x

47. 11.55%

48. log2 64 � 6

49. 16 � 11i

50. center: (�1, 0);foci: (�1 ��1�0�, 0); vertices: (�1 ��2�, 0); asymptotes: y � �2(x � 1)



�53

�

415-418 U03-0-02-834179 10/10/00 10:58 AM Page 418


UNIT4

Solve.

1. Find the 20th term in the arithmeticsequence for which a1 � 3 and d � �2.

2. Find the sum of the first nine terms ofthe geometric series 2 � 4 � 8 � . . . .

3. Write an arithmetic sequence thathas two arithmetic meansbetween �3 and 9.

4. Find the sixth term of the geometricsequence �13�, �1

45�, �17

65�, . . . .

Find each limit or state that the limitdoes not exist.

5. lim �4n3�n

1� 6. lim �n2

n� 1�

7. lim �42n� 8. lim �(4n � 5

n)2(n � 3)�

Find the sum of each series or state thatthe sum does not exist.

9. 1 � �12� � �14� � . . .

10. 1 � �14� � �116� � . . .


11. 1 � 4 � 7 � 10 � . . .

12. 6 � 2 � �23� � �29� � . . .

13. �13

1� � �23

2� � �33

3� � . . .

Write each expression in expanded formand then find the sum.

14. �5

k�12k 15. �

7

k�1(2k2 � 1)

16. �8

a�2(3a � 6) 17. �

6

k�03��12��k

Find the designated term of each binomial expansion.

18. 4th term of (3x � 1)9

19. 7th term of (x � 2y)12

Use the first five terms of the exponentialseries and a calculator to approximateeach value to the nearest hundredth.20. e1.72 21. e�0.3

Find the first three iterates of each function using the given initial value. Ifnecessary, round your answers to thenearest hundredth.22. ƒ(x) � 3x � 1, x0 � 1

23. ƒ(x) � x2 � 5, x0 � �2

24. ƒ(x) � �1x�, x0 � 2

Find the first three iterates of the function ƒ(z) � 2z � 3i for each initialvalue.25. z0 � �i 26. z0 � 3 � i

Use mathematical induction to provethat each proposition is valid for all positive integral values of n.27. 3 � 9 � 15 � . . . � (6n � 3) � 3n2

28. 5n � 1 is divisible by 4

Find each value.29. P (9, 6) 30. P (7, 4)

31. C(8, 2) 32. C(6, 5)

Solve.33. The letters a, b, c, d, and e are to be

used to form 5-letter patterns. Howmany patterns can be formed if repetitions are not allowed?

34. How many different 5-member teamscan be formed from 10 players?

35. How many different ways can the let-ters of the word color be arranged?

36. From a group of 3 men and 5 women,how many different committees of 2men and 2 women can be formed?

37. How many different ways can 8 keysbe arranged on a circular key ring?


NAME _____________________________ DATE _______________ PERIOD ________

n→� n→�

n→� n→�



State the odds of an event occurring,given the probability of the event.38. �27� 39. �1

14�

40. �121� 41. �1

33�

Solve.42. Three cards are drawn at random

from a standard deck of 52 cards.What is the probability that all threeare clubs?

43. Find the probability of getting a sumof 6 on the first throw of two numbercubes and a sum of 2 on the secondthrow.

44. Find the probability of getting a sum of 7 or 9 on a single throw of two number cubes.

45. One card is drawn from a standarddeck of 52 cards. What is the probability that it is a king if it is known to be a face card?

Renee is a forward on her school’ssoccer team. The probability of her making a goal is �1

4�. Find each probability

if Renee makes 5 attempts on the goal.46. P(3 goals)47. P(at least 2 goals)

The test scores for Ms. Humphrey’shumanities class are listed below.89 95 65 70 77 82 66 69 91 8277 99 65 89 72 80 42 76 86 77

48. Find the range of the data.49. Find the mean of the data.50. Find the median of the data.51. Find the mode of the data.52. Find the mean deviation of the data.53. Find the semi-interquartile range of

the data.54. Find the standard deviation of the data.Solve.55. A set of data is normally distributed

with a mean of 30 and a standard deviation of 7. What percent of thedata is between 9 and 51?

56. A population is normally distributedwith a mean of 60 and a standard deviation of 5. What is the probabilitythat a randomly selected value will begreater than 65?

57. A set of data is normally distributedwith a mean of 70 and a standard deviation of 6. What is the probabilitythat a randomly selected value liesbetween 65 and 75?

Solve.58. Find the interval about the mean

within which 77% of the values of aset of normally distributed data canbe found if X� � 67 and � � 3.2.

59. Find the interval about the meanwithin which 16% of the values of aset of normally distributed data canbe found if X� � 0.25 and � � 0.12.

Find the standard error of the mean foreach sample.60. � � 8, N � 10061. � � 3.23, N � 3062. � � 5, N � 3863. � � 12.3, N � 89

Solve.64. A set of data of size N � 100 is normally

distributed with a mean of 50 and astandard deviation of 5. Determine theinterval about the sample mean thathas a 1% level of confidence.

65. A set of data of size N � 30 is normally distributed with a mean of100 and a standard deviation of 3.3.Determine the interval about thesample mean that has a 1% level of confidence.

66. A set of data of size N � 36 is normally distributed with a mean of20 and a standard deviation of 4.2.Determine the interval about thesample mean that has 5% level of confidence.

Unit 4 Review, Chapters 12-14 (continued)




UNIT4 Unit 4 Test, Chapters 12–14

NAME _____________________________ DATE _______________ PERIOD ________

1. Express 23 � 29 � 35 � 41 � 47 using sigma notation. 1. __________________

2. A set of data is normally distributed with a mean of 16 and 2. __________________a standard deviation of 0.3. What percent of the data is between 15.2 and 16?

3. Use mathematical induction to prove that 3. __________________12 � 22 � 32 � . . . � n2 � �n(n � 1

6)(2n � 1)� for

all positive integral values of n.

4. Find the twelfth term of the geometric sequence �116�, �18�, �14�, . . . . 4. __________________

5. There are 21 wrapped packages in a grab bag at an office 5. __________________holiday party. Five of the packages contain $20 bills,7 packages contain $5 bills, and 9 packages contain $1 bills.What is the probability that the first two people will choose packages with $20 bills inside?

6. Find the sum of the infinite series 1 � �15� � �215� � . . . . 6. __________________

7. Find P(5, 2). 7. __________________

8. True or false: Choosing an entrée and choosing an appetizer 8. __________________from a dinner menu are independent events.

9. How many ways can 9 keys be arranged on a circular 9. __________________key ring?

10. Determine whether the series 1 � �21

5� � �31

5� � �41

5� � . . . 10. __________________is convergent or divergent.

11. What is the probability of getting an even number on a 11. __________________single roll of a number cube if you roll a 3 or greater?

12. How many eight-letter patterns can be formed from the 12. __________________letters of the word circular?




NAME _____________________________ DATE _______________ PERIOD ________

13. Use the ratio test to determine whether the series 13. __________________�31� � �3

12� � �3

13� � . . . is convergent or divergent.

14. Find the first three iterates of the function 14. __________________ƒ(z) � z2 � z � i , if the initial value is i.

15. The probability of being named captain of the basketball 15. __________________team is �15�. What are the odds of being named captain of the team?

16. Eight out of ten people surveyed prefer to observe Veterans 16. __________________Day on November 11 rather than on the second Monday of November. Use the Binomial Theorem to determine the probability that each of the first three people surveyed prefer to observe Veterans Day on November 11.

17. Use the Binomial Theorem to expand (x � 2y)3. 17. __________________

18. Find the probability of tossing 2 heads on 3 tosses of a 18. __________________fair coin.

19. Find n for the arithmetic sequence for which an � 129, 19. __________________a1 � 15, and d � 6.

20. The set of class marks in a frequency distribution is 20. __________________{25.5, 35.5, 45.5, 55.5}. Find the class interval and the class limits.

21. Use the first f ive terms of the exponential series and a 21. __________________calculator to approximate the value of e0.67 to the nearest hundredth.

22. Find the first three iterates of the function ƒ(x) � x2 � 2, 22. __________________if the initial value is �2.

23. Find limn→�

�n2

n� 1�, or state that the limit does not exist. 23. __________________

UNIT4




NAME _____________________________ DATE _______________ PERIOD ________

24. Twenty slips of paper are numbered 1 to 20 and placed 24. __________________in a box. What is the probability of drawing a number that is odd or a multiple of 5?

25. Find C(10, 4). 25. __________________

The table below shows the number of gallons of heating oil delivered to a residential customer each December from 1986 to 1993. Use the table for Exercises 26-28.

26. Make a stem-and-leaf plot of the data. 26. __________________

27. Find the mean, median, and mode of the data. 27. __________________

28. Find the interquartile range and the semi-interquartile 28. __________________range of the data.

29. Two cards are drawn at random from a standard deck of 29. __________________52 cards. What is the probability that both cards are queens?


31. Find the 50th term in the arithmetic sequence 1, 5, 9, . . . . 31. __________________

32. A set of data is normally distributed with a mean 32. __________________of 50 points and a standard deviation of 10 points. What percent of the data is greater than 60?

Year 1986 1987 1988 1989 1990 1991 1992 1993

Gallons of Oil 42 61 53 59 53 51 75 100

UNIT4



33. A red number cube and a white number cube are rolled. 33. __________________Find the probability that the red number cube shows a 2,given that the sum showing on the two number cubes is less than or equal to 5.

34. Find the standard error of the mean for a sample in which 34. __________________� � 3.6, N � 100, and X� � 36. Then use t � 2.58 to f ind the interval about the sample mean that has a 1% level of confidence. Round your answer to the nearest hundredth.

The table below shows a frequency distribution of the number of pickup trucks sold at 85 truck dealerships in Maine over an 18-month period.

35. Find the interval about the sample mean such that the 35. __________________probability is 0.90 that the true mean lies within the interval. (When P � 90%, t � 1.65.)

A set of data is normally distributed with a mean of 500 and a standard deviation of 40.


37. Find the probability that a value selected at random is 37. __________________less than 420.

Suppose that the respondents in a survey of 100 teenagers watchan average of 10 hours of television per week. The standarddeviation of the sample is 2.5 hours.


39. What is the interval about the sample mean that has a 39. __________________1% level of confidence?

40. Find the interval about the sample mean that gives a 90% 40. __________________chance that the true mean lies within the interval.


NAME _____________________________ DATE _______________ PERIOD ________

Number of70�90 90�110 110�130 130�150 150�170 170�190Trucks Sold

Number of2 11 39 17 9 7Dealerships

UNIT4



Unit 4 Review

1. �35 2. 1022

3. �3, 1, 5, 9

4. �19032745

� 5. �34�

6. does not exist

7. 0 8. 4

9. 2 10. �34�

11. divergent

12. convergent

13. convergent

14. 2(1) � 2(2) � 2(3) � 2(4) �2(5) or 30

15. [2(1)2 � 1] � [2(2)2 � 1] �[2(3)2 � 1] � [2(4)2 � 1] �[2(5)2 � 1] � [2(6)2 � 1] �[2(7)2 � 1] or 287

16. [3(2) � 6] � [3(3) � 6] �[3(4) � 6] � [3(5) � 6] �[3(6) � 6] � [3(7) � 6] �[3(8) � 6] or 63

17. 3��12��0 � 3��12��1 � 3��12��2 � 3��12��3 �

3��12��4 � 3��12��5 � 3��12��6 or �36841�

18. 61,236x6

19. 59,136x6y6

20. 5.41 21. 0.74

22. 4, 13, 40

23. �1, �4, 11

24. �12

�, 2, �12

�

25. i, 5i, 13i

26. 6 � i, 12 � 5i, 24 � 13i

27. Proof: Sn is defined as 3 � 9 � 15 � ... � (6n � 3) � 3n2. Step 1: Verify that Sn is valid for n � 1. Since S1 � 3 and 3(1)2 � 3, the formula is valid for n � 1.

Step 2: Assume that Sn is valid for n � kand derive a formula for n � k � 1.Sk ⇒ 3 � 9 � 15 � ... � (6k � 3) � 3k2

Sk�1 ⇒ 3 � 9 � 15 � ... � (6k � 3) � [6(k � 1) �3]

� 3k2 � [6(k � 1) � 3]� 3k2 � 6k � 3 � 3(k2 � 2k � 1) � 3(k � 1)2

The formula gives the same result asadding the (k � 1) term directly. Thus, ifthe formula is valid for n � k, it is alsovalid for n � k � 1. Since the formula isvalid for n � 1, it is also valid for n � 2.Since it is valid for n � 2, it is also validfor n � 3, and so on, indefinitely. Thus,the formula is valid for all positiveintegral values of n.

28. Proof: Sn ⇒ 5n � 1 � 4r for some integer rStep 1: Verify that Sn is valid for n � 1.S1 � 51 � 1 or 4. Since 4 � 4 � 1, Sn is valid for n � 1.

Step 2: Assume that Sn is valid for n � kand show that it is also valid for n � k � 1.Sk ⇒ 5k � 1 � 4r for some integer r.Sk � 1 ⇒ 5k�1 � 1 � 4t for some integer t.

5k � 1 � 4r5(5k � 1) � 5(4r)5k�1 � 5 � 20r5k�1 � 1 � 20r � 45k�1 � 1 � 4(5r � 1)Thus, 5k�1 � 1 � 4t, where t � (5r � 1), isan integer, and we have shown that if Skis valid, then Sk�1 is also valid. Since Snis valid for n � 1, it is also valid for n � 2, n � 3, and so on, indefinitely.Hence 5n � 1 is divisible by 4 for allpositive integral values of n.

29. 60,480 30. 840

31. 28 32. 6

33. 120 patterns

34. 252 teams

35. 60 ways

36. 30 committees

37. 2520 ways

38. �25

� 39. �113�

40. �29

� 41. �130�

42. �81510

� 43. �12

596�

44. �158� 45. �

31�

46. �54152

� 47. �14278

�

48. 57 49. 77.45

50. 77 51. 77

52. 9.595 53. 9

54. 12.65 55. 99.7%

56. 0.1585

57. 0.576

58. 63.16�70.84

59. 0.226�0.274

60. 0.8 61. about 0.59

62. about 0.81

63. about 1.30

64. 48.71–51.29

65. 98.45–101.55

66. 18.628–21.372

Unit 4 Answer Key

425-426 U04-0-02-834179 10/10/00 10:59 AM Page 425

Unit 4 Test

1. �4

k�0(23 � 6k)

2. 49.85%

3. Proof: Sn is defined as 12 � 22 � 32 � . . . � n2 � �

n(n � 1)6(2n � 1)�

Step 1: Verify that Sn is valid for n � 1.

Since S1 � 1 and � 1,

the formula is valid for n � 1.Step 2: Assume that Sn is valid for n � kand derive a formula for n � k � 1.

Sk ⇒ 12 � 22 � 32 � . . . � k2 �

�k(k � 1)

6(2k � 1)�

Sk�1 ⇒ 12 � 22 � 32 � . . . � k2 � (k � 1)2

� �k(k � 1)

6(2k � 1)� � (k � 1)2

�

�

�

�

�

The formula gives the same result asadding the (k � 1) term directly. Thus, ifthe formula is valid for n � k, it is alsovalid for n � k � 1. Since the formula isvalid for n � 1, it is also valid for n � 2.Since it is valid for n � 2, it is also validfor n � 3, and so on, indefinitely. Thus,the formula is valid for all positiveintegral values of n.

4. 128

5. �211�

6. �45�

7. 20

8. true

9. 20,160

10. convergent

11. 0.5

12. 10,080

13. convergent

14. �1, �i, �1 � 2i

15. �41�

16. �16245

�

17. x3 � 6x2y � 12xy2 � 8y3

18. 0.375

19. 20

20. 10; 20.5, 30.5, 40.5, 50.5, 60.5

21. 1.95

22. 2, 2, 2

23. does not exist

24. �53�

25. 210

26. stem leaf

4 25 1 3 3 96 17 5

10 04|2 � 42

27. mean: 61.75; median: 56; mode: 53

28. 16; 8

29. �22

11

�

30. �141�

31. 197

32. 15.85%

33. �130�

34. 0.36; 35.07�36.93

35. 125.48�133.81

36. 68.3%

37. 0.0225

38. 0.25

39. 9.355�10.645

40. about 9.59�10.41

(k � 1)[(k � 1) � 1][2(k � 1) � 1]��

6

(k � 1)(2k � 3)(k � 2)��

6

(k � 1)(2k2 � 7k � 6)��

6

2k3 � 9k2 � 13k � 6��

6

(k2 � k)(2k � 1) � 6(k2 � 2k � 1)��

6

1(1 � 1)(2 � 1 � 1)��

6



425-426 U04-0-02-834179 10/10/00 10:59 AM Page 426


Semester Test, Chapters 4–9

NAME _____________________________ DATE _______________ PERIOD ________


1. Which angle is not coterminal with �30°? 1. ________A. � �

�6� B. �750� C. �

356

�� D. 750�

2. Which ordered triple represents CD� for C(5, 0, �1) and D(3, �2, 6)? 2. ________A. �8, �2, 5� B. ��2, �2, 7�C. ��2, 2, �7� D. �2, 2, �5�

3. Evaluate cos �Sin�1 ��23��. 3. ________

A. ��23�

� B. �12� C. �3� D. ��2

2��

4. Write the polynomial equation of least degree with roots 7i and �7i. 4. ________A. x2 � 49 � 0 B. x2 � 49x � 0C. x2 � 7 � 0 D. x2 � 7 � 0

5. Find the angle to the nearest degree that the normal to the line 5. ________with equation 3x � y � 4 � 0 makes with the positive x-axis.A. �18° B. 18° C. 162° D. 108°

6. Find the x-intercepts of the graph of the function 6. ________ƒ(x) � (x � 3)(x2 � 4x � 3).A. 3, 1 B. �9 C. �3, �1, 3 D. 9, �9

7. Find the discriminant of 4m2 � 2m � 1 � 0 and describe the nature of 7. ________the roots of the equation.A. �12, imaginary B. 12, realC. 4, imaginary D. 2, real

8. Solve sin � � �1 for all values of �. Assume k is any integer. 8. ________A. 90° � 360k° B. 180° � 360k° C. 360k° D. 270° � 360k°

9. List all possible rational roots of ƒ(x) � 2x3 � 5x2 � 4x � 3. 9. ________A. �1, �2 B. �1, �3, � �12�, � �2

3�

C. �1, �2, �3, � �23� D. �1, �3

10. Find the rectangular coordinates of the point with polar 10. ________coordinates �1, �

�4��.

A. �1, �12�� B. ��

�22��, �

�22�� C. ��2

2��, �

�22�� D. ��

�23��, �

�22��



11. A section of highway is 4.2 kilometers long and rises at a uniform 11. ________grade making a 3.2° angle with the horizontal. What is the change in elevation of this section of highway to the nearest thousandth?A. 0.235 km B. 0.013 km C. 4.193 km D. 0.234 km

12. Choose the graph of the point with polar coordinates �3, ��4��. 12. ________

A. B. C. D.

13. Use the Remainder Theorem to find the remainder for 13. ________(2x3 � 5x2 � 3x � 4) � (x � 2).A. �6 B. 6 C. 2 D. 0

14. Find the polar coordinates of the point with rectangular 14. ________coordinates (2, 2).A. �3�2�, �

�3�� B. �2�2�, �

�4�� C. (�2�, �) D. ��2�, �

32��

15. If v� has magnitude 6 kilometers, w� has magnitude 18 kilometers, 15. ________and both vectors have the same direction, which of the following is true?A. vv� � 3w� B. 3v� � w� C. vv� � w� D. 3v� � 18w�

16. Find the magnitude of AB� for A(8, 8) and B(�7, 3). 16. ________A. 5�1�0� B. �2�6� C. 10�2� D. �1�2�3�

17. Change 54° to radian measure in terms of �. 17. ________A. �

54�� B. �

31

�0� C. �

�4� D. �

49��

18. Find one positive and one negative angle that are coterminal with an 18. ________angle measuring ��6�.

A. ��4�, ��

32�� B. �

136

��, ��

116

�� C. �

76��, ��

56�� D. �

23��, ��

23��

19. Simplify sec � � tan � sin �. 19. ________A. cos � B. sin � C. sec � D. csc �

Semester Test, Chapters 4–9 (continued)

NAME _____________________________ DATE _______________ PERIOD ________




NAME _____________________________ DATE _______________ PERIOD ________

20. Express 3�cos �32�� i sin �

32�� in rectangular form. 20. ________

A. �3i B. 3i C. ��23�� i D. i

21. If sin � � ��12� and � lies in Quadrant III, find cot �. 21. ________

A. ��33�� B. �

�33�� C. �3� D. ��3�

22. State the amplitude, period, and phase shift of the function 22. ________y � 2 sin �3x � ��3��.A. 2, 3, �

�2� B. 3, 3, � C. 2, �23

��, ��9� D. 2, �23��, �9

��

23. Find the value of Cos�1 �sin ��2��. 23. ________A. 0 B. �

�2� C. � D. �

32��

24. Which equation is a trigonometric identity? 24. ________A. cos 2� � cos2 � � sin2 � B. cos2 � � sin2 � � 1C. sin 2� � sin � cos � D. cos (� �) � �cos �

25. If � is a first quadrant angle and cos � � ��11�00��, find sin 2�. 25. ________

A. �3�

51�0�� B. �

35� C. � �

45� D. � �4

3�

26. Which expression is equivalent to sin (90° � �)? 26. ________A. �sin � B. tan � C. cos � D. �cos �

27. Simplify i17. 27. ________A. �i B. i C. 1 D. �1

28. Write the rectangular equation y � 1 in polar form. 28. ________A. r cos � � 1 B. r � sin � C. r sin � � 1 D. 2r sin � � 1

29. Simplify i5 � i3. 29. ________A. 0 B. 2i C. i D. �2i



30. Find the distance from P(1, 3) to the line with 30. __________________equation 3x � 2y � 4.

31. Solve 2 cos x � sec x � 1 for 0° x 180°. 31. __________________

32. If vv� has a magnitude of 20 and a direction of 140°, find the 32. __________________magnitude of its vertical and horizontal components.

33. Solve 5x2 � 10x � 6 � 3 by using the Quadratic Formula. 33. __________________

34. Describe the transformation that relates the graph of 34. __________________y � sin �x � ��2�� to the parent graph y � sin x.

35. Graph y � tan ��2�� 4�� 1. 35.

36. Given a central angle of 56°, find the length of its 36. __________________intercepted arc in a circle of radius 6 centimeters.Round your answer to the nearest thousandth.

37. If vv� � ��5, 1� and w� � �4, �6�, find vv� �2w�. 37. __________________

38. Write an equation in slope-intercept form of the line with 38. __________________parametric equations x � �3t � 2 and y � 4t � 5.

39. Find the distance between the lines with equations 39. __________________6y � 8x � 18 and 4x � 3y � 7.


NAME _____________________________ DATE _______________ PERIOD ________




NAME _____________________________ DATE _______________ PERIOD ________

40. Determine the rational zeros of ƒ(x) � 2x3 � 3x2 � 18x � 8. 40. __________________

41. State the amplitude and period for y � �4 cos x. 41. __________________

42. Write an equation of a cosine function with amplitude 42. __________________5 and period 6.

43. If sin � � �13� and cos � � �34�, find cos (� � �) if � is a first 43. __________________quadrant angle and � is a fourth quadrant angle.

44. Approximate the positive real zeros of the function 44. __________________ƒ(x) � x3 � 3x � 8 to the nearest tenth.

45. Evaluate �1, 5, �3� �2, 1, 1�. 45. __________________

46. Use the Law of Cosines to solve �ABC if a � 10, b � 40, 46. __________________and C � 120°. Round answers to the nearest tenth.

47. Simplify (3 � i)(4 � 2i). 47. __________________

48. Simplify (�1 � 5i) � (2 � 3i). 48. __________________

49. Write the rectangular form of the polar equation r � 3. 49. __________________

50. Express �3� � i in polar form. 50. __________________




1. Solve 62x�3 � 34.7. Round your answer to the nearest hundredth. 1. ________A. 2.49 B. �1.37 C. �0.51 D. �1.25

2. Evaluate limx→5

�xx3

2

��

12255�. 2. ________

A. �125� B. 0 C. undefined D. �5

2�

3. If the probability that it will rain on the day of the senior class 3. ________picnic is 0.4, what are the odds that it will not rain on that day?A. �35� B. �23� C. �32� D. �2

1�

4. Find the distance between the points at (�3, 0) and (1, 4). 4. ________A. 2 B. 4�2� C. 4 D. 3�3�

5. Find the fourth iterate of ƒ(x) � (x � 1)2 for x0 � �1. 5. ________A. 25 B. 1 C. 36 D. 9

6. Find the standard error of the mean for a sample with N � 400 6. ________and � � 15.A. 0.75 B. 0.0375 C. 26.67 D. 1.33

7. Which equation represents an ellipse? 7. ________A. 4x � 2y2 � 6y � 66 B. x2 � 2y2 � 8C. 2x2 � 5x � y � 19 � 0 D. 3x2 � 5x � y � 17

8. Solve 3x�1 � 17. Round your answer to the nearest hundredth. 8. ________A. 5.23 B. 1.58 C. 3.00 D. 3.38

9. Find the twenty-first term in the arithmetic sequence 9. ________8, 3, �2, �7, ....A. �97 B. �95 C. �105 D. �92


NAME _____________________________ DATE _______________ PERIOD ________




NAME _____________________________ DATE _______________ PERIOD ________

10. A band director checking the overnight rates at six area hotels 10. ________found the following prices: $80, $51, $72, $89, $68, and $60. What is the standard deviation of the hotel prices?A. $18.60 B. $12.45 C. $83.64 D. $56.36

11. Find the midpoint of the segment that has endpoints at (�2, �1) 11. ________and (2, 3).A. (�1, 1) B. (0, 0) C. (0, 1) D. (1, 0)

12. A landscaper has 4 rose bushes and 4 peony bushes that are all 12. ________different colors. In how many different ways can she plant the 8 bushes in a circular bed if she plants roses and peonies alternately?A. 288 B. 2520 C. 5040 D. 144

13. Find the rectangular equation of the curve whose parametric 13. ________equations are x � 3 sin 2t and y � �2 cos 2t, where 0 t 2�.

A. �x92� � �

y4

2

� � 1 B. �x42� � �

y9

2

� � 2

C. x2 � y2 � 4 D. �x92� � �

y4

2

� � 1

14. Use a calculator to evaluate 5��

4�

to the nearest ten-thousandth. 14. ________A. 6.8792 B. 4.3532 C. 25.2535 D. 3.5397

15. Which series is divergent? 15. ________

A. 1 � �21

3� � �31

3� � �41

3� � . . . B. �15� � �51

2� � �51

3� � . . .

C. �44� � �45� � �46� � �47� � . . . D. 2 � �23� � �29� � �227� � . . .

16. Find the median of the set {175, 235, 210, 256, 215, 198}. 16. ________A. 212.5 B. 235.5 C. 214.8 D. 210

17. Find the sum of the series 36 � 24 � 16 � . . . . 17. ________A. 72 B. 108 C. 162 D. 76



18. If you are proving “for all positive integers n, if n � 4, 18. ________then n! � 2n” by mathematical induction, what must you prove in the second part of the proof?A. If k! � 2k, then (k � 1)! � 2k�1.B. If 1! � 2!, then 4! � 24.C. If 2! � 24, then 5! � 25.D. If n! � 2n, then k! � 2k.

19. From a group of 5 science teachers and 3 mathematics teachers, 19. ________3 teachers will be selected to attend a conference on chaos theory.Determine the probability that exactly 1 mathematics teacher is chosen.A. �1

14� B. �12

58� C. �15

56� D. �5

16�

20. Choose the equation of a circle with center (�2, 1) and radius 3. 20. ________A. x2 � 6x � y2 � 8y � �9 B. x2 � y2 � 3C. (x � 2)2 � ( y � 1)2 � 3 D. x2 � 4x � y2 � 2y � 4

21. Find �1

0(x2 � 3x � 5)dx. 21. ________

A. ��169� B. 5 C. �1 D. ��2

3�

22. Express (x3y2)�n7

�

using radicals. 22. ________

A. �n

x�21�y�14� B. �n

x�10�y�9� C. �7

x�3n�y�2n� D. �7n

x�3y�2�


A. 7

k�1

(25 � 3k) B. 8

k�0

(25 � 3k)

C. 8

k�1

(22 � 3k) D. 7

k�0

(25 � 3k)

24. Identify the conic section represented by the equation 24. ________x2 � y2 � 2x � 3y � 6 � 0.A. parabola B. circle C. hyperbola D. ellipse

25. Solve log6 5(2y � 3) � log6 25. 25. ________A. 2 B. 4 C. �

7 �23�5�� D. 1


NAME _____________________________ DATE _______________ PERIOD ________




NAME _____________________________ DATE _______________ PERIOD ________

26. Use the Binomial Theorem to find the fourth term 26. __________________of ( p � 2q)9.

27. Determine the number of ways to choose 4 objects from 7. 27. __________________

28. Use a calculator to f ind antilog 0.0783 to the 28. __________________nearest hundredth.

29. A coin is tossed 8 times. What is the probability 29. __________________that exactly 3 heads will occur?

30. A set of data has a normal distribution with a mean of 100 30. __________________and a standard deviation of 10. What percent of the data is between 70 and 130?

31. Simplify ��8zx

�

3

3

y6

��13�

. 31. __________________

32. Find the first four terms of the geometric sequence for 32. __________________which a10 � 128 and r � 2.

33. In how many ways can 5 people be seated around a 33. __________________circular table?

34. Write the equation of a parabola that passes through the 34. __________________point at (0, 3), has a vertex at (1, 4), and opens downward.

35. Identify the conic section represented by the equation 35. __________________x2 � 2x � 3y � 2 � 0.

36. Find the interquartile range and the semi-interquartile 36. __________________range of {49, 28, 41, 33, 38, 43, 29, 40, 27, 35, 30, 41}.

37. What is the probability that a card drawn from a standard 37. __________________deck of 52 cards is a jack, a king, or a queen?

38. Graph the system of inequalities. 9x2 � 4y2 36 38.y2 � x � 1



39. Find the derivative of 3x3 � 2x2 � x � 1. 39. __________________

40. If $500 is invested in a savings account providing an 40. __________________annual interest rate of 5.6% compounded continuously,how much money will be in the account after 12 years?

41. Evaluate C(6, 2) � C(4, 1) � C(3, 2). 41. __________________

42. Find the center and foci of the ellipse �(x �

91)2

� � �( y �

43)2

� � 1. 42. __________________

43. Evaluate 10

n�12n. 43. __________________

44. A set of data has a normal distribution with a mean of 25 44. __________________and a standard deviation of 5. Find the probability that a value selected at random is less than 20.

45. Solve 250 � 5e0.45t. Round your answer to the 45. __________________nearest hundredth.

46. The annual salaries obtained from a random sample of 46. __________________50 engineers ranged from $51,000 to $135,000. Name the class limits of a frequency distribution with 5 class intervals drawn to represent the data.

47. Evaluate 7

k�2(3 � 5k). 47. __________________

48. The probability that Joseph will buy his lunch at school 48. __________________is �23�. Find the probability that Joseph will buy his lunch at least 3 days out of a 5-day school week.

49. Write an equation of the hyperbola whose vertices are 49. __________________at (�3, 5) and (�3, 1) with a focus at (�3, �2).

50. Graph the equation y � 2x�1. 50.


NAME _____________________________ DATE _______________ PERIOD ________



Final Test, Chapters 4–15

NAME _____________________________ DATE _______________ PERIOD ________


1. A 10-pound force and a 10�3�-pound force act on the same object. 1. ________The angle between the forces measures 90°. Find the measure of the angle that the resultant force makes with the 10�3�-pound force.A. 270° B. 30° C. 300° D. 60°

2. Find the eighth term of the geometric sequence 2. ________432, �216, 108, . . . .A. �3.375 B. �1.6875 C. 3.375 D. 1.6875

3. Write an equation of the cosine function with amplitude 2, period �, 3. ________and phase shift ��4�.

A. y � 2 cos 2�x � ��4�� B. y � 2 cos 2�x � ��2��C. y � 2 cos �2x � ��4�� D. y � 2 cos �2x � ��2��

4. Identify the conic section represented by the equation 4. ________x2 � 2x � y2 � 6y � 26 � 0.A. circle B. hyperbola C. ellipse D. parabola

5. For which measures does � A BC have no solution? 5. ________A. A � 30°, a � 5, b � 10B. B � 126°, b � 12, c � 7C. B � 40°, a � 12, b � 6D. A � 75°, B � 45°, c � 3

6. Find �(x5 � 1) dx. 6. ________

A. 5x4 � C B. x6 � x � C C. �16�x6 � C D. �16�x6 � x � C

7. The vector u� has a magnitude of 4.3 centimeters and a direction 7. ________of 45°. Find the magnitude of its vertical component.A. 2.45 cm B. 3.04 cm C. 3.25 cm D. 6.53 cm



8. Find one positive angle and one negative angle that are coterminal 8. ________with the angle �

34��.

A. �11

4��, ��

74�� B. �

114

��, ��54

�� C. 2�, � �4�

� D. �74��, � �4

��

9. How many ways can 5 students be chosen from a group of 4 juniors 9. ________and 6 seniors to be on the prom committee if exactly 3 students must be seniors?A. 720 B. 1440 C. 60 D. 120

10. Solve sin 2x � 3 cos x for all values of x. Assume that k is any integer. 10. ________A. 180� � 360k� B. 180° � 180k° C. 90� � 360k� D. 90° � 180k°

11. Write log5 �1125� � �3 in exponential form. 11. ________

A. 5�3 � �1125� B. ��15��

�3� �1

125� C. �

35� � �1

125� D. ��

�15��

3� �1

125�

12. Which statement is true for a polynomial function with odd degree? 12. ________A. Its graph must cross the x-axis exactly once.B. Its graph must cross the x-axis at least once.C. Its graph must cross the x-axis more than once.D. Its graph does not cross the x-axis.

13. Find the degree measure of the central angle whose intercepted 13. ________arc measures 8 centimeters in a circle of radius 15 centimeters.A. 45.3° B. 3.0° C. 30.6° D. 25.2°

14. Identify the polar form of the linear equation 4x � 3y � 10. 14. ________A. 5 � r cos (� � 37°) B. 2 � r cos (� � 53°)C. 2 � r cos (� � 53°) D. 2 � r cos (� � 37°)

Final Test, Chapters 4–15 (continued)

NAME _____________________________ DATE _______________ PERIOD ________




NAME _____________________________ DATE _______________ PERIOD ________

15. Identify the equation of a parabola whose focus is at (5, �1) and 15. ________whose directrix is x � 3.A. x2 � 2x � 4y � 17 � 0 B. x2 � 2x � 4y � 5 � 0C. y2 � 2y � 4x � 17 � 0 D. y2 � 2y � 4x � 5 � 0

16. Find the area of � ABC if B � 30�, C � 120�, and a � 4. 16. ________A. 4 units2 B. 6.9 units2

C. 3.5 units2 D. 13.9 units2

17. Change 37.92° to radian measure to the nearest thousandth. 17. ________A. 1.987 B. 2.925 C. 8.245 D. 0.662

18. Find the value of cos 105° without using a calculator. 18. ________

A. ��2� �

4�3�

� B. ��3� �

2�2�

� C. ��2� �

4�6�

� D. ��2� �

4�6�

�

19. Find the interquartile range of {38, 26, 37, 61, 38, 51, 28, 32, 9, 20}. 19. ________A. 33.75 B. 12 C. 10 D. 34

20. Write the rectangular equation y � 1 in polar form. 20. ________A. r � cos � B. r � sec �C. r � csc � D. r � sin �

21. Which is the graph of y � sec x on the interval of 0° x 360°? 21. ________

A. B.

C. D.

22. Find the probability of getting a sum of 5 on the first toss of two 22. ________number cubes and a sum of 6 on the second toss.A. �3

524� B. �5

14� C. �14� D. �1

58�



23. Which graph represents the point at ��2, �56��? 23. ________

A. B.

C. D.

24. Find the fifth term in the binomial expansion of (2x � y3)6. 24. ________A. 15x2y12 B. 60x2y12 C. 60x2y7 D. �12xy15

25. Evaluate limx→0

�(x � 2x)2 � 4� or state that the limit does not exist. 25. ________

A. �14� B. 0 C. 4 D. Does not exist

26. Find the magnitude of AB� from A(2, 3, 5) to B(�1, 0, 4). 26. ________A. �1�1� B. �1�9� C. 3�1�1� D. �9�1�

27. Solve e�x� � 4. 27. ________A. 1.18 B. 1.92 C. 0.69 D. 7.39

28. State the period, phase shift, and vertical shift of the function 28. ________

y � tan ��2�

� � ��4�� 3.

A. 2�, ��2�, 3 B. �, ��

�2�, 3 C. 2�, ��

�4�, 3 D. 4�, ��2�, 3

29. Approximate the greatest real root of ƒ(x) � x3 � 2x � 5 to 29. ________the nearest tenth.A. 1.8 B. 2.0 C. 2.1 D. �0.6


A. 11

n�1(n2 � 2) B.

13

n�3(n2 � n) C.

13

n�3(n2 � 2) D.

9

n�22n2


NAME _____________________________ DATE _______________ PERIOD ________




NAME _____________________________ DATE _______________ PERIOD ________

31. Find the values of � for which the equation 31. __________________tan � � 1 is true.

32. Find the area of a sector if the central angle measures 24° 32. __________________and the radius of the circle is 36 centimeters. Round to the nearest tenth.

33. Write the equation of the circle that passes through the 33. __________________point at (3, 4) and has its center at (1, 2).

34. Evaluate �(x4 � 3x2) dx. 34. __________________

35. Write the equation x � y � 3�2� in normal form. Then, 35. __________________find the length p of the normal and the angle � it makes with the positive x-axis.

36. The weekly salaries of a sample of 100 recent graduates of 36. __________________a private women's college are normally distributed with a mean of $600 and a standard deviation of $80. Determine the interval about the sample mean that has a 1% level of confidence. Use t � 2.58.

37. Find the median of the data set 37. __________________{75, 200, 67, 101, 150, 2, 300, 45, 454, 53}.

38. Write the equation of the conic section with eccentricity 38. __________________0.8 and foci at (1, �3) and (9, �3).

39. Find the area of the shaded 39. __________________region in the graph at the right.

40. Does Sin�1 (�x) � �Sin�1 x? Write true or false. 40. __________________

41. A serial number is formed from the digits 1, 2, and 3, and 41. __________________the letters A and B. Given that the serial number ends in a letter, what is the probability that it ends in 2A?



42. Find the number of possible positive real zeros and the 42. __________________number of possible negative real zeros for ƒ(x) � x3 � 2x2 � 13x � 10. Then determine all the rational zeros.

43. During one year, the cost of tuition, room, and board at a 43. __________________state university increased 5%. If the cost continues to increase at a rate of 5% per year, how long will it take for the cost of tuition, room, and board to double? Round your answer to the nearest year.

44. A football is kicked with an initial velocity of 42 feet 44. __________________per second at an angle of 35° with the horizontal. How far has the football traveled horizontally after 0.5 second? Round your answer to the nearest tenth.

45. Find �3

8�i�. 45. __________________

46. Name four different pairs 46. __________________of polar coordinates that represent the point A.

47. Find the distance between points at (4, �1) and (�2, �5). 47. __________________

48. If csc � � 3.65, find sin � to the nearest thousandth. 48. __________________

49. List all possible rational zeros of ƒ(x) � 3x3 � x2 � 2x � 2. 49. __________________

50. Write the equation of the hyperbola for which the length 50. __________________of the transverse axis is 6 units and the foci are at (0, 5) and (0, �5).

51. Find the values of x in the interval 0° x 360° that 51. __________________

satisfy the equation x � arcsin ��22��.


NAME _____________________________ DATE _______________ PERIOD ________




NAME _____________________________ DATE _______________ PERIOD ________

52. Find the area of the region between the graph of y � x3 52. __________________and the x-axis from x � 1 to x � 4.

53. Write a polynomial equation of least degree with 53. __________________roots 1, i, and �i.

54. Find the rectangular coordinates of the point with polar 54. __________________

coordinates �2, �76��.

55. Use a sum or difference identity to find the value of 55. __________________cos 255°.

56. How many different ways can nine people line up to 56. __________________take turns hitting a piñata?

57. Write �1 � i in exponential form. 57. __________________

58. Find the first three iterates of the function 58. __________________ƒ(z) � z � (2 � i) if the initial value is z0 � �2i.

59. Express 5 � 2i in polar form. Express your answer in 59. __________________radians to the nearest hundredth.

60. Simplify i51. 60. __________________

61. Graph y � 2 sin (2x � �) 61.on the interval 0 x 2�.

62. Evaluate P(4, 2) � C(5, 3). 62. __________________

63. Given that x is an integer, state the relation represented 63. __________________by y � �2

x� and 0 x 2 by listing a set of ordered pairs.Then state whether the relation is a function. Write yes or no.



64. Write MN� as a sum of unit vectors for M(�1, 3, �2) 64. __________________and N(�7, 5, 1).

65. How many ways can 6 pins be arranged around a circular 65. __________________hat band?

66. Evaluate �5

2(4x3 � x2 � 2) dx. 66. __________________

67. Two cards are drawn from a standard deck of 52 cards. 67. __________________What is the probability of drawing 2 queens or 2 red cards?


69. Use a calculator to evaluate 3�0�.5�

to the nearest hundredth. 69. __________________

70. Simplify sin x � �sceoct x

x�. 70. __________________

71. Find the distance between the lines with equations 71. __________________3x � 2y � 1 � 0 and y � �32� x � 3.

72. Find the mode of the data set 72. __________________{85, 98, 3, 104, 28, 3, 1, 3, 100, 206, 79}.

73. A publisher is liquidating a supply of 40,000 sheets of 73. __________________children’s stickers by including a sheet of stickers in each subscription request. If the publisher estimates that 20% of the remaining supply is used each year,how many sheets of stickers are remaining at the end of 4 years?

74. Express tan 315° as a function of an angle in Quadrant I. 74. __________________

75. Write an equation in slope-intercept form of the line 75. __________________whose parametric equations are x � 3 � 6t and y � �5 � 2t.


NAME _____________________________ DATE _______________ PERIOD ________




NAME _____________________________ DATE _______________ PERIOD ________


1. Given that x is an integer and 0 x 4, state the relation represented 1. ________by y � x2 � 1 by listing a set of ordered pairs. Then state whether the relation is a function.A. {(0, �1), (1, 0), (2, 3), (3, 8)}; noB. {(0, �1), (1, 0), (2, 3), (3, 8)}; yesC. {(�1, 0), (0, 1), (3, 2), (8, 3)}; noD. {(0, 0), (1, 1), (2, 4), (3, 9)}; yes

2. Find the zero of the function ƒ(x) � 2x � 3. 2. ________A. � �23� B. �23� C. �32� D. � �2

3�

3. Which angle is not coterminal with �45°? 3. ________

A. ��4� B. 315° C. �

72�� D. 675°

4. Evaluate cos �Cos�1 �12��. 4. ________

A. ��22�� B. �

�23�� C. �

12� D. 1

5. Which function is an even function? 5. ________A. y � x3 B. y � x2 � x C. y � x|x| D. y � �x6 � 5

6. Find the slope of the line passing through (�1, 1) and (1, 3). 6. ________A. �1 B. 1 C. 2 D. undefined

7. Write the equation of the line that passes through (1, 2) and is parallel 7. ________to the line x � 3y � 1 � 0.A. 3x � y � 5 � 0 B. x � 3y � 7 � 0

C. x � 3y � 5 � 0 D. �13�x � y � 3 � 0

8. Choose the graph of ƒ(x) � �|x � 2| � 2. 8. ________A. B. C. D.


9. Write the slope-intercept form of the equation of the line passing 9. ________through (�1, 0) and (2, 3).A. y � x � 1 B. y � 2x � 2 C. y � �x � 1 D. y � 3x � 1

10. Write the polynomial equation of least degree with a leading 10. ________coefficient of 1 and roots 0, 2i, and �2i.A. x2 � 4 � 0 B. x3 � 4x � 0 C. 4x2 � 0 D. 4x3 � 0

11. Find the inverse of y � (x � 1)3. 11. ________A. y � �

3x� � 1 B. y � �

3x� � 1 C. y � �

3x� �� 1� D. y � �

3x� �� 1�

12. Choose the graph of y � 2 � |x � 1|. 12. ________A. B. C. D.

13. Find the x-intercept(s) of the graph of the function 13. ________ƒ(x) � (x � 2)(x2 � 25).A. 2, 5 B. 2, 25 C. �2, �25 D. �5, 2, 5

14. Find A � B if A � � and B � �. 14. ________

15. Find the discriminant of 2m2 � 3m � 1 � 0 and describe the nature 15. ________of the roots.A. �6, imaginary B. 3, realC. 4, imaginary D. 1, real

16. Which best describes the graph of ƒ(x) � �xx2

��

24�? 16. ________

A. The graph has infinite discontinuity.B. The graph has jump discontinuity.C. The graph has point discontinuity.D. The graph is continuous.

17. Solve sin � � 1 for all values of �. Assume k is any integer. 17. ________A. 90° � 360k° B. 360k°C. 180° � 360k° D. 270° � 360k°

23

21

42

53



NAME _____________________________ DATE _______________ PERIOD ________

A. � B. � C. � D. �97

�26

�13

23

2�1

32

65

74


A. � B. � C. � D. �11

2�32�

�12

1��32�

2�1

32

�24

2�3



NAME _____________________________ DATE _______________ PERIOD ________

18. Find the value of � �. 18. ________

A. �5 B. 13 C. 5 D. 2

19. List all possible rational zeros of ƒ(x) � 2x3 � 3x2 � 2x � 5. 19. ________

A. �1, �2, �5 B. �1, �5, �3, � �12�, � �23�

C. �1, �5, � �12�, � �52� D. �1, �5

20. A section of highway is 5.1 kilometers long and rises at a uniform 20. ________grade, making a 2.9° angle with the horizontal. What is the change in elevation of this section of highway to the nearest thousandth?A. 5.093 km B. 0.258 km C. 4.193 km D. 0.276 km

21. Use the Remainder Theorem to find the remainder for 21. ________(x3 � 2x2 � 2x � 3) � (x � 1).A. 4 B. 1 C. 2 D. 5

22. Find the inverse of �. 22. ________

23. The graph of y � x4 � 1 is symmetric with respect to 23. ________A. the x-axis. B. the y-axis.C. the line y � x. D. the line y � �x.

24. Find one positive and one negative angle that are coterminal with 24. ________an angle measuring �

34��.

A. �34��, ��

112

�� B. �

54��, ��

114

�� C. �

114

�� , ��

54�� D. �

32��, � �4

��

25. If sin � � ��22�� and � lies in Quadrant III, find cot �. 25. ________

A. ��33�� B. 1 C. �3� D. �1

22

43

23

32



26. State the amplitude, period, and phase shift of the 26. __________________function y � 3 sin (2x � �).

27. Find the value of Cos�1 �sin ��6��. 27. __________________

28. Find the values of x and y for which �� . 28. __________________

29. If � is a first quadrant angle and cos � � �13�, find sin �. 29. __________________

30. Find the value of sec � for angle � in standard position 30. __________________if a point with coordinates (�3, 4) lies on its terminal side.

31. Given ƒ(x) � x2 � |x|, find ƒ(�2). 31. __________________

32. Find the slope and y-intercept of the line passing 32. __________________through (�2, 4) and (3, 2).

33. Determine the slant asymptote for the graph 33. __________________

of ƒ(x) � �x2

x�

�x

2�1�.

34. Determine whether the system is consistent and 34. __________________independent, consistent and dependent,or inconsistent.

�6x � 3y � 0�4x � 2y � 2

35. Solve 2 sin x � csc x � 0 for 0° x 180°. 35. __________________

36. If ƒ(x) � x2 � 1 and g(x) � x � 1, find [ ƒ � g](x). 36. __________________

37. Write the slope-intercept form of the equation of the line 37. __________________passing through (1, 7) and (�3, �1).

38. Solve 2x2 � 4x � 2 � 1 by using the Quadratic Formula. 38. __________________

39. Describe the transformation that relates the graph of 39. __________________y � tan �x � �

�4�� to the parent graph y � tan x.

3 � y62

xx � 2

2y


NAME _____________________________ DATE _______________ PERIOD ________




NAME _____________________________ DATE _______________ PERIOD ________

40. Find AB if A � � and B � �. 40. __________________

41. Write the equation for the inverse of y � arctan x. Then 41. __________________graph the function and its inverse.

42. Given a central angle of 75°, find the length of the angle’s 42. __________________intercepted arc in a circle of radius 8 centimeters. Round to the nearest thousandth.

43. Determine whether the graph of y � �x23� � 3 has infinite 43. __________________

discontinuity, jump discontinuity, point discontinuity,or is continuous.

44. Determine the rational zeros of ƒ(x) � 6x3 � 11x2 � 6x � 1. 44. __________________

45. State the amplitude and period of y � �3 cos 2x. 45. __________________

46. Find an equation for a sine function with amplitude 2, 46. __________________period �, phase shift 0, and vertical shift 1.

47. Find the value of � �. 47. __________________

48. Find the inverse of �, if it exists. 48. __________________

49. The function ƒ(x) � �2x2 � 4x � 1 has a critical point 49. __________________when x � 1. Identify the point as a maximum, a minimum,or a point of inflection, and state its coordinates.

50. Use the Law of Cosines to solve � ABC with a � 15, 50. __________________b � 20, and C � 95°. Round to the nearest tenth.

12

23

35

22

�15

32

2�3

14




1. Solve 2x�1 � 17.6. Round your answer to the nearest hundredth. 1. ________A. 1.99 B. 3.14 C. �2.45 D. �3.84

2. Simplify i29 � i20. 2. ________A. 0 B. �2 C. �1 � i D. �2i

3. Find the third iterate of the function ƒ(x) � 2x � 1, if the initial 3. ________value is x0 � 3.A. 15 B. 31 C. 7 D. 12

4. Find the nineteenth term in the arithmetic sequence 10, 7, 4, 1, .... 4. ________A. 102 B. 0 C. �47 D. �44

5. If tan � � � �14� and � has its terminal side in Quandrant II, find the 5. ________exact value of sin �.A. �4 B. ��1�7� C. 2 D. �

�11�77�

�

6. Which expression is equivalent to tan ��2�

� ��? 6. ________

A. sin � B. cot � C. �cos � D. sec �

7. Find the ordered pair that represents the vector from A(1, �2) 7. ________to B(2, 3).A. �2, 3� B. �1, 2� C. �1, 5� D. �3, 1�

8. Which series is divergent? 8. ________A. 1 � �

21

3� � �

31

3� � �

41

3� � . . . B. 1 � �1

2� � �1

3� � . . .

C. 1 � �12� � �14� � �18� � . . . D. 2 � �23� � �29� � �227� � . . .

9. Which represents the graph of (3, 45°)? 9. ________A. B.

C. D.

Semester Test, Chapters 7–9, 11, and 12

NAME _____________________________ DATE _______________ PERIOD ________



Semester Test, Chapters 7–9, 11, and 12(continued)

NAME _____________________________ DATE _______________ PERIOD ________

10. Express x�29� y�

13� using radicals. 10. ________

A. �3

x�2y� B. �9

x�2y� C. �9

x�2y�3� �3

x�3y�

11. Which expression is equivalent to sin2 � for all values of �? 11. ________A. sin ��

�2� � �� B. 1 C. 2 sin � cos � D. 1 � cos2 �

12. Find the inner product of v� and w� if v� � �1, 2, 0� and w� � �3, �2, 1�. 12. ________A. 3 B. 2 C. �1 D. 1

13. Simplify (1 � i)(�2 � 2i). 13. ________A. 3 � 2i B. 2 � i C. �2 � 3i D. �4

14. Which equation is a trigonometric identity? 14. ________A. sin 2� � sin � cos � B. tan � � �s

cions��

C. tan 2� � �1

2�

ttaann

�2 �

� D. cos 2� � 4 cos2 � � 1

15. Write MN� as the sum of unit vectors for M(�11, 6, �7) and 15. ________N(4, �3, �15).A. �7i� � 3j� � 22k�� B. 15i� � 9j� � 8k��

C. �15i� � 9j�� 8k�� D. 7i� � 9j� � 8k��

16. Express �5

x�25�y�2� using rational exponents. 16. ________

A. x5y�52� B. x2y2 C. x5y�5

2� D. x�

12�y�5

2�

17. Express (x2 y3)�23� using radicals. 17. ________

A. �2

x�4y�6� B. �2

x�6y�9� C. �3

x�2y�3� D. �3

x�4y�6�

18. Which polar equation represents a rose? 18. ________A. r � 3� B. r � 3 � 3 sin �C. r � 3 cos 2� D. r2 � 4 cos 2�

19. Use a sum or difference identity to find the exact value of cos 105°. 19. ________

A. ��42�� B. �

�2� �4

�6�� C. �

�2� �4

�6�� D. �

�2� �

2�3�

�



20. Write the equation of the line 2x � 3y � 6 in parametric form. 20. ________

A. x � t; y � ��23�t � 2 B. x � 2t; y � 2t � 3

C. x � t; y � ��32�t � 2 D. x � 3t; y � t � 2

21. Find the sum of the geometric series �13� � �19� � �217� � . . . . 21. ________

A. �27� B. �12� C. �23� D. 1

22. Which is the graph of the equation � � �34��? 22. ________

A. B.

C. D.

23. Solve log6 x � 2. 23. ________A. 3 B. 36 C. 12 D. 4

24. Express 0.3 � 0.03 � 0.003 � . . . using sigma notation. 24. ________

A. ∞

k�1

3 � 10�k B. ∞

k�1

0.3 � 10�k

C. ∞

k�0

3 � 10�k D. ∞

k�1

3 � 10�2k

25. Andre kicks a soccer ball with an initial velocity of 48 feet per 25. ________second at an angle of 17° with the horizontal. After 0.35 second,what is the height of the ball?A. 16.07 ft B. 4.91 ft C. 14.11 ft D. 2.95 ft

Semester Test, Chapters 7–9, 11, and 12 (continued)

NAME _____________________________ DATE _______________ PERIOD ________



Semester Test, Chapters 7–9, 11 and 12(continued)

NAME _____________________________ DATE _______________ PERIOD ________

26. Find v� w� if v� � �2, �4, 6� and w� � �2, 1, �1�. 26. __________________

27. Write an equation in slope-intercept form of the line whose 27. __________________parametric equations are x � 4t � 3 and y � �2t � 7.

28. Use a calculator to find antiln (�0.23) to the nearest 28. __________________hundredth.

29. Find the first four terms of the geometric sequence for 29. __________________which a9 � 6561 and r � 3.

30. Find the polar coordinates of the point with rectangular 30. __________________coordinates (2, 2).

31. Find the rectangular coordinates of the point with polar 31. __________________coordinates ��2�, �

34��.

32. Express cos 840° as a trigonometric function of an angle 32. __________________in Quadrant II.

33. Solve 2 sin2 x � sin x � 0 for principle values of x. 33. __________________Express in degrees.

34. Find the distance between the point P(1, 0) and the line 34. __________________with equation 2x � 3y � �2.

35. If cos � � �34� and � has its terminal side in Quadrant I, find 35. __________________the exact value of sin �.

36. Express 2 � 4 � 6 � 8 � 10 using sigma notation. 36. __________________

37. Find limn→∞

�(3n � 2

n)2(n � 5)�. 37. __________________

38. Use a calculator to evaluate 3�3

�� to the nearest 38. __________________ten-thousandth.



39. Evaluate ��19��

12�

. 39. __________________

40. Use the Binomial Theorem to find the fourth term in 40. __________________the expansion of (2c � 3d)8.

41. Find the ordered triple that represents the vector 41. __________________from C(2, 0, 1) to D(6, 4, 3).

42. Find the distance between the lines with equations 42. __________________3x � 2y � 1 � 0 and 3x � y � 1 � 0.

43. Find an ordered pair to represent u� in 43. __________________u� � 2w� � v� if w� � �2, 3� and v� � �5, 5�.

44. Write the rectangular equation y � 1 in polar form. 44. __________________

45. Write the polar equation � � ��2� in rectangular form. 45. __________________

46. If CD� is a vector from C(1, 2, �1) to D(2, 3, 2), find 46. __________________the magnitude of CD�.

47. Are the vectors �2, �2, 1� and �3, 2, �2� perpendicular? 47. __________________Write yes or no.

48. Solve 235 � 4e0.35t. Round your answer to the 48. __________________nearest hundredth.


50. Graph the equation y � 3x�1. 50.

Semester Test, Chapters 7–9, 11, and 12 (continued)

NAME _____________________________ DATE _______________ PERIOD ________



Final Test, Chapters 1–9, 11, and 12

NAME _____________________________ DATE _______________ PERIOD ________


1. Given ƒ(x) � 2x � 4 and g(x) � x3 � x � 2, find ( ƒ � g)(x). 1. ________A. x3 � x � 2 B. x3 � 3x � 6C. �x3 � 3x � 2 D. �x3 � x � 2

2. Describe the graph of ƒ(x) � �xx2 �

�255�. 2. ________

A. The graph has infinite discontinuity.B. The graph has jump discontinuity.C. The graph has point discontinuity.D. The graph is continuous.

3. Choose the graph of the relation whose inverse is a function. 3. ________A. B. C. D.

4. Which is the graph of the system? x � 0 4. ________y � 02x � y � 2

A. B. C. D.

5. For which measures does � ABC have no solution? 5. ________A. A � 60°, a � 8, b � 8B. A � 70°, b � 10, c � 10C. A � 45°, a � 1, b � 20D. A � 60°, B � 30°, c � 2



6. Choose the amplitude, period, and phase shift of the function 6. ________y � �13� cos (2x � �).

A. �13�, �, ��2� B. �13�, �, � C. 3, �, ��2� D. �13�, �, �4��

7. Given ƒ(x) � x � 1 and g(x) � x2 � 2, find [ g ° ƒ ](x). 7. ________A. x2 � 2x � 3 B. x2 � 2x � 1 C. x2 � 3 D. x3 � x2 � 2x � 2

8. Find one positive angle and one negative angle that are coterminal 8. ________with the angle �

34��.

A. �54��, ��

114

�� B. �

23��, ��

54�� C. �

114

��, � �

�4� D. �

114

��, ��

54��

9. Simplify �1 �si

cno�s2 ��. 9. ________

A. �csoins

�� B. tan � C. sin � D. cos �

10. Find an ordered triple to represent u� in u� � 2v� � 3w� if v� � ��1, 0, 2� 10. ________and w� � �2, 3, 1�.A. �2, 3, 1� B. �4, 3, 2� C. �3, 1, �2� D. �4, 9, 7�

11. Write log7 49 � 2 in exponential form. 11. ________A. 27 � 49 B. 72 � 49

C. 7�12

�� 49 D. 49

�12

�� 7

12. Write parametric equations of �4x � 5y � 10. 12. ________A. x � t; y � �45�t � 2 B. x � t; y � 4t � 10C. x � t; y � �4t � 5 D. x � t; y � �54�t � 2

13. Describe the transformation that relates the graph of y � 100x3 13. ________to the parent graph y � x3.A. The parent graph is ref lected over the line y � x.B. The parent graph is stretched horizontally.C. The parent graph is stretched vertically.D. The parent graph is translated horizontally right 1 unit.

14. Identify the polar form of the linear equation �3�x � y � 4. 14. ________A. 4 � r cos (� � 30°) B. 2 � r cos (� � 30°)C. 3 � r sin (� � 70°) D. 2 � r cos (� � 42°)

Final Test, Chapters 1–9, 11, and 12 (continued)

NAME _____________________________ DATE _______________ PERIOD ________




NAME _____________________________ DATE _______________ PERIOD ________

15. Which function is an even function? 15. ________A. y � x4 � x2 B. y � x3

C. y � x4 � x5 D. y � 2x4 � 3x

16. Find the area of � ABC to the nearest tenth if B � 30°, C � 70°, 16. ________and a � 10.A. 15.2 units2 B. 30.3 units2

C. 19.2 units2 D. 23.9 units2

17. The line with equation 9x � y � 2 is perpendicular to the line that 17. ________passes throughA. (0, 0) and (1, 3). B. (1, 2) and (10, 3).C. (2, �5) and (3, 4). D. (10, 4) and (1, 5).

18. Find the 17th term of the arithmetic sequence �13�, 1, �53�, . . . . 18. ________

A. 9 B. �137� C. 11 D. 7

19. Solve 2x2 � 4x � 1 � 0 by using the Quadratic Formula. 19. ________

A. 1 � ��22�� B. �1 � �

�22�� C. 2, 4 D. �1 � �6�

20. What are the dimensions of matrix AB if A is a 2 3 matrix 20. ________and B is a 3 7 matrix?A. 7 2 B. 3 3 C. 3 2 D. 2 7

21. Find the polar coordinates of the point whose rectangular 21. ________coordinates are (�1, 1).

A. ��2�, ��4�� B. ��2�, ��

34��

C. ��2�, � ��4�� D. ��2�, �

34��

22. Find the fifteenth term of the geometric sequence �53�, �56�, �152�, . . . . 22. ________

A. �1965,608� B. �49,

5152� C. �98,

5304� D. �2

1�

23. Find the magnitude of AB� from A(2, 4, 0) to B(1, 2, 2). 23. ________A. �7� B. 2�3� C. 3 D. �5�



24. Choose the graph of y � csc x on the interval 0 � x � 2�. 24. ________A. B.

C. D.

25. Without graphing, describe the end behavior of y � �x4 � 2x3 � x2 � 1. 25. ________A. y → ∞ as x → ∞; y → ∞ as x → �∞B. y →�∞ as x → ∞; y → �∞ as x → �∞C. y →�∞ as x → ∞; y → ∞ as x → �∞D. y → ∞ as x → ∞; y → �∞ as x → �∞

26. Which expression is equivalent to sin (90° � �)? 26. ________A. sin � B. cos � C. tan � D. sec �

27. Describe the transformation that relates the graph of y � ��x� �� 3� 27. ________to the parent graph y � �x�.A. The parent graph is reflected over the x-axis and translated left 3 units.B. The parent graph is compressed vertically and translated right 3 units.C. The parent graph is reflected over the x-axis and translated up 3 units.D. The parent graph is reflected over the y-axis and translated left 3 units.

28. Express cos 854° as a function of an angle in Quandrant II. 28. ________A. tan 134° B. sin 64° C. cos 134° D. �cos 134°

29. Express �3

2�7�a�9b�2� by using rational exponents.

A. 3a3b�32� B. 9a6b�23

�C. 27a3b

�23

�D. 3a3b

�23

�29. ___________________________________

30. Identify the equation of the tangent function with period 2�, 30. ________phase shift �

�2�, and vertical shift �5.

A. y � tan (x � �) � 5 B. y � tan ��12�x � ��4�� 5

C. y � �5 tan ��12�x � ��4�� D. y � tan �2x � �

�2��


NAME _____________________________ DATE _______________ PERIOD ________




NAME _____________________________ DATE _______________ PERIOD ________

31. Given ƒ(x) � x3 � 2x, find ƒ(�2). 31. __________________

32. Find the area of a sector if the central angle measures 32. __________________35° and the radius of the circle is 45 centimeters. Round to the nearest tenth.

33. Find A � B if A � � and B � � . 33. __________________

34. Solve the system of equations algebraically. 34. __________________3x � 2y � 3�2x � 5y � 17

35. Find the inverse of y � (x � 1)3. 35. __________________

36. State the domain and range of the relation 36. __________________{(�2, 5), (0, 3), (4, 5), (9, �3)}. Then state whether the relation is a function.

37. Graph y �|x � 1| � 2. 37.

38. Find the sum of the infinite geometric 38. __________________series 35 � 7 � 1.4 � . . . .

39. Find AB if A � � and B � � . 39. __________________

40. Simplify sin � csc � (sin2 � � cos2 �). 40. __________________

41. A football is kicked with an initial velocity of 38 feet per 41. __________________second at an angle of 32° to the horizontal. How far has the football traveled horizontally after 0.25 second? Round your answer to the nearest tenth.

42. Find the sum of the first 23 terms of the arithmetic 42. __________________series �10 � �3 � 4 � . . . .

23

3�1

�20

31

�14

22

33

1�2



43. Graph the polar equation � � ��3�. 43.

44. During one year, the cost of tuition, room, and board at a 44. __________________state university increased 6%. If the cost continues to increase at a rate of 6% per year, how long will it take for the cost of tuition, room, and board to triple? Round your answer to the nearest year.

45. Name four different pairs 45. __________________of polar coordinates thatrepresent the point A.

46. Evaluate log7 15 to four decimal places. 46. __________________

47. Use a calculator to approximate the value of sec (�137°) 47. __________________to four decimal places.

48. Find the sum of the first 13 terms of the geometric series 48. __________________2 � 6 � 18 � . . . .

49. Find the exact value of sin 105°. 49. __________________

50. Find the inverse of � . 50. __________________21

52


NAME _____________________________ DATE _______________ PERIOD ________

51. Find the value of . 51. __________________�1

423




NAME _____________________________ DATE _______________ PERIOD ________

52. The vector u� has a magnitude of 4.5 centimeters and a 52. __________________direction of 56°. Find the magnitude of its horizontal component. Round to the nearest hundredth.

53. Graph the equation y � 2 sin (2x � �). 53.


55. Use a sum or difference identity to f ind the value of 55. __________________cos 285°.

56. Solve e�1x

�� 5. Round your answer to the nearest hundredth. 56. __________________

57. Write MN� as a sum of unit vectors for M(2, 1, 0) 57. __________________and N(5, �3, 2).

58. Express 3 � 5 � 9 � 17 � . . . � 513 using sigma notation. 58. __________________

59. Express 2 � 3i in polar form. Express your answer in 59. __________________radians to the nearest hundredth.

60. Find the third iterate of the function ƒ(z) � 2z � 1, if the 60. __________________initial value is z0 � 1 � i.

61. Find the rectangular coordinates of the point whose polar 61. __________________

coordinates are �2�2�, �54��.

62. Find the discriminant of w2 � 4w � 5 � 0 and describe the 62. __________________nature of the roots of the equation.

63. Find the value of Cos�1 ��12�� in degrees. 63. __________________

64. State the degree and leading coefficient of the polynomial 64. __________________function ƒ(x) � �6x3 � 2x4 � 3x5 � 2.



65. Write an equation in slope-intercept form of the line with 65. __________________the given parametric equations. x � �3t � 5

y � 1 � t

66. Solve the system of equations. 66. __________________x � 5y � z � 12x � y � 2z � 2x � 3y � 4z � 6

67. Use the Remainder Theorem to find the remainder when 67. __________________2x3 � x2 � 2x is divided by x � 1.

68. Find the value of cos �Sin�1 ��23��. 68. __________________

69. Solve log5 x � log5 9 � log5 4. 69. __________________

70. Find the rational zero(s) of ƒ(x) � 3x3 � 2x2 � 6x � 4. 70. __________________

71. Find the number of possible positive real zeros of 71. __________________ƒ(x) � 3x4 � 2x3 � 3x2 � x � 3.

72. List all possible rational zeros of ƒ(x) � 3x3 � bx2 � cx � 2, 72. __________________where b and c are integers.

73. Given a central angle of 87°, f ind the length of its 73. __________________intercepted arc in a circle of radius 19 centimeters.Round to the nearest tenth.

74. Solve � ABC if A � 27.2°, B � 76.1°, and a � 31.2. Round 74. __________________to the nearest tenth.

75. Solve � ABC if A � 40°, b � 10.2, and c � 5.7. Round to 75. __________________the nearest tenth.


NAME _____________________________ DATE _______________ PERIOD ________



Semester Test Chapters 4–9Page 427

1. D

2. B

3. B

4. A

5. C

6. C

7. A

8. D

9. B

10. C

Page 428

11. D

12. D

13. B

14. B

15. B

16. A

17. B

18. B

19. A

Page 429

20. A

21. C

22. C

23. A

24. A

25. B

26. C

27. B

28. C

29. A

Answer Key

463-476 A&E 0-02-834178-1 10/10/00 11:04 AM Page 463

Page 430

30. �5�13

1�3��

31. 0°, 120°

32. 12.86, �15.32

33. ��5 �5

�1�0��

34.

35.

36. 5.864 cm

37. ��13, 13�

38. y � ��43

�x � �37�

39. 3.2 units

Page 431

40. �2, �0.5, 4

41. 4, 2�

42.

43. �6�2�1�2

�7��

44. 1.5

45. �8, �7, �9�

A � 10.9�, B � 49.1�, 46. c � 45.8

47. 14 � 2i

48. 1 � i

49. x2 � y2 � 9

50. 2�cos ��6

� � i sin ��6

��

Semester Test Chapters 10–15Page 432

1. C

2. A

3. C

4. B

5. A

6. A

7. B

8. B

9. D


Answer Key

translated ��2

� units to theright

y � �5 cos ��3��

463-476 A&E 0-02-834178-1 10/10/00 11:04 AM Page 464


Page 433

10. B

11. C

12. D

13. A

14. D

15. C

16. A

17. B

Page 434

18. A

19. B

20. D

21. A

22. A

23. D

24. B

25. B

Page 435

26. �672p6q3

27. 35

28. 1.20

29. �372�

30. 99.7%

31. 2xy2z

32. �14

�, �12

�, 1, 2

33. 24

34. (x � 1)2 � �(y � 4)

35. parabola

36. 11.5; 5.75

37. �133�

38.

Answer Key

463-476 A&E 0-02-834178-1 10/10/00 11:04 AM Page 465

Page 436

39. 9x2 � 4x � 1

40. $979.07

41. 180

42.

43. 110

44. 15.85%

45. 8.69

46.

47. �117

48. �86

14�

49.�(y �4

3)2� � �

(x �21

3)2� � 1

50.

Final Test Chapters 4–15Page 437

1. B

2. A

3. A

4. A

5. C

6. D

7. B

Page 438

8. B

9. D

10. D

11. A

12. B

13. C

14. D


Answer Key

center: (1, �3) foci: (1 � �5�, �3)

Sample answer: 50,000, 70,000, 90,000, 110,000, 130,000, 150,000

463-476 A&E 0-02-834178-1 10/10/00 11:04 AM Page 466


Page 439

15. C

16. B

17. D

18. D

19. B

20. C

21. D

22. A

Page 440

23. D

24. B

25. A

26. B

27. B

28. A

29. C

30. C

Page 441

31. ��4

� � �k

32. 271.4 cm2

33. (x � 1)2 � (y � 2)2 � 8

34. �15

�x5 � x3 � C

35. ��22��x � ��2

2��y � 3 � 0; 3; 45�

36. $579.36 � $620.64

37. 88

38. �(x �

255)2� � �

(y �9

3)2� � 1

39. �536� units2

40. true

41. �81�

Answer Key

463-476 A&E 0-02-834178-1 10/10/00 11:04 AM Page 467

Page 442

42. 0 or 2; 1; 1, 2, �5

43. 14 yrs

44. 17.2 ft

45. �3� � i

46.

47. 2�1�3�

48. 0.274

49. �1, �2, ��13

�, ��32�

50. �y9

2� � �

1x62� � 1

51. 45°, 135°

Page 443

52. �2545� units2

53. x3 � x2 � x � 1 � 0

54. (��3�, � 1)

55. ��2� �

4�6��

56. 362,880

57. �2�ei �

54��

58.2 � 3i, 4 � 4i, 6 � 5i

59.�2�9�(cos 5.90 � i sin 5.90)

60. �i

61.

62. 22

63.

Page 444

64. �6i� � 2j� � 3k�

65. 120

66. 576

67.

68. 1

69. 2.17

70. csc x

71. �7�13

1�3��

72. 3

73. 16,384

74. �tan 45�

75. y � ��13

�x � 4

55�221


Answer Key

Sample answers: (�8, 60�), (8, �120�), (8, 240�), (�8, �300�)

{(0, 0), (1, 0.5), (1, �0.5), (2, 1), (2, �1)}; no

463-476 A&E 0-02-834178-1 10/10/00 11:04 AM Page 468


Semester TestChapters 1–6Page 445

1. B

2. C

3. C

4. C

5. D

6. B

7. B

8. C

Page 446

9. A

10. B

11. B

12. C

13. D

14. B

15. D

16. C

17. A

Page 447

18. C

19. C

20. B

21. A

22. C

23. B

24. C

25. B

Answer Key

463-476 A&E 0-02-834178-1 10/10/00 11:04 AM Page 469

Page 448

26. 3, �, �2��

27. �3��

28. (4, 1)

29. �2�3

2��

30. ��35�

31. 6

32. ��25

�, �156�

33. y � x � 1

34. inconsistent

35. 45°, 135°

36. x2 � 2x � 2

37. y � 2x � 5

38. ��2 �2

�2��

39.

Page 449

40. � �41. y � tan x

42. 10.472 cm

43. infinite discontinuity

44. 1, �12

�, �31�

45. 3, �

46. y � �2 sin 2� � 1

47. 4

48. � �49. maximum, (1, 1)

50. c � 26.0, A � 35.0�, B � 50.0�

Semester TestChapters 7–9, 11, and 12

Page 450

1. B

2. C

3. B

4. D

5. D

6. B

7. C

8. B

9. A

�12

2�3

9�19

7 6


Answer Key

translated ��4

� unitto the right

3

463-476 A&E 0-02-834178-1 10/10/00 11:04 AM Page 470


Page 451

10. C

11. D

12. C

13. D

14. C

15. B

16. C

17. D

18. C

19. C

Page 452

20. A

21. B

22. A

23. B

24. A

25. D

Page 453

26. ��2, 14, 10�

27. y � ��21�x � �1

21�

28. 0.79

29. 1, 3, 9, 27

30. �2�2�, ��4

��

31. (�1, 1)

32. cos 120�

33. 0�, 30�

34. �4�13

1�3��

35. ��47��

36. �5

k�12k

37. 3

38. 4.3938

Answer Key

463-476 A&E 0-02-834178-1 10/10/00 11:04 AM Page 471

Page 45439. 3

40. �48,384c5d3

41. �4, 4, 2�

42. �3�13

1�3��

43. �9, 11�

r � csc � or44. 1 � r cos ��

2��

45. x � 0

46. �1�1�

47. yes

48. 11.64

49. �171�

50.

Final TestChapters 1–9, 11, and 12

Page 455

1. D

2. C

3. C

4. B

5. C

Page 4566. A

7. A

8. D

9. C

10. D

11. B

12. A

13. C

14. B


Answer Key

463-476 A&E 0-02-834178-1 10/10/00 11:04 AM Page 472


Page 457

15. A

16. D

17. B

18. C

19. B

20. D

21. D

22. B

23. C

Page 458

24. B

25. B

26. B

27. A

28. C

29. D

30. B

Page 459

31. �4

32. 618.5 cm2

33. � �34. (�1, 3)

35. y � �3x�� 1

36.

37.

38. �1475�

39. � �40. 1

41. 8.1 ft

42. 1541

02

113

27

30

Answer Key

yesD � {�2, 0, 4, 9}R � {�3, 3, 5}

463-476 A&E 0-02-834178-1 10/10/00 11:04 AM Page 473

Page 460

43.

44. 19 yrs

45.

46. 1.3917

47. �1.3673

48. 1,594,322

49. ��6� �4

�2��

50. � �

51. 11

Page 461

52. 2.52 cm

53.

54. log3 �19

� � �2

55. ��6� �4

�2��

56. 0.62

57. 3i� � 4j� � 2k�

58. �9

n�1(2n � 1)

59. �1�3� (cos 5.30 � i sin 5.30)

60. 15 � 8i

61. (�2, �2)

62. �4; imaginary

63. 60°

64. 5; 3

Page 462

65. y � �13

�x � �23

�

66. (2, 0, �1)

67. 5

68. �21�

69. 36

70. �32�

71. 3 or 1

72. ��13

�, ��32�, �1, �2

73. 28.9 cm

74.

75.

�25

1�2


Answer Key

Sample answer: (2, 135°), (�2, �45°),(2, �225°), (�2, 315°)

c � 66.4, C � 76.7°,b � 66.3

a � 6.9, B � 107.9°,C � 32.1°

463-476 A&E 0-02-834178-1 10/10/00 11:04 AM Page 474

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Advanced Math Cocepts Evaluation