Chapter 2 Resource Masters - Math Class · PDF file©Glencoe/McGraw-Hill v Glencoe Algebra 2 Assessment Options The assessment masters in the Chapter 2 Resource Masters offer a wide

Chapter 2Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9

ANSWERS FOR WORKBOOKS The answers for Chapter 2 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-828005-2 Algebra 2Chapter 2 Resource Masters

1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02

Glencoe/McGraw-Hill

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 2-1Study Guide and Intervention . . . . . . . . . 57–58Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 59Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Reading to Learn Mathematics . . . . . . . . . . . 61Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 62







Chapter 2 AssessmentChapter 2 Test, Form 1 . . . . . . . . . . . . . 99–100Chapter 2 Test, Form 2A . . . . . . . . . . . 101–102Chapter 2 Test, Form 2B . . . . . . . . . . . 103–104Chapter 2 Test, Form 2C . . . . . . . . . . . 105–106Chapter 2 Test, Form 2D . . . . . . . . . . . 107–108Chapter 2 Test, Form 3 . . . . . . . . . . . . 109–110Chapter 2 Open-Ended Assessment . . . . . . 111Chapter 2 Vocabulary Test/Review . . . . . . . 112Chapter 2 Quizzes 1 & 2 . . . . . . . . . . . . . . . 113Chapter 2 Quizzes 3 & 4 . . . . . . . . . . . . . . . 114Chapter 2 Mid-Chapter Test . . . . . . . . . . . . . 115Chapter 2 Cumulative Review . . . . . . . . . . . 116Chapter 2 Standardized Test Practice . . 117–118

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 2 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 2 Resource Masters includes the core materials neededfor Chapter 2. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 2-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 2

Assessment OptionsThe assessment masters in the Chapter 2Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 106–107. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

22

© Glencoe/McGraw-Hill vii Glencoe Algebra 2

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 2.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

absolute value function

boundary

constant function

family of graphs

function

greatest integer function

identity function

linear equation

line of fit

one-to-one function

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

parent graph

piecewise function

PEES·WYZ

point-slope form

prediction equation

pree·DIHK·shuhn

relation

scatter plot

slope

slope-intercept form

IHN·tuhr·SEHPT

standard form

step function

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

22

Study Guide and InterventionRelations and Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

© Glencoe/McGraw-Hill 57 Glencoe Algebra 2

Less

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Graph Relations A relation can be represented as a set of ordered pairs or as anequation; the relation is then the set of all ordered pairs (x, y) that make the equation true.The domain of a relation is the set of all first coordinates of the ordered pairs, and therange is the set of all second coordinates.A function is a relation in which each element of the domain is paired with exactly oneelement of the range. You can tell if a relation is a function by graphing, then using thevertical line test. If a vertical line intersects the graph at more than one point, therelation is not a function.

Graph the equation y ! 2x " 3 and find the domain and range. Doesthe equation represent a function?

Make a table of values to find ordered pairs that satisfy the equation. Then graph the ordered pairs.

The domain and range are both all real numbers. Thegraph passes the vertical line test, so it is function.

Graph each relation or equation and find the domain and range. Then determinewhether the relation or equation is a function.

1. {(1, 3), (!3, 5), 2. {(3, !4), (1, 0), 3. {(0, 4), (!3, !2),(!2, 5), (2, 3)} (2, !2), (3, 2)} (3, 2), (5, 1)}

D ! {"3, "2, 1, 2}, D ! {1, 2, 3}, D ! {"3, 0, 3, 5},R ! {3, 5}; yes R ! {"4, "2, 0, 2}; no R ! {"2, 1, 2, 4}; yes

4. y " x2 ! 1 5. y " x ! 4 6. y " 3x # 2

D ! all reals, D ! all reals, D ! all reals,R ! {y⏐y # "1}; yes R ! all reals; yes R ! all reals; yes

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Equations of Functions and Relations Equations that represent functions areoften written in functional notation. For example, y " 10 ! 8x can be written as f(x) " 10 ! 8x. This notation emphasizes the fact that the values of y, the dependentvariable, depend on the values of x, the independent variable.

To evaluate a function, or find a functional value, means to substitute a given value in thedomain into the equation to find the corresponding element in the range.

Given the function f(x) ! x2 $ 2x, find each value.

a. f(3)

f(x) " x2 # 2x Original function

f(3) " 32 # 2(3) Substitute.

" 15 Simplify.

b. f(5a)

f(x) " x2 # 2x Original function

f(5a) " (5a)2 # 2(5a) Substitute.

" 25a2 # 10a Simplify.

Find each value if f(x) ! "2x $ 4.

1. f(12) "20 2. f(6) "8 3. f(2b) "4b $ 4

Find each value if g(x) ! x3 " x.

4. g(5) 120 5. g(!2) "6 6. g(7c) 343c3 " 7c

Find each value if f(x) ! 2x $ and g(x) ! 0.4x2 " 1.2.

7. f(0.5) 5 8. f(!8) "16 9. g(3) 2.4

10. g(!2.5) 1.3 11. f(4a) 8a $ 12. g! " " 1.2

13. f ! " 6 14. g(10) 38.8 15. f(200) 400.01

Let f(x) ! 2x2 " 1.

16. Find the values of f(2) and f(5). f (2) ! 7, f (5) ! 49

17. Compare the values of f(2) $ f(5) and f(2 $ 5). f (2) % f (5) ! 343, f (2 % 5) ! 199

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Study Guide and Intervention (continued)

Relations and Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

ExampleExample

ExercisesExercises

Skills PracticeRelations and Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1


Less

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Determine whether each relation is a function. Write yes or no.

1. yes 2. no

3. yes 4. no


5. {(2, !3), (2, 4), (2, !1)} 6. {(2, 6), (6, 2)}

D ! {2}, R ! {"3, "1, 4}; no D ! {2, 6}, R ! {2, 6}; yes7. {(!3, 4), (!2, 4), (!1, !1), (3, !1)} 8. x " !2

D ! {"3, "2, "1, 3}, D ! {"2}, R ! all reals; no R ! {"1, 4}; yes

Find each value if f(x) ! 2x " 1 and g(x) ! 2 " x2.

9. f(0) "1 10. f(12) 23 11. g(4) "1412. f(!2) "5 13. g(!1) 1 14. f(d) 2d " 1

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Determine whether each relation is a function. Write yes or no.

1. no 2. yes

3. yes 4. no


5. {(!4, !1), (4, 0), (0, 3), (2, 0)} 6. y " 2x ! 1

D ! {"4, 0, 2, 4}, D ! all reals, R ! all reals; yesR ! {"1, 0, 3}; yes

Find each value if f(x) ! and g(x) ! "2x $ 3.

7. f(3) 1 8. f(!4) " 9. g! " 2

10. f(!2) undefined 11. g(!6) 15 12. f(m ! 2)

13. MUSIC The ordered pairs (1, 16), (2, 16), (3, 32), (4, 32), and (5, 48) represent the cost ofbuying various numbers of CDs through a music club. Identify the domain and range ofthe relation. Is the relation a function? D ! {1, 2, 3, 4, 5}, R ! {16, 32, 48}; yes

14. COMPUTING If a computer can do one calculation in 0.0000000015 second, then thefunction T(n) " 0.0000000015n gives the time required for the computer to do ncalculations. How long would it take the computer to do 5 billion calculations? 7.5 s

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Practice (Average)

Relations and Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

Reading to Learn MathematicsRelations and Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1


Less

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2-1

Pre-Activity How do relations and functions apply to biology?

Read the introduction to Lesson 2-1 at the top of page 56 in your textbook.

• Refer to the table. What does the ordered pair (8, 20) tell you? For adeer, the average longevity is 8 years and the maximumlongevity is 20 years.

• Suppose that this table is extended to include more animals. Is it possibleto have an ordered pair for the data in which the first number is largerthan the second? Sample answer: No, the maximum longevitymust always be greater than the average longevity.

Reading the Lesson

1. a. Explain the difference between a relation and a function. Sample answer: Arelation is any set of ordered pairs. A function is a special kind ofrelation in which each element of the domain is paired with exactlyone element in the range.

b. Explain the difference between domain and range. Sample answer: The domainof a relation is the set of all first coordinates of the ordered pairs. Therange is the set of all second coordinates.

2. a. Write the domain and range of the relation shown in the graph.

D: {"3, "2, "1, 0, 3}; R: {"5, "4, 0, 1, 2, 4}b. Is this relation a function? Explain. Sample answer: No, it is not a function

because one of the elements of the domain, 3, is paired with twoelements of the range.

Helping You Remember

3. Look up the words dependent and independent in a dictionary. How can the meaning ofthese words help you distinguish between independent and dependent variables in afunction? Sample answer: The variable whose values depend on, or aredetermined by, the values of the other variable is the dependent variable.

(0, 4)

(3, 1)

(3, –4)(–1, –5)

(–2, 0)

(–3, 2)

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MappingsThere are three special ways in which one set can be mapped to another. A setcan be mapped into another set, onto another set, or can have a one-to-onecorrespondence with another set.

State whether each set is mapped into the second set, onto the second set, or has a one-to-one correspondence with the second set.

1. 2. 3. 4.

into, onto into, onto into, onto, into, ontoone-to-one

5. 6. 7. 8.

into into, onto into, onto into, onto,one-to-one

9. Can a set be mapped onto a set with fewer elements than it has? yes

10. Can a set be mapped into a set that has more elements than it has? yes

11. If a mapping from set A into set B is a one-to-one correspondence, what can you conclude about the number of elements in A and B?The sets have the same number of elements.

–29

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Into mapping A mapping from set A to set B where every element of A is mapped to one or more elements of set B, but never to an element not in B.

Onto mapping A mapping from set A to set B where each element of set B has at least one element of set A mapped to it.

One-to-one A mapping from set A onto set B where each element of set A is mapped to exactly one correspondence element of set B and different elements of A are never mapped to the same element of B.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

Study Guide and InterventionLinear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2


Less

on

2-2

Identify Linear Equations and Functions A linear equation has no operationsother than addition, subtraction, and multiplication of a variable by a constant. Thevariables may not be multiplied together or appear in a denominator. A linear equation doesnot contain variables with exponents other than 1. The graph of a linear equation is a line.A linear function is a function whose ordered pairs satisfy a linear equation. Any linearfunction can be written in the form f(x) " mx # b, where m and b are real numbers.If an equation is linear, you need only two points that satisfy the equation in order to graphthe equation. One way is to find the x-intercept and the y-intercept and connect these twopoints with a line.

Is f(x) ! 0.2 " alinear function? Explain.

Yes; it is a linear function because it canbe written in the formf(x) " ! x # 0.2.

Is 2x $ xy " 3y ! 0 alinear function? Explain.

No; it is not a linear function becausethe variables x and y are multipliedtogether in the middle term.

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Find the x-intercept and they-intercept of the graph of 4x " 5y ! 20.Then graph the equation.

The x-intercept is the value of x when y " 0.

4x ! 5y " 20 Original equation

4x ! 5(0) " 20 Substitute 0 for y.

x " 5 Simplify.

So the x-intercept is 5.Similarly, the y-intercept is !4. x

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Example 1Example 1 Example 3Example 3

Example 2Example 2

ExercisesExercises

State whether each equation or function is linear. Write yes or no. If no, explain.

1. 6y ! x " 7 yes 2. 9x " No; the 3. f(x) " 2 ! yesvariable y appears in the denominator.

Find the x-intercept and the y-intercept of the graph of each equation. Then graphthe equation.

4. 2x # 7y " 14 5. 5y ! x " 10 6. 2.5x ! 5y # 7.5 " 0

x-int: 7; y-int: 2 x-int: "10; y-int: 2 x-int: "3; y-int: 1.5

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Standard Form The standard form of a linear equation is Ax # By " C, where A, B, and C are integers whose greatest common factor is 1.

Write each equation in standard form. Identify A, B, and C.


Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2

ExampleExample

a. y ! 8x " 5

y " 8x ! 5 Original equation!8x # y " !5 Subtract 8x from each side.

8x ! y " 5 Multiply each side by !1.

So A " 8, B " !1, and C " 5.

b. 14x ! "7y $ 21

14x " !7y # 21 Original equation14x # 7y " 21 Add 7y to each side.

2x # y " 3 Divide each side by 7.

So A " 2, B " 1, and C " 3.

ExercisesExercises


1. 2x " 4y !1 2. 5y " 2x # 3 3. 3x " !5y # 22x " 4y ! "1; A ! 2, 2x " 5y ! "3; A ! 2, 3x $ 5y ! 2; A ! 3,B ! "4, C ! "1 B ! "5, C ! "3 B ! 5, C ! 2

4. 18y " 24x ! 9 5. y " x # 5 6. 6y ! 8x # 10 " 0

8x " 6y ! 3; A ! 8, 8x " 9y ! "60; A ! 8, 4x " 3y ! 5; A ! 4,B ! "6, C ! 3 B ! "9, C ! "60 B ! "3, C ! 5

7. 0.4x # 3y " 10 8. x " 4y ! 7 9. 2y " 3x # 62x $ 15y ! 50; A ! 2, x " 4y ! "7; A ! 1, 3x " 2y ! "6; A ! 3,B ! 15, C ! 50 B ! "4, C! "7 B ! "2, C ! "6

10. x # y !2 " 0 11. 4y # 4x # 12 " 0 12. 3x " !18

6x $ 5y ! 30; A ! 6, x $ y ! "3; A ! 1, x ! "6; A ! 1,B ! 5, C ! 30 B ! 1, C ! "3 B ! 0, C ! "6

13. x " # 7 14. 3y " 9x ! 18 15. 2x " 20 ! 8y

9x " y ! 63; A ! 9, 3x " y ! 6; A ! 3, x $ 4y ! 10; A ! 1,B ! "1, C ! 63 B ! "1, C ! 6 B ! 4, C ! 10

16. ! 3 " 2x 17. ! " " y # 8 18. 0.25y " 2x ! 0.75

8x " y ! "12; A ! 8, 10x " 3y ! 32; A ! 10, 8x " y ! 3; A ! 8,B ! "1, C! "12 B ! "3, C ! 32 B ! "1, C ! 3

19. 2y! ! 4 " 0 20. 1.6x ! 2.4y " 4 21. 0.2x " 100 ! 0.4y

x " 12y ! "24; A ! 1, 2x " 3y ! 5; A ! 2, x $ 2y ! 500; A ! 1,B ! "12, C ! "24 B ! "3, C ! 5 B ! 2, C ! 500

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Skills PracticeLinear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2


Less

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State whether each equation or function is linear. Write yes or no. If no, explainyour reasoning.

1. y " 3x 2. y " !2 # 5x

yes yes3. 2x # y " 10 4. f(x) " 4x2

yes No; the exponent of x is not 1.5. ! # y " 15 6. x " y # 8

No; x is in a denominator. yes7. g(x) " 8 8. h(x) " #x$ # 3

yes No; x is inside a square root.


9. y " x x " y ! 0; 1, "1, 0 10. y " 5x # 1 5x " y ! "1; 5, "1, "1

11. 2x " 4 ! 7y 2x $ 7y ! 4; 2, 7, 4 12. 3x " !2y ! 2 3x $ 2y ! "2; 3, 2, "2

13. 5y ! 9 " 0 5y ! 9; 0, 5, 9 14. !6y # 14 " 8x 4x $ 3y ! 7; 4, 3, 7


15. y " 3x ! 6 2, "6 16. y " !2x 0, 0

17. x # y " 5 5, 5 18. 2x # 5y " 10 5, 2

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State whether each equation or function is linear. Write yes or no. If no, explainyour reasoning.

1. h(x) " 23 yes 2. y " x yes

3. y " No; x is a denominator. 4. 9 ! 5xy " 2 No; x and y are multiplied.


5. y " 7x ! 5 7x " y ! 5; 7, "1, 5 6. y " x # 5 3x " 8y ! "40; 3, "8, "40

7. 3y ! 5 " 0 3y ! 5; 0, 3, 5 8. x " ! y # 28x $ 8y ! 21; 28, 8, 21


9. y " 2x # 4 "2, 4 10. 2x # 7y " 14 7, 2

11. y " !2x ! 4 "2, "4 12. 6x # 2y " 6 1, 3

13. MEASURE The equation y " 2.54x gives the length in centimeters corresponding to alength x in inches. What is the length in centimeters of a 1-foot ruler? 30.48 cm

LONG DISTANCE For Exercises 14 and 15, use the following information.

For Meg’s long-distance calling plan, the monthly cost C in dollars is given by the linearfunction C(t) " 6 # 0.05t, where t is the number of minutes talked.

14. What is the total cost of talking 8 hours? of talking 20 hours? $30; $66

15. What is the effective cost per minute (the total cost divided by the number of minutestalked) of talking 8 hours? of talking 20 hours? $0.0625; $0.055

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Practice (Average)

Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2

Reading to Learn MathematicsLinear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2


Less

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Pre-Activity How do linear equations relate to time spent studying?


• If Lolita spends 2 hours studying math, how many hours will she have

to study chemistry? 1 hours• Suppose that Lolita decides to stay up one hour later so that she now has

5 hours to study and do homework. Write a linear equation that describesthis situation. x $ y ! 5

Reading the Lesson

1. Write yes or no to tell whether each linear equation is in standard form. If it is not,explain why it is not.

a. !x # 2y " 5 No; A is negative.

b. 9x ! 12y " !5 yes

c. 5x ! 7y " 3 yes

d. 2x ! y " 1 No; B is not an integer.

e. 0x # 0y " 0 No; A and B are both 0.

f. 2x # 4y " 8 No; The greatest common factor of 2, 4, and 8 is 2, not 1.

2. How can you use the standard form of a linear equation to tell whether the graph is ahorizontal line or a vertical line? If A ! 0, then the graph is a horizontal line. IfB ! 0, then the graph is a vertical line.


3. One way to remember something is to explain it to another person. Suppose that you are studying this lesson with a friend who thinks that she should let x " 0 to find the x-intercept and let y " 0 to find the y-intercept. How would you explain to her how toremember the correct way to find intercepts of a line? Sample answer: The x-intercept is the x-coordinate of a point on the x-axis. Every point on the x-axis has y-coordinate 0, so let y ! 0 to find an x-intercept. The y-intercept is the y-coordinate of a point on the y-axis. Every point on the y-axis has x-coordinate 0, so let x ! 0 to find a y-intercept.

4%7

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Greatest Common FactorSuppose we are given a linear equation ax # by " c where a, b, and c are nonzerointegers, and we want to know if there exist integers x and y that satisfy theequation. We could try guessing a few times, but this process would be timeconsuming for an equation such as 588x # 432y " 72. By using the EuclideanAlgorithm, we can determine not only if such integers x and y exist, but also find them. The following example shows how this algorithm works.

Find integers x and y that satisfy 588x $ 432y ! 72.

Divide the greater of the two coefficients by the lesser to get a quotient andremainder. Then, repeat the process by dividing the divisor by the remainderuntil you get a remainder of 0. The process can be written as follows.

588 " 432(1) # 156 (1)432 " 156(2) # 120 (2)156 " 120(1) # 36 (3)120 " 36(3) # 12 (4)36 " 12(3)

The last nonzero remainder is the GCF of the two coefficients. If the constantterm 72 is divisible by the GCF, then integers x and y do exist that satisfy theequation. To find x and y, work backward in the following manner.

72 " 6 $ 12" 6 $ [120 ! 36(3)] Substitute for 12 using (4)" 6(120) ! 18(36)" 6(120) ! 18[156 ! 120(1)] Substitute for 36 using (3)" !18(156) # 24(120)" !18(156) # 24[432 ! 156(2)] Substitute for 120 using (2)" 24(432) ! 66(156)" 24(432) ! 66[588 ! 432(1)] Substitute for 156 using (1)" 588(!66) # 432(90)

Thus, x " !66 and y " 90.

Find integers x and y, if they exist, that satisfy each equation.

1. 27x # 65y " 3 2. 45x # 144y " 36

3. 90x # 117y " 10 4. 123x # 36y " 15

5. 1032x # 1001y " 1 6. 3125x # 3087y " 1

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2

ExampleExample

Study Guide and InterventionSlope

NAME ______________________________________________ DATE ____________ PERIOD _____

2-32-3


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Slope

Slope m of a Line For points (x1, y1) and (x2, y2), where x1 & x2, m " "y2 ! y1%x2 ! x1

change in y%%change in x

Determine the slope ofthe line that passes through (2, "1) and("4, 5).

m " Slope formula

" (x1, y1) " (2, !1), (x2, y2) " (!4, 5)

" " !1 Simplify.

The slope of the line is !1.

6%!6

5 ! (!1)%%!4 ! 2

y2 ! y1%x2 ! x1

Graph the line passingthrough ("1, "3) with a slope of .

Graph the ordered pair (!1, !3). Then,according to the slope, go up 4 unitsand right 5 units.Plot the new point(4,1). Connect thepoints and draw the line.

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4&5


ExercisesExercises

Find the slope of the line that passes through each pair of points.

1. (4, 7) and (6, 13) 3 2. (6, 4) and (3, 4) 0 3. (5, 1) and (7, !3) "2

4. (5, !3) and (!4, 3) " 5. (5, 10) and (!1,!2) 2 6. (!1, !4) and (!13, 2) "

7. (7, !2) and (3, 3) " 8. (!5, 9) and (5, 5) " 9. (4, !2) and (!4, !8)

Graph the line passing through the given point with the given slope.

10. slope " ! 11. slope " 2 12. slope " 0

passes through (0, 2) passes through (1, 4) passes through (!2, !5)

13. slope " 1 14. slope " ! 15. slope "

passes through (!4, 6) passes through (!3, 0) passes through (0, 0)

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Parallel and Perpendicular Lines


Slope

NAME ______________________________________________ DATE ____________ PERIOD _____

2-32-3

In a plane, nonvertical lines with thesame slope are parallel. All verticallines are parallel.

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slope m

slope m

In a plane, two oblique lines are perpendicular ifand only if the product of their slopes is !1. Anyvertical line is perpendicular to any horizontal line.

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slope m

slope 1m

ExampleExample Are the line passing through (2, 6) and ("2, 2) and the line passingthrough (3, 0) and (0, 4) parallel, perpendicular, or neither?

Find the slopes of the two lines.

The slope of the first line is " 1.

The slope of the second line is " ! .

The slopes are not equal and the product of the slopes is not !1, so the lines are neitherparallel nor perpendicular.

Are the lines parallel, perpendicular, or neither?

1. the line passing through (4, 3) and (1. !3) and the line passing through (1, 2) and (!1, 3)perpendicular

2. the line passing through (2, 8) and (!2, 2) and the line passing through (0, 9) and (6, 0)neither

3. the line passing through (3, 9) and (!2, !1) and the graph of y " 2x parallel

4. the line with x-intercept !2 and y-intercept 5 and the line with x-intercept 2 and y-intercept !5 parallel

5. the line with x-intercept 1 and y-intercept 3 and the line with x-intercept 3 and y-intercept 1 neither

6. the line passing through (!2, !3) and (2, 5) and the graph of x # 2y " 10perpendicular

7. the line passing through (!4, !8) and (6, !4) and the graph of 2x ! 5y " 5 parallel

4%3

4 ! 0%0 ! 3

6 ! 2%%2 ! (!2)

ExercisesExercises

Skills PracticeSlope

NAME ______________________________________________ DATE ____________ PERIOD _____

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1. (1, 5), (!1, !3) 4 2. (0, 2), (3, 0) " 3. (1, 9), (0, 6) 3

4. (8, !5), (4, !2) " 5. (!3, 5), (!3, !1) undefined 6. (!2, !2), (10, !2) 0

7. (4, 5), (2, 7) "1 8. (!2, !4), (3, 2) 9. (5, 2), (!3, 2) 0


10. (0, 4), m " 1 11. (2, !4), m " !1

12. (!3, !5), m " 2 13. (!2, !1), m " !2

Graph the line that satisfies each set of conditions.

14. passes through (0, 1), perpendicular to 15. passes through (0, !5), parallel to the

a line whose slope is graph of y " 1

16. HIKING Naomi left from an elevation of 7400 feet at 7:00 A.M. and hiked to an elevationof 9800 feet by 11:00 A.M. What was her rate of change in altitude? 600 ft /h

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1. (3, !8), (!5, 2) " 2. (!10, !3), (7, 2) 3. (!7, !6), (3, !6) 0

4. (8, 2), (8, !1) undefined 5. (4, 3), (7, !2) " 6. (!6, !3), (!8, 4) "


7. (0, !3), m " 3 8. (2, 1), m " !

9. (0, 2), m " 0 10. (2, !3), m "

Graph the line that satisfies each set of conditions.

11. passes through (3, 0), perpendicular 12. passes through (!3, !1), parallel to a line

to a line whose slope is whose slope is !1

DEPRECIATION For Exercises 13–15, use the following information.A machine that originally cost $15,600 has a value of $7500 at the end of 3 years. The samemachine has a value of $2800 at the end of 8 years.

13. Find the average rate of change in value (depreciation) of the machine between itspurchase and the end of 3 years. "$2700 per year

14. Find the average rate of change in value of the machine between the end of 3 years andthe end of 8 years. "$940 per year

15. Interpret the sign of your answers. It is negative because the value is decreasing.

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Practice (Average)

Slope

NAME ______________________________________________ DATE ____________ PERIOD _____

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Reading to Learn MathematicsSlope

NAME ______________________________________________ DATE ____________ PERIOD _____

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Pre-Activity How does slope apply to the steepness of roads?


• What is the grade of a road that rises 40 feet over a horizontal distanceof 1000 feet? 4%

• What is the grade of a road that rises 525 meters over a horizontaldistance of 10 kilometers? (1 kilometer " 1000 meters) 5.25%

Reading the Lesson

1. Describe each type of slope and include a sketch.

Type of Slope Description of Graph Sketch

Positive The line rises to the right.

Zero The line is horizontal.

Negative The line falls to the right.

Undefined The line is vertical.

2. a. How are the slopes of two nonvertical parallel lines related? They are equal.b. How are the slopes of two oblique perpendicular lines related? Their product is "1.


3. Look up the terms grade, pitch, slant, and slope. How can everyday meanings of thesewords help you remember the definition of slope? Sample answer: All these wordscan be used when you describe how much a thing slants upward ordownward. You can describe this numerically by comparing rise to run.


Aerial Surveyors and AreaMany land regions have irregular shapes. Aerial surveyors supply aerial mappers with lists of coordinates and elevations for the areas that need to be photographed from the air. These maps provide information about the horizontal and vertical features of the land.

Step 1 List the ordered pairs for the vertices in counterclockwise order, repeating the first ordered pair at the bottom of the list.

Step 2 Find D, the sum of the downward diagonal products (from left to right).D " (5 $ 5) # (2 $ 1) # (2 $ 3) # (6 $ 7)

" 25 # 2 # 6 # 42 or 75

Step 3 Find U, the sum of the upward diagonal products (from left to right).U " (2 $ 7) # (2 $ 5) # (6 $ 1) # (5 $ 3)

" 14 # 10 # 6 # 15 or 45

Step 4 Use the formula A " %12%(D ! U) to find the area.

A " %12%(75 ! 45)

" %12%(30) or 15

The area is 15 square units. Count the number of square units enclosed by the polygon. Does this result seem reasonable?

Use the coordinate method to find the area of each region in square units.

1. 2. 3.

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(5, 7)

(2, 5)

(2, 1)

(6, 3)

(5, 7)

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(2, 1)

(2, 5)

(5, 7)

(6, 3)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Study Guide and InterventionWriting Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

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Forms of Equations

Slope-Intercept Form of a Linear Equation y " mx # b, where m is the slope and b is the y-intercept

Point-Slope Form y ! y1 " m(x ! x1), where (x1, y1) are the coordinates of a point on the line and of a Linear Equation m is the slope of the line

Write an equation inslope-intercept form for the line thathas slope "2 and passes through thepoint (3, 7).

Substitute for m, x, and y in the slope-intercept form.

y " mx # b Slope-intercept form

7 " (!2)(3) # b (x, y ) " (3, 7), m " !2

7 " !6 # b Simplify.

13 " b Add 6 to both sides.

The y-intercept is 13. The equation in slope-intercept form is y " !2x # 13.

Write an equation inslope-intercept form for the line thathas slope and x-intercept 5.

y " mx # b Slope-intercept form

0 " ! "(5) # b (x, y ) " (5, 0), m "

0 " # b Simplify.

! " b Subtract from both sides.

The y-intercept is ! . The slope-intercept

form is y " x ! .5%3

1%3

5%3

5%35%3

5%3

1%31%3

1&3


ExercisesExercises

Write an equation in slope-intercept form for the line that satisfies each set ofconditions.

1. slope !2, passes through (!4, 6) 2. slope , y-intercept 4

y ! "2x " 2 y ! x $ 4

3. slope 1, passes through (2, 5) 4. slope ! , passes through (5, !7)

y ! x $ 3 y ! " x $ 6

Write an equation in slope-intercept form for each graph.

5. 6. 7.

y ! "3x $ 9 y ! x y ! x $ 1 4&9

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(–4, 1)(5, 2)

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(4, 5)

(0, 0)

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(1, 6)

(3, 0)

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Parallel and Perpendicular Lines Use the slope-intercept or point-slope form to findequations of lines that are parallel or perpendicular to a given line. Remember that parallellines have equal slope. The slopes of two perpendicular lines are negative reciprocals, thatis, their product is !1.


Writing Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-42-4

Write an equation of theline that passes through (8, 2) and isperpendicular to the line whose equation is y ! " x $ 3.

The slope of the given line is ! . Since the

slopes of perpendicular lines are negativereciprocals, the slope of the perpendicularline is 2.Use the slope and the given point to writethe equation.y ! y1 " m(x ! x1) Point-slope formy ! 2 " 2(x ! 8) (x1, y1) " (8, 2), m " 2y ! 2 " 2x ! 16 Distributive Prop.

y " 2x ! 14 Add 2 to each side.

An equation of the line is y " 2x ! 14.

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Write an equation of theline that passes through ("1, 5) and isparallel to the graph of y ! 3x $ 1.

The slope of the given line is 3. Since theslopes of parallel lines are equal, the slopeof the parallel line is also 3.Use the slope and the given point to writethe equation.y !y1 " m(x ! x1) Point-slope formy ! 5 " 3(x ! (!1)) (x1, y1) " (!1, 5), m " 3y ! 5 " 3x # 3 Distributive Prop.

y " 3x # 8 Add 5 to each side.

An equation of the line is y " 3x # 8.


ExercisesExercises


1. passes through (!4, 2), parallel to the line whose equation is y " x # 5 y ! x $ 4

2. passes through (3, 1), perpendicular to the graph of y " !3x # 2 y ! x

3. passes through (1, !1), parallel to the line that passes through (4, 1) and (2, !3)y ! 2x " 3

4. passes through (4, 7), perpendicular to the line that passes through (3, 6) and (3, 15)y ! 7

5. passes through (8, !6), perpendicular to the graph of 2x ! y " 4 y ! " x " 2

6. passes through (2, !2), perpendicular to the graph of x # 5y " 6 y ! 5x " 12

7. passes through (6, 1), parallel to the line with x-intercept !3 and y-intercept 5

y ! x " 9

8. passes through (!2, 1), perpendicular to the line y " 4x ! 11 y ! " x $ 1&2

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Skills PracticeWriting Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

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State the slope and y-intercept of the graph of each equation.

1. y " 7x ! 5 7, "5 2. y " ! x # 3 " , 3

3. y " x , 0 4. 3x # 4y " 4 " , 1

5. 7y " 4x ! 7 , "1 6. 3x ! 2y # 6 " 0 , 3

7. 2x ! y " 5 2, "5 8. 2y " 6 ! 5x " , 3


9. 10. 11.

y ! 3x " 1 y ! "1 y ! "2x $ 3


12. slope 3, passes through (1, !3) 13. slope !1, passes through (0, 0)

y ! 3x " 6 y ! "x

14. slope !2, passes through (0, !5) 15. slope 3, passes through (2, 0)

y ! "2x " 5 y ! 3x " 6

16. passes through (!1, !2) and (!3, 1) 17. passes through (!2, !4) and (1, 8)

y ! " x " y ! 4x $ 4

18. x-intercept 2, y-intercept !6 19. x-intercept , y-intercept 5

y ! 3x " 6 y ! "2x $ 5

20. passes through (3, !1), perpendicular to the graph of y " ! x ! 4. y ! 3x " 101%3

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(0, 3)

(3, –3)

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O(–3, –1) (4, –1)

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(–1, –4)

(1, 2)

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State the slope and y-intercept of the graph of each equation.

1. y " 8x # 12 8, 12 2. y " 0.25x ! 1 0.25, "1 3. y " ! x " , 0

4. 3y " 7 0, 5. 3x " !15 # 5y , 3 6. 2x ! 3y " 10 , "


7. 8. 9.

y ! 2 y ! x " 2 y ! " x $ 1


10. slope !5, passes through (!3, !8) 11. slope , passes through (10, !3)

y ! "5x " 23 y ! x " 11

12. slope 0, passes through (0, !10) 13. slope ! , passes through (6, !8)

y ! "10 y ! " x " 4

14. passes through (3, 11) and (!6, 5) 15. passes through (7, !2) and (3, !1)

y ! x $ 9 y ! " x "

16. x-intercept 3, y-intercept 2 17. x-intercept !5, y-intercept 7

y ! " x $ 2 y ! x $ 7

18. passes through (!8, !7), perpendicular to the graph of y " 4x ! 3 y ! " x " 919. RESERVOIRS The surface of Grand Lake is at an elevation of 648 feet. During the

current drought, the water level is dropping at a rate of 3 inches per day. If this trendcontinues, write an equation that gives the elevation in feet of the surface of Grand Lakeafter x days. y ! "0.25x $ 648

20. BUSINESS Tony Marconi’s company manufactures CD-ROM drives. The company willmake $150,000 profit if it manufactures 100,000 drives, and $1,750,000 profit if itmanufactures 500,000 drives. The relationship between the number of drivesmanufactured and the profit is linear. Write an equation that gives the profit P when n drives are manufactured. P ! 4n " 250,000

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(–3, 3)

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(4, 4)

(0, –2)

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(0, 2)

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Practice (Average)

Writing Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

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Reading to Learn MathematicsWriting Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

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Pre-Activity How do linear equations apply to business?


• If the total cost of producing a product is given by the equation y " 5400 # 1.37x, what is the fixed cost? What is the variable cost (for each item produced)? $5400; $1.37

• Write a linear equation that describes the following situation:A company that manufactures computers has a fixed cost of $228,750 anda variable cost of $852 to produce each computer.y ! 228,750 $ 852x

Reading the Lesson

1. a. Write the slope-intercept form of the equation of a line. Then explain the meaning ofeach of the variables in the equation. y ! mx $ b; m is the slope and b is they-intercept. The variables x and y are the coordinates of any point onthe line.

b. Write the point-slope form of the equation of a line. Then explain the meaning of eachof the variables in the equation. y " y1 ! m(x " x1); m is the slope. x and yare the coordinates of any point on the line. x1 and y1 are the coordinates of one specific point on the line.

2. Suppose that your algebra teacher asks you to write the point-slope form of the equationof the line through the points (!6, 7) and (!3, !2). You write y # 2 " !3(x # 3) andyour classmate writes y ! 7 " !3(x # 6). Which of you is correct? Explain. You areboth correct. Either point may be used as (x1, y1) in the point-slope form.You used ("3, "2), and your classmate used ("6, 7).

3. You are asked to write an equation of two lines that pass through (3, !5), one of themparallel to and one of them perpendicular to the line whose equation is y " !3x # 4.The first step in finding these equations is to find their slopes. What is the slope of theparallel line? What is the slope of the perpendicular line? "3;


4. Many students have trouble remembering the point-slope form for a linear equation.How can you use the definition of slope to remember this form? Sample answer:Write the definition of slope: m ! . Multiply both sides of this

equation by x2 " x1. Drop the subscripts in y2 and x2. This gives thepoint-slope form of the equation of a line.

y2 " y1&x2 " x1

1&3


Two-Intercept Form of a Linear EquationYou are already familiar with the slope-intercept form of a linear equation,

y " mx # b. Linear equations can also be written in the form %ax

% # %by

% " 1 with x-intercept a and y-intercept b. This is called two-intercept form.

Draw the graph of &"x3& $ &6

y& ! 1.

The graph crosses the x-axis at !3 and the y-axis at 6. Graph (!3, 0) and (0, 6), then draw a straight line through them.

Write 3x $ 4y ! 12 in two-intercept form.

%132x% # %1

42y% " %

1122% Divide by 12 to obtain 1 on the right side.

%4x

% # %3y

% " 1 Simplify.

The x-intercept is 4; the y-intercept is 3.

Use the given intercepts a and b, to write an equation in two-intercept form. Then draw the graph. See students’ graphs.1. a " !2, b " !4 2. a " 1, b " 8

3. a " 3, b " 5 4. a " 6, b " 9

Write each equation in two-intercept form. Then draw the graph.

5. 3x ! 2y " !6 6. %12%x # %

14%y " 1 7. 5x # 2y " !10

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Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Example 1Example 1

Example 2Example 2

Study Guide and InterventionModeling Real-World Data: Using Scatter Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

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Scatter Plots When a set of data points is graphed as ordered pairs in a coordinateplane, the graph is called a scatter plot. A scatter plot can be used to determine if there isa relationship among the data.

BASEBALL The table below shows the number of home runs andruns batted in for various baseball players who won the Most Valuable PlayerAward during the 1990s. Make a scatter plot of the data.

Source: New York Times Almanac

Make a scatter plot for the data in each table below.

1. FUEL EFFICIENCY The table below shows the average fuel efficiency in miles per gallon of new cars manufactured during the years listed.


2. CONGRESS The table below shows the number of women serving in the United States Congress during the years 1987!1999.

Source: Wall Street Journal Almanac

Congressional Session Number of Women

100 25

101 31

102 33

103 55

104 58

105 62

Session of Congress

Nu

mb

er o

f W

om

en

100 102 104

70

56

42

28

14

0

Women in Congress

Year Fuel Efficiency (mpg)

1960 15.5

1970 14.1

1980 22.6

1990 26.9 Year

Mile

s p

er G

allo

n

1960 1970 1980 1990

36

30

24

18

12

6

0

Average Fuel Efficiency

Home Runs

MVP HRs and RBIs

Ru

ns

Bat

ted

In

1260 24 3618 30 42 48

150

125

100

75

50

25

Home Runs Runs Batted In

33 114

39 116

40 130

28 61

41 128

47 144

ExampleExample

ExercisesExercises


Prediction Equations A line of fit is a line that closely approximates a set of datagraphed in a scatter plot. The equation of a line of fit is called a prediction equationbecause it can be used to predict values not given in the data set.

To find a prediction equation for a set of data, select two points that seem to represent thedata well. Then to write the prediction equation, use what you know about writing a linearequation when given two points on the line.

STORAGE COSTS According to a certain prediction equation, thecost of 200 square feet of storage space is $60. The cost of 325 square feet ofstorage space is $160.

a. Find the slope of the prediction equation. What does it represent?Since the cost depends upon the square footage, let x represent the amount of storagespace in square feet and y represent the cost in dollars. The slope can be found using the

formula m " . So, m " " " 0.8

The slope of the prediction equation is 0.8. This means that the price of storage increases80¢ for each one-square-foot increase in storage space.

b. Find a prediction equation.Using the slope and one of the points on the line, you can use the point-slope form to finda prediction equation.y ! y1 " m(x ! x1) Point-slope formy ! 60 " 0.8(x ! 200) (x1, y1) " (200, 60), m " 0.8y ! 60 " 0.8x ! 160 Distributive Property

y " 0.8x ! 100 Add 60 to both sides.

A prediction equation is y " 0.8x ! 100.

SALARIES The table below shows the years of experience for eight technicians atLewis Techomatic and the hourly rate of pay each technician earns. Use the datafor Exercises 1 and 2.

Experience (years) 9 4 3 1 10 6 12 8

Hourly Rate of Pay $17 $10 $10 $7 $19 $12 $20 $15

1. Draw a scatter plot to show how years of experience are related to hourly rate of pay. Draw a line of fit. See graph.

2. Write a prediction equation to show how years of experience(x) are related to hourly rate of pay (y). Sample answerusing (1, 7) and (9, 17): y ! 1.25x $ 5.75

Experience (years)

Ho

url

y Pa

y ($

)

20 6 104 8 12 14

24

20

16

12

8

4

Technician Salaries

100%125

160 ! 60%%325 ! 200

y2 ! y1%x2 ! x1


Modeling Real-World Data: Using Scatter Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5

ExampleExample

ExercisesExercises

Skills PracticeModeling Real-World Data: Using Scatter Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5


Less

on

2-5

For Exercises 1–3, complete parts a–c for each set of data.

a. Draw a scatter plot.b. Use two ordered pairs to write a prediction equation.c. Use your prediction equation to predict the missing value.

1. 1a.

1b. Sample answer using (1, 1) and (8, 15): y ! 2x " 11c. Sample answer: 19

2. 2a.

2b. Sample answer using (5, 9) and (40, 44): y ! x $ 42c. Sample answer: 54

3. 3a.

3b. Sample answer using (2, 16) and (7, 34): y ! 3.6x $ 8.83c. Sample answer: 19.6

1 3 5 72 4 6 8

36

30

24

18

12

6

0 x

yx y

1 16

2 16

3 ?

4 22

5 30

7 34

8 36

5 15 25 3510 20 30 40

40

32

24

16

8

0 x

yx y

5 9

10 17

20 22

25 30

35 38

40 44

50 ?

1 3 5 72 4 6 8

15

12

9

6

3

0 x

yx y

1 1

3 5

4 7

6 11

7 12

8 15

10 ?

Brad and Anissa Stuckey

Text

Use Graphing Calculator to find equation


For Exercises 1–3, complete parts a–c for each set of data.a. Draw a scatter plot.b. Use two ordered pairs to write a prediction equation.c. Use your prediction equation to predict the missing value.

1. FUEL ECONOMY The table gives the approximate weights in tons and estimates for overall fuel economy in miles per gallon for several cars.1b. Sample answer using (1.4, 24) and

(2.4, 15): y ! "9x $ 36.61c. Sample answer: 18.6 mi/gal

2. ALTITUDE In most cases, temperature decreases with increasing altitude. As Ancharadrives into the mountains, her car thermometer registers the temperatures (°F) shownin the table at the given altitudes (feet).

2b. Sample answer using (7500, 61) and (9700, 50): y ! "0.005x $ 98.5

2c. Sample answer: 38.5°F

3. HEALTH Alton has a treadmill that uses the time on the treadmill and the speed of walking or running to estimate the number of Calories he burns during a workout. Thetable gives workout times and Calories burned for several workouts.

3b. Sample answer using (24, 280) and(48, 440): y ! 6.67x $ 119.92

3c. Sample answer: about 520 calories

Time (min) 18 24 30 40 42 48 52 60

Calories Burned 260 280 320 380 400 440 475 ?

Altitude (ft)

Tem

per

atu

re ( F

)

0 7,000 8,000 9,000 10,000

65

60

55

50

45

TemperatureVersus Altitude

Altitude (ft) 7500 8200 8600 9200 9700 10,400 12,000

Temperature ('F) 61 58 56 53 50 46 ?

Weight (tons)

Fuel

Eco

no

my

(mi/

gal

)

0 0.5 1.0 1.5 2.0 2.5

30

25

20

15

10

5

Fuel Economy Versus Weight

Weight (tons) 1.3 1.4 1.5 1.8 2 2.1 2.4

Miles per Gallon 29 24 23 21 ? 17 15

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5

Brad and Anissa Stuckey

Use Graphing Calculatorto find equation

Reading to Learn MathematicsModeling Real-World Data: Using Scatter Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5


Less

on

2-5

Pre-Activity How can a linear equation model the number of Calories you burnexercising?


• If a woman runs 5.5 miles per hour, about how many Calories will sheburn in an hour? Sample answer: 572 Calories

• If a man runs 7.5 miles per hour, about how many Calories will he burnin half an hour? Sample answer: 397 Calories

Reading the Lesson

1. Suppose that a set of data can be modeled by a linear equation. Explain the differencebetween a scatter plot of the data and a graph of the linear equation that models thatdata.Sample answer: The scatter plot is a discrete graph. It is made up just ofthe individual points that represent the data points. The linear equationhas a continuous graph that is the line that best fits the data points.

2. Suppose that tuition at a state college was $3500 per year in 1995 and has beenincreasing at a rate of $225 per year.

a. Write a prediction equation that expresses this information.y ! 3500 $ 225x

b. Explain the meaning of each variable in your prediction equation.x represents the number of year since 1995 and y represents thetuition in that year.

3. Use this model to predict the tuition at this college in 2007. $6200


4. Look up the word scatter in a dictionary. How can its definition help you to rememberthe meaning of the difference between a scatter plot and the graph of a linear equation?Sample answer: To scatter means to break up and go in many directions.The points on a scatter plot are broken up. In a scatter plot, the pointsare scattered or broken up. In the graph of a linear equation, the pointsare connected to form a continuous line.


Median-Fit Lines A median-fit line is a particular type of line of fit. Follow the steps below to find the equation of the median-fit line for the data.

Approximate Percentage of Violent Crimes Committed by Juveniles That Victims Reported to Law Enforcement

Year 1980 1982 1984 1986 1988 1990 1992 1994 1996

Offenders 36 35 33 32 31 30 29 29 30Source: U.S. Bureau of Justice Statistics

1. Divide the data into three approximately equal groups. There should always be the same number of points in the first and third groups. In this case, there will be three data points in each group.

Group 1 Group 2 Group 3enders

2. Find x1, x2, and x3, the medians of the x values in groups 1, 2, and 3,respectively. Find y1, y2, and y3, the medians of the y values in groups 1, 2, and 3, respectively. 1982, 1988, 1994; 35, 31, 29

3. Find an equation of the line through (x1, y1) and (x3, y3). y ! "0.5x $ 1026

4. Find Y, the y-coordinate of the point on the line in Exercise 2 with an x-coordinate of x2. 32

5. The median-fit line is parallel to the line in Exercise 2, but is one-third

closer to (x2, y2). This means it passes through !x2, %23%Y # %

13%y2". Find this

ordered pair. about (1988, 31.67)

6. Write an equation of the median-fit line. y ! "0.5x $ 1025.67

7. Use the median-fit line to predict the percentage of juvenile violent crime offenders in 2010 and 2020. 2010: about 21%; 2020: about16%

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5

Study Guide and InterventionSpecial Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6


Less

on

2-6

Step Functions, Constant Functions, and the Identity Function The chartbelow lists some special functions you should be familiar with.

Function Written as Graph

Constant f(x) " c horizontal line

Identity f(x) " x line through the origin with slope 1

Greatest Integer Function f(x) " %x& one-unit horizontal segments, with right endpoints missing, arranged like steps

The greatest integer function is an example of a step function, a function with a graph thatconsists of horizontal segments.

Identify each function as a constant function, the identity function,or a step function.

a. b.

a constant function a step function

Identify each function as a constant function, the identity function, a greatestinteger function, or a step function.

1. 2. 3.

a constant function a step function the identity function

x

f (x)

Ox

f (x)

Ox

f (x)

O

x

f (x)

Ox

f (x)

O

ExampleExample

ExercisesExercises


Absolute Value and Piecewise Functions Another special function is theabsolute value function, which is also called a piecewise function.

Absolute Value Function f(x ) " x two rays that are mirror images of each other and meet at a point, the vertex

To graph a special function, use its definition and your knowledge of the parent graph. Findseveral ordered pairs, if necessary.

Graph f(x) ! 3⏐x⏐ " 4.

Find several ordered pairs. Graph the points andconnect them. You would expect the graph to looksimilar to its parent function, f(x) " x.

Graph f(x) ! !2x if x ( 2x " 1 if x # 2.

First, graph the linear function f(x) " 2x for x ' 2. Since 2 does notsatisfy this inequality, stop with a circle at (2, 4). Next, graph thelinear function f(x) " x ! 1 for x ( 2. Since 2 does satisfy thisinequality, begin with a dot at (2, 1).

Graph each function. Identify the domain and range.

1. g(x) " % & 2. h(x) " 2x # 1 3. h(x ) "

domain: all real domain: all real domain: all real numbers; range: numbers; range: numbers; range:all integers {y⏐y # 0} {y⏐y ) 1}

x

y

O

x

y

O

x

y

O

x%3

x

f (x)

O

x

f (x)

O

x 3⏐x⏐ " 4

0 !4

1 !1

2 2

!1 !1

!2 2


Special Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6

ExercisesExercises

Example 1Example 1

Example 2Example 2

if x ) 0

2x ! 6 if 0 ' x ' 21 if x ( 2

x%3

Skills PracticeSpecial Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6


Less

on

2-6

Identify each function as S for step, C for constant, A for absolute value, or P forpiecewise.

1. 2. 3.

S C A


4. f(x) " %x # 1& 5. f(x) " %x ! 3&

D ! all reals, R ! all integers D ! all reals, R ! all integers6. g(x) " 2x 7. f(x) " x # 1

D ! all reals, D ! all reals, R ! {y⏐y # 1}R ! nonnegative reals

8. f(x) " 'x if x ' 0 9. h(x) " '3 if x ' !12 if x ( 0 x # 1 if x > 1

D ! all reals, D ! {x⏐x ( "1 or x * 1},R ! {y⏐y ( 0 or y ! 2} R ! {y⏐y * 2}

x

h(x)

O

x

f (x)

O

x

f (x)

Ox

g(x)

O

x

f (x)

O

x

f (x)

O

x

y

O

x

y

Ox

y

O



1. f(x) " %0.5x& 2. f(x) " %x& ! 2

D ! all reals, R ! all integers D ! all reals, R ! all integers3. g(x) " !2x 4. f(x) " x # 1

D ! all reals, D ! all reals,R ! nonpositive reals R ! nonnegative reals

5. f(x) " 'x # 2 if x ) ! 2 6. h(x) " '4 ! x if x * 03x if x * !2 !2x ! 2 if x ' 0

D ! all reals, R ! all reals D ! all nonzero reals, R ! all reals7. BUSINESS A Stitch in Time charges 8. BUSINESS A wholesaler charges a store $3.00

$40 per hour or any fraction thereof per pound for less than 20 pounds of candy andfor labor. Draw a graph of the step $2.50 per pound for 20 or more pounds. Draw afunction that represents this situation. graph of the function that represents this

situation.

Hours

Tota

l Co

st (

$)

10 3 52 4 6 7

280

240

200

160

120

80

40

Labor Costs

x

f (x)

O

x

g(x)

O

x

f (x)

Ox

f (x)

O

Practice (Average)

Special Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6

Reading to Learn MathematicsSpecial Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6


Less

on

2-6

Pre-Activity How do step functions apply to postage rates?


• What is the cost of mailing a letter that weighs 0.5 ounce?$0.34 or 34 cents

• Give three different weights of letters that would each cost 55 cents tomail. Answers will vary. Sample answer: 1.1 ounces,1.9 ounces, 2.0 ounces

Reading the Lesson

1. Find the value of each expression.

a. !3 " %!3& "

b. 6.2 " %6.2& "

c. !4.01 " %!4.01& "

2. Tell how the name of each kind of function can help you remember what the graph looks like.

a. constant function Sample answer: Something is constant if it does notchange. The y-values of a constant function do not change, so thegraph is a horizontal line.

b. absolute value function Sample answer: The absolute value of a numbertells you how far it is from 0 on the number line. It makes no differencewhether you go to the left or right so long as you go the samedistance each time.

c. step function Sample answer: A step function’s graph looks like stepsthat go up or down.

d. identity function Sample answer: The x- and y-values are alwaysidentically the same for any point on the graph. So the graph is a linethrough the origin that has slope 1.


3. Many students find the greatest integer function confusing. Explain how you can use anumber line to find the value of this function for any real number. Answers will vary.Sample answer: Draw a number line that shows the integers. To find thevalue of the greatest integer function for any real number, place thatnumber on the number line. If it is an integer, the value of the function isthe number itself. If not, move to the integer directly to the left of thenumber you chose. This integer will give the value you need.

"54.01

66.2

"33


Greatest Integer FunctionsUse the greatest integer function % x& to explore some unusual graphs. It will be helpful to make a chart of values for each functions and to use a colored pen or pencil.

Graph each function.

1. y " 2x ! % x& 2. y " %%%xx

&&

%

3. y " %%%00..55xx

#

#

11

&&

% 4. y " %%xx&%

x

y

O 1–1–2–3–4 2 3 4

4

3

2

1

–1

–2

–3

–4

x

y

O 1–1–2–3–4 2 3 4

4

3

2

1

–1

–2

–3

–4

x

y

O 1–1–2–3–4 2 3 4

4

3

2

1

–1

–2

–3

–4

x

y

O 1–1–2–3–4 2 3 4

4

3

2

1

–1

–2

–3

–4

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6

Study Guide and InterventionGraphing Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7


Less

on

2-7

Graph Linear Inequalities. A linear inequality, like y ( 2x ! 1, resembles a linearequation, but with an inequality sign instead of an equals sign. The graph of the relatedlinear equation separates the coordinate plane into two half-planes. The line is theboundary of each half-plane.

To graph a linear inequality, follow these steps.

1. Graph the boundary, that is, the related linear equation. If the inequality symbol is ) or (, the boundary is solid. If the inequality symbol is ' or *, the boundary is dashed.

2. Choose a point not on the boundary and test it in the inequality. (0, 0) is a good point tochoose if the boundary does not pass through the origin.

3. If a true inequality results, shade the half-plane containing your test point. If a falseinequality results, shade the other half-plane.

Graph x $ 2y # 4.

The boundary is the graph of x # 2y " 4.

Use the slope-intercept form, y " ! x # 2, to graph the boundary line.

The boundary line should be solid.

Now test the point (0, 0).

0 # 2(0) (? 4 (x, y ) " (0, 0)

0 ( 4 false

Shade the region that does not contain (0, 0).

Graph each inequality.

1. y ' 3x # 1 2. y ( x ! 5 3. 4x # y ) !1

4. y ' ! 4 5. x # y * 6 6. 0.5x ! 0.25y ' 1.5

x

y

O

x

y

O

x

y

O

x%2

x

y

O

x

y

O

x

y

O

1%2

x

y

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ExercisesExercises

ExampleExample


Graph Absolute Value Inequalities Graphing absolute value inequalities is similarto graphing linear inequalities. The graph of the related absolute value equation is theboundary. This boundary is graphed as a solid line if the inequality is ) or (, and dashed ifthe inequality is ' or *. Choose a test point not on the boundary to determine which regionto shade.

Graph y ) 3⏐x " 1⏐.

First graph the equation y " 3x ! 1.Since the inequality is ), the graph of the boundary is solid.Test (0, 0).0 )? 30 ! 1 (x, y) " (0, 0)

0 )? 3!1 !1 " 1

0 ) 3 true

Shade the region that contains (0, 0).


1. y ( x # 1 2. y ) 2x ! 1 3. y ! 2x * 3

4. y ' !x ! 3 5. x # y ( 4 6. x # 1 # 2y ' 0

7. 2 ! x # y * !1 8. y ' 3x ! 3 9. y ) 1 ! x # 4

x

y

O

x

y

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x

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x

y

O

x

y

O

x

y

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x

y

Ox

y

Ox

y

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x

y

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Graphing Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7

ExercisesExercises

ExampleExample

Skills PracticeGraphing Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7


Less

on

2-7


1. y * 1 2. y ) x # 2 3. x # y ) 4

4. x # 3 ' y 5. 2 ! y ' x 6. y ( !x

7. x ! y * !2 8. 9x # 3y ! 6 ) 0 9. y # 1 ( 2x

10. y ! 7 ) !9 11. x * !5 12. y * x

x

y

Ox

y

Ox

y

O

x

y

Ox

y

Ox

y

O

x

y

Ox

y

O

x

y

O

x

y

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x

y

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y

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1. y ) !3 2. x * 2 3. x # y ) !4

4. y ' !3x # 5 5. y ' x # 3 6. y ! 1 ( !x

7. x ! 3y ) 6 8. y * x ! 1 9. y * !3x # 1 ! 2

COMPUTERS For Exercises 10–12, use the following information.

A school system is buying new computers. They will buy desktop computers costing $1000 per unit, andnotebook computers costing $1200 per unit. The total cost of the computers cannot exceed $80,000.

10. Write an inequality that describes this situation.1000d $ 1200n ) 80,000

11. Graph the inequality.

12. If the school wants to buy 50 of the desktop computers and 25 of the notebook computers,will they have enough money? yes

Desktops

No

teb

oo

ks

100 30 5020 40 60 70 80 90 100

80

70

60

50

40

30

20

10

Computers Purchased

x

y

Ox

y

O

x

y

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1%2

x

y

O

x

y

O

x

y

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Practice (Average)

Graphing Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7

Reading to Learn MathematicsGraphing Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7


Less

on

2-7

Pre-Activity How do inequalities apply to fantasy football?


• Which of the combinations of yards and touchdowns listed would Danaconsider a good game? The first one: 168 yards and 3 touchdowns

• Suppose that in one of the games Dana plays, Moss gets 157 receivingyards. What is the smallest number of touchdowns he must get in orderfor Dana to consider this a good game? 3

Reading the Lesson

1. When graphing a linear inequality in two variables, how do you know whether to makethe boundary a solid line or a dashed line? If the symbol is # or ), the line issolid. If the symbol is * or (, the line is dashed.

2. How do you know which side of the boundary to shade? Sample answer: If the testpoint gives a true inequality, shade the region containing the test point. Ifthe test point gives a false inequality, shade the region not containingthe test point.

3. Match each inequality with its graph.

a. y * 2x ! 3 iii b. y ' !2x # 3 iv c. y ( 2x ! 3 ii d. y ( !2x # 3 i

i. ii. iii. iv.


4. Describe some ways in which graphing an inequality in one variable on a number line issimilar to graphing an inequality in two variables in a coordinate plane. How can whatyou know about graphing inequalities on a number line help you to graph inequalities ina coordinate plane? Sample answer: A boundary on a coordinate graph issimilar to an endpoint on a number line graph. A dashed line is similar toa circle on a number line: both are open and mean not included; theyrepresent the symbols * and (. A solid line is similar to a dot on anumber line: both are closed and mean included; they represent thesymbols # and ).

x

y

O

x

y

Ox

y

O

x

y

O


Algebraic ProofThe following paragraph states a result you might be asked to prove in amathematics course. Parts of the paragraph are numbered.

01 Let n be a positive integer.

02 Also, let n1 " s(n1) be the sum of the squares of the digits in n.

03 Then n2 " s(n1) is the sum of the squares of the digits of n1, and n3 " s(n2)is the sum of the squares of the digits of n2.

04 In general, nk " s(nk ! 1) is the sum of the squares of the digits of nk ! 1.

05 Consider the sequence: n, n1, n2, n3, …, nk, ….

06 In this sequence either all the terms from some k on have the value 1,

07 or some term, say nj, has the value 4, so that the eight terms 4, 16, 37, 58, 89, 145, 42, and 20 keep repeating from that point on.

Use the paragraph to answer these questions.

1. Use the sentence in line 01. List the first five values of n.

2. Use 9246 for n and give an example to show the meaning of line 02.

3. In line 02, which symbol shows a function? Explain the function in a sentence.

4. For n " 9246, find n2 and n3 as described in sentence 03.

5. How do the first four sentences relate to sentence 05?

6. Use n " 31 and find the first four terms of the sequence.

7. Which sentence of the paragraph is illustrated by n " 31?

8. Use n " 61 and find the first ten terms.

9. Which sentence is illustrated by n " 61?

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7

© Glencoe/McGraw-Hill A2 Glencoe Algebra 2

Answers (Lesson 2-1)

Stu

dy

Gu

ide

and I

nte

rven

tion

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____

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;yes

R !

all r

eals

;yesx

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

xy

!1

!5

0!

3

1!

1

21

33

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-Hi

ll58

Gle

ncoe

Alg

ebra

2

Equ

atio

ns

of

Fun

ctio

ns

and

Rel

atio

ns

Equ

atio

ns t

hat

repr

esen

t fu

ncti

ons

are

ofte

n w

ritt

en in

fu

nct

ion

al n

otat

ion

.For

exa

mpl

e,y

"10

!8x

can

be w

ritt

en a

s f(

x) "

10 !

8x.T

his

nota

tion

em

phas

izes

the

fac

t th

at t

he v

alue

s of

y,t

he d

epen

den

tva

riab

le,d

epen

d on

the

val

ues

of x

,the

in

dep

end

ent

vari

able

.

To e

valu

ate

a fu

ncti

on,o

r fi

nd a

fun

ctio

nal v

alue

,mea

ns t

o su

bsti

tute

a g

iven

val

ue in

the

dom

ain

into

the

equ

atio

n to

fin

d th

e co

rres

pond

ing

elem

ent

in t

he r

ange

.

Giv

en t

he

fun

ctio

n f

(x)

!x2

$2x

,fin

d e

ach

val

ue.

a.f(

3)

f(x)

"x2

#2x

Orig

inal

func

tion

f(3)

"32

#2(

3)Su

bstit

ute.

"15

Sim

plify

.

b.f(

5a)

f(x)

"x2

#2x

Orig

inal

func

tion

f(5a

) "(5

a)2

#2(

5a)

Subs

titut

e.

"25

a2#

10a

Sim

plify

.

Fin

d e

ach

val

ue

if f

(x)

!"

2x$

4.

1.f(

12)

"20

2.f(

6)"

83.

f(2b

)"

4b$

4

Fin

d e

ach

val

ue

if g

(x)

!x3

"x.

4.g(

5)12

05.

g(!

2)"

66.

g(7c

)34

3c3

"7c

Fin

d e

ach

val

ue

if f

(x)

!2x

$an

d g

(x)

!0.

4x2

"1.

2.

7.f(

0.5)

58.

f(!

8)"

169.

g(3)

2.4

10.g

(!2.

5)1.

311

.f(4

a)8a

$12

.g!

""

1.2

13.f

!"6

14.g

(10)

38.8

15.f

(200

)40

0.01

Let

f(x

) !

2x2

"1.

16.F

ind

the

valu

es o

f f(2

) an

d f(

5).

f(2)

!7,

f(5)

!49

17.C

ompa

re t

he v

alue

s of

f(2

) $

f(5)

and

f(2

$5)

.f(

2) %

f(5)

!34

3,f(

2 %

5) !

199

2 & 31 % 3

b2& 10

b % 21 & 2a1 & 4

2 & x

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Rel

atio

ns a

nd F

unct

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-1

2-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Rel

atio

ns a

nd F

unct

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-1

2-1

©G

lenc

oe/M

cGra

w-Hi

ll59

Gle

ncoe

Alg

ebra

2

Lesson 2-1

Det

erm

ine

wh

eth

er e

ach

rel

atio

n i

s a

fun

ctio

n.W

rite

yes

or n

o.

1.ye

s2.

no

3.ye

s4.

no

Gra

ph

eac

h r

elat

ion

or

equ

atio

n a

nd

fin

d t

he

dom

ain

an

d r

ange

.Th

en d

eter

min

ew

het

her

th

e re

lati

on o

r eq

uat

ion

is

a fu

nct

ion

.

5.{(

2,!

3),(

2,4)

,(2,

!1)

}6.

{(2,

6),(

6,2)

}

D !

{2},

R !

{"3,

"1,

4};n

oD

!{2

,6},

R !

{2,6

};ye

s7.

{(!

3,4)

,(!

2,4)

,(!

1,!

1),(

3,!

1)}

8.x

"!

2

D !

{"3,

"2,

"1,

3},

D !

{"2}

,R !

all r

eals

;no

R !

{"1,

4};y

es

Fin

d e

ach

val

ue

if f

(x)

!2x

"1

and

g(x

) !

2 "

x2.

9.f(

0)"

110

.f(1

2)23

11.g

(4)

"14

12.f

(!2)

"5

13.g

(!1)

114

.f(d

)2d

"1

x

y

O

( –2,

4)

( –3,

4)

( –1,

–1)

( 3, –

1)x

y

O

( 2, 6

) ( 6, 2

)

x

y

O

( 2, 4

)

( 2, –

1)

( 2, –

3)

x

y

O

x

y

O

xy

12

24

36

D 3

R 1 5

D 100

200

300

R 50

100

150

©G

lenc

oe/M

cGra

w-Hi

ll60

Gle

ncoe

Alg

ebra

2

Det

erm

ine

wh

eth

er e

ach

rel

atio

n i

s a

fun

ctio

n.W

rite

yes

or n

o.

1.no

2.ye

s

3.ye

s4.

no

Gra

ph

eac

h r

elat

ion

or

equ

atio

n a

nd

fin

d t

he

dom

ain

an

d r

ange

.Th

en d

eter

min

ew

het

her

th

e re

lati

on o

r eq

uat

ion

is

a fu

nct

ion

.

5.{(

!4,

!1)

,(4,

0),(

0,3)

,(2,

0)}

6.y

"2x

!1

D !

{"4,

0,2,

4},

D !

all r

eals

,R !

all r

eals

;yes

R !

{"1,

0,3}

;yes

Fin

d e

ach

val

ue

if f

(x)

!an

d g

(x)

!"

2x$

3.

7.f(

3)1

8.f(

!4)

"9.

g !"2

10.f

(!2)

unde

fined

11.g

(!6)

1512

.f(m

!2)

13.M

USI

CT

he o

rder

ed p

airs

(1,

16),

(2,1

6),(

3,32

),(4

,32)

,and

(5,

48)

repr

esen

t th

e co

st o

fbu

ying

var

ious

num

bers

of

CD

s th

roug

h a

mus

ic c

lub.

Iden

tify

the

dom

ain

and

rang

e of

the

rela

tion

.Is

the

rela

tion

a f

unct

ion?

D !

{1,2

,3,4

,5},

R !

{16,

32,4

8};y

es

14.C

OM

PUTI

NG

If a

com

pute

r ca

n do

one

cal

cula

tion

in 0

.000

0000

015

seco

nd,t

hen

the

func

tion

T(n

) "0.

0000

0000

15n

give

s th

e ti

me

requ

ired

for

the

com

pute

r to

do

nca

lcul

atio

ns.H

ow lo

ng w

ould

it t

ake

the

com

pute

r to

do

5 bi

llion

cal

cula

tion

s?7.

5 s

5 & m

1 % 25 & 2

5& x

$2

x

y

O

( –4,

–1)

( 2, 0

)

( 0, 3

)

( 4, 0

) x

y

O

x

y

O

xy

!3

0

!1

!1

00

2!

2

34

D 5 10 15

R 105

110

D 2 8

R 21 25 30

Pra

ctic

e (A

vera

ge)

Rel

atio

ns a

nd F

unct

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

__PE

RIO

D__

___

2-1

2-1



Rea

din

g t

o L

earn

Math

emati

csR

elat

ions

and

Fun

ctio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-1

2-1

©G

lenc

oe/M

cGra

w-Hi

ll61

Gle

ncoe

Alg

ebra

2

Lesson 2-1

Pre-

Act

ivit

yH

ow d

o re

lati

ons

and

fu

nct

ion

s ap

ply

to

biol

ogy?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

1 at

the

top

of

page

56

in y

our

text

book

.

•R

efer

to

the

tabl

e.W

hat

does

the

ord

ered

pai

r (8

,20)

tel

l you

?Fo

r ade

er,t

he a

vera

ge lo

ngev

ity is

8 y

ears

and

the

max

imum

long

evity

is 2

0 ye

ars.

•Su

ppos

e th

at t

his

tabl

e is

ext

ende

d to

incl

ude

mor

e an

imal

s.Is

it p

ossi

ble

to h

ave

an o

rder

ed p

air

for

the

data

in w

hich

the

fir

st n

umbe

r is

larg

erth

an t

he s

econ

d?Sa

mpl

e an

swer

:No,

the

max

imum

long

evity

mus

t alw

ays

be g

reat

er th

an th

e av

erag

e lo

ngev

ity.

Rea

din

g t

he

Less

on

1.a.

Exp

lain

the

dif

fere

nce

betw

een

a re

lati

on a

nd a

fun

ctio

n.Sa

mpl

e an

swer

:Are

latio

n is

any

set

of o

rder

ed p

airs

.A fu

nctio

n is

a s

peci

al k

ind

ofre

latio

n in

whi

ch e

ach

elem

ent o

f the

dom

ain

is p

aire

d w

ith e

xact

lyon

e el

emen

t in

the

rang

e.b.

Exp

lain

the

dif

fere

nce

betw

een

dom

ain

and

rang

e.Sa

mpl

e an

swer

:The

dom

ain

of a

rela

tion

is th

e se

t of a

ll fir

st c

oord

inat

es o

f the

ord

ered

pai

rs.T

hera

nge

is th

e se

t of a

ll se

cond

coo

rdin

ates

.

2.a.

Wri

te t

he d

omai

n an

d ra

nge

of t

he r

elat

ion

show

n in

the

gra

ph.

D:{"

3,"

2,"

1,0,

3};R

:{"

5,"

4,0,

1,2,

4}b.

Is t

his

rela

tion

a f

unct

ion?

Exp

lain

.Sa

mpl

e an

swer

:No,

it is

not

a fu

nctio

nbe

caus

e on

e of

the

elem

ents

of t

he d

omai

n,3,

is p

aire

d w

ith tw

oel

emen

ts o

f the

rang

e.

Hel

pin

g Y

ou

Rem

emb

er

3.L

ook

up t

he w

ords

dep

ende

ntan

d in

depe

nden

tin

a d

icti

onar

y.H

ow c

an t

he m

eani

ng o

fth

ese

wor

ds h

elp

you

dist

ingu

ish

betw

een

inde

pend

ent

and

depe

nden

t va

riab

les

in a

func

tion

?Sa

mpl

e an

swer

:The

var

iabl

e w

hose

val

ues

depe

nd o

n,or

are

dete

rmin

ed b

y,th

e va

lues

of t

he o

ther

var

iabl

e is

the

depe

nden

t var

iabl

e.

( 0, 4

)

( 3, 1

)

( 3, –

4)( –

1, –

5)

( –2,

0)

( –3,

2)

x

y

O

©G

lenc

oe/M

cGra

w-Hi

ll62

Gle

ncoe

Alg

ebra

2

Map

ping

sT

here

are

thr

ee s

peci

al w

ays

in w

hich

one

set

can

be

map

ped

to a

noth

er.A

set

can

be m

appe

d in

toan

othe

r se

t,on

toan

othe

r se

t,or

can

hav

e a

one-

to-o

neco

rres

pond

ence

wit

h an

othe

r se

t.

Sta

te w

het

her

eac

h s

et i

s m

app

ed i

nto

th

e se

con

d s

et,o

nto

th

e se

con

d

set,

or h

as a

on

e-to

-on

e co

rres

pon

den

ce w

ith

th

e se

con

d s

et.

1.2.

3.4.

into

,ont

oin

to,o

nto

into

,ont

o,in

to,o

nto

one-

to-o

ne

5.6.

7.8.

into

into

,ont

oin

to,o

nto

into

,ont

o,on

e-to

-one

9.C

an a

set

be

map

ped

onto

a se

t w

ith

few

er e

lem

ents

tha

n it

has

?ye

s

10.C

an a

set

be

map

ped

into

a se

t th

at h

as m

ore

elem

ents

tha

n it

has

?ye

s

11.I

f a

map

ping

fro

m s

et A

into

set

Bis

a o

ne-t

o-on

e co

rres

pond

ence

,wha

t ca

n yo

u co

nclu

de a

bout

the

num

ber

of e

lem

ents

in A

and

B?

The

sets

hav

e th

e sa

me

num

ber o

f ele

men

ts.

–2 9 12 5

1 4 –7 0

–2 9 12 5

1 4 –7 0

–315 10 2

–2 9 12 5

1 4 –7 0

10 –6 24 2

3

1 3 7 9 –5

a g k l q

0 –3 9 7

4 12 6

2 4 –1 –4

7 0 2

Into

map

ping

Am

appi

ng fr

om s

et A

to s

et B

wher

e ev

ery

elem

ent o

f Ais

map

ped

to o

ne o

r mor

e el

emen

ts o

f set

B,b

ut n

ever

to a

n el

emen

t not

in B

.

Ont

o m

appi

ngA

map

ping

from

set

Ato

set

Bwh

ere

each

ele

men

t of s

et B

has

at le

ast o

ne e

lem

ent o

f se

t Am

appe

d to

it.

One

-to-o

ne

Am

appi

ng fr

om s

et A

onto

set

Bwh

ere

each

ele

men

t of s

et A

is m

appe

d to

exa

ctly

one

corr

espo

nden

ceel

emen

t of s

et B

and

diffe

rent

ele

men

ts o

f Aar

e ne

ver m

appe

d to

the

sam

e el

emen

t of B

.

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-1

2-1


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Line

ar E

quat

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-2

2-2

©G

lenc

oe/M

cGra

w-Hi

ll63

Gle

ncoe

Alg

ebra

2

Lesson 2-2

Iden

tify

Lin

ear

Equ

atio

ns

and

Fu

nct

ion

sA

lin

ear

equ

atio

nha

s no

ope

rati

ons

othe

r th

an a

ddit

ion,

subt

ract

ion,

and

mul

tipl

icat

ion

of a

var

iabl

e by

a c

onst

ant.

The

vari

able

s m

ay n

ot b

e m

ulti

plie

d to

geth

er o

r ap

pear

in a

den

omin

ator

.A li

near

equ

atio

n do

esno

t co

ntai

n va

riab

les

wit

h ex

pone

nts

othe

r th

an 1

.The

gra

ph o

f a

linea

r eq

uati

on is

a li

ne.

A l

inea

r fu

nct

ion

is a

fun

ctio

n w

hose

ord

ered

pai

rs s

atis

fy a

line

ar e

quat

ion.

Any

line

arfu

ncti

on c

an b

e w

ritt

en in

the

for

m f

(x) "

mx

#b,

whe

re m

and

bar

e re

al n

umbe

rs.

If a

n eq

uati

on is

line

ar,y

ou n

eed

only

tw

o po

ints

tha

t sa

tisf

y th

e eq

uati

on in

ord

er t

o gr

aph

the

equa

tion

.One

way

is t

o fi

nd t

he x

-int

erce

pt a

nd t

he y

-int

erce

pt a

nd c

onne

ct t

hese

tw

opo

ints

wit

h a

line. Is

f(x

) !

0.2

"a

lin

ear

fun

ctio

n?

Exp

lain

.

Yes;

it is

a li

near

fun

ctio

n be

caus

e it

can

be w

ritt

en in

the

for

mf(

x) "

!x

#0.

2. Is 2

x$

xy"

3y!

0 a

lin

ear

fun

ctio

n?

Exp

lain

.

No;

it is

not

a li

near

fun

ctio

n be

caus

eth

e va

riab

les

xan

d y

are

mul

tipl

ied

toge

ther

in t

he m

iddl

e te

rm.

1 % 5

x & 5F

ind

th

e x-

inte

rcep

t an

d t

he

y-in

terc

ept

of t

he

grap

h o

f 4x

"5y

!20

.T

hen

gra

ph

th

e eq

uat

ion

.

The

x-i

nter

cept

is t

he v

alue

of x

whe

n y

"0.

4x!

5y"

20O

rigin

al e

quat

ion

4x!

5(0)

"20

Subs

titut

e 0

for y

.

x"

5Si

mpl

ify.

So t

he x

-int

erce

pt is

5.

Sim

ilarl

y,th

e y-

inte

rcep

t is

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E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-2

2-2

Exam

ple

Exam

ple



Stu

dy

Gu

ide

and I

nte

rven

tion

Slop

e

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-3

2-3

©G

lenc

oe/M

cGra

w-Hi

ll69

Gle

ncoe

Alg

ebra

2

Lesson 2-3

Slo

pe

Slop

e m

of a

Lin

eFo

r poi

nts

(x1,

y 1) a

nd (x

2, y 2

), wh

ere

x 1&

x 2, m

""

y 2!

y 1% x 2

!x 1

chan

ge in

y%

%ch

ange

in x

Det

erm

ine

the

slop

e of

the

lin

e th

at p

asse

s th

rou

gh (

2,"

1) a

nd

("4,

5).

m"

Slop

e fo

rmul

a

"(x

1, y 1

) "(2

, !1)

, (x 2

, y2)

"(!

4, 5

)

""

!1

Sim

plify

.

The

slo

pe o

f th

e lin

e is

!1.

6% !

6

5 !

(!1)

%%

!4

!2

y 2!

y 1% x 2

!x 1

Gra

ph

th

e li

ne

pas

sin

gth

rou

gh (

"1,

"3)

wit

h a

slo

pe

of

.

Gra

ph t

he o

rder

ed

pair

(!

1,!

3).T

hen,

acco

rdin

g to

the

sl

ope,

go u

p 4

unit

san

d ri

ght

5 un

its.

Plo

t th

e ne

w p

oint

(4,1

).C

onne

ct t

hepo

ints

and

dra

w

the

line.

x

y

O

4 & 5

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d t

he

slop

e of

th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts.

1.(4

,7)

and

(6,1

3)3

2.(6

,4)

and

(3,4

)0

3.(5

,1)

and

(7,!

3)"

2

4.(5

,!3)

and

(!

4,3)

"5.

(5,1

0) a

nd (

!1,

!2)

26.

(!1,

!4)

and

(!13

,2)"

7.(7

,!2)

and

(3,

3)"

8.(!

5,9)

and

(5,

5)"

9.(4

,!2)

and

(!

4,!

8)

Gra

ph

th

e li

ne

pas

sin

g th

rou

gh t

he

give

n p

oin

t w

ith

th

e gi

ven

slo

pe.

10.s

lope

"!

11.s

lope

"2

12.s

lope

"0

pass

es t

hrou

gh (

0,2)

pass

es t

hrou

gh (

1,4)

pass

es t

hrou

gh (!

2,!

5)

13.s

lope

"1

14.s

lope

"!

15.s

lope

"

pass

es t

hrou

gh (!

4,6)

pass

es t

hrou

gh (

!3,

0)pa

sses

thr

ough

(0,

0) x

y

O

x

y

O

x

y

O

1 % 53 % 4

x

y

O

x

y

Ox

y

O

1 % 3

3 & 42 & 5

5 & 4

1 & 22 & 3

©G

lenc

oe/M

cGra

w-Hi

ll70

Gle

ncoe

Alg

ebra

2

Para

llel a

nd

Per

pen

dic

ula

r Li

nes

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Slop

e

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-3

2-3

In a

pla

ne,n

onve

rtic

al li

nes

wit

h th

esa

me

slop

e ar

e p

aral

lel.

All

vert

ical

lines

are

par

alle

l.

x

y

O

slop

e !

m

slop

e !

m

In a

pla

ne,t

wo

obliq

ue li

nes

are

per

pen

dic

ula

rif

and

only

if t

he p

rodu

ct o

f th

eir

slop

es is

!1.

Any

vert

ical

line

is p

erpe

ndic

ular

to

any

hori

zont

al li

ne.

x

y

O

slop

e !

m

slop

e !

"1 m

Exam

ple

Exam

ple

Are

th

e li

ne

pas

sin

g th

rou

gh (

2,6)

an

d (

"2,

2) a

nd

th

e li

ne

pas

sin

gth

rou

gh (

3,0)

an

d (

0,4)

par

alle

l,p

erp

end

icu

lar,

or n

eith

er?

Fin

d th

e sl

opes

of

the

two

lines

.

The

slo

pe o

f th

e fi

rst

line

is

"1.

The

slo

pe o

f th

e se

cond

line

is

"!

.

The

slo

pes

are

not

equa

l and

the

pro

duct

of

the

slop

es is

not

!1,

so t

he li

nes

are

neit

her

para

llel n

or p

erpe

ndic

ular

.

Are

th

e li

nes

par

alle

l,p

erp

end

icu

lar,

or n

eith

er?

1.th

e lin

e pa

ssin

g th

roug

h (4

,3)

and

(1.!

3) a

nd t

he li

ne p

assi

ng t

hrou

gh (

1,2)

and

(!1,

3)pe

rpen

dicu

lar

2.th

e lin

e pa

ssin

g th

roug

h (2

,8)

and

(!2,

2) a

nd t

he li

ne p

assi

ng t

hrou

gh (

0,9)

and

(6,

0)ne

ither

3.th

e lin

e pa

ssin

g th

roug

h (3

,9)

and

(!2,

!1)

and

the

gra

ph o

f y"

2xpa

ralle

l

4.th

e lin

e w

ith

x-in

terc

ept

!2

and

y-in

terc

ept

5 an

d th

e lin

e w

ith

x-in

terc

ept

2 an

d y-

inte

rcep

t !

5pa

ralle

l

5.th

e lin

e w

ith

x-in

terc

ept

1 an

d y-

inte

rcep

t 3

and

the

line

wit

h x-

inte

rcep

t 3

and

y-in

terc

ept

1ne

ither

6.th

e lin

e pa

ssin

g th

roug

h (!

2,!

3) a

nd (

2,5)

and

the

gra

ph o

f x#

2y"

10pe

rpen

dicu

lar

7.th

e lin

e pa

ssin

g th

roug

h (!

4,!

8) a

nd (

6,!

4) a

nd t

he g

raph

of

2x!

5y"

5pa

ralle

l

4 % 34

!0

% 0 !

3

6 !

2%

%2

!(!

2)

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Slop

e

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-3

2-3

©G

lenc

oe/M

cGra

w-Hi

ll71

Gle

ncoe

Alg

ebra

2

Lesson 2-3

Fin

d t

he

slop

e of

th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts.

1.(1

,5),

(!1,

!3)

42.

(0,2

),(3

,0)

"3.

(1,9

),(0

,6)

3

4.(8

,!5)

,(4,

!2)

"5.

(!3,

5),(

!3,

!1)

unde

fined

6.(!

2,!

2),(

10,!

2)0

7.(4

,5),

(2,7

)"

18.

(!2,

!4)

,(3,

2)9.

(5,2

),(!

3,2)

0

Gra

ph

th

e li

ne

pas

sin

g th

rou

gh t

he

give

n p

oin

t w

ith

th

e gi

ven

slo

pe.

10.(

0,4)

,m"

111

.(2,

!4)

,m"

!1

12.(

!3,

!5)

,m"

213

.(!

2,!

1),m

"!

2

Gra

ph

th

e li

ne

that

sat

isfi

es e

ach

set

of

con

dit

ion

s.

14.p

asse

s th

roug

h (0

,1),

perp

endi

cula

r to

15.p

asse

s th

roug

h (0

,!5)

,par

alle

l to

the

a lin

e w

hose

slo

pe is

gr

aph

of y

"1

16.H

IKIN

GN

aom

i lef

t fr

om a

n el

evat

ion

of 7

400

feet

at

7:00

A.M

.and

hik

ed t

o an

ele

vati

onof

980

0 fe

et b

y 11

:00

A.M

.Wha

t w

as h

er r

ate

of c

hang

e in

alt

itud

e?60

0 ft

/h

(0 ,–

5)

x

y

O(0

,1)

x

y

O

1 % 3

(–2,

–1)

x

y

O

(–3,

–5)

x

y

O

( 2, –

4)x

y

O( 0

, 4)

x

y

O

6 & 5

3 & 4

2 & 3

©G

lenc

oe/M

cGra

w-Hi

ll72

Gle

ncoe

Alg

ebra

2

Fin

d t

he

slop

e of

th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts.

1.(3

,!8)

,(!

5,2)

"2.

(!10

,!3)

,(7,

2)3.

(!7,

!6)

,(3,

!6)

0

4.(8

,2),

(8,!

1)un

defin

ed5.

(4,3

),(7

,!2)

"6.

(!6,

!3)

,(!

8,4)

"

Gra

ph

th

e li

ne

pas

sin

g th

rou

gh t

he

give

n p

oin

t w

ith

th

e gi

ven

slo

pe.

7.(0

,!3)

,m"

38.

(2,1

),m

"!

9.(0

,2),

m"

010

.(2,

!3)

,m"

Gra

ph

th

e li

ne

that

sat

isfi

es e

ach

set

of

con

dit

ion

s.

11.p

asse

s th

roug

h (3

,0),

perp

endi

cula

r12

.pas

ses

thro

ugh

(!3,

!1)

,par

alle

l to

a lin

e

to a

line

who

se s

lope

is

who

se s

lope

is !

1

DEP

REC

IATI

ON

For

Exe

rcis

es 1

3–15

,use

th

e fo

llow

ing

info

rmat

ion

.A

mac

hine

tha

t or

igin

ally

cos

t $1

5,60

0 ha

s a

valu

e of

$75

00 a

t th

e en

d of

3 y

ears

.The

sam

em

achi

ne h

as a

val

ue o

f $2

800

at t

he e

nd o

f 8

year

s.

13.F

ind

the

aver

age

rate

of

chan

ge in

val

ue (

depr

ecia

tion

) of

the

mac

hine

bet

wee

n it

spu

rcha

se a

nd t

he e

nd o

f 3

year

s."

$270

0 pe

r yea

r14

.Fin

d th

e av

erag

e ra

te o

f ch

ange

in v

alue

of

the

mac

hine

bet

wee

n th

e en

d of

3 y

ears

and

the

end

of 8

yea

rs.

"$9

40 p

er y

ear

15.I

nter

pret

the

sig

n of

you

r an

swer

s.It

is n

egat

ive

beca

use

the

valu

e is

dec

reas

ing.

( –3,

–1)

x

y

O

(3, 0

)x

y

O

3 % 2

( 2, –

3)

x

y

O( 0

, 2)

x

y

O

4 % 5

(2, 1

)

x

y

O

(0, –

3)

x

y

O

3 % 4

7 & 25 & 35 & 17

5 & 4

Pra

ctic

e (A

vera

ge)

Slop

e

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-3

2-3



Rea

din

g t

o L

earn

Math

emati

csSl

ope

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-3

2-3

©G

lenc

oe/M

cGra

w-Hi

ll73

Gle

ncoe

Alg

ebra

2

Lesson 2-3

Pre-

Act

ivit

yH

ow d

oes

slop

e ap

ply

to

the

stee

pn

ess

of r

oad

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

3 at

the

top

of

page

68

in y

our

text

book

.

•W

hat

is t

he g

rade

of

a ro

ad t

hat

rise

s 40

fee

t ov

er a

hor

izon

tal d

ista

nce

of 1

000

feet

?4%

•W

hat

is t

he g

rade

of

a ro

ad t

hat

rise

s 52

5 m

eter

s ov

er a

hor

izon

tal

dist

ance

of

10 k

ilom

eter

s? (

1 ki

lom

eter

"10

00 m

eter

s)5.

25%

Rea

din

g t

he

Less

on

1.D

escr

ibe

each

typ

e of

slo

pe a

nd in

clud

e a

sket

ch.

Type

of S

lope

Desc

riptio

n of

Gra

phSk

etch

Posit

iveTh

e lin

e ris

es to

the

right

.

Zero

The

line

is h

oriz

onta

l.

Nega

tive

The

line

falls

to th

e rig

ht.

Unde

fined

The

line

is v

ertic

al.

2.a.

How

are

the

slo

pes

of t

wo

nonv

erti

cal p

aral

lel l

ines

rel

ated

?Th

ey a

re e

qual

.b.

How

are

the

slo

pes

of t

wo

obliq

ue p

erpe

ndic

ular

line

s re

late

d?Th

eir p

rodu

ct is

"1.

Hel

pin

g Y

ou

Rem

emb

er

3.L

ook

up t

he t

erm

s gr

ade,

pitc

h,sl

ant,

and

slop

e.H

ow c

an e

very

day

mea

ning

s of

the

sew

ords

hel

p yo

u re

mem

ber

the

defi

niti

on o

f sl

ope?

Sam

ple

answ

er:A

ll th

ese

wor

dsca

n be

use

d w

hen

you

desc

ribe

how

muc

h a

thin

g sl

ants

upw

ard

ordo

wnw

ard.

You

can

desc

ribe

this

num

eric

ally

by

com

parin

g ris

e to

run.

x

y

O

x

y

O

x

y

O

x

y

O

©G

lenc

oe/M

cGra

w-Hi

ll74

Gle

ncoe

Alg

ebra

2

Aer

ial S

urve

yors

and

Are

aM

any

land

reg

ions

hav

e ir

regu

lar

shap

es.A

eria

l sur

veyo

rs

supp

ly a

eria

l map

pers

wit

h lis

ts o

f co

ordi

nate

s an

d el

evat

ions

fo

r th

e ar

eas

that

nee

d to

be

phot

ogra

phed

fro

m t

he a

ir.T

hese

m

aps

prov

ide

info

rmat

ion

abou

t th

e ho

rizo

ntal

and

ver

tica

l fe

atur

es o

f th

e la

nd.

Ste

p 1

Lis

t th

e or

dere

d pa

irs

for

the

vert

ices

in

coun

terc

lock

wis

e or

der,

repe

atin

g th

e fi

rst

orde

red

pair

at

the

bott

om o

f th

e lis

t.

Ste

p 2

Fin

d D

,the

sum

of

the

dow

nwar

d di

agon

al p

rodu

cts

(fro

m le

ft t

o ri

ght)

.D

"(5

$5)

#(2

$1)

#(2

$3)

#(6

$7)

"25

#2

#6

#42

or

75

Ste

p 3

Fin

d U

,the

sum

of

the

upw

ard

diag

onal

pro

duct

s (f

rom

left

to

righ

t).

U"

(2 $

7) #

(2 $

5) #

(6 $

1) #

(5 $

3)"

14 #

10 #

6 #

15 o

r 45

Ste

p 4

Use

the

for

mul

a A

"%1 2% (

D!

U)

to f

ind

the

area

.

A"

%1 2% (75

!45

)

"%1 2% (

30)

or 1

5

The

are

a is

15

squa

re u

nits

.Cou

nt t

he n

umbe

r of

squ

are

unit

s en

clos

ed b

y th

e po

lygo

n.D

oes

this

res

ult

seem

rea

sona

ble?

Use

th

e co

ord

inat

e m

eth

od t

o fi

nd

th

e ar

ea o

f ea

ch r

egio

n i

n s

quar

e u

nit

s.

1.2.

3.

20 u

nits

214

uni

ts2

34 u

nits

2

x

y

O

x

y

Ox

y

O

(5, 7

)

(2, 5

)

(2, 1

)

(6, 3

)

(5, 7

)

x

y

O

(2, 1

)

(2, 5

)

(5, 7

) (6, 3

)

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-3

2-3


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Writ

ing

Line

ar E

quat

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-4

2-4

©G

lenc

oe/M

cGra

w-Hi

ll75

Gle

ncoe

Alg

ebra

2

Lesson 2-4

Form

s o

f Eq

uat

ion

s

Slop

e-In

terc

ept F

orm

of

a L

inea

r Equ

atio

ny

"m

x#

b, w

here

mis

the

slope

and

bis

the

y-in

terc

ept

Poin

t-Slo

pe F

orm

y

!y 1

"m

(x!

x 1),

wher

e (x

1, y 1

) are

the

coor

dina

tes

of a

poi

nt o

n th

e lin

e an

d of

a L

inea

r Equ

atio

nm

is th

e slo

pe o

f the

line

Wri

te a

n e

quat

ion

in

slop

e-in

terc

ept

form

for

th

e li

ne

that

has

slo

pe

"2

and

pas

ses

thro

ugh

th

ep

oin

t (3

,7).

Subs

titu

te f

or m

,x,a

nd y

in t

he

slop

e-in

terc

ept

form

.y

"m

x#

bSl

ope-

inte

rcep

t for

m

7 "

(!2)

(3) #

b(x

, y) "

(3, 7

), m

"!

2

7 "

!6

#b

Sim

plify

.

13 "

bAd

d 6

to b

oth

sides

.

The

y-i

nter

cept

is 1

3.T

he e

quat

ion

in

slop

e-in

terc

ept

form

is y

"!

2x#

13.

Wri

te a

n e

quat

ion

in

slop

e-in

terc

ept

form

for

th

e li

ne

that

has

slo

pe

and

x-i

nte

rcep

t 5.

y"

mx

#b

Slop

e-in

terc

ept f

orm

0 "

!"(5

) #

b(x

, y) "

(5, 0

), m

"

0 "

#b

Sim

plify

.

!"

bSu

btra

ct

from

bot

h sid

es.

The

y-i

nter

cept

is !

.The

slo

pe-i

nter

cept

form

is y

"x

!.

5 % 31 % 3

5 % 3

5 % 35 % 3

5 % 3

1 % 31 % 3

1 & 3

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm f

or t

he

lin

e th

at s

atis

fies

eac

h s

et o

fco

nd

itio

ns.

1.sl

ope

!2,

pass

es t

hrou

gh (!

4,6)

2.sl

ope

,y-i

nter

cept

4

y!

"2x

"2

y!

x$

4

3.sl

ope

1,pa

sses

thr

ough

(2,

5)4.

slop

e !

,pas

ses

thro

ugh

(5,!

7)

y!

x$

3y

!"

x$

6

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm f

or e

ach

gra

ph

.

5.6.

7.

y!

"3x

$9

y!

xy

!x

$14 & 9

1 & 95 & 4

x

y

O

( –4,

1)

( 5, 2

)

x

y O

( 4, 5

)

( 0, 0

)

x

y

O

( 1, 6

)

( 3, 0

)

13 & 513 % 5

3 & 23 % 2

©G

lenc

oe/M

cGra

w-Hi

ll76

Gle

ncoe

Alg

ebra

2

Para

llel a

nd

Per

pen

dic

ula

r Li

nes

Use

the

slo

pe-i

nter

cept

or

poin

t-sl

ope

form

to

find

equa

tion

s of

line

s th

at a

re p

aral

lel o

r pe

rpen

dicu

lar

to a

giv

en li

ne.R

emem

ber

that

par

alle

llin

es h

ave

equa

l slo

pe.T

he s

lope

s of

tw

o pe

rpen

dicu

lar

lines

are

neg

ativ

e re

cipr

ocal

s,th

atis

,the

ir p

rodu

ct is

!1.

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Writ

ing

Line

ar E

quat

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-4

2-4

Wri

te a

n e

quat

ion

of

the

lin

e th

at p

asse

s th

rou

gh (

8,2)

an

d i

sp

erp

end

icu

lar

to t

he

lin

e w

hos

e eq

uat

ion

is

y!

"x

$3.

The

slo

pe o

f th

e gi

ven

line

is !

.Sin

ce t

he

slop

es o

f pe

rpen

dicu

lar

lines

are

neg

ativ

ere

cipr

ocal

s,th

e sl

ope

of t

he p

erpe

ndic

ular

line

is 2

.U

se t

he s

lope

and

the

giv

en p

oint

to

wri

teth

e eq

uati

on.

y!

y1

"m

(x!

x 1)

Poin

t-slo

pe fo

rmy

!2

"2(

x!

8)(x

1, y 1

) "(8

, 2),

m"

2y

!2

"2x

!16

Dist

ribut

ive P

rop.

y"

2x!

14Ad

d 2

to e

ach

side.

An

equa

tion

of

the

line

is y

"2x

!14

.

1 % 2

1 & 2

Wri

te a

n e

quat

ion

of

the

lin

e th

at p

asse

s th

rou

gh (

"1,

5) a

nd

is

par

alle

l to

the

grap

h o

f y

!3x

$1.

The

slo

pe o

f th

e gi

ven

line

is 3

.Sin

ce t

hesl

opes

of

para

llel l

ines

are

equ

al,t

he s

lope

of t

he p

aral

lel l

ine

is a

lso

3.U

se t

he s

lope

and

the

giv

en p

oint

to

wri

teth

e eq

uati

on.

y!

y 1"

m(x

!x 1

)Po

int-s

lope

form

y!

5 "

3(x

!(!

1))

(x1,

y 1) "

(!1,

5),

m"

3y

!5

"3x

#3

Dist

ribut

ive P

rop.

y"

3x#

8Ad

d 5

to e

ach

side.

An

equa

tion

of

the

line

is y

"3x

#8.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm f

or t

he

lin

e th

at s

atis

fies

eac

h s

et o

fco

nd

itio

ns.

1.pa

sses

thr

ough

(!4,

2),p

aral

lel t

o th

e lin

e w

hose

equ

atio

n is

y"

x#

5y

!x

$4

2.pa

sses

thr

ough

(3,

1),p

erpe

ndic

ular

to

the

grap

h of

y"

!3x

#2

y!

x

3.pa

sses

thr

ough

(1,

!1)

,par

alle

l to

the

line

that

pas

ses

thro

ugh

(4,1

) an

d (2

,!3)

y!

2x"

3

4.pa

sses

thr

ough

(4,

7),p

erpe

ndic

ular

to

the

line

that

pas

ses

thro

ugh

(3,6

) an

d (3

,15)

y!

7

5.pa

sses

thr

ough

(8,

!6)

,per

pend

icul

ar t

o th

e gr

aph

of 2

x!

y"

4y

!"

x"

2

6.pa

sses

thr

ough

(2,

!2)

,per

pend

icul

ar t

o th

e gr

aph

of x

#5y

"6

y!

5x"

12

7.pa

sses

thr

ough

(6,

1),p

aral

lel t

o th

e lin

e w

ith

x-in

terc

ept

!3

and

y-in

terc

ept

5

y!

x"

9

8.pa

sses

thr

ough

(!2,

1),p

erpe

ndic

ular

to

the

line

y"

4x!

11y

!"

x$

1 & 21 & 4

5 & 3

1 & 2

1 & 3

1 & 21 % 2



Skil

ls P

ract

ice

Writ

ing

Line

ar E

quat

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-4

2-4

©G

lenc

oe/M

cGra

w-Hi

ll77

Gle

ncoe

Alg

ebra

2

Lesson 2-4

Sta

te t

he

slop

e an

d y

-in

terc

ept

of t

he

grap

h o

f ea

ch e

quat

ion

.

1.y

"7x

!5

7,"

52.

y"

!x

#3

",3

3.y

"x

,04.

3x#

4y"

4"

,1

5.7y

"4x

!7

,"1

6.3x

!2y

#6

"0

,3

7.2x

!y

"5

2,"

58.

2y"

6 !

5x"

,3

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm f

or e

ach

gra

ph

.

9.10

.11

.

y!

3x"

1y

!"

1y

!"

2x$

3

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm f

or t

he

lin

e th

at s

atis

fies

eac

h s

et o

fco

nd

itio

ns.

12.s

lope

3,p

asse

s th

roug

h (1

,!3)

13.s

lope

!1,

pass

es t

hrou

gh (

0,0)

y!

3x"

6y

!"

x

14.s

lope

!2,

pass

es t

hrou

gh (

0,!

5)15

.slo

pe 3

,pas

ses

thro

ugh

(2,0

)

y!

"2x

"5

y!

3x"

6

16.p

asse

s th

roug

h (!

1,!

2) a

nd (

!3,

1)17

.pas

ses

thro

ugh

(!2,

!4)

and

(1,

8)

y!

"x

"y

!4x

$4

18.x

-int

erce

pt 2

,y-i

nter

cept

!6

19.x

-int

erce

pt

,y-i

nter

cept

5

y!

3x"

6y

!"

2x$

5

20.p

asse

s th

roug

h (3

,!1)

,per

pend

icul

ar t

o th

e gr

aph

of y

"!

x!

4.y

!3x

"10

1 % 3

5 % 2

7 & 23 & 2

x

y

O

( 0, 3

)

( 3, –

3)

x

y

O( –

3, –

1)( 4

, –1)

x

y

O

( –1,

–4)

( 1, 2

)

5 & 2

3 & 24 & 7

3 & 42 & 3

2 % 3

3 & 53 % 5

©G

lenc

oe/M

cGra

w-Hi

ll78

Gle

ncoe

Alg

ebra

2

Sta

te t

he

slop

e an

d y

-in

terc

ept

of t

he

grap

h o

f ea

ch e

quat

ion

.

1.y

"8x

#12

8,12

2.y

"0.

25x

!1

0.25

,"1

3.y

"!

x"

,0

4.3y

"7

0,5.

3x"

!15

#5y

,36.

2x!

3y"

10,"

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm f

or e

ach

gra

ph

.

7.8.

9.

y!

2y

!x

"2

y!

"x

$1

Wri

te a

n e

quat

ion

in

slo

pe-

inte

rcep

t fo

rm f

or t

he

lin

e th

at s

atis

fies

eac

h s

et o

fco

nd

itio

ns.

10.s

lope

!5,

pass

es t

hrou

gh (!

3,!

8)11

.slo

pe

,pas

ses

thro

ugh

(10,

!3)

y!

"5x

"23

y!

x"

11

12.s

lope

0,p

asse

s th

roug

h (0

,!10

)13

.slo

pe !

,pas

ses

thro

ugh

(6,!

8)

y!

"10

y!

"x

"4

14.p

asse

s th

roug

h (3

,11)

and

(!6,

5)15

.pas

ses

thro

ugh

(7,!

2) a

nd (

3,!

1)

y!

x$

9y

!"

x"

16.x

-int

erce

pt 3

,y-i

nter

cept

217

.x-i

nter

cept

!5,

y-in

terc

ept

7

y!

"x

$2

y!

x$

7

18.p

asse

s th

roug

h (!

8,!

7),p

erpe

ndic

ular

to

the

grap

h of

y"

4x!

3y

!"

x"

919

.RES

ERV

OIR

ST

he s

urfa

ce o

f G

rand

Lak

e is

at

an e

leva

tion

of

648

feet

.Dur

ing

the

curr

ent

drou

ght,

the

wat

er le

vel i

s dr

oppi

ng a

t a

rate

of

3 in

ches

per

day

.If

this

tre

ndco

ntin

ues,

wri

te a

n eq

uati

on t

hat

give

s th

e el

evat

ion

in f

eet

of t

he s

urfa

ce o

f G

rand

Lak

eaf

ter

xda

ys.

y!

"0.

25x

$64

820

.BU

SIN

ESS

Tony

Mar

coni

’s c

ompa

ny m

anuf

actu

res

CD

-RO

M d

rive

s.T

he c

ompa

ny w

illm

ake

$150

,000

pro

fit

if it

man

ufac

ture

s 10

0,00

0 dr

ives

,and

$1,

750,

000

prof

it if

itm

anuf

actu

res

500,

000

driv

es.T

he r

elat

ions

hip

betw

een

the

num

ber

of d

rive

sm

anuf

actu

red

and

the

prof

it is

line

ar.W

rite

an

equa

tion

tha

t gi

ves

the

prof

it P

whe

n n

driv

es a

re m

anuf

actu

red.

P!

4n"

250,

000

1 & 4

7 & 52 & 3

1 & 41 & 4

2 & 3

2 & 32 % 3

4 & 54 % 5

2 & 33 & 2

x

y

O( 3

, –1)

( –3,

3)

x

y

O

( 4, 4

)

( 0, –

2)

x

y

O

( 0, 2

)

10 & 32 & 3

3 & 57 & 3

3 & 53 % 5

Pra

ctic

e (A

vera

ge)

Writ

ing

Line

ar E

quat

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-4

2-4


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csW

ritin

g Li

near

Equ

atio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-4

2-4

©G

lenc

oe/M

cGra

w-Hi

ll79

Gle

ncoe

Alg

ebra

2

Lesson 2-4

Pre-

Act

ivit

yH

ow d

o li

nea

r eq

uat

ion

s ap

ply

to

busi

nes

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

4 at

the

top

of

page

75

in y

our

text

book

.

•If

the

tot

al c

ost

of p

rodu

cing

a p

rodu

ct is

giv

en b

y th

e eq

uati

on

y"

5400

#1.

37x,

wha

t is

the

fix

ed c

ost?

Wha

t is

the

var

iabl

e co

st

(for

eac

h it

em p

rodu

ced)

?$5

400;

$1.3

7•

Wri

te a

line

ar e

quat

ion

that

des

crib

es t

he f

ollo

win

g si

tuat

ion:

A c

ompa

ny t

hat

man

ufac

ture

s co

mpu

ters

has

a f

ixed

cos

t of

$22

8,75

0 an

da

vari

able

cos

t of

$85

2 to

pro

duce

eac

h co

mpu

ter.

y!

228,

750

$85

2x

Rea

din

g t

he

Less

on

1.a.

Wri

te t

he s

lope

-int

erce

pt f

orm

of

the

equa

tion

of

a lin

e.T

hen

expl

ain

the

mea

ning

of

each

of

the

vari

able

s in

the

equ

atio

n.y

!m

x$

b;m

is th

e sl

ope

and

bis

the

y-in

terc

ept.

The

varia

bles

xan

d y

are

the

coor

dina

tes

of a

ny p

oint

on

the

line.

b.W

rite

the

poi

nt-s

lope

for

m o

f th

e eq

uati

on o

f a

line.

The

n ex

plai

n th

e m

eani

ng o

f ea

chof

the

var

iabl

es in

the

equ

atio

n.y

"y 1

!m

(x"

x 1);

mis

the

slop

e.x

and

yar

e th

e co

ordi

nate

s of

any

poi

nt o

n th

e lin

e.x 1

and

y 1ar

e th

e co

ordi

nate

s of

one

spe

cific

poi

nt o

n th

e lin

e.

2.Su

ppos

e th

at y

our

alge

bra

teac

her

asks

you

to

wri

te t

he p

oint

-slo

pe f

orm

of

the

equa

tion

of t

he li

ne t

hrou

gh t

he p

oint

s (!

6,7)

and

(!

3,!

2).Y

ou w

rite

y#

2 "

!3(

x#

3) a

ndyo

ur c

lass

mat

e w

rite

s y

!7

"!

3(x

#6)

.Whi

ch o

f yo

u is

cor

rect

?E

xpla

in. Y

ou a

rebo

th c

orre

ct.E

ither

poi

nt m

ay b

e us

ed a

s (x

1,y 1

) in

the

poin

t-slo

pe fo

rm.

You

used

("3,

"2)

,and

you

r cla

ssm

ate

used

("6,

7).

3.Yo

u ar

e as

ked

to w

rite

an

equa

tion

of

two

lines

tha

t pa

ss t

hrou

gh (

3,!

5),o

ne o

f th

empa

ralle

l to

and

one

of t

hem

per

pend

icul

ar t

o th

e lin

e w

hose

equ

atio

n is

y"

!3x

#4.

The

fir

st s

tep

in f

indi

ng t

hese

equ

atio

ns is

to

find

the

ir s

lope

s.W

hat

is t

he s

lope

of

the

para

llel l

ine?

Wha

t is

the

slo

pe o

f th

e pe

rpen

dicu

lar

line?

"3;

Hel

pin

g Y

ou

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____

____

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____

____

____

____

____

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____

____

____

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2-4

2-4

Exam

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Exam

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Exam

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©G

lencoe/McG

raw-HillA14

Glencoe Algebra 2

Answers

(Lesson 2-5)

Study Guide and InterventionModeling Real-World Data: Using Scatter Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5


Less

on

2-5

Scatter Plots When a set of data points is graphed as ordered pairs in a coordinateplane, the graph is called a scatter plot. A scatter plot can be used to determine if there isa relationship among the data.

BASEBALL The table below shows the number of home runs andruns batted in for various baseball players who won the Most Valuable PlayerAward during the 1990s. Make a scatter plot of the data.


Make a scatter plot for the data in each table below.

1. FUEL EFFICIENCY The table below shows the average fuel efficiency in miles per gallon of new cars manufactured during the years listed.


2. CONGRESS The table below shows the number of women serving in the United States Congress during the years 1987!1999.

Source: Wall Street Journal Almanac

Congressional Session Number of Women

100 25

101 31

102 33

103 55

104 58

105 62

Session of Congress

Nu

mb

er o

f W

om

en

100 102 104

70

56

42

28

14

0

Women in Congress

Year Fuel Efficiency (mpg)

1960 15.5

1970 14.1

1980 22.6

1990 26.9 Year

Mile

s p

er G

allo

n

1960 1970 1980 1990

36

30

24

18

12

6

0

Average Fuel Efficiency

Home Runs

MVP HRs and RBIs

Ru

ns

Bat

ted

In

1260 24 3618 30 42 48

150

125

100

75

50

25

Home Runs Runs Batted In

33 114

39 116

40 130

28 61

41 128

47 144

ExampleExample

ExercisesExercises


Prediction Equations A line of fit is a line that closely approximates a set of datagraphed in a scatter plot. The equation of a line of fit is called a prediction equationbecause it can be used to predict values not given in the data set.

To find a prediction equation for a set of data, select two points that seem to represent thedata well. Then to write the prediction equation, use what you know about writing a linearequation when given two points on the line.

STORAGE COSTS According to a certain prediction equation, thecost of 200 square feet of storage space is $60. The cost of 325 square feet ofstorage space is $160.

a. Find the slope of the prediction equation. What does it represent?Since the cost depends upon the square footage, let x represent the amount of storagespace in square feet and y represent the cost in dollars. The slope can be found using the

formula m " . So, m " " " 0.8

The slope of the prediction equation is 0.8. This means that the price of storage increases80¢ for each one-square-foot increase in storage space.

b. Find a prediction equation.Using the slope and one of the points on the line, you can use the point-slope form to finda prediction equation.y ! y1 " m(x ! x1) Point-slope formy ! 60 " 0.8(x ! 200) (x1, y1) " (200, 60), m " 0.8y ! 60 " 0.8x ! 160 Distributive Property

y " 0.8x ! 100 Add 60 to both sides.

A prediction equation is y " 0.8x ! 100.

SALARIES The table below shows the years of experience for eight technicians atLewis Techomatic and the hourly rate of pay each technician earns. Use the datafor Exercises 1 and 2.

Experience (years) 9 4 3 1 10 6 12 8

Hourly Rate of Pay $17 $10 $10 $7 $19 $12 $20 $15

1. Draw a scatter plot to show how years of experience are related to hourly rate of pay. Draw a line of fit. See graph.

2. Write a prediction equation to show how years of experience(x) are related to hourly rate of pay (y). Sample answerusing (1, 7) and (9, 17): y ! 1.25x $ 5.75

Experience (years)

Ho

url

y Pa

y ($

)

20 6 104 8 12 14

24

20

16

12

8

4

Technician Salaries

100%125

160 ! 60%%325 ! 200

y2 ! y1%x2 ! x1



NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5

ExampleExample

ExercisesExercises

©G

lencoe/McG

raw-HillA15

Glencoe Algebra 2

Answers

Answers

(Lesson 2-5)

Skills PracticeModeling Real-World Data: Using Scatter Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5


Less

on

2-5

For Exercises 1–3, complete parts a–c for each set of data.

a. Draw a scatter plot.b. Use two ordered pairs to write a prediction equation.c. Use your prediction equation to predict the missing value.

1. 1a.

1b. Sample answer using (1, 1) and (8, 15): y ! 2x " 11c. Sample answer: 19

2. 2a.

2b. Sample answer using (5, 9) and (40, 44): y ! x $ 42c. Sample answer: 54

3. 3a.

3b. Sample answer using (2, 16) and (7, 34): y ! 3.6x $ 8.83c. Sample answer: 19.6

1 3 5 72 4 6 8

36

30

24

18

12

6

0 x

yx y

1 16

2 16

3 ?

4 22

5 30

7 34

8 36

5 15 25 3510 20 30 40

40

32

24

16

8

0 x

yx y

5 9

10 17

20 22

25 30

35 38

40 44

50 ?

1 3 5 72 4 6 8

15

12

9

6

3

0 x

yx y

1 1

3 5

4 7

6 11

7 12

8 15

10 ?


For Exercises 1–3, complete parts a–c for each set of data.a. Draw a scatter plot.b. Use two ordered pairs to write a prediction equation.c. Use your prediction equation to predict the missing value.

1. FUEL ECONOMY The table gives the approximate weights in tons and estimates for overall fuel economy in miles per gallon for several cars.1b. Sample answer using (1.4, 24) and

(2.4, 15): y ! "9x $ 36.61c. Sample answer: 18.6 mi/gal

2. ALTITUDE In most cases, temperature decreases with increasing altitude. As Ancharadrives into the mountains, her car thermometer registers the temperatures (°F) shownin the table at the given altitudes (feet).

2b. Sample answer using (7500, 61) and (9700, 50): y ! "0.005x $ 98.5

2c. Sample answer: 38.5°F

3. HEALTH Alton has a treadmill that uses the time on the treadmill and the speed of walking or running to estimate the number of Calories he burns during a workout. Thetable gives workout times and Calories burned for several workouts.

3b. Sample answer using (24, 280) and(48, 440): y ! 6.67x $ 119.92

3c. Sample answer: about 520 calories

Time (min)

Cal

ori

es B

urn

ed

0 10 20 30 40 50 555 15 25 35 45

500

400

300

200

100

Burning Calories

Time (min) 18 24 30 40 42 48 52 60

Calories Burned 260 280 320 380 400 440 475 ?

Altitude (ft)

Tem

per

atu

re (

'F)

0 7,000 8,000 9,000 10,000

65

60

55

50

45

TemperatureVersus Altitude

Altitude (ft) 7500 8200 8600 9200 9700 10,400 12,000

Temperature ('F) 61 58 56 53 50 46 ?

Weight (tons)

Fuel

Eco

no

my

(mi/

gal

)

0 0.5 1.0 1.5 2.0 2.5

30

25

20

15

10

5

Fuel Economy Versus Weight

Weight (tons) 1.3 1.4 1.5 1.8 2 2.1 2.4

Miles per Gallon 29 24 23 21 ? 17 15

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5



Rea

din

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o L

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Math

emati

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Rea

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Usi

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catte

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NAM

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____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-5

2-5

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Lesson 2-5

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Act

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lin

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the

top

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in y

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.

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our?

Sam

ple

answ

er:3

97 C

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Rea

din

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he

Less

on

1.Su

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e th

at a

set

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data

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.Exp

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a di

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raph

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s m

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a co

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ph th

at is

the

line

that

bes

t fits

the

data

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nts.

2.Su

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at t

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on a

t a

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How

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the

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.In

the

grap

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a c

ontin

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.

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Med

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A

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the

ste

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to

find

the

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atio

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the

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.

Appr

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Perc

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at V

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Law

Enf

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t

Year

1980

1982

1984

1986

1988

1990

1992

1994

1996

Offe

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3533

3231

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2930

Sour

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.S. B

urea

u of

Justi

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1.D

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to t

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ups.

The

re s

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in t

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and

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s.In

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w

ill b

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poi

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Gro

up 1

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Year

Offe

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Offe

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2.F

ind

x 1,x

2,an

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,the

med

ians

of

the

xva

lues

in g

roup

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2,an

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vely

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d y 1

,y2,

and

y 3,t

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edia

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f th

e y

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es in

gro

ups

1,2,

and

3,re

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ly.

1982

,198

8,19

94;3

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3.F

ind

an e

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of t

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2 w

ith

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2.32

5.T

he m

edia

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e is

par

alle

l to

the

line

in E

xerc

ise

2,bu

t is

one

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clos

er t

o (x

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).T

his

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thro

ugh

!x 2,%

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Fin

d th

is

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pair

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31.6

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6.W

rite

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tion

of

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ian-

fit

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y!

"0.

5x$

1025

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7.U

se t

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e to

pre

dict

the

per

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age

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veni

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En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-5

2-5


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Spec

ial F

unct

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-6

2-6

©G

lenc

oe/M

cGra

w-Hi

ll87

Gle

ncoe

Alg

ebra

2

Lesson 2-6

Step

Fu

nct

ion

s, C

on

stan

t Fu

nct

ion

s, a

nd

th

e Id

enti

ty F

un

ctio

nT

he c

hart

belo

w li

sts

som

e sp

ecia

l fun

ctio

ns y

ou s

houl

d be

fam

iliar

wit

h.

Func

tion

Writ

ten

asG

raph

Cons

tant

f(x) "

cho

rizon

tal l

ine

Iden

tity

f(x) "

xlin

e th

roug

h th

e or

igin

with

slo

pe 1

Gre

ates

t Int

eger

Fun

ctio

nf(x

) "%x

&on

e-un

it ho

rizon

tal s

egm

ents

, with

righ

t end

poin

ts m

issin

g, a

rrang

ed

like

step

s

The

gre

ates

t in

tege

r fu

ncti

on is

an

exam

ple

of a

ste

p fu

nct

ion

,a fu

ncti

on w

ith

a gr

aph

that

cons

ists

of

hori

zont

al s

egm

ents

.

Iden

tify

eac

h f

un

ctio

n a

s a

con

stan

t fu

nct

ion

,th

e id

enti

ty f

un

ctio

n,

or a

ste

p f

un

ctio

n.

a.b.

a co

nsta

nt f

unct

ion

a st

ep f

unct

ion

Iden

tify

eac

h f

un

ctio

n a

s a

con

stan

t fu

nct

ion

,th

e id

enti

ty f

un

ctio

n,a

gre

ates

tin

tege

r fu

nct

ion

,or

a st

ep f

un

ctio

n.

1.2.

3.

a co

nsta

nt fu

nctio

na

step

func

tion

the

iden

tity

func

tionx

f (x)

Ox

f (x)

Ox

f (x)

O

x

f (x)

Ox

f (x)

O

Exam

ple

Exam

ple

Exer

cises

Exer

cises

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2

Ab

solu

te V

alu

e an

d P

iece

wis

e Fu

nct

ion

sA

noth

er s

peci

al f

unct

ion

is t

heab

solu

te v

alu

e fu

nct

ion

,whi

ch is

als

o ca

lled

a p

iece

wis

e fu

nct

ion

.

Abso

lute

Val

ue F

unct

ion

f(x) "

x

two

rays

that

are

mirr

or im

ages

of e

ach

othe

r and

mee

t at a

poi

nt, t

he v

erte

x

To g

raph

a s

peci

al f

unct

ion,

use

its

defi

niti

on a

nd y

our

know

ledg

e of

the

par

ent

grap

h.F

ind

seve

ral o

rder

ed p

airs

,if

nece

ssar

y.

Gra

ph

f(x

) !

3⏐x⏐

"4.

Fin

d se

vera

l ord

ered

pai

rs.G

raph

the

poi

nts

and

conn

ect

them

.You

wou

ld e

xpec

t th

e gr

aph

to lo

oksi

mila

r to

its

pare

nt f

unct

ion,

f(x)

"x

.

Gra

ph

f(x

) !

!2xif

x(

2x

"1

if x

#2.

Fir

st,g

raph

the

line

ar f

unct

ion

f(x)

"2x

for

x'

2.Si

nce

2 do

es n

otsa

tisf

y th

is in

equa

lity,

stop

wit

h a

circ

le a

t (2

,4).

Nex

t,gr

aph

the

linea

r fu

ncti

on f

(x) "

x!

1 fo

r x

(2.

Sinc

e 2

does

sat

isfy

thi

sin

equa

lity,

begi

n w

ith

a do

t at

(2,

1).

Gra

ph

eac

h f

un

ctio

n.I

den

tify

th

e d

omai

n a

nd

ran

ge.

1.g(

x) "

%&2.

h(x)

"2

x#

13.

h(x

) "

dom

ain:

all r

eal

dom

ain:

all r

eal

dom

ain:

all r

eal

num

bers

;ran

ge:

num

bers

;ran

ge:

num

bers

;ran

ge:

all i

nteg

ers

{y⏐y

#0}

{y⏐y

)1}

x

y

O

x

y

O

x

y

Ox % 3

x

f (x)

O

x

f (x)

O

x3⏐

x⏐"

4

0!

4

1!

1

22

!1

!1

!2

2

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Spec

ial F

unct

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-6

2-6

Exer

cises

Exer

cises

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

if x

)0

2x!

6 if

0 '

x'

21

if x

(2

x % 3

©G

lencoe/McG

raw-HillA18

Glencoe Algebra 2

Answers

(Lesson 2-6)

Skills PracticeSpecial Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6


Less

on

2-6

Identify each function as S for step, C for constant, A for absolute value, or P forpiecewise.

1. 2. 3.

S C A


4. f(x) " %x # 1& 5. f(x) " %x ! 3&

D ! all reals, R ! all integers D ! all reals, R ! all integers6. g(x) " 2x 7. f(x) " x # 1

D ! all reals, D ! all reals, R ! {y⏐y # 1}R ! nonnegative reals

8. f(x) " 'x if x ' 0 9. h(x) " '3 if x ' !12 if x ( 0 x # 1 if x > 1

D ! all reals, D ! {x⏐x ( "1 or x * 1},R ! {y⏐y ( 0 or y ! 2} R ! {y⏐y * 2}

x

h(x)

O

x

f(x)

O

x

f(x)

Ox

g(x)

O

x

f(x)

O

x

f(x)

O

x

y

O

x

y

Ox

y

O



1. f(x) " %0.5x& 2. f(x) " %x& ! 2

D ! all reals, R ! all integers D ! all reals, R ! all integers3. g(x) " !2x 4. f(x) " x # 1

D ! all reals, D ! all reals,R ! nonpositive reals R ! nonnegative reals

5. f(x) " 'x # 2 if x ) ! 2 6. h(x) " '4 ! x if x * 03x if x * !2 !2x ! 2 if x ' 0

D ! all reals, R ! all reals D ! all nonzero reals, R ! all reals7. BUSINESS A Stitch in Time charges 8. BUSINESS A wholesaler charges a store $3.00

$40 per hour or any fraction thereof per pound for less than 20 pounds of candy andfor labor. Draw a graph of the step $2.50 per pound for 20 or more pounds. Draw afunction that represents this situation. graph of the function that represents this

situation.

Hours

Tota

l Co

st (

$)

10 3 52 4 6 7

280

240

200

160

120

80

40

Labor Costs

Pounds

Co

st (

$)

50 15 2510 20 30 35

105

90

75

60

45

30

15

Candy Costs

x

h(x)

O

f(x)

xO

x

f(x)

O

x

g(x)

O

x

f(x)

Ox

f(x)

O

Practice (Average)

Special Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csSp

ecia

l Fun

ctio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-6

2-6

©G

lenc

oe/M

cGra

w-Hi

ll91

Gle

ncoe

Alg

ebra

2

Lesson 2-6

Pre-

Act

ivit

yH

ow d

o st

ep f

un

ctio

ns

app

ly t

o p

osta

ge r

ates

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

6 at

the

top

of

page

89

in y

our

text

book

.

•W

hat

is t

he c

ost

of m

ailin

g a

lett

er t

hat

wei

ghs

0.5

ounc

e?$0

.34

or 3

4 ce

nts

•G

ive

thre

e di

ffer

ent

wei

ghts

of

lett

ers

that

wou

ld e

ach

cost

55

cent

s to

mai

l.An

swer

s w

ill v

ary.

Sam

ple

answ

er:1

.1 o

unce

s,1.

9 ou

nces

,2.0

oun

ces

Rea

din

g t

he

Less

on

1.F

ind

the

valu

e of

eac

h ex

pres

sion

.

a.!

3"

%!3&

"

b.6

.2

"%6

.2&"

c.!

4.01

"

%!4.

01&"

2.Te

ll ho

w t

he n

ame

of e

ach

kind

of

func

tion

can

hel

p yo

u re

mem

ber

wha

t th

e gr

aph

look

s lik

e.

a.co

nsta

nt f

unct

ion

Sam

ple

answ

er:S

omet

hing

is c

onst

ant i

f it d

oes

not

chan

ge.T

he y

-val

ues

of a

con

stan

t fun

ctio

n do

not

cha

nge,

so th

egr

aph

is a

hor

izon

tal l

ine.

b.ab

solu

te v

alue

fun

ctio

nSa

mpl

e an

swer

:The

abs

olut

e va

lue

of a

num

ber

tells

you

how

far i

t is

from

0 o

n th

e nu

mbe

r lin

e.It

mak

es n

o di

ffere

nce

whe

ther

you

go

to th

e le

ft or

righ

t so

long

as

you

go th

e sa

me

dist

ance

eac

h tim

e.c.

step

fun

ctio

nSa

mpl

e an

swer

:A s

tep

func

tion’

s gr

aph

look

s lik

e st

eps

that

go

up o

r dow

n.

d.id

enti

ty f

unct

ion

Sam

ple

answ

er:T

he x

- and

y-v

alue

s ar

e al

way

sid

entic

ally

the

sam

e fo

r any

poi

nt o

n th

e gr

aph.

So th

e gr

aph

is a

line

thro

ugh

the

orig

in th

at h

as s

lope

1.

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stud

ents

fin

d th

e gr

eate

st in

tege

r fu

ncti

on c

onfu

sing

.Exp

lain

how

you

can

use

anu

mbe

r lin

e to

fin

d th

e va

lue

of t

his

func

tion

for

any

rea

l num

ber.

Answ

ers

will

var

y.Sa

mpl

e an

swer

:Dra

w a

num

ber l

ine

that

sho

ws

the

inte

gers

.To

find

the

valu

e of

the

grea

test

inte

ger f

unct

ion

for a

ny re

al n

umbe

r,pl

ace

that

num

ber o

n th

e nu

mbe

r lin

e.If

it is

an

inte

ger,

the

valu

e of

the

func

tion

isth

e nu

mbe

r its

elf.

If no

t,m

ove

to th

e in

tege

r dire

ctly

to th

e le

ft of

the

num

ber y

ou c

hose

.Thi

s in

tege

r will

giv

e th

e va

lue

you

need

.

"5

4.01

66.

2

"3

3

©G

lenc

oe/M

cGra

w-Hi

ll92

Gle

ncoe

Alg

ebra

2

Gre

ates

t Int

eger

Fun

ctio

nsU

se t

he g

reat

est

inte

ger

func

tion

%x&

to e

xplo

re s

ome

unus

ual g

raph

s.It

will

be

hel

pful

to

mak

e a

char

t of

val

ues

for

each

fun

ctio

ns a

nd t

o us

e a

colo

red

pen

or p

enci

l.

Gra

ph

eac

h f

un

ctio

n.

1.y

"2x

!%x

&2.

y"

%% %x x& &%

3.y

"%% %0 0. .5 5x x

# #

1 1& &%

4.y

"% %x x&%

x

y

O1

–1–2

–3–4

23

4

4 3 2 1 –1 –2 –3 –4

x

y

O1

–1–2

–3–4

23

4

4 3 2 1 –1 –2 –3 –4

x

y

O1

–1–2

–3–4

23

4

4 3 2 1 –1 –2 –3 –4

x

y

O1

–1–2

–3–4

23

4

4 3 2 1 –1 –2 –3 –4

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-6

2-6



Stu

dy

Gu

ide

and I

nte

rven

tion

Gra

phin

g In

equa

litie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-7

2-7

©G

lenc

oe/M

cGra

w-Hi

ll93

Gle

ncoe

Alg

ebra

2

Lesson 2-7

Gra

ph

Lin

ear

Ineq

ual

itie

s.A

lin

ear

ineq

ual

ity,

like

y(

2x!

1,re

sem

bles

a li

near

equa

tion

,but

wit

h an

ineq

ualit

y si

gn in

stea

d of

an

equa

ls s

ign.

The

gra

ph o

f th

e re

late

dlin

ear

equa

tion

sep

arat

es t

he c

oord

inat

e pl

ane

into

tw

o ha

lf-p

lane

s.T

he li

ne is

the

boun

dary

of

each

hal

f-pl

ane.

To g

raph

a li

near

ineq

ualit

y,fo

llow

the

se s

teps

.

1.G

raph

the

bou

ndar

y,th

at is

,the

rel

ated

line

ar e

quat

ion.

If t

he in

equa

lity

sym

bol i

s )

or (

,the

bou

ndar

y is

sol

id.I

f th

e in

equa

lity

sym

bol i

s '

or *

,the

bou

ndar

y is

das

hed.

2.C

hoos

e a

poin

t no

t on

the

bou

ndar

y an

d te

st it

in t

he in

equa

lity.

(0,0

) is

a g

ood

poin

t to

choo

se if

the

bou

ndar

y do

es n

ot p

ass

thro

ugh

the

orig

in.

3.If

a t

rue

ineq

ualit

y re

sult

s,sh

ade

the

half

-pla

ne c

onta

inin

g yo

ur t

est

poin

t.If

a f

alse

ineq

ualit

y re

sult

s,sh

ade

the

othe

r ha

lf-p

lane

.

Gra

ph

x$

2y#

4.

The

bou

ndar

y is

the

gra

ph o

f x#

2y"

4.

Use

the

slo

pe-i

nter

cept

for

m,y

"!

x#

2,to

gra

ph t

he b

ound

ary

line.

The

bou

ndar

y lin

e sh

ould

be

solid

.

Now

tes

t th

e po

int

(0,0

).

0 #

2(0)

(?

4(x

, y) "

(0, 0

)

0 (

4fa

lse

Shad

e th

e re

gion

tha

t do

es n

otco

ntai

n (0

,0).

Gra

ph

eac

h i

neq

ual

ity.

1.y

'3x

#1

2.y

(x

!5

3.4x

#y

)!

1

4.y

'!

45.

x#

y*

66.

0.5x

!0.

25y

'1.

5

x

y

O

x

y

O

x

y

O

x % 2

x

y

O

x

y

O

x

y

O

1 % 2x

y O

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-Hi

ll94

Gle

ncoe

Alg

ebra

2

Gra

ph

Ab

solu

te V

alu

e In

equ

alit

ies

Gra

phin

g ab

solu

te v

alue

ineq

ualit

ies

is s

imila

rto

gra

phin

g lin

ear

ineq

ualit

ies.

The

gra

ph o

f th

e re

late

d ab

solu

te v

alue

equ

atio

n is

the

boun

dary

.Thi

s bo

unda

ry is

gra

phed

as

a so

lid li

ne if

the

ineq

ualit

y is

)or

(,a

nd d

ashe

d if

the

ineq

ualit

y is

'or

*.C

hoos

e a

test

poi

nt n

ot o

n th

e bo

unda

ry t

o de

term

ine

whi

ch r

egio

nto

sha

de.

Gra

ph

y)

3⏐x

"1⏐

.

Fir

st g

raph

the

equ

atio

n y

"3

x!

1.

Sinc

e th

e in

equa

lity

is )

,the

gra

ph o

f th

e bo

unda

ry is

sol

id.

Test

(0,

0).

0 )?

30

!1

(x, y

) "(0

, 0)

0 )?

3!

1!

1"

1

0 )

3tru

e

Shad

e th

e re

gion

tha

t co

ntai

ns (

0,0)

.

Gra

ph

eac

h i

neq

ual

ity.

1.y

(x

#

12.

y)

2x

!1

3.y

!2

x*

3

4.y

'!x

!

35.

x

#y

(4

6.x

#1

#2y

'0

7.2

!x

#y

*!

18.

y'

3x

!3

9.y

)1

!x

#4 x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

Ox

y

Ox

y

O

x

y

O

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Gra

phin

g In

equa

litie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-7

2-7

Exer

cises

Exer

cises

Exam

ple

Exam

ple


An

swer

s


Skil

ls P

ract

ice

Gra

phin

g In

equa

litie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-7

2-7

©G

lenc

oe/M

cGra

w-Hi

ll95

Gle

ncoe

Alg

ebra

2

Lesson 2-7

Gra

ph

eac

h i

neq

ual

ity.

1.y

*1

2.y

)x

#2

3.x

#y

)4

4.x

#3

'y

5.2

!y

'x

6.y

(!

x

7.x

!y

*!

28.

9x#

3y!

6 )

09.

y#

1 (

2x

10.y

!7

)!

911

.x*

!5

12.y

*x

x

y

Ox

y

Ox

y

O

x

y

Ox

y

Ox

y

O

x

y

Ox

y

O

x

y

O

x

y

O

x

y

Ox

y

O

©G

lenc

oe/M

cGra

w-Hi

ll96

Gle

ncoe

Alg

ebra

2

Gra

ph

eac

h i

neq

ual

ity.

1.y

)!

32.

x*

23.

x#

y)

!4

4.y

'!

3x#

55.

y'

x#

36.

y!

1 (

!x

7.x

!3y

)6

8.y

*x

!

19.

y*

!3

x#

1!

2

CO

MPU

TER

SF

or E

xerc

ises

10–

12,u

se t

he

foll

owin

g in

form

atio

n.

A s

choo

l sys

tem

is b

uyin

g ne

w c

ompu

ters

.The

y w

ill

buy

desk

top

com

pute

rs c

osti

ng $

1000

per

uni

t,an

dno

tebo

ok c

ompu

ters

cos

ting

$12

00 p

er u

nit.

The

tot

al

cost

of

the

com

pute

rs c

anno

t ex

ceed

$80

,000

.

10.W

rite

an

ineq

ualit

y th

at d

escr

ibes

thi

s si

tuat

ion.

1000

d$

1200

n)

80,0

00

11.G

raph

the

ineq

ualit

y.

12.I

f th

e sc

hool

wan

ts t

o bu

y 50

of

the

desk

top

com

pute

rs a

nd 2

5 of

the

not

eboo

k co

mpu

ters

,w

ill t

hey

have

eno

ugh

mon

ey?

yes

Des

kto

ps

Notebooks

100

3050

2040

6070

8090

100

80 70 60 50 40 30 20 10

Co

mp

ute

rs P

urc

has

edx

y

O

x

y

Ox

y O

x

y

Ox

y

O

x

y

O

1 % 2

x

y

O

x

y

O

x

y

OPra

ctic

e (A

vera

ge)

Gra

phin

g In

equa

litie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-7

2-7



Rea

din

g t

o L

earn

Math

emati

csG

raph

ing

Ineq

ualit

ies

NAM

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2-7

2-7

©G

lenc

oe/M

cGra

w-Hi

ll97

Gle

ncoe

Alg

ebra

2

Lesson 2-7

Pre-

Act

ivit

yH

ow d

o in

equ

alit

ies

app

ly t

o fa

nta

sy f

ootb

all?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 2-

7 at

the

top

of

page

96

in y

our

text

book

.

•W

hich

of

the

com

bina

tion

s of

yar

ds a

nd t

ouch

dow

ns li

sted

wou

ld D

ana

cons

ider

a g

ood

gam

e?Th

e fir

st o

ne:1

68 y

ards

and

3

touc

hdow

ns•

Supp

ose

that

in o

ne o

f th

e ga

mes

Dan

a pl

ays,

Mos

s ge

ts 1

57 r

ecei

ving

yard

s.W

hat

is t

he s

mal

lest

num

ber

of t

ouch

dow

ns h

e m

ust

get

in o

rder

for

Dan

a to

con

side

r th

is a

goo

d ga

me?

3

Rea

din

g t

he

Less

on

1.W

hen

grap

hing

a li

near

ineq

ualit

y in

tw

o va

riab

les,

how

do

you

know

whe

ther

to

mak

eth

e bo

unda

ry a

sol

id li

ne o

r a

dash

ed li

ne?

If th

e sy

mbo

l is

#or

),t

he li

ne is

solid

.If

the

sym

bol i

s *

or (

,the

line

is d

ashe

d.

2.H

ow d

o yo

u kn

ow w

hich

sid

e of

the

bou

ndar

y to

sha

de?

Sam

ple

answ

er:I

f the

test

poin

t giv

es a

true

ineq

ualit

y,sh

ade

the

regi

on c

onta

inin

g th

e te

st p

oint

.If

the

test

poi

nt g

ives

a fa

lse

ineq

ualit

y,sh

ade

the

regi

on n

otco

ntai

ning

the

test

poi

nt.

3.M

atch

eac

h in

equa

lity

wit

h it

s gr

aph.

a.y

*2x

!3

iiib.

y'

!2x

#3

ivc.

y(

2x!

3ii

d.y

(!

2x#

3i

i.ii

.ii

i.iv

.

Hel

pin

g Yo

u R

emem

ber

4.D

escr

ibe

som

e w

ays

in w

hich

gra

phin

g an

ineq

ualit

y in

one

var

iabl

e on

a n

umbe

r lin

e is

sim

ilar

to g

raph

ing

an in

equa

lity

in t

wo

vari

able

s in

a c

oord

inat

e pl

ane.

How

can

wha

tyo

u kn

ow a

bout

gra

phin

g in

equa

litie

s on

a n

umbe

r lin

e he

lp y

ou t

o gr

aph

ineq

ualit

ies

ina

coor

dina

te p

lane

?Sa

mpl

e an

swer

:A b

ound

ary

on a

coo

rdin

ate

grap

h is

sim

ilar t

o an

end

poin

t on

a nu

mbe

r lin

e gr

aph.

A da

shed

line

is s

imila

r to

a ci

rcle

on

a nu

mbe

r lin

e:bo

th a

re o

pen

and

mea

n no

t inc

lude

d;th

eyre

pres

ent t

he s

ymbo

ls *

and

(.A

sol

id li

ne is

sim

ilar t

o a

dot o

n a

num

ber l

ine:

both

are

clo

sed

and

mea

n in

clud

ed;t

hey

repr

esen

t the

sym

bols

#an

d )

.

x

y

O

x

y

Ox

y

O

x

y

O

©G

lenc

oe/M

cGra

w-Hi

ll98

Gle

ncoe

Alg

ebra

2

Alg

ebra

ic P

roof

The

fol

low

ing

para

grap

h st

ates

a r

esul

t yo

u m

ight

be

aske

d to

pro

ve in

am

athe

mat

ics

cour

se.P

arts

of

the

para

grap

h ar

e nu

mbe

red.

01L

et n

be a

pos

itiv

e in

tege

r.

02A

lso,

let

n 1"

s(n 1

) be

the

sum

of

the

squa

res

of t

he d

igit

s in

n.

03T

hen

n 2"

s(n 1

) is

the

sum

of

the

squa

res

of t

he d

igit

s of

n1,

and

n 3"

s(n 2

)is

the

sum

of

the

squa

res

of t

he d

igit

s of

n2.

04In

gen

eral

,nk

"s(

n k!

1) is

the

sum

of

the

squa

res

of t

he d

igit

s of

nk

!1.

05C

onsi

der

the

sequ

ence

:n,n

1,n 2

,n3,

…,n

k,…

.

06In

thi

s se

quen

ce e

ithe

r al

l the

ter

ms

from

som

e k

on h

ave

the

valu

e 1,

07or

som

e te

rm,s

ay n

j,ha

s th

e va

lue

4,so

tha

t th

e ei

ght

term

s 4,

16,3

7,58

,89,

145,

42,a

nd 2

0 ke

ep r

epea

ting

fro

m t

hat

poin

t on

.

Use

th

e p

arag

rap

h t

o an

swer

th

ese

ques

tion

s.

1.U

se t

he s

ente

nce

in li

ne 0

1.L

ist

the

firs

t fi

ve v

alue

s of

n.

1,2,

3,4,

5

2.U

se 9

246

for

nan

d gi

ve a

n ex

ampl

e to

sho

w t

he m

eani

ng o

f lin

e 02

.n 1

!s(

9246

) !13

7,be

caus

e 13

7 !

81 #

4 #

16 #

36

3.In

line

02,

whi

ch s

ymbo

l sho

ws

a fu

ncti

on?

Exp

lain

the

func

tion

in a

sen

tenc

e.s(

n);t

he s

um o

f the

squ

ares

of t

he d

igits

of a

num

ber i

s a

func

tion

of th

e nu

mbe

r

4.Fo

r n

"92

46,f

ind

n 2an

d n 3

as d

escr

ibed

in s

ente

nce

03.

n 2!

59,n

3!

106

5.H

ow d

o th

e fi

rst

four

sen

tenc

es r

elat

e to

sen

tenc

e 05

?Th

ey e

xpla

in h

ow to

com

pute

the

term

s of

the

sequ

ence

.

6.U

se n

"31

and

fin

d th

e fi

rst

four

ter

ms

of t

he s

eque

nce.

31,1

0,1,

1

7.W

hich

sen

tenc

e of

the

par

agra

ph is

illu

stra

ted

by n

"31

?se

nten

ce 0

6

8.U

se n

"61

and

fin

d th

e fi

rst

ten

term

s.61

,37,

58,8

9,14

5,42

,20,

4,16

,37

9.W

hich

sen

tenc

e is

illu

stra

ted

by n

"61

?se

nten

ce 0

7

En

rich

men

t

NAM

E__

____

____

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____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

2-7

2-7

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