Chapter 8 Resource Masters - ktlmathclass.weebly.com · ©Glencoe/McGraw-Hill iv Glencoe Algebra 2 Teacher’s Guide to Using the Chapter 8 Resource Masters The Fast FileChapter Resource

Chapter 8Resource 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 8 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: 0-07-828011-7 Algebra 2Chapter 8 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 8-1Study Guide and Intervention . . . . . . . . 455–456Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 457Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 458Reading to Learn Mathematics . . . . . . . . . . 459Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 460




Lesson 8-5Study Guide and Intervention . . . . . . . 479–480Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 481Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Reading to Learn Mathematics . . . . . . . . . . 483Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 484



Chapter 8 AssessmentChapter 8 Test, Form 1 . . . . . . . . . . . . 497–498Chapter 8 Test, Form 2A . . . . . . . . . . . 499–500Chapter 8 Test, Form 2B . . . . . . . . . . . 501–502Chapter 8 Test, Form 2C . . . . . . . . . . . 503–504Chapter 8 Test, Form 2D . . . . . . . . . . . 505–506Chapter 8 Test, Form 3 . . . . . . . . . . . . 507–508Chapter 8 Open-Ended Assessment . . . . . . 509Chapter 8 Vocabulary Test/Review . . . . . . . 510Chapter 8 Quizzes 1 & 2 . . . . . . . . . . . . . . . 511Chapter 8 Quizzes 3 & 4 . . . . . . . . . . . . . . . 512Chapter 8 Mid-Chapter Test . . . . . . . . . . . . 513Chapter 8 Cumulative Review . . . . . . . . . . . 514Chapter 8 Standardized Test Practice . . 515–516

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

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

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 8 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 8 Resource Masters includes the core materials neededfor Chapter 8. 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 8-1.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 8Resource 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 468–469. 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 _____

88

© 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 8.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

asymptote

A·suhm(p)·TOHT

center of a circle

center of an ellipse

circle

conic section

conjugate axis

KAHN·jih·guht

directrix

duh·REHK·trihks

distance formula

ellipse

ih·LIHPS

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

foci of an ellipse

focus of a parabola

FOH·kuhs

hyperbola

hy·PUHR·buh·luh

latus rectum

LA·tuhs REHK·tuhm

major axis

midpoint formula

minor axis

parabola

puh·RA·buh·luh

tangent

TAN·juhnt

transverse axis

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

88

Study Guide and InterventionMidpoint and Distance Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

© Glencoe/McGraw-Hill 455 Glencoe Algebra 2

Less

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8-1

The Midpoint Formula

Midpoint Formula The midpoint M of a segment with endpoints (x1, y1) and (x2, y2) is � , �.y1 � y2�2

x1 � x2�2

Find the midpoint of theline segment with endpoints at (4, �7) and (�2, 3).

� , � � � , �� , � or (1, �2)

The midpoint of the segment is (1, �2).

�4�2

2�2

�7 � 3�2

4 � (�2)��2

y1 � y2�2x1 � x2�2

A diameter A�B� of a circlehas endpoints A(5, �11) and B(�7, 6).What are the coordinates of the centerof the circle?

The center of the circle is the midpoint of allof its diameters.

� , � � � , �� , � or ��1, �2 �

The circle has center ��1, �2 �.1�2

1�2

�5�2

�2�2

�11 � 6��2

5 � (�7)��2

y1 � y2�2

x1 � x2�2

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the midpoint of each line segment with endpoints at the given coordinates.

1. (12, 7) and (�2, 11) 2. (�8, �3) and (10, 9) 3. (4, 15) and (10, 1)

(5, 9) (1, 3) (7, 8)

4. (�3, �3) and (3, 3) 5. (15, 6) and (12, 14) 6. (22, �8) and (�10, 6)

(0, 0) (13.5, 10) (6, �1)

7. (3, 5) and (�6, 11) 8. (8, �15) and (�7, 13) 9. (2.5, �6.1) and (7.9, 13.7)

�� , 8� � , �1� (5.2, 3.8)

10. (�7, �6) and (�1, 24) 11. (3, �10) and (30, �20) 12. (�9, 1.7) and (�11, 1.3)

(�4, 9) � , �15� (�10, 1.5)

13. Segment M�N� has midpoint P. If M has coordinates (14, �3) and P has coordinates (�8, 6), what are the coordinates of N? (�30, 15)

14. Circle R has a diameter S�T�. If R has coordinates (�4, �8) and S has coordinates (1, 4),what are the coordinates of T? (�9, �20)

15. Segment A�D� has midpoint B, and B�D� has midpoint C. If A has coordinates (�5, 4) and C has coordinates (10, 11), what are the coordinates of B and D?

B is �5, 8 �, D is �15, 13 �.1�

2�

33�

1�

3�


The Distance Formula

Distance FormulaThe distance between two points (x1, y1) and (x2, y2) is given by

d � �(x2 ��x1)2 �� (y2 �� y1)2�.

What is the distance between (8, �2) and (�6, �8)?

d � �(x2 ��x1)2 ��( y2 �� y1)2� Distance Formula

� �(�6 �� 8)2 �� [�8 �� (�2)]�2� Let (x1, y1) � (8, �2) and (x2, y2) � (�6, �8).

� �(�14)�2 � (��6)2� Subtract.

� �196 �� 36� or �232� Simplify.

The distance between the points is �232� or about 15.2 units.

Find the perimeter and area of square PQRS with vertices P(�4, 1),Q(�2, 7), R(4, 5), and S(2, �1).

Find the length of one side to find the perimeter and the area. Choose P�Q�.

d � �(x2 ��x1)2 ��( y2 �� y1)2� Distance Formula

� �[�4 �� (�2)]�2 � (1� � 7)2� Let (x1, y1) � (�4, 1) and (x2, y2) � (�2, 7).

� �(�2)2�� (�6�)2� Subtract.

� �40� or 2�10� Simplify.

Since one side of the square is 2�10�, the perimeter is 8�10� units. The area is (2�10�)2, or40 units2.

Find the distance between each pair of points with the given coordinates.

1. (3, 7) and (�1, 4) 2. (�2, �10) and (10, �5) 3. (6, �6) and (�2, 0)

5 units 13 units 10 units

4. (7, 2) and (4, �1) 5. (�5, �2) and (3, 4) 6. (11, 5) and (16, 9)

3�2� units 10 units �41� units

7. (�3, 4) and (6, �11) 8. (13, 9) and (11, 15) 9. (�15, �7) and (2, 12)

3�34� units 2�10� units 5�26� units

10. Rectangle ABCD has vertices A(1, 4), B(3, 1), C(�3, �2), and D(�5, 1). Find theperimeter and area of ABCD. 2�13� � 6�5� units; 3�65� units2

11. Circle R has diameter S�T� with endpoints S(4, 5) and T(�2, �3). What are thecircumference and area of the circle? (Express your answer in terms of �.)10� units; 25� units2

Study Guide and Intervention (continued)

Midpoint and Distance Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeMidpoint and Distance Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1


Less

on

8-1


1. (4, �1), (�4, 1) (0, 0) 2. (�1, 4), (5, 2) (2, 3)

3. (3, 4), (5, 4) (4, 4) 4. (6, 2), (2, �1) �4, �

5. (3, 9), (�2, �3) � , 3� 6. (�3, 5), (�3, �8) ��3, � �

7. (3, 2), (�5, 0) (�1, 1) 8. (3, �4), (5, 2) (4, �1)

9. (�5, �9), (5, 4) �0, � � 10. (�11, 14), (0, 4) �� , 9�

11. (3, �6), (�8, �3) �� , � � 12. (0, 10), (�2, �5) ��1, �


13. (4, 12), (�1, 0) 13 units 14. (7, 7), (�5, �2) 15 units

15. (�1, 4), (1, 4) 2 units 16. (11, 11), (8, 15) 5 units

17. (1, �6), (7, 2) 10 units 18. (3, �5), (3, 4) 9 units

19. (2, 3), (3, 5) �5� units 20. (�4, 3), (�1, 7) 5 units

21. (�5, �5), (3, 10) 17 units 22. (3, 9), (�2, �3) 13 units

23. (6, �2), (�1, 3) �74� units 24. (�4, 1), (2, �4) �61� units

25. (0, �3), (4, 1) 4�2� units 26. (�5, �6), (2, 0) �85� units

5�

9�

5�

11�

5�

3�

1�

1�



1. (8, �3), (�6, �11) (1, �7) 2. (�14, 5), (10, 6) ��2, �3. (�7, �6), (1, �2) (�3, �4) 4. (8, �2), (8, �8) (8, �5)

5. (9, �4), (1, �1) �5, � � 6. (3, 3), (4, 9) � , 6�7. (4, �2), (3, �7) � , � � 8. (6, 7), (4, 4) �5, �9. (�4, �2), (�8, 2) (�6, 0) 10. (5, �2), (3, 7) �4, �

11. (�6, 3), (�5, �7) �� , �2� 12. (�9, �8), (8, 3) �� , � �13. (2.6, �4.7), (8.4, 2.5) (5.5, �1.1) 14. �� , 6�, � , 4� � , 5�15. (�2.5, �4.2), (8.1, 4.2) (2.8, 0) 16. � , �, �� , � � �� , 0�


17. (5, 2), (2, �2) 5 units 18. (�2, �4), (4, 4) 10 units

19. (�3, 8), (�1, �5) �173� units 20. (0, 1), (9, �6) �130� units

21. (�5, 6), (�6, 6) 1 unit 22. (�3, 5), (12, �3) 17 units

23. (�2, �3), (9, 3) �157� units 24. (�9, �8), (�7, 8) 2�65� units

25. (9, 3), (9, �2) 5 units 26. (�1, �7), (0, 6) �170� units

27. (10, �3), (�2, �8) 13 units 28. (�0.5, �6), (1.5, 0) 2�10� units

29. � , �, �1, � 1 unit 30. (�4�2�, ��5�), (�5�2�, 4�5�) �127� units

31. GEOMETRY Circle O has a diameter A�B�. If A is at (�6, �2) and B is at (�3, 4), find thecenter of the circle and the length of its diameter. �� , 1�; 3�5� units

32. GEOMETRY Find the perimeter of a triangle with vertices at (1, �3), (�4, 9), and (�2, 1).18 � 2�17� units

9�

7�5

3�5

2�5

1�

1�2

5�8

1�2

1�8

1�

2�3

1�3

5�

1�

11�

5�

11�

9�

7�

7�

5�

11�

Practice (Average)

Midpoint and Distance Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

Reading to Learn MathematicsMidpoint and Distance Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1


Less

on

8-1

Pre-Activity How are the Midpoint and Distance Formulas used in emergencymedicine?

Read the introduction to Lesson 8-1 at the top of page 412 in your textbook.

How do you find distances on a road map?

Sample answer: Use the scale of miles on the map. You mightalso use a ruler.

Reading the Lesson

1. a. Write the coordinates of the midpoint of a segment with endpoints (x1, y1) and (x2, y2).

� , �b. Explain how to find the midpoint of a segment if you know the coordinates of the

endpoints. Do not use subscripts in your explanation.

Sample answer: To find the x-coordinate of the midpoint, add the x-coordinates of the endpoints and divide by two. To find the y-coordinate of the midpoint, do the same with the y-coordinates ofthe endpoints.

2. a. Write an expression for the distance between two points with coordinates (x1, y1) and(x2, y2). �(x2 ��x1)2 �� (y2 �� y1)2�

b. Explain how to find the distance between two points. Do not use subscripts in yourexplanation.

Sample answer: Find the difference between the x-coordinates and square it. Find the difference between the y-coordinates and square it. Add the squares. Then find the squareroot of the sum.

3. Consider the segment connecting the points (�3, 5) and (9, 11).

a. Find the midpoint of this segment. (3, 8)

b. Find the length of the segment. Write your answer in simplified radical form. 6�5�

Helping You Remember

4. How can the “mid” in midpoint help you remember the midpoint formula?

Sample answer: The midpoint is the point in the middle of a segment. Itis halfway between the endpoints. The coordinates of the midpoint arefound by finding the average of the two x-coordinates (add them anddivide by 2) and the average of the two y-coordinates.

y1 � y2�2

x1 � x2�2


Quadratic FormConsider two methods for solving the following equation.

(y � 2)2 � 5(y � 2) � 6 � 0

One way to solve the equation is to simplify first, then use factoring.

y2 � 4y � 4 � 5y � 10 � 6 � 0y2 � 9y � 20 � 0

( y � 4)( y � 5) � 0

Thus, the solution set is {4, 5}.

Another way to solve the equation is first to replace y � 2 by a single variable.This will produce an equation that is easier to solve than the original equation.Let t � y � 2 and then solve the new equation.

( y � 2)2 � 5( y � 2) � 6 � 0t2 � 5t � 6 � 0

(t � 2)(t � 3) � 0

Thus, t is 2 or 3. Since t � y � 2, the solution set of the original equation is {4, 5}.

Solve each equation using two different methods.

1. (z � 2)2 � 8(z � 2) � 7 � 0 2. (3x � 1)2 � (3x � 1) � 20 � 0

3. (2t � 1)2 � 4(2t � 1) � 3 � 0 4. ( y2 � 1)2 � ( y2 � 1) � 2 � 0

5. (a2 � 2)2 � 2(a2 � 2) � 3 � 0 6. (1 � �c�)2 � (1 � �c�) � 6 � 0

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

Study Guide and InterventionParabolas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2


Less

on

8-2

Equations of Parabolas A parabola is a curve consisting of all points in thecoordinate plane that are the same distance from a given point (the focus) and a given line(the directrix). The following chart summarizes important information about parabolas.

Standard Form of Equation y � a(x � h)2 � k x � a(y � k)2 � h

Axis of Symmetry x � h y � k

Vertex (h, k ) (h, k )

Focus �h, k � � �h � , k�Directrix y � k � x � h �

Direction of Opening upward if a � 0, downward if a � 0 right if a � 0, left if a � 0

Length of Latus Rectum units units

Identify the coordinates of the vertex and focus, the equations ofthe axis of symmetry and directrix, and the direction of opening of the parabolawith equation y � 2x2 � 12x � 25.

y � 2x2 � 12x � 25 Original equation

y � 2(x2 � 6x) � 25 Factor 2 from the x-terms.

y � 2(x2 � 6x � ■ ) � 25 � 2(■ ) Complete the square on the right side.

y � 2(x2 � 6x � 9) � 25 � 2(9) The 9 added to complete the square is multiplied by 2.

y � 2(x � 3)2 � 43 Write in standard form.

The vertex of this parabola is located at (3, �43), the focus is located at �3, �42 �, the

equation of the axis of symmetry is x � 3, and the equation of the directrix is y � �43 .The parabola opens upward.

Identify the coordinates of the vertex and focus, the equations of the axis ofsymmetry and directrix, and the direction of opening of the parabola with thegiven equation.

1. y � x2 � 6x � 4 2. y � 8x � 2x2 � 10 3. x � y2 � 8y � 6

(�3, �13), (2, 18), �2, 17 �, (�10, 4), ��9 , 4�,��3, �12 �, x � �3, x � 2, y � 18 , y � 4, x � �10 ,

y � �13 , up down right

Write an equation of each parabola described below.

4. focus (�2, 3), directrix x � �2 5. vertex (5, 1), focus �4 , 1�x � 6(y � 3)2 � 2 x � �3(y � 1)2 � 51

�

11�12

1�12

1�

1�

1�

3�

3�

1�

1�8

7�8

1�a

1�a

1�4a

1�4a

1�4a

1�4a

ExampleExample

ExercisesExercises


Graph Parabolas To graph an equation for a parabola, first put the given equation instandard form.

y � a(x � h)2 � k for a parabola opening up or down, orx � a(y � k)2 � h for a parabola opening to the left or right

Use the values of a, h, and k to determine the vertex, focus, axis of symmetry, and length ofthe latus rectum. The vertex and the endpoints of the latus rectum give three points on theparabola. If you need more points to plot an accurate graph, substitute values for pointsnear the vertex.

Graph y � (x � 1)2 � 2.

In the equation, a � , h � 1, k � 2.

The parabola opens up, since a � 0.vertex: (1, 2)axis of symmetry: x � 1

focus: �1, 2 � � or �1, 2 �

length of latus rectum: or 3 units

endpoints of latus rectum: �2 , 2 �, �� , 2 �

The coordinates of the focus and the equation of the directrix of a parabola aregiven. Write an equation for each parabola and draw its graph.

1. (3, 5), y � 1 2. (4, �4), y� �6 3. (5, �1), x � 3

y � (x � 3)2 � 3 y � (x � 4)2 � 5 x � (y � 1)2 � 41�

1�

1�

x

y

Ox

y

O

x

y

O

3�4

1�2

3�4

1�2

1�

�13

�

3�4

1�

4��13��

x

y

O

1�3

1�3


Parabolas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

ExampleExample

ExercisesExercises

Skills PracticeParabolas

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8-2

Write each equation in standard form.

1. y � x2 � 2x � 2 2. y � x2 � 2x � 4 3. y � x2 � 4x � 1

y � [x � (�1)]2 � 1 y � (x � 1)2 � 3 y � [x � (�2)]2 � (�3)

Identify the coordinates of the vertex and focus, the equations of the axis ofsymmetry and directrix, and the direction of opening of the parabola with thegiven equation. Then find the length of the latus rectum and graph the parabola.

4. y � (x � 2)2 5. x � (y � 2)2 � 3 6. y � �(x � 3)2 � 4

vertex: (2, 0); vertex: (3, 2); vertex: (�3, 4);

focus: �2, �; focus: �3 , 2�; focus: ��3, 3 �;axis of symmetry: axis of symmetry: axis of symmetry: x � 2; y � 2; x � �3;directrix: y � � ; directrix: x � 2 ; directrix: y � 4 ;

opens up; opens right; opens down;latus rectum: 1 unit latus rectum: 1 unit latus rectum: 1 unit

Write an equation for each parabola described below. Then draw the graph.

7. vertex (0, 0), 8. vertex (5, 1), 9. vertex (1, 3),

focus �0, � � focus �5, � directrix x �

y � �3x2 y � (x � 5)2 � 1 x � 2(y � 3)2 � 1

x

y

Ox

y

O

x

y

O

7�8

5�4

1�12

1�

3�

1�

3�

1�

1�

x

y

Ox

y

O

x

y

O


Write each equation in standard form.

1. y � 2x2 � 12x � 19 2. y � x2 � 3x � 3. y � �3x2 � 12x � 7

y � 2(x � 3)2 � 1 y � [x � (�3)]2 � (�4) y � �3[x � (�2)]2 � 5

Identify the coordinates of the vertex and focus, the equations of the axis ofsymmetry and directrix, and the direction of opening of the parabola with thegiven equation. Then find the length of the latus rectum and graph the parabola.

4. y � (x � 4)2 � 3 5. x � � y2 � 1 6. x � 3(y � 1)2 � 3

vertex: (4, 3); vertex: (1, 0); vertex: (�3, �1);

focus: �4, 3 �; focus: � , 0�; focus: ��2 , �1�;axis: x � 4; axis: y � 0; axis: y � �1;directrix: y � 2 ; directrix: x � 1 ; directrix: x � �3 ;

opens up; opens left; opens right;latus rectum: 1 unit latus rectum: 3 units latus rectum: unit

Write an equation for each parabola described below. Then draw the graph.

7. vertex (0, �4), 8. vertex (�2, 1), 9. vertex (1, 3),

focus �0, �3 � directrix x � �3 axis of symmetry x � 1,latus rectum: 2 units,a � 0

y � 2x2 � 4 x � (y � 1)2 � 2 y � � (x � 1)2 � 3

10. TELEVISION Write the equation in the form y � ax2 for a satellite dish. Assume that thebottom of the upward-facing dish passes through (0, 0) and that the distance from thebottom to the focus point is 8 inches. y � x21

�

x

y

Ox

y

O

1�

1�

7�8

1�

1�

3�

3�

11�

1�

1�

x

y

O

x

y

O

1�3

1�

1�2

1�2

Practice (Average)

Parabolas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

Reading to Learn MathematicsParabolas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2


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on

8-2

Pre-Activity How are parabolas used in manufacturing?


Name at least two reflective objects that might have the shape of aparabola.

Sample answer: telescope mirror, satellite dish

Reading the Lesson

1. In the parabola shown in the graph, the point (2, �2) is called

the and the point (2, 0) is called the

. The line y � �4 is called the

, and the line x � 2 is called the

.

2. a. Write the standard form of the equation of a parabola that opens upward ordownward. y � a (x � h)2 � k

b. The parabola opens downward if and opens upward if . The

equation of the axis of symmetry is , and the coordinates of the vertex are

.

3. A parabola has equation x � � ( y � 2)2 � 4. This parabola opens to the .

It has vertex and focus . The directrix is . The length

of the latus rectum is units.


4. How can the way in which you plot points in a rectangular coordinate system help you toremember what the sign of a tells you about the direction in which a parabola opens?Sample answer: In plotting points, a positive x-coordinate tells you tomove to the right and a negative x-coordinate tells you to move to theleft. This is like a parabola whose equation is of the form “x � …”; itopens to the right if a � 0 and to the left if a � 0. Likewise, a positive y-coordinate tells you to move up and a negative y-coordinate tells youto move down. This is like a parabola whose equation is of the form “y � …”; it opens upward if a � 0 and downward if a � 0.

8

x � 6(2, 2)(4, 2)

left1�8

(h, k)

x � h

a � 0a � 0

axis of symmetry

directrix

focus

vertex

x

y

O

(2, –2)

(2, 0)

y � –4


Tangents to ParabolasA line that intersects a parabola in exactly one point without crossing the curve is a tangent to the parabola. The point where a tangent line touches a parabola is the point of tangency. The line perpendicular to a tangent to a parabola at the point of tangency is called the normal to the parabola at that point. In the diagram, line � is tangent to the

parabola that is the graph of y � x2 at ��32�, �

94��. The

x-axis is tangent to the parabola at O, and the y-axis is the normal to the parabola at O.

Solve each problem.

1. Find an equation for line � in the diagram. Hint: A nonvertical line with anequation of the form y � mx � b will be tangent to the graph of y � x2 at

��32�, �

94�� if and only if ��

32�, �

94�� is the only pair of numbers that satisfies both

y � x2 and y � mx � b.

2. If a is any real number, then (a, a2) belongs to the graph of y � x2. Express m and b in terms of a to find an equation of the form y � mx � b for the linethat is tangent to the graph of y � x2 at (a, a2).

3. Find an equation for the normal to the graph of y � x2 at ��32�, �

94��.

4. If a is a nonzero real number, find an equation for the normal to the graph ofy � x2 at (a, a2).

x

y

O

�

y � x2

1–1–2–3 2

6

5

4

3

2

1

3

�3–2, 9–4�

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

Study Guide and InterventionCircles

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8-38-3


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8-3

Equations of Circles The equation of a circle with center (h, k) and radius r units is (x � h)2 � (y � k)2 � r2.

Write an equation for a circle if the endpoints of a diameter are at(�4, 5) and (6, �3).

Use the midpoint formula to find the center of the circle.

(h, k) � � , � Midpoint formula

� � , � (x1, y1) � (�4, 5), (x2, y2) � (6, �3)

� � , � or (1, 1) Simplify.

Use the coordinates of the center and one endpoint of the diameter to find the radius.

r � �(x2 �x�1)2 ��( y2 �� y1)2� Distance formula

r � �(�4 �� 1)2 �� (5 ��1)2� (x1, y1) � (1, 1), (x2, y2) � (�4, 5)

� �(�5)2� � 42� � �41� Simplify.

The radius of the circle is �41�, so r2 � 41.

An equation of the circle is (x � 1)2 � (y � 1)2 � 41.

Write an equation for the circle that satisfies each set of conditions.

1. center (8, �3), radius 6 (x � 8)2 � (y � 3)2 � 36

2. center (5, �6), radius 4 (x � 5)2 � (y � 6)2 � 16

3. center (�5, 2), passes through (�9, 6) (x � 5)2 � (y � 2)2 � 32

4. endpoints of a diameter at (6, 6) and (10, 12) (x � 8)2 � (y � 9)2 � 13

5. center (3, 6), tangent to the x-axis (x � 3)2 � (y � 6)2 � 36

6. center (�4, �7), tangent to x � 2 (x � 4)2 � (y � 7)2 � 36

7. center at (�2, 8), tangent to y � �4 (x � 2)2 � (y � 8)2 � 144

8. center (7, 7), passes through (12, 9) (x � 7)2 � (y � 7)2 � 29

9. endpoints of a diameter are (�4, �2) and (8, 4) (x � 2)2 � (y � 1)2 � 45

10. endpoints of a diameter are (�4, 3) and (6, �8) (x � 1)2 � (y � 2.5)2 � 55.25

2�2

2�2

5 � (�3)��2

�4 � 6�2

y1 � y2�2

x1 � x2�2

ExampleExample

ExercisesExercises


Graph Circles To graph a circle, write the given equation in the standard form of theequation of a circle, (x � h)2 � (y � k)2 � r2.

Plot the center (h, k) of the circle. Then use r to calculate and plot the four points (h � r, k),(h � r, k), (h, k � r), and (h, k � r), which are all points on the circle. Sketch the circle thatgoes through those four points.

Find the center and radius of the circle whose equation is x2 � 2x � y2 � 4y � 11. Then graph the circle.

x2 � 2x � y2 � 4y � 11x2 � 2x � ■ � y2 � 4y � ■ � 11 �■

x2 � 2x � 1 � y2 � 4y � 4 � 11 � 1 � 4(x � 1)2 � ( y � 2)2 � 16

Therefore, the circle has its center at (�1, �2) and a radius of �16� � 4. Four points on the circle are (3, �2), (�5, �2), (�1, 2),and (�1, �6).

Find the center and radius of the circle with the given equation. Then graph thecircle.

1. (x � 3)2 � y2 � 9 2. x2 � (y � 5)2 � 4 3. (x � 1)2 � (y � 3)2 � 9

(3, 0), r � 3 (0, �5), r � 2 (1, �3), r � 3

4. (x � 2)2 � (y � 4)2 � 16 5. x2 � y2 � 10x � 8y � 16 � 0 6. x2 � y2 � 4x � 6y � 12

(2, �4), r � 4 (5, �4), r � 5 (2, �3), r � 5

x

y

Ox

y

Ox

y

O

x

y

Ox

y

O

x

y

O

x

y

O

x2 � 2x � y2 � 4y � 11


Circles

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8-38-3

ExampleExample

ExercisesExercises

Skills PracticeCircles

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1. center (0, 5), radius 1 unit 2. center (5, 12), radius 8 unitsx2 � (y � 5)2 � 1 (x � 5)2 � (y � 12)2 � 64

3. center (4, 0), radius 2 units 4. center (2, 2), radius 3 units(x � 4)2 � y2 � 4 (x � 2)2 � (y � 2)2 � 9

5. center (4, �4), radius 4 units 6. center (�6, 4), radius 5 units(x � 4)2 � (y � 4)2 � 16 (x � 6)2 � (y � 4)2 � 25

7. endpoints of a diameter at (�12, 0) and (12, 0) x2 � y2 � 144

8. endpoints of a diameter at (�4, 0) and (�4, �6) (x � 4)2 � (y � 3)2 � 9

9. center at (7, �3), passes through the origin (x � 7)2 � (y � 3)2 � 58

10. center at (�4, 4), passes through (�4, 1) (x � 4)2 � (y � 4)2 � 9

11. center at (�6, �5), tangent to y-axis (x � 6)2 � (y � 5)2 � 36

12. center at (5, 1), tangent to x-axis (x � 5)2 � (y � 1)2 � 1


13. x2 � y2 � 9 14. (x � 1)2 � (y � 2)2 � 4 15. (x � 1)2 � y2 � 16

(0, 0), 3 units (1, 2), 2 units (�1, 0), 4 units

16. x2 � (y � 3)2 � 81 17. (x � 5)2 � (y � 8)2 � 49 18. x2 � y2 � 4y � 32 � 0

(0, �3), 9 units (5, �8), 7 units (0, 2), 6 units

x

y

O 4 8

8

4

–4

–8

–4–8

x

y

O 4 8 12

–4

–8

–12

x

y

O 6 12

12

6

–6

–12

–6–12

x

y

Ox

y

Ox

y

O



1. center (�4, 2), radius 8 units 2. center (0, 0), radius 4 units(x � 4)2 � (y � 2)2 � 64 x2 � y2 � 16

3. center �� , ��3��, radius 5�2� units 4. center (2.5, 4.2), radius 0.9 unit

�x � �2 � (y � �3�)2 � 50 (x � 2.5)2 � (y � 4.2)2 � 0.81

5. endpoints of a diameter at (�2, �9) and (0, �5) (x � 1)2 � (y � 7)2 � 5

6. center at (�9, �12), passes through (�4, �5) (x � 9)2 � (y � 12)2 � 74

7. center at (�6, 5), tangent to x-axis (x � 6)2 � (y � 5)2 � 25


8. (x � 3)2 � y2 � 16 9. 3x2 � 3y2 � 12 10. x2 � y2 � 2x � 6y � 26(�3, 0), 4 units (0, 0), 2 units (�1, �3), 6 units

11. (x � 1)2 � y2 � 4y � 12 12. x2 � 6x � y2 � 0 13. x2 � y2 � 2x � 6y � �1(1, �2), 4 units (3, 0), 3 units (�1, �3), 3 units

WEATHER For Exercises 14 and 15, use the following information.On average, the circular eye of a hurricane is about 15 miles in diameter. Gale winds canaffect an area up to 300 miles from the storm’s center. In 1992, Hurricane Andrew devastatedsouthern Florida. A satellite photo of Andrew’s landfall showed the center of its eye on onecoordinate system could be approximated by the point (80, 26).

14. Write an equation to represent a possible boundary of Andrew’s eye.(x � 80)2 � (y � 26)2 � 56.25

15. Write an equation to represent a possible boundary of the area affected by gale winds.(x � 80)2 � (y � 26)2 � 90,000

x

y

O 4 8

4

–4

–8

–4–8x

y

Ox

y

O

1�

1�4

Practice (Average)

Circles

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3

Reading to Learn MathematicsCircles

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8-38-3


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on

8-3

Pre-Activity Why are circles important in air traffic control?


A large home improvement chain is planning to enter a new metropolitanarea and needs to select locations for its stores. Market research has shownthat potential customers are willing to travel up to 12 miles to shop at oneof their stores. How can circles help the managers decide where to placetheir store?

Sample answer: A store will draw customers who live inside acircle with center at the store and a radius of 12 miles. The management should select locations for whichas many people as possible live within a circle of radius 12 miles around one of the stores.

Reading the Lesson

1. a. Write the equation of the circle with center (h, k) and radius r.(x � h)2 � (y � k)2 � r 2

b. Write the equation of the circle with center (4, �3) and radius 5.(x � 4)2 � (y � 3)2 � 25

c. The circle with equation (x � 8)2 � y2 � 121 has center and radius

.

d. The circle with equation (x � 10)2 � ( y � 10)2 � 1 has center and

radius .

2. a. In order to find center and radius of the circle with equation x2 � y2 � 4x � 6y �3 � 0,

it is necessary to . Fill in the missing parts of thisprocess.

x2 � y2 � 4x � 6y � 3 � 0

x2 � y2 � 4x � 6y �

x2 � 4x � � y2 � 6y � � � �

(x � )2 � ( y � )2 �

b. This circle has radius 4 and center at .


3. How can the distance formula help you to remember the equation of a circle?Sample answer: Write the distance formula. Replace (x1, y1) with (h, k)and (x2, y2) with (x, y ). Replace d with r. Square both sides. Now youhave the equation of a circle.

(�2, 3)

163294394

3

complete the square

1(10, �10)

11(�8, 0)


Tangents to CirclesA line that intersects a circle in exactly one point is a tangent to the circle. In the diagram, line � is tangent to the circle with equation x2 � y2 � 25 at the point whose coordinates are (3, 4).

A line is tangent to a circle at a point P on the circle if and only if the line is perpendicular to the radius from the center of the circle to point P. This fact enables you to find an equation of the tangent to a circle at a point P if you know an equation for the circle and the coordinates of P.

Use the diagram above to solve each problem.

1. What is the slope of the radius to the point with coordinates (3, 4)? What isthe slope of the tangent to that point?

2. Find an equation of the line � that is tangent to the circle at (3, 4).

3. If k is a real number between �5 and 5, how many points on the circle have x-coordinate k? State the coordinates of these points in terms of k.

4. Describe how you can find equations for the tangents to the points you namedfor Exercise 3.

5. Find an equation for the tangent at (�3, 4).

5

–5

–5

5

(3, 4)

y

xO

�x2 � y2 � 25

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3

Study Guide and InterventionEllipses

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4


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on

8-4

Equations of Ellipses An ellipse is the set of all points in a plane such that the sumof the distances from two given points in the plane, called the foci, is constant. An ellipsehas two axes of symmetry which contain the major and minor axes. In the table, thelengths a, b, and c are related by the formula c2 � a2 � b2.

Standard Form of Equation � � 1 � � 1

Center (h, k) (h, k)

Direction of Major Axis Horizontal Vertical

Foci (h � c, k ), (h � c, k ) (h, k � c), (h, k � c)

Length of Major Axis 2a units 2a units

Length of Minor Axis 2b units 2b units

Write an equation for the ellipse shown.

The length of the major axis is the distance between (�2, �2) and (�2, 8). This distance is 10 units.

2a � 10, so a � 5The foci are located at (�2, 6) and (�2, 0), so c � 3.

b2 � a2 � c2

� 25 � 9� 16

The center of the ellipse is at (�2, 3), so h � �2, k � 3,a2 � 25, and b2 � 16. The major axis is vertical.

An equation of the ellipse is � � 1.

Write an equation for the ellipse that satisfies each set of conditions.

1. endpoints of major axis at (�7, 2) and (5, 2), endpoints of minor axis at (�1, 0) and (�1, 4)

� � 1

2. major axis 8 units long and parallel to the x-axis, minor axis 2 units long, center at (�2, �5)

� (y � 5)2 � 1

3. endpoints of major axis at (�8, 4) and (4, 4), foci at (�3, 4) and (�1, 4)

� � 1

4. endpoints of major axis at (3, 2) and (3, �14), endpoints of minor axis at (�1, �6) and (7, �6)

� � 1

5. minor axis 6 units long and parallel to the x-axis, major axis 12 units long, center at (6, 1)

� � 1(x � 6)2�

9(y � 1)2�

36

(x � 3)2�

16(y � 6)2�

64

(y � 4)2�

35(x � 2)2�

36

(x � 2)2�

16

(y � 2)2�

4(x � 1)2�

36

(x � 2)2�16

( y � 3)2�25

x

F1

F2O

y

(x � h)2

�b2

(y � k)2

�a2

(y � k)2�

b2(x � h)2�

a2

ExampleExample

ExercisesExercises


Graph Ellipses To graph an ellipse, if necessary, write the given equation in thestandard form of an equation for an ellipse.

� � 1 (for ellipse with major axis horizontal) or

� � 1 (for ellipse with major axis vertical)

Use the center (h, k) and the endpoints of the axes to plot four points of the ellipse. To makea more accurate graph, use a calculator to find some approximate values for x and y thatsatisfy the equation.

Graph the ellipse 4x2 � 6y2 � 8x � 36y � �34.

4x2 � 6y2 � 8x � 36y � �344x2 � 8x � 6y2 � 36y � � 34

4(x2 � 2x � ■ ) � 6( y2 � 6y � ■ ) � �34 � ■4(x2 � 2x � 1) � 6( y2 � 6y � 9) � �34 � 58

4(x � 1)2 � 6( y � 3)2 � 24

� � 1

The center of the ellipse is (�1, 3). Since a2 � 6, a � �6�.Since b2 � 4, b � 2.The length of the major axis is 2�6�, and the length of the minor axis is 4. Since the x-termhas the greater denominator, the major axis is horizontal. Plot the endpoints of the axes.Then graph the ellipse.

Find the coordinates of the center and the lengths of the major and minor axesfor the ellipse with the given equation. Then graph the ellipse.

1. � � 1 (0, 0), 4�3�, 6 2. � � 1 (0, 0), 10, 4

3. x2 � 4y2 � 24y � �32 (0, �3), 4, 2 4. 9x2 � 6y2 � 36x � 12y � 12 (2, �1), 6, 2�6�

x

y

Ox

y

O

x

y

Ox

y

O

y2�4

x2�25

x2�9

y2�12

( y � 3)2�4

(x � 1)2�6 xO

y

4x2 � 6y2 � 8x � 36y � �34

(x � h)2�

b2( y � k)2�

a2

( y � k)2�

b2(x � h)2�

a2


Ellipses

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4

ExampleExample

ExercisesExercises

Skills PracticeEllipses

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Write an equation for each ellipse.

1. 2. 3.

� � 1 � � 1 � � 1


4. endpoints of major axis 5. endpoints of major axis 6. endpoints of major axis at (0, 6) and (0, �6), at (2, 6) and (8, 6), at (7, 3) and (7, 9),endpoints of minor axis endpoints of minor axis endpoints of minor axis at (�3, 0) and (3, 0) at (5, 4) and (5, 8) at (5, 6) and (9, 6)

� � 1 � � 1 � � 1

7. major axis 12 units long 8. endpoints of major axis 9. endpoints of major axis atand parallel to x-axis, at (�6, 0) and (6, 0), foci (0, 12) and (0, �12), foci atminor axis 4 units long, at (��32�, 0) and (�32�, 0) (0, �23� ) and (0, ��23� )center at (0, 0)

� � 1 � � 1 � � 1

Find the coordinates of the center and foci and the lengths of the major andminor axes for the ellipse with the given equation. Then graph the ellipse.

10. � � 1 11. � � 1 12. � � 1

(0, 0); (0, ��19�); (0, 0); (�6�2�, 0); (0, 0), (0, �2�6�); 20; 18 18; 6 14; 10

x

y

O 4 8

8

4

–4

–8

–4–8x

y

O 4 8

8

4

–4

–8

–4–8x

y

O 4 8

8

4

–4

–8

–4–8

x2�25

y2�49

y2�9

x2�81

x2�81

y2�100

x2�

y2�

y2�

x2�

y2�

x2�

(x � 7)2�

4(y � 6)2�

9(y � 6)2�

4(x � 5)2�

9x2�

y2�

(y � 2)2�

9x2�

x2�

y2�

y2�

x2�

xO

y(0, 5)

(0, –1)

(–4, 2) (4, 2)

xO

y

(0, 3)

(0, –3)

(0, –5)

(0, 5)

xO

y

(0, 2)

(0, –2)

(–3, 0)(3, 0)


Write an equation for each ellipse.

1. 2. 3.

� � 1 � � 1 � � 1


4. endpoints of major axis 5. endpoints of major axis 6. major axis 20 units long at (�9, 0) and (9, 0), at (4, 2) and (4, �8), and parallel to x-axis,endpoints of minor axis endpoints of minor axis minor axis 10 units long,at (0, 3) and (0, �3) at (1, �3) and (7, �3) center at (2, 1)

� � 1 � � 1 � � 1

7. major axis 10 units long, 8. major axis 16 units long, 9. endpoints of minor axis minor axis 6 units long center at (0, 0), foci at at (0, 2) and (0, �2), foci and parallel to x-axis, (0, 2�15� ) and (0, �2�15� ) at (�4, 0) and (4, 0)center at (2, �4)

� � 1 � � 1 � � 1

Find the coordinates of the center and foci and the lengths of the major andminor axes for the ellipse with the given equation. Then graph the ellipse.

10. � � 1 11. � � 1 12. � � 1

(0, 0); (0, ��7�); 8; 6 (3, 1); (3, 1 � �35� ); (�4, �3); 12; 2 (�4 � 2�6�, �3); 14;

10

13. SPORTS An ice skater traces two congruent ellipses to form a figure eight. Assume that thecenter of the first loop is at the origin, with the second loop to its right. Write an equationto model the first loop if its major axis (along the x-axis) is 12 feet long and its minoraxis is 6 feet long. Write another equation to model the second loop.

x

y

O 4 8

8

4

–4

–8

–4–8x

y

O

( y � 3)2�25

(x � 4)2�49

(x � 3)2�1

( y � 1)2�36

x2�9

y2�16

y2�

x2�

x2�

y2�

(x � 2)2�

9(y � 4)2�

25

(y � 1)2�

25(x � 2)2�

100(x � 4)2�

9(y � 3)2�

25y2�

x2�

(y � 3)2�

9(x � 1)2�

25x2�

(y � 2)2�

9y2�

x2�

xO

y

(–5, 3)

(–6, 3)

(3, 3)

(4, 3)

xO

y

(0, 2 � ��5)

(0, 2 � ��5)

(0, –1)

(0, 5)

xO

y(0, 3)

(0, –3)

(–11, 0) (11, 0)6 12

2

–2

–6–12

Practice (Average)

Ellipses

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8-48-4

Reading to Learn MathematicsEllipses

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8-48-4


Less

on

8-4

Pre-Activity Why are ellipses important in the study of the solar system?


Is the Earth always the same distance from the Sun? Explain your answerusing the words circle and ellipse. No; if the Earth’s orbit were acircle, it would always be the same distance from the Sunbecause every point on a circle is the same distance from thecenter. However, the Earth’s orbit is an ellipse, and the pointson an ellipse are not all the same distance from the center.

Reading the Lesson1. An ellipse is the set of all points in a plane such that the of the

distances from two fixed points is . The two fixed points are called the

of the ellipse.

2. Consider the ellipse with equation � � 1.

a. For this equation, a � and b � .

b. Write an equation that relates the values of a, b, and c. c2 � a2 � b2

c. Find the value of c for this ellipse. �5�

3. Consider the ellipses with equations � � 1 and � � 1. Complete the

following table to describe characteristics of their graphs.


Direction of Major Axis vertical horizontal

Direction of Minor Axis horizontal vertical

Foci (0, 3), (0, �3) (�5�, 0), (��5�, 0)

Length of Major Axis 10 units 6 units

Length of Minor Axis 8 units 4 units

Helping You Remember4. Some students have trouble remembering the two standard forms for the equation of an

ellipse. How can you remember which term comes first and where to place a and b inthese equations? The x-axis is horizontal. If the major axis is horizontal, the first term is . The y-axis is vertical. If the major axis is vertical, the

first term is . a is always the larger of the numbers a and b.y2�

x2�

y2�4

x2�9

x2�16

y2�25

y2�4

x2�9

x2�16

y2�25

23

y2�4

x2�9

fociconstant

sum


Eccentricity In an ellipse, the ratio �d

c� is called the eccentricity and is denoted by the

letter e. Eccentricity measures the elongation of an ellipse. The closer e is to 0,the more an ellipse looks like a circle. The closer e is to 1, the more elongated

it is. Recall that the equation of an ellipse is �ax2

2� � �by2

2� � 1 or �bx2

2� � �ay2

2� � 1

where a is the length of the major axis, and that c � �a2 � b�2�.

Find the eccentricity of each ellipse rounded to the nearesthundredth.

1. �x9

2� � �3

y6

2� � 1 2. �8

x1

2� � �

y9

2� � 1 3. �

x4

2� � �

y9

2� � 1

0.87 0.94 0.75

4. �1x6

2� � �

y9

2� � 1 5. �3

x6

2� � �1

y6

2� � 1 6. �

x4

2� � �3

y6

2� � 1

0.66 0.75 0.94

7. Is a circle an ellipse? Explain your reasoning.

Yes; it is an ellipse with eccentricity 0.

8. The center of the sun is one focus of Earth's orbit around the sun. Thelength of the major axis is 186,000,000 miles, and the foci are 3,200,000miles apart. Find the eccentricity of Earth's orbit.

approximately 0.17

9. An artificial satellite orbiting the earth travels at an altitude that variesbetween 132 miles and 583 miles above the surface of the earth. If thecenter of the earth is one focus of its elliptical orbit and the radius of theearth is 3950 miles, what is the eccentricity of the orbit?

approximately 0.052

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4

Study Guide and InterventionHyperbolas

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8-58-5


Less

on

8-5

Equations of Hyperbolas A hyperbola is the set of all points in a plane such thatthe absolute value of the difference of the distances from any point on the hyperbola to anytwo given points in the plane, called the foci, is constant.

In the table, the lengths a, b, and c are related by the formula c2 � a2 � b2.


Equations of the Asymptotes y � k � � (x � h) y � k � � (x � h)

Transverse Axis Horizontal Vertical

Foci (h � c, k), (h � c, k) (h, k � c), (h, k � c)

Vertices (h � a, k), (h � a, k) (h, k � a), (h, k � a)

Write an equation for the hyperbola with vertices (�2, 1) and (6, 1)and foci (�4, 1) and (8, 1).

Use a sketch to orient the hyperbola correctly. The center of the hyperbola is the midpoint of the segment joining the two

vertices. The center is ( , 1), or (2, 1). The value of a is the

distance from the center to a vertex, so a � 4. The value of c is the distance from the center to a focus, so c � 6.

c2 � a2 � b2

62 � 42 � b2

b2 � 36 � 16 � 20

Use h, k, a2, and b2 to write an equation of the hyperbola.

� � 1

Write an equation for the hyperbola that satisfies each set of conditions.

1. vertices (�7, 0) and (7, 0), conjugate axis of length 10 � � 1

2. vertices (�2, �3) and (4, �3), foci (�5, �3) and (7, �3) � � 1

3. vertices (4, 3) and (4, �5), conjugate axis of length 4 � � 1

4. vertices (�8, 0) and (8, 0), equation of asymptotes y � � x � � 1

5. vertices (�4, 6) and (�4, �2), foci (�4, 10) and (�4, �6) � � 1(x � 4)2�

48(y � 2)2�

16

9y2�

x2�

1�6

(x � 4)2�

4(y � 1)2�

16

(y � 3)2�

27(x � 1)2�

9

y2�

x2�

( y � 1)2�20

(x � 2)2�16

�2 � 6�2

x

y

O

a�b

b�a

(x � h)2�

b2(y � k)2�

a2(y � k)2�

b2(x � h)2�

a2

ExampleExample

ExercisesExercises


Graph Hyperbolas To graph a hyperbola, write the given equation in the standardform of an equation for a hyperbola

� � 1 if the branches of the hyperbola open left and right, or

� � 1 if the branches of the hyperbola open up and down

Graph the point (h, k), which is the center of the hyperbola. Draw a rectangle withdimensions 2a and 2b and center (h, k). If the hyperbola opens left and right, the verticesare (h � a, k) and (h � a, k). If the hyperbola opens up and down, the vertices are (h, k � a)and (h, k � a).

Draw the graph of 6y2 � 4x2 � 36y � 8x � �26.

Complete the squares to get the equation in standard form.6y2 � 4x2 � 36y � 8x � �266( y2 � 6y � ■ ) � 4(x2 � 2x � ■ ) � �26 � ■6( y2 � 6y � 9) � 4(x2 � 2x � 1) � �26 � 506( y � 3)2 � 4(x � 1)2 � 24

� � 1

The center of the hyperbola is (�1, 3).According to the equation, a2 � 4 and b2 � 6, so a � 2 and b � �6�.The transverse axis is vertical, so the vertices are (�1, 5) and (�1, 1). Draw a rectangle withvertical dimension 4 and horizontal dimension 2�6� � 4.9. The diagonals of this rectangleare the asymptotes. The branches of the hyperbola open up and down. Use the vertices andthe asymptotes to sketch the hyperbola.

Find the coordinates of the vertices and foci and the equations of the asymptotesfor the hyperbola with the given equation. Then graph the hyperbola.

1. � � 1 2. ( y � 3)2 � � 1 3. � � 1

(2, 0), (�2, 0); (�2, 4), (�2, 2); (0, 4), (0, �4); (2�5�, 0), (�2�5�, 0); (�2, 3 � �10� ), (0, 5), (0, �5); y � �2x (�2, 3 � �10� ); y � � x

y � x � 3 ,

y � � x � 2 1�

1�

2�

1�

xO

y

4�

x2�9

y2�16

(x � 2)2�9

y2�16

x2�4

(x � 1)2�6

( y � 3)2�4 xO

y

(x � h)2�

b2( y � k)2�

a2

( y � k)2��

b2(x � h)2�

a2


Hyperbolas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5

ExampleExample

ExercisesExercises

xO

y

xO

y

Skills PracticeHyperbolas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5


Less

on

8-5

Write an equation for each hyperbola.

1. 2. 3.

� � 1 � � 1 � � 1






8. vertices (�3, 0) and (3, 0), foci (�5, 0) � � 1

9. vertices (0, 2) and (0, �2), foci (0, �3) � � 1

10. vertices (0, �2) and (6, �2), foci (3 � �13�, �2) � � 1


11. � � 1 12. � � 1 13. � � 1

(�3, 0); (�3�5�, 0); (0, �7); (0, ��58� ); (�4, 0); (��17�, 0);y � �2x y � � x y � � x

xO

y

4 8

8

4

–4

–8

–4–8xO

y

4 8

8

4

–4

–8

–4–8xO

y

1�

7�

y2�1

x2�16

x2�9

y2�49

y2�36

x2�9

(y � 2)2�

4(x � 3)2�

9

x2�

y2�

y2�

x2�

y2�

x2�

x2�

y2�

x2�

y2�

y2�

x2�

y2�

x2�

x2�

y2�

y2�

x2�

x

y

O

(��29, 0)(–��29, 0)

(2, 0)(–2, 0)

4 8

8

4

–4

–8

–4–8x

y

O

(0, ��61)

(0, –��61)

(0, 6)

(0, –6)

4 8

8

4

–4

–8

–4–8x

y

O

(��41, 0)(–��41, 0)

(5, 0)

(–5, 0)

4 8

8

4

–4

–8

–4–8


Write an equation for each hyperbola.

1. 2. 3.

� � 1 � � 1 � � 1


4. vertices (0, 7) and (0, �7), conjugate axis of length 18 units � � 1

5. vertices (1, �1) and (1, �9), conjugate axis of length 6 units � � 1

6. vertices (�5, 0) and (5, 0), foci (��26�, 0) � � 1

7. vertices (1, 1) and (1, �3), foci (1, �1 � �5�) � � 1


8. � � 1 9. � � 1 10. � � 1

(0, �4); (0, �2�5�); (1, 3), (1, 1); (3, 0), (3, �4); y � �2x (1, 2 � �5�); (3, �2 � 2�2�);

y � 2 � � (x � 1) y � 2 � �(x � 3)

11. ASTRONOMY Astronomers use special X-ray telescopes to observe the sources ofcelestial X rays. Some X-ray telescopes are fitted with a metal mirror in the shape of ahyperbola, which reflects the X rays to a focus. Suppose the vertices of such a mirror arelocated at (�3, 0) and (3, 0), and one focus is located at (5, 0). Write an equation thatmodels the hyperbola formed by the mirror.

� � 1y2�

x2�

xO

y

xO

y

4 8

8

4

–4

–8

–4–8

1�

(x � 3)2�4

( y � 2)2�4

(x � 1)2�4

( y � 2)2�1

x2�4

y2�16

(x � 1)2�

1(y � 1)2�

4

y2�

x2�

(x � 1)2�

9(y � 5)2�

16

x2�

y2�

(y � 2)2�

16(x � 1)2�

4(x � 3)2�

25(y � 2)2�

9x2�

y2�

x

y

O(–1, –2)

(1, –2)

(3, –2)x

y

O

(–3, 2 � ��34)

(–3, 2 � ��34)

(–3, –1)(–3, 5)

4

8

4

–4

–4–8x

y

O

(0, 3��5)

(0, –3��5)

(0, 3)

(0, –3)

4 8

8

4

–4

–8

–4–8

Practice (Average)

Hyperbolas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5

Reading to Learn MathematicsHyperbolas

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5


Less

on

8-5

Pre-Activity How are hyperbolas different from parabolas?


Look at the sketch of a hyperbola in the introduction to this lesson. Listthree ways in which hyperbolas are different from parabolas.Sample answer: A hyperbola has two branches, while aparabola is one continuous curve. A hyperbola has two foci,while a parabola has one focus. A hyperbola has two vertices,while a parabola has one vertex.

Reading the Lesson

1. The graph at the right shows the hyperbola whose

equation in standard form is � � 1.

The point (0, 0) is the of the hyperbola.

The points (4, 0) and (�4, 0) are the of the hyperbola.

The points (5, 0) and (�5, 0) are the of the hyperbola.

The segment connecting (4, 0) and (�4, 0) is called the axis.

The segment connecting (0, 3) and (0, �3) is called the axis.

The lines y � x and y � � x are called the .

2. Study the hyperbola graphed at the right.

The center is .

The value of a is .

The value of c is .

To find b2, solve the equation � � .

The equation in standard form for this hyperbola is .


3. What is an easy way to remember the equation relating the values of a, b, and c for ahyperbola? This equation looks just like the Pythagorean Theorem,although the variables represent different lengths in a hyperbola than ina right triangle.

�x4

2� � �

1y2

2� � 1

b2a2c2

4

2

(0, 0)

x

y

O

asymptotes3�4

3�4

conjugate

transverse

foci

vertices

center

y2�9

x2�16

x

y

O(–4, 0) (4, 0)(–5, 0) (5, 0)

y � 34xy � – 34x


Rectangular Hyperbolas A rectangular hyperbola is a hyperbola with perpendicular asymptotes.For example, the graph of x2 � y2 � 1 is a rectangular hyperbola. A hyperbolawith asymptotes that are not perpendicular is called a nonrectangularhyperbola. The graphs of equations of the form xy � c, where c is a constant,are rectangular hyperbolas.

Make a table of values and plot points to graph each rectangularhyperbola below. Be sure to consider negative values for thevariables. See students’ tables.

1. xy � �4 2. xy � 3

3. xy � �1 4. xy � 8

5. Make a conjecture about the asymptotes of rectangular hyperbolas.

The coordinate axes are the asymptotes.

x

y

Ox

y

O

x

y

Ox

y

O

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5

Study Guide and InterventionConic Sections

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6


Less

on

8-6

Standard Form Any conic section in the coordinate plane can be described by anequation of the form

Ax2 � Bxy � Cy2 � Dx � Ey � F � 0, where A, B, and C are not all zero.One way to tell what kind of conic section an equation represents is to rearrange terms andcomplete the square, if necessary, to get one of the standard forms from an earlier lesson.This method is especially useful if you are going to graph the equation.

Write the equation 3x2 � 4y2 � 30x � 8y � 59 � 0 in standard form.State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola.

3x2 � 4y2 � 30x � 8y � 59 � 0 Original equation

3x2 � 30x � 4y2 � 8y � �59 Isolate terms.

3(x2 � 10x � ■ ) � 4( y2 � 2y � ■ ) � �59 � ■ � ■ Factor out common multiples.

3(x2 � 10x � 25) � 4( y2 � 2y � 1) � �59 � 3(25) � (�4)(1) Complete the squares.

3(x � 5)2 � 4( y � 1)2 � 12 Simplify.

� � 1 Divide each side by 12.

The graph of the equation is a hyperbola with its center at (5, �1). The length of the transverse axis is 4 units and the length of the conjugate axis is 2�3� units.

Write each equation in standard form. State whether the graph of the equation isa parabola, circle, ellipse, or hyperbola.

1. x2 � y2 � 6x � 4y � 3 � 0 2. x2 � 2y2 � 6x � 20y � 53 � 0

(x � 3)2 � (y � 2)2 � 10; circle � � 1; ellipse

3. 6x2 � 60x � y � 161 � 0 4. x2 � y2 � 4x �14y � 29 � 0

y � 6(x � 5)2 � 11; parabola (x � 2)2 � (y � 7)2 � 24; circle

5. 6x2 � 5y2 � 24x � 20y � 56 � 0 6. 3y2 � x � 24y � 46 � 0

� � 1; hyperbola x � �3(y � 4)2 � 2; parabola

7. x2 � 4y2 � 16x � 24y � 36 � 0 8. x2 � 2y2 � 8x � 4y � 2 � 0

� � 1; hyperbola � � 1; ellipse

9. 4x2 � 48x � y � 158 � 0 10. 3x2 � y2 � 48x � 4y � 184 � 0

y � �4(x � 6)2 � 14; parabola � � 1; ellipse

11. �3x2 � 2y2 � 18x � 20y � 5 � 0 12. x2 � y2 � 8x � 2y � 8 � 0

� � 1; hyperbola (x � 4)2 � (y � 1)2 � 9; circle(x � 3)2�

6(y � 5)2�

9

(y � 2)2�

12(x � 8)2�

4

(y � 1)2�

8(x � 4)2�

16(y � 3)2�

16(x � 8)2�

64

(y � 2)2�

12(x � 2)2�

10

(y � 5)2�

3(x � 3)2�

6

( y � 1)2�3

(x � 5)2�4

ExampleExample

ExercisesExercises


Identify Conic Sections If you are given an equation of the formAx2 � Bxy � Cy2 � Dx � Ey � F � 0, with B � 0,

you can determine the type of conic section just by considering the values of A and C. Referto the following chart.

Relationship of A and C Type of Conic Section

A � 0 or C � 0, but not both. parabola

A � C circle

A and C have the same sign, but A C. ellipse

A and C have opposite signs. hyperbola

Without writing the equation in standard form, state whether thegraph of each equation is a parabola, circle, ellipse, or hyperbola.


Conic Sections

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6

ExampleExample

a. 3x2 � 3y2 � 5x � 12 � 0A � 3 and C � �3 have opposite signs, sothe graph of the equation is a hyperbola.

b. y2 � 7y � 2x � 13A � 0, so the graph of the equation isa parabola.

ExercisesExercises

Without writing the equation in standard form, state whether the graph of eachequation is a parabola, circle, ellipse, or hyperbola.

1. x2 � 17x � 5y � 8 2. 2x2 � 2y2 � 3x � 4y � 5parabola circle

3. 4x2 � 8x � 4y2 � 6y � 10 4. 8(x � x2) � 4(2y2 � y) � 100hyperbola circle

5. 6y2 � 18 � 24 � 4x2 6. y � 27x � y2

ellipse parabola7. x2 � 4( y � y2) � 2x � 1 8. 10x � x2 � 2y2 � 5y

ellipse ellipse9. x � y2 � 5y � x2 � 5 10. 11x2 � 7y2 � 77

circle hyperbola

11. 3x2 � 4y2 � 50 � y2 12. y2 � 8x � 11circle parabola

13. 9y2 � 99y � 3(3x � 3x2) 14. 6x2 � 4 � 5y2 � 3circle hyperbola

15. 111 � 11x2 � 10y2 16. 120x2 � 119y2 � 118x � 117y � 0ellipse hyperbola

17. 3x2 � 4y2 � 12 18. 150 � x2 � 120 � yhyperbola parabola

Skills PracticeConic Sections

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6


Less

on

8-6

Write each equation in standard form. State whether the graph of the equation isa parabola, circle, ellipse, or hyperbola. Then graph the equation.

1. x2 � 25y2 � 25 hyperbola 2. 9x2 � 4y2 � 36 ellipse 3. x2 � y2 � 16 � 0 circle� � 1 � � 1 x2 � y2 � 16

4. x2 � 8x � y2 � 9 circle 5. x2 � 2x � 15 � y parabola 6. 100x2 � 25y2 � 400ellipse(x � 4)2 � y2 � 25 y � (x � 1)2 � 16 � � 1


7. 9x2 � 4y2 � 36 ellipse 8. x2 � y2 � 25 circle

9. y � x2 � 2x parabola 10. y � 2x2 � 4x � 4 parabola

11. 4y2 � 25x2 � 100 hyperbola 12. 16x2 � y2 � 16 ellipse

13. 16x2 � 4y2 � 64 hyperbola 14. 5x2 � 5y2 � 25 circle

15. 25y2 � 9x2 � 225 ellipse 16. 36y2 � 4x2 � 144 hyperbola

17. y � 4x2 � 36x � 144 parabola 18. x2 � y2 � 144 � 0 circle

19. (x � 3)2 � ( y � 1)2 � 4 circle 20. 25y2 � 50y � 4x2 � 75 ellipse

21. x2 � 6y2 � 9 � 0 hyperbola 22. x � y2 � 5y � 6 parabola

23. (x � 5)2 � y2 � 10 circle 24. 25x2 � 10y2 � 250 � 0 ellipse

x

y

O

xy

O 4 8

–4

–8

–12

–16

–4–8

x

y

O 4 8

8

4

–4

–8

–4–8

y2�

x2�

x

y

Ox

y

OxO

y

4 8

4

2

–2

–4

–4–8

y2�

x2�

y2�

x2�


Write each equation in standard form. State whether the graph of the equation isa parabola, circle, ellipse, or hyperbola. Then graph the equation.

1. y2 � �3x 2. x2 � y2 � 6x � 7 3. 5x2 � 6y2 � 30x � 12y � �9parabola circle hyperbola

x � � y 2 (x � 3)2 � y2 � 16 � � 1

4. 196y2 � 1225 � 100x2 5. 3x2 � 9 � 3y2 � 6y 6. 9x2 � y2 � 54x � 6y � �81ellipse circle ellipse

� � 1 x2 � (y � 1)2 � 4 � � 1


7. 6x2 � 6y2 � 36 8. 4x2 � y2 � 16 9. 9x2 � 16y2 � 64y � 80 � 0 circle hyperbola ellipse

10. 5x2 � 5y2 � 45 � 0 11. x2 � 2x � y 12. 4y2 � 36x2 � 4x � 144 � 0circle parabola hyperbola

13. ASTRONOMY A satellite travels in an hyperbolic orbit. It reaches the vertex of its orbit

at (5, 0) and then travels along a path that gets closer and closer to the line y � x.

Write an equation that describes the path of the satellite if the center of its hyperbolicorbit is at (0, 0).

� � 1y2�

x2�

2�5

(y � 3)2�

9(x � 3)2�

1y2

�x2

�

xO

y

x

y

Ox

y

O

(y � 1)2�

5(x � 3)2�

61�

Practice (Average)

Conic Sections

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6

Reading to Learn MathematicsConic Sections

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on

8-6

Pre-Activity How can you use a flashlight to make conic sections?


The figures in the introduction show how a plane can slice a double cone toform the conic sections. Name the conic section that is formed if the planeslices the double cone in each of the following ways:

• The plane is parallel to the base of the double cone and slices throughone of the cones that form the double cone. circle

• The plane is perpendicular to the base of the double cone and slicesthrough both of the cones that form the double cone. hyperbola

Reading the Lesson

1. Name the conic section that is the graph of each of the following equations. Give thecoordinates of the vertex if the conic section is a parabola and of the center if it is acircle, an ellipse, or a hyperbola.

a. � � 1 ellipse; (3, �5)

b. x � �2( y � 1)2 � 7 parabola; (7, �1)

c. (x � 5)2 � ( y � 5)2 � 1 hyperbola; (5, �5)

d. (x � 6)2 � ( y � 2)2 � 1 circle; (�6, 2)

2. Each of the following is the equation of a conic section. For each equation, identify thevalues of A and C. Then, without writing the equation in standard form, state whetherthe graph of each equation is a parabola, circle, ellipse, or hyperbola.

a. 2x2 � y2 � 6x � 8y � 12 � 0 A � ; C � ; type of graph:

b. 2x2 � 3x � 2y � 5 � 0 A � ; C � ; type of graph:

c. 5x2 � 10x � 5y2 � 20y � 1 � 0 A � ; C � ; type of graph:

d. x2 � y2 � 4x � 2y � 5 � 0 A � ; C � ; type of graph:


3. What is an easy way to recognize that an equation represents a parabola rather thanone of the other conic sections?

If the equation has an x2 term and y term but no y2 term, then the graphis a parabola. Likewise, if the equation has a y2 term and x term but nox2 term, then the graph is a parabola.

hyperbola�11

circle55

parabola02

ellipse12

( y � 5)2�15

(x � 3)2�36


LociA locus (plural, loci) is the set of all points, and only those points, that satisfya given set of conditions. In geometry, figures often are defined as loci. Forexample, a circle is the locus of points of a plane that are a given distancefrom a given point. The definition leads naturally to an equation whose graphis the curve described.

Write an equation of the locus of points that are thesame distance from (3, 4) and y � �4.

Recognizing that the locus is a parabola with focus (3, 4) and directrix y � �4,you can find that h � 3, k � 0, and a � 4 where (h, k) is the vertex and 4 unitsis the distance from the vertex to both the focus and directrix.

Thus, an equation for the parabola is y � �116�(x � 3)2.

The problem also may be approached analytically as follows:

Let (x, y) be a point of the locus.

The distance from (3, 4) to (x, y) � the distance from y � �4 to (x, y).

�(x � 3�)2 � (�y � 4)�2� � �(x � x�)2 � (�y � (��4))2�(x � 3)2 � y2 � 8y � 16 � y2 � 8y � 16

(x � 3)2 � 16y

�116�(x � 3)2 � y

Describe each locus as a geometric figure. Then write an equationfor the locus.

1. All points that are the same distance from (0, 5) and (4, 5).

2. All points that are 4 units from the origin.

3. All points that are the same distance from (�2, �1) and x � 2.

4. The locus of points such that the sum of the distances from (�2, 0) and (2, 0) is 6.

5. The locus of points such that the absolute value of the difference of the distances from (�3, 0) and (3, 0) is 2.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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ExampleExample

Study Guide and InterventionSolving Quadratic Systems

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Less

on

8-7

Systems of Quadratic Equations Like systems of linear equations, systems ofquadratic equations can be solved by substitution and elimination. If the graphs are a conicsection and a line, the system will have 0, 1, or 2 solutions. If the graphs are two conicsections, the system will have 0, 1, 2, 3, or 4 solutions.

Solve the system of equations. y � x2 � 2x � 15x � y � �3

Rewrite the second equation as y � �x � 3 and substitute into the first equation.

�x � 3 � x2 � 2x � 150 � x2 � x � 12 Add x � 3 to each side.

0 � (x � 4)(x � 3) Factor.

Use the Zero Product property to getx � 4 or x � �3.

Substitute these values for x in x � y � �3:

4 � y � �3 or �3 � y � �3y � �7 y � 0

The solutions are (4, �7) and (�3, 0).

Find the exact solution(s) of each system of equations.

1. y� x2 � 5 2. x2 � ( y � 5)2 � 25y� x � 3 y � �x2

(2, �1), (�1, �4) (0, 0)

3. x2 � ( y � 5)2 � 25 4. x2 � y2 � 9y � x2 x2 � y � 3

(0, 0), (3, 9), (�3, 9) (0, 3), (�5�, �2), (��5�, �2)

5. x2 � y2 � 1 6. y � x � 3x2 � y2 � 16 x � y2 � 4

� , �, � , � �, � , �, �� , �, �� , � � � , �1 � �29��

27 � �29��

2�30��34��30��34��

1 � �29��2

7 � �29��2

�30��34��30��34��

ExampleExample

ExercisesExercises


Systems of Quadratic Inequalities Systems of quadratic inequalities can be solvedby graphing.

Solve the system of inequalities by graphing.x2 � y2 25

�x � �2� y2

The graph of x2 � y2 25 consists of all points on or inside the circle with center (0, 0) and radius 5.The graph of

�x � �2� y2 � consists of all points on or outside the

circle with center � , 0� and radius . The solution of the

system is the set of points in both regions.

Solve the system of inequalities by graphing.x2 � y2 25

� � 1

The graph of x2 � y2 25 consists of all points on or inside the circle with center (0, 0) and radius 5.The graph of

� � 1 are the points “inside” but not on the branches of

the hyperbola shown. The solution of the system is the set ofpoints in both regions.

Solve each system of inequalities below by graphing.

1. � 1 2. x2 � y2 169 3. y � (x � 2)2

y � x � 2x2 � 9y2 � 225 (x � 1)2 � ( y � 1)2 16

x

y

Ox

y

O 6 12

12

6

–6

–12

–6–12x

y

O

1�2

y2�4

x2�16

x2�9

y2�4

x2�9

y2�4

x

y

O

5�2

5�2

25�4

5�2

25�4

5�2

x

y

O


Solving Quadratic Systems

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeSolving Quadratic Systems

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7


Less

on

8-7


1. y � x � 2 (0, �2), (1, �1) 2. y � x � 3 (�1, 2), 3. y � 3x (0, 0)y � x2 � 2 y � 2x2 (1.5, 4.5) x � y2

4. y � x (�2�, �2�), 5. x � �5 (�5, 0) 6. y � 7 no solutionx2 � y2 � 4 (��2�, ��2�) x2 � y2 � 25 x2 � y2 � 9

7. y � �2x � 2 (2, �2), 8. x � y � 1 � 0 (1, 2) 9. y � 2 � x (0, 2), (3,�1)y2 � 2x � , 1� y2 � 4x y � x2 � 4x � 2

10. y � x � 1 no solution 11. y � 3x2 (0, 0) 12. y � x2 � 1 (�1, 2), y � x2 y � �3x2 y � �x2 � 3 (1, 2)

13. y � 4x (�1, �4), (1, 4) 14. y � �1 (0, �1) 15. 4x2 � 9y2 � 36 (�3, 0), 4x2 � y2 � 20 4x2 � y2 � 1 x2 � 9y2 � 9 (3, 0)

16. 3( y � 2)2 � 4(x � 3)2 � 12 17. x2 � 4y2 � 4 (�2, 0), 18. y2 � 4x2 � 4 no y � �2x � 2 (0, 2), (3, �4) x2 � y2 � 4 (2, 0) y � 2x solution

Solve each system of inequalities by graphing.

19. y 3x � 2 20. y x 21. 4y2 � 9x2 � 144x2 � y2 � 16 y � �2x2 � 4 x2 � 8y2 � 16

x

y

O 4 8

8

4

–4

–8

–4–8x

y

Ox

y

O

1�



1. (x � 2)2 � y2 � 5 2. x � 2( y � 1)2 � 6 3. y2 � 3x2 � 6 4. x2 � 2y2 � 1x � y � 1 x � y � 3 y � 2x � 1 y � �x � 1

(0, �1), (3, 2) (2, 1), (6.5, �3.5) (�1, �3), (5, 9) (1, 0), � , �5. 4y2 � 9x2 � 36 6. y � x2 � 3 7. x2 � y2 � 25 8. y2 � 10 � 6x2

4x2 � 9y2 � 36 x2 � y2 � 9 4y � 3x 4y2 � 40 � 2x2

no solution (0, �3), (��5�, 2) (4, 3), (�4, �3) (0, ��10� )

9. x2 � y2 � 25 10. 4x2 � 9y2 � 36 11. x � �( y � 3)2 � 2 12. � � 1x � 3y � 5 2x2 � 9y2 � 18 x � ( y � 3)2 � 3

x2 � y2 � 9

(�5, 0), (4, 3) (�3, 0) no solution (�3, 0)

13. 25x2 � 4y2 � 100 14. x2 � y2 � 4 15. x2 � y2 � 3

x � � � � 1 y2 � x2 � 3

no solution (�2, 0) no solution

16. � � 1 17. x � 2y � 3 18. x2 � y2 � 64

3x2 � y2 � 9x2 � y2 � 9 x2 � y2 � 8

(�2, ��3�) (3, 0), �� , � (�6, �2�7�)

Solve each system of inequalities by graphing.

19. y � x2 20. x2 � y2 � 36 21. � 1y � �x � 2 x2 � y2 � 16

(x � 1)2 � ( y � 2)2 4

22. GEOMETRY The top of an iron gate is shaped like half an ellipse with two congruent segments from the center of theellipse to the ellipse as shown. Assume that the center ofthe ellipse is at (0, 0). If the ellipse can be modeled by theequation x2 � 4y2 � 4 for y � 0 and the two congruent

segments can be modeled by y � x and y � � x,

what are the coordinates of points A and B?

�3��2

�3��2

BA

(0, 0)

x

y

O

x

y

O 4 8

8

4

–4

–8

–4–8

x

y

O

(x � 2)2�4

( y � 3)2�16

12�

9�

y2�7

x2�7

y2�8

x2�4

5�2

y2�16

x2�9

2�

1�

Practice (Average)

Solving Quadratic Systems

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7

��1, � and �1, ��3��3��

Reading to Learn MathematicsSolving Quadratic Systems

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7


Less

on

8-7

Pre-Activity How do systems of equations apply to video games?


The figure in your textbook shows that the spaceship hits the circular forcefield in two points. Is it possible for the spaceship to hit the force field ineither fewer or more than two points? State all possibilities and explainhow these could happen. Sample answer: The spaceship could hitthe force field in zero points if the spaceship missed the forcefield all together. The spaceship could also hit the force fieldin one point if the spaceship just touched the edge of theforce field.

Reading the Lesson

1. Draw a sketch to illustrate each of the following possibilities.

a. a parabola and a line b. an ellipse and a circle c. a hyperbola and athat intersect in that intersect in line that intersect in2 points 4 points 1 point

2. Consider the following system of equations.

x2 � 3 � y2

2x2 � 3y2 � 11

a. What kind of conic section is the graph of the first equation? hyperbola

b. What kind of conic section is the graph of the second equation? ellipse

c. Based on your answers to parts a and b, what are the possible numbers of solutionsthat this system could have? 0, 1, 2, 3, or 4


3. Suppose that the graph of a quadratic inequality is a region whose boundary is a circle.How can you remember whether to shade the interior or the exterior of the circle?Sample answer: The solutions of an inequality of the form x2 � y2 � r2

are all points that are less than r units from the origin, so the graph isthe interior of the circle. The solutions of an inequality of the form x2 � y2 � r2 are the points that are more than r units from the origin, sothe graph is the exterior of the circle.


Graphing Quadratic Equations with xy-TermsYou can use a graphing calculator to examine graphs of quadratic equations that contain xy-terms.

Use a graphing calculator to display the graph of x2 � xy � y2 � 4.

Solve the equation for y in terms of x by using the quadratic formula.

y2 � xy � (x2 � 4) � 0

To use the formula, let a � 1, b � x, and c � (x2 � 4).

y �

y �

To graph the equation on the graphing calculator, enter the two equations:

y � and y �

Use a graphing calculator to graph each equation. State the type of curve each graph represents.

1. y2 � xy � 8 2. x2 � y2 � 2xy � x � 0

3. x2 � xy � y2 � 15 4. x2 � xy � y2 � �9

5. 2x2 � 2xy � y2 � 4x � 20 6. x2 � xy � 2y2 � 2x � 5y � 3 � 0

�x � �16 ��3x2��2

�x � �16 ��3x2��2

�x � �16 ��3x2��2

�x � �x2 � 4�(1)(x2�� 4)��2

x

y

O 1–1–2 2

2

1

–1

–2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7

ExampleExample

Chapter 8 Test, Form 1

NAME DATE PERIOD

SCORE


Ass

essm

ent

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

1. What is the midpoint of the line segment with endpoints at (12, 7) and (18, 19)?A. (30, 26) B. (15, 13) C. (�6, �12) D. (3, 6) 1.

2. Choose the midpoint of the line segment with endpoints at (5, 9) and (11, 15).A. (8, 12) B. (16, 24) C. (6, 6) D. (�6, �6) 2.

3. Find the distance between A(12, 8) and B(4, 2).A. 14 units B. 100 units C. 10 units D. �10 units 3.

4. What is the distance between C(4, 3) and D(7, 7)?A. �5 units B. 7 units C. 25 units D. 5 units 4.

5. Write the equation of the parabola y � x2 � 10x � 16 in standard form.A. y � (x � 5)2 � 9 B. y � (x � 5)2 � 41C. y � (x � 5)2 � 16 D. y � (x � 8)(x � 2) 5.

6. Write an equation for the parabola with vertex (1, 0) if the length of the

latus rectum is �12� and the parabola opens down.

A. y � ��12�(x � 1)2 B. y � �2(x � 1)2 C. x � �2(y � 1)2 D. x � ��

12�(y � 1)2 6.

7. Which is the equation of a parabola that opens downward and has axis of symmetry x � �1?A. y � (x � 1)2 � 2 B. y � (x � 1)2 � 2C. y � �(x � 1)2 � 2 D. y � �(x � 1)2 � 2 7.

8. Find the center and radius of the circle with equation (x � 2)2 � y2 � 9.A. (�2, 0); 9 B. (0, 2); 9 C. (2, 0); 3 D. (0, �2); 3 8.

9. Write an equation for the circle with center (2, �3) that is tangent to the y-axis.A. (x � 2)2 � (y � 3)2 � 9 B. (x � 2)2 � (y � 3)2 � 9C. (x � 2)2 � (y � 3)2 � 4 D. (x � 2)2 � (y � 3)2 � 4 9.

10. Which is the equation of a circle with center (2, 1) that passes through (2, 4)?A. (x � 2)2 � (y � 1)2 � 9 B. (x � 2)2 � (y � 1)2 � 3C. (x � 2)2 � (y � 1)2 � 9 D. (x � 2)2 � (y � 1)2 � 3 10.

11. Which is the equation of an ellipse with foci at (0, 3) and (0, �3) that has the endpoints of its major axis at (0, 4) and (0, �4)?

A. �1y62� � �

x92� � 1 B. x2 � y2 � 16 C. �1

x62� � �

y72� � 1 D. �1

y62� � �

x72� � 1 11.

88


Chapter 8 Test, Form 1 (continued)

12. Which equation is graphed at the right?

A. �1x62� � �

y42� � 1 B. �1

y62� � �

x42� � 1

C. �1x62� � �

y42� � 1 D. �1

y62� � �

x42� � 1 12.

13. Which is the equation of a hyperbola with vertices (0, 2) and (0, �2) and foci (0, 3) and (0, �3)?

A. �y52� � �

x42� � 1 B. �

y42� � �

x52� � 1 C. �

x42� � �

y52� � 1 D. �

x52� � �

y42� � 1 13.

14. Which equation is graphed at the right?

A. �1x62� � �

y42� � 1 B. �

x42� � �1

y62� � 1

C. �1y62� � �

x42� � 1 D. �

y42� � �1

x62� � 1 14.

15. What is the standard form of the equation 5x2 � 5y2 � 20 � 0?A. 5x2 � 5y2 � 20 B. y2 � �x2 � 4 C. x2 � y2 � 4 � 0 D. x2 � y2 � 4 15.

16. What is the graph of x2 � 4y2 � 2y � 8?A. parabola B. circle C. ellipse D. hyperbola 16.

17. Which equation has a hyperbola as its graph?A. 4x2 � 4y2 � 16 B. 4x2 � 4y � 16 C. 4x2 � 4y2 � 16 D. x2 � 4y2 � 16 17.

18. Find the exact solution(s) of the system of equations x2 � y2 � 16 and x � y � 4.A. (�4, 0) and (0, 4) B. (4, 0) and (�4, 0)C. (0, 4) and (0, �4) D. (4, 0) and (0, �4) 18.

19. Solve the system of equations by graphing y � x2 and y � 2x.A. (0, 0) and (4, �2) B. (0, 0) and (�2, 4)C. (0, 0) and (2, 4) D. (0, �1) and (2, 2) 19.

20. Which system of inequalities is graphed at the right?A. x2 � y2 � 9 B. x2 � y2 � 9

y � x � 1 y � x � 1

C. x2 � y2 � 9 D. x2 � y2 � 9y � x � 1 y � x � 1 20.

Bonus For the equation 4x2 � ky2 � 8x � 17y � 3, find a value of k so that the graph of the equation is a. a circle b. an ellipse c. a hyperbola d. a parabola B:

NAME DATE PERIOD

88

y

xO

y

xO

y

xO

Chapter 8 Test, Form 2A

NAME DATE PERIOD

SCORE


Ass

essm

ent


For Questions 1 and 2, refer to the figure at the right showing six city locations. The origin is at the lower left corner of the grid.

1. What is the location of the point halfway between the wharf and library?

A. (7, 2) B. ��125�, �

52�� C. �7, �

52�� D. ��

125�, 2� 1.

2. What is the distance between the library and zoo?

A. 11 units B. �61� units C. 61 units D. �621� units 2.

3. Write the equation of the parabola y � 2x2 � 8x � 1 in standard form.A. y � 2(x � 2)2 � 9 B. y � (x � 4)2 � 15C. y � 2(x � 2)2 � 7 D. y � 2(x � 4)2 � 15 3.

4. Write an equation for the parabola with focus (4, 0) and vertex (2, 0).

A. x � �18�y2 � 2 B. x � ��

18�y2 � 2 C. y � �

18�x2 � 2 D. y � ��

18�x2 � 2 4.

5. Which equation is graphed at the right?A. y � 4x2 � 16x � 16 B. x � 4y2 � 16y � 16

C. y � �14�x2 � x � 1 D. x � �

14�y2 � y � 1 5.

6. Write an equation for a circle if the endpoints of a diameter are at (�7, 1) and (5, 1).A. x2 � (y � 1)2 � 6 B. (x � 1)2 � (y � 1)2 � 36C. (x � 1)2 � y2 � 6 D. (x � 1)2 � (y � 1)2 � 36 6.

7. Which is the equation of a circle with center (2, 0) and radius 2 units?A. x2 � y2 � 4x � 0 B. x2 � y2 � 4x � 0C. x2 � y2 � 4y � 0 D. x2 � y2 � 4y � 0 7.

8. Write an equation for an ellipse if the endpoints of the major axis are at (�1, 5) and (�1, �3) and the endpoints of the minor axis are at (�4, 1) and (2, 1).

A. �(y �

161)2� � �

(x �9

1)2� � 1 B. �

(x �16

1)2� � �

(y �9

1)2� � 1

C. �(x �

161)2� � �

(y �9

1)2� � 1 D. �

(y �16

1)2� � �

(x �9

1)2� � 1 8.

9. Which is the equation of an ellipse with center (1, �2) and a vertical major axis?

A. �(y �

92)2� � �

(x �4

1)2� � 1 B. �

(x �9

1)2� � �

(y �4

2)2� � 1

C. �(y �

92)2� � �

(x �4

1)2� � 1 D. �

(x �9

1)2� � �

(y �4

2)2� � 1 9.

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Chapter 8 Test, Form 2A (continued)

10. Find the center and radius of the circle with equation x2 � (y � 4)2 � 9.A. (0, 4); 9 B. (4, 0); 3 C. (�4, 0); 9 D. (0, 4); 3 10.

11. Write an equation for the hyperbola with vertices (�10, 1) and (6, 1) and foci (�12, 1) and (8, 1).

A. �(x �

642)2� � �

(y �36

1)2� � 1 B. �

(x �36

2)2� � �

(y �64

1)2� � 1

C. �(x �

642)2� � �

(y �36

1)2� � 1 D. �

(x �36

2)2� � �

(y �64

1)2� � 1 11.

12. Which equation is graphed at the right?A. x2 � 9y2 � 9 B. 9y2 � x2 � 9C. 9x2 � y2 � 9 D. y2 � 9x2 � 9 12.

13. Write the equation x2 � 2x � y2 � 4y � 11 in standard form.A. (x � 1)2 � (y � 2)2 � 16 B. (x � 1)2 � (y � 2)2 � 16

C. �(x �

11)2� � �

(y �4

2)2� � 1 D. �

(x �4

1)2� � �

(y �4

1)2� � 1 13.

14. Write the equation 4x2 � 24x � y � 34 � 0 in standard form.A. y � 4(x � 3)2 � 2 B. x � 4y2 � 2C. y � 4(x � 3)2 � 2 D. x � 4(y � 3)2 � 2 14.

15. What is the graph of 4x2 � y2 � 8y � 32?A. parabola B. circle C. ellipse D. hyperbola 15.

16. The graph of which equation is a circle?A. 5x2 � 10x � 9 � 5y2 B. 5x2 � 10x � 9 � 5y2

C. 5x2 � 5x � y2 � 9 D. 5x2 � 10x � 5y � 9 16.

17. Solve the system of equations by graphing x2 � y2 � 16 and y � �x � 4.A. (4, 0), (0, �4) B. (0, �4), (�4, 0) C. (�4, 0), (0, �4) D. (0, 4), (4, 0) 17.


x � y � �3 x � y � �3C. x2 � y2 � 16 D. x2 � y2 � 16

x � y � �3 x � y � �3 18.


19. x2 � y2 � 25 and 9y � 4x2

A. (4, 3), (�4, 3) B. (3, 4), (3, �4) C. (4, 3), (4, �3) D. (3, 4), (�3, 4) 19.

20. y � x2 � 1 and y � 2xA. (1, 2), (�1, 2) B. (�1, 2) C. (1, 2) D. (�1, 2), (0, 2) 20.

Bonus Solve the system of equations (x � 2)2 � y2 � 1 and B:(x � 2)2 � y2 � 1.

NAME DATE PERIOD

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Chapter 8 Test, Form 2B

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For Questions 1 and 2, refer to the figure at the right showing six city locations. The origin is at the lower left corner of the grid.

1. What is the location of the point halfway between the hospital and arena?

A. (2, 6) B. �6, �52�� C. ��

52�, 6� D. ��

12�, 3� 1.

2. What is the distance between the museum and sailing club?

A. �53� units B. 53 units C. �523� units D. 9 units 2.

3. Write the equation of the parabola y � 4x2 � 8x � 1 in standard form.A. y � (x � 4)2 � 15 B. y � 4(x � 1)2 � 5C. y � 4(x � 1)2 � 3 D. y � 4(x � 4)2 � 15 3.

4. Write an equation for the parabola with focus (1, 3) and vertex (0, 3).

A. y � 4(x � 3)2 B. x � ��14�(y � 3)2 C. y � �4�x � �

34��2

D. x � �14�(y � 3)2 4.

5. Which equation is graphed at the right?A. y � 2x2 � 8x � 7 B. x � �2y2 � 8y � 7C. y � �2x2 � 8x � 7 D. y � �2x2 � 8x � 7 5.

6. Write an equation for a circle if the endpoints of a diameter are at (1, 1) and (1, �9).A. (x � 1)2 � (y � 4)2 � 5 B. (x � 1)2 � (y � 4)2 � 25C. (x � 1)2 � (y � 4)2 � 5 D. (x � 1)2 � (y � 4)2 � 25 6.

7. Which is the equation of a circle with center (0, 1) and radius 2 units?A. x2 � y2 � 2y � 3 B. x2 � y2 � 2y � 1C. x2 � y2 � 2y � 4 D. x2 � y2 � 2y � 3 7.

8. Write an equation for an ellipse if the endpoints of the major axis are at (1, 6) and (1, �6) and the endpoints of the minor axis are at (5, 0) and (�3, 0).

A. �(x �

361)2� � �1

y62� � 1 B. �

(x �36

1)2� � �1

y62� � 1

C. �3y62� � �

(x �16

1)2� � 1 D. �3

y62� � �1

x62� � 1 8.

9. Which is the equation of an ellipse with center (�4, 2) and a horizontal major axis?

A. �(x �

164)2� � �

(y �4

2)2� � 1 B. �

(x �16

4)2� � �

(y �4

2)2� � 1

C. �(y �

162)2� � �

(x �4

4)2� � 1 D. �

(y �16

2)2� � �

(x �4

4)2� � 1 9.

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ARENA SAILING CLUB

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Chapter 8 Test, Form 2B (continued)

10. Find the center and radius of the circle with equation (x � 1)2 � y2 � 16.A. (1, 0); 4 B. (�1, 0); 16 C. (0, 1); 4 D. (0, �1); 16 10.

11. Write an equation for the hyperbola with vertices (0, �1) and (0, 3) and foci (0, �3) and (0, 5).

A. �(y �

41)2� � �1

x22� � 1 B. �

(x �12

1)2� � �

y42� � 1

C. �(y �

41)2� � �1

x22� � 1 D. �

x42� � �

(y �12

1)2� � 1 11.

12. Which equation is graphed at the right?A. 9x2 � 4y2 � 36 B. 4x2 � 9y2 � 36C. 9y2 � 4x2 � 36 D. 4y2 � 9x2 � 36 12.

13. Write the equation 4x2 � 8x � y2 � 4y � 4 � 0 in standard form.

A. �(x �

11)2� � �

(y �4

2)2� � 1 B. (x � 1)2 � (y � 2)2 � 4

C. �(x �

11)2� � �

(y �4

2)2� � 1 D. �

(x �4

1)2� � �

(y �1

2)2� � 1 13.

14. Write the equation 2y2 � 4y � x � 12 � 0 in standard form.A. y � 2(x � 1)2 � 6 B. x � 2(y � 1)2 � 10C. y � (x � 1)2 � 10 D. x � 2(y � 1)2 � 6 14.

15. The graph of which equation is a circle?A. 6x2 � 12x � 6y2 � 1 B. 6x2 � 12x � 6y2 � 1C. 6x2 � 6y2 � 12x � 1 D. 6x2 � 6y � 12x � 1 15.

16. What is the graph of x2 � 25y2 � 50?A. parabola B. circle C. ellipse D. hyperbola 16.

17. Solve the system of equations by graphing y � x2 � 2 and y � 2x � 2.A. (�2, 0), (2, 2) B. (2, 0), (0, �2) C. (�2, 0), (�2, 2) D. (0, �2), (2, 2) 17.


y2 � x � 0 y2 � x � 0C. x2 � y2 � 9 D. x2 � y2 � 9

y2 � x � 0 y2 � x � 0 18.


19. x2 � 4y2 � 16 and x � 2y � �4A. (0, �2), (0, 2) B. (0, �2), (4, 0) C. (0, �2), (�4, 0) D. (0, 2), (4, 0) 19.

20. x2 � y2 � 36 and y � x � 6A. (0, �6), (6, 0) B. (0, 6), (6, 0) C. (6, 0), (�6, 0) D. (�6, 0), (0, 6) 20.

Bonus Solve the system of equations x2 � (y � 3)2 � 4 and B:x2 � (y � 3)2 � 4.

NAME DATE PERIOD

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Chapter 8 Test, Form 2C


1. Find the midpoint of the line segment with endpoints at 1.(�2, 3) and (14, 6).

2. Find the distance between A(4, �2) and B(10, �7). 2.

3. Write an equation for the parabola with focus (4, 4) and 3.directrix x � �2.

4. Write the y � 3x2 � 6x � 2 in standard form. 4.

5. Identify the coordinates of the vertex and focus, the 5.equations of the axis of symmetry and directrix, and the direction of opening of the parabola with equation y2 � 8y � 18 � x.

6. Write an equation for the circle with center (�4, 2) that is 6.tangent to the y-axis.

Graph each equation.

7. x2 � y2 � 4x � 6y � 3 � 0 7.

8. 9x2 � 4y2 � 36 8.

For Questions 9 and 10, write an equation for the ellipse that satisfies each set of conditions.

9. endpoints of major axis at (9, 3) and (�11, 3), 9.endpoints of minor axis at (�1, 8) and (�1, �2)

10. major axis 12 units long and parallel to the y-axis, 10.minor axis 8 units long, center at (�2, 5)

11. Find the exact solution(s) of the system of equations. 11.x2 � y � 44x2 � y2 � 12

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Chapter 8 Test, Form 2C (continued)

For Questions 12 and 13, write an equation for the hyperbola that satisfies each set of conditions.

12. vertices (9, 0) and (�9, 0), conjugate axis of length 10 units 12.

13. vertices (�1, 4) and (�1, �8), foci ( � 1, � 2 �39�) 13.

14. Find the coordinates of the vertices and foci and the 14.equations of the asymptotes for the hyperbola (x � 3)2 � (y � 1)2 � 4.

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola.

15. x2 � y2 � 2x � 2y � 23 15.

16. 4x2 � 9y2 � 24x � 18y � 9 � 0 16.

For Questions 17 and 18, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. State the values used to identify each conic section without writing each equation in standard form.

17. 3(x � 5)2 � 3y � 15 � 0 17.

18. 4x2 � 8x � 4(y2 � 2y) � 7 18.

19. Graph the system of equations. Use the graph to solve the 19.system.y � x2 � 4xy � x � 4

20. Solve the system of inequalities by graphing. 20.x2 � y2 � 16y � �2x2 � 1

Bonus Write an equation for the circle with the same center as B:

the graph of �(x �4

3)2� � �

(y �16

1)2� � 1 and the same radius

as the graph of x2 � y2 � 4x � 10y � 9.

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Chapter 8 Test, Form 2D


1. Find the midpoint of the line segment with endpoints at 1.(�4, 5) and (7, �3).

2. Find the distance between A(�7, 3) and B(4, �6). 2.

3. Write an equation for the parabola with focus (�1, 1) and 3.directrix y � �7.

4. Write the equation of the parabola x � 5y2 � 10y � 2 in 4.standard form.

5. Identify the coordinates of the vertex and focus, the 5.equations of the axis of symmetry and directrix, and the direction of opening of the parabola with equation y � 2x2 � 4x � 5.

6. Write an equation for the circle with center ��12�, �2� that is 6.

tangent to the x-axis.

Graph each equation.

7. x2 � y2 � 2x � 4y � 4 7.

8. 9x2 � 16y2 � 144 8.

For Questions 9 and 10, write an equation for the ellipse thatsatisfies each set of conditions.

9. endpoints of major axis at (2, �5) and (2, 9), 9.endpoints of minor axis at (�4, 2) and (6, 2)

10. major axis 16 units long and parallel to the x-axis, 10.minor axis 6 units long, center at (1, �4)

11. Find the exact solution(s) of the system of equations. 11.x2 � 2y � 113x2 � y2 � 24

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NAME DATE PERIOD

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Chapter 8 Test, Form 2D (continued)

For Questions 12 and 13, write an equation for the hyperbola that satisfies each set of conditions.

12. vertices (0, 12) and (0, �12), conjugate axis of length 8 units 12.

13. vertices (�10, 1) and (4, 1), foci (�3 �70�, 1) 13.

14. Find the coordinates of the vertices and foci and the 14.equations of the asymptotes for the hyperbola (x � 1)2 � (y � 3)2 � 4.


15. 4x2 � 16x � y � 21 � 0 15.

16. y2 � 6y � 4x2 � 8x � 95 16.


17. 2x2 � 10x � 8y � 2y2 � 5 17.

18. 3(y � 2)2 � 8 � 9x � 10x2 18.

19. Graph the system of equations. Use the graph to solve the 19.system.y2 � 9 � x2

y � ��34�x � 4

20. Solve the system of inequalities by graphing. 20.x2 � 4y2 � 1x � 4(y � 2)2

Bonus Write an equation for the circle with the same center as B:

the graph of �(x �16

5)2� � �

(y �9

2)2� � 1 and the same radius

as the graph of x2 � y2 � 2y � 16x � 1.

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88

Chapter 8 Test, Form 3


1. Find the midpoint of the line segment with endpoints at 1.(�12, 3.5) and (5.1, 4.8).

2. Find the distance between A(4�5�, �2) and B(�5�, 9). 2.

3. Write the equation x � �y2 � 6y � 7 in standard form. 3.

4. Write an equation for the parabola with vertex (�5, 1) and 4.

directrix x � ��72�.

5. The path traveled by Pati’s remote-controlled model 5.airplane is shaped like a parabola. It took off from the ground and landed on the ground 160 feet away from where it took off. If the airplane reached a maximum height of 40 feet, write an equation for the parabola that models the path of the plane. Assume that the point of take-off is the origin.

6. Identify the coordinates of the vertex and focus, the 6.equations of the axis of symmetry and directrix, and the direction of opening of the parabola with equation x � �y2 � 2y � 9.

7. Write an equation for a circle if its center is in the first 7.quadrant, and it is tangent to x � �2, x � 8 and the x-axis.

8. Graph x2 � y2 � 4x � 2y � 3 � 0. 8.

9. Graph 5x2 � 2y2 � 4y � 22. 9.

For Questions 10 and 11, write an equation for the ellipse that satisfies each set of conditions.

10. major axis 14 units long and parallel to the x-axis, minor 10.

axis 10 units long, center at �5, ��12��

11. endpoints of major axis at (3, �8) and (3, 4), foci at 11.(3, �2 � 2�5�) and (3, �2 � 2�5�)

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Chapter 8 Test, Form 3 (continued)

12. Find the coordinates of the center and foci and the lengths 12.of the major and minor axes for the ellipse with equation 6x2 � 5y2 � 24x � 30y � �39.

13. Write an equation for the hyperbola with vertices (4, �5) 13.and (4, 1) and foci (4, 3) and (4, �7).


14. 2x2 � 3y2 � 15 � 4(x � 2y) 14.

15. �18�x � y2 � �(y � 12) 15.


16. 3x2 � 9x � y2 � 2(24y � y2 � 27) 16.

17. 34x2 � 40y2 � 18x � 25y � 17(2x2 � 1) 17.

18. Find the exact solution(s) of the system of equations. 18.

�2x52� � �1

y62� � 1

x � y

19. Solve the system of equations by graphing. 19.x2 � y2 � 4x � 6y � 4 � 0x2 � 4x � 3y � 4 � 0

20. Solve the system of inequalities by graphing. 20.x2 � y2 � 4x � �4 � 102 � y � (x � 1.75)2

Bonus The parabolic curve of a certain camera lens can be B:represented by the equation y � 10x2 � 50x � 63.2.What are the coordinates of the focus?

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88

Chapter 8 Open-Ended Assessment


Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solution in more thanone way or investigate beyond the requirements of the problem.

1. Harry was asked to determine whether the graph of the equationx2 � y2 � 8x � 6y � 30 � 0 was a parabola, circle, ellipse, or hyperbola.At first glance, he identified the equation as that of a circle.a. What made Harry think he was looking at the equation of a circle?b. When Harry attempted to find the center and radius of the circle,

he ran into a problem. What was the problem?c. Change the equation so that Harry’s problem no longer exists,

then find the center and radius of the circle represented by yourequation.

2. Do the graphs of any of the conic sections you have studied in this chapter represent relations that are functions? Explain your reasoning.

3. What do the graphs of the parabolas y � (x � 2)2 � 1 andx � (y � 1)2 � 2 have in common? How are the graphs different?

4. The graphs of the equations (x � 4)2 � (y � 3)2 � 4 and y � (x � 4)2 � 3 are shown. For parts a and b,replace each of the �s with one of the inequality symbols (�, �, � , �) so that the solution of the system is the region indicated. Explain your choices.a. (x � 4)2 � (y � 3)2 � 4

y � (x � 4)2 � 3The solution of the system is region 2.

b. (x � 4)2 � (y � 3)2 � 4y � (x � 4)2 � 3The solution of the system is region 3.

c. What region is represented by the system (x � 4)2 � (y � 3)2 � 4 and y � (x � 4)2 � 3? Explain.

5. The graph of the equation �(x �4

1)2� � �

(y �9

2)2� � 1

is shown. Find values of k for which the given system of equations has the given number of solutions. Explain the reasoning for your choices.

�(x �

41)2� � �

(y �9

2)2� � 1

y � k

a. For k � ____ and k � ____, the system has two solutions.b. For k � ____ and k � ____, the system has one solution.c. For k � ____ and k � ____, the system has no solutions.

NAME DATE PERIOD

SCORE 88

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1

4

2

3

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Chapter 8 Vocabulary Test/Review

Choose from the terms above to complete each sentence.

1. A is the set of all points in a plane that are the same distance from a given point and a given line. The given point is called the

and the given line is called the .

2. The set of all points in a plane the sum of whose distances from two fixed points is

constant is a(n) . The two fixed points are called the

.

3. The set of all points in a plane such that the absolute value of the difference of their

distances from the two given points is constant is a(n) .

4. The points at which an ellipse intersects its axes of symmetry determine two segments

on the ellipse. The shorter of these segments is called the

and the longer one is called the .

5. The segment that connects the two vertices of a hyperbola is called the

.

6. A line that intersects a circle in exactly one point is to the circle.

7. The line segment through the focus of a parabola and perpendicular

to the line of symmetry is called the .

8. A line that the branches of a hyperbola approach but do not intersect

is called a(n) .

9. The segment of length 2b units that is perpendicular to the transverse axis

of a hyperbola at its center is called the .

10. The formula that can be used to find the length of a line segment if you know

the coordinates of its endpoints is called the .

In your own words—Define each term.

11. circle

12. vertex of a hyperbola

asymptotecenter of a circlecenter of an ellipsecenter of a hyperbolacircleconic section

conjugate axisdirectrixDistance Formulaellipsefoci of an ellipsefoci of a hyperbola

focus of a parabolahyperbolalatus rectummajor axisMidpoint Formulaminor axis

parabolatangenttransverse axisvertex of a hyperbola

NAME DATE PERIOD

SCORE 88

Chapter 8 Quiz (Lessons 8–1 and 8–2)

88


1. Find the midpoint of the line segment with endpoints at 1.(�7, �3) and (5, 10).

2. Standardized Test Practice Which point is farthest from (2, �1)?A. (3, 3) B. (�2, �1) C. (4, 0) D. (�1, 0) 2.

3. Write an equation for the parabola with focus (1, 4) and 3.directrix y � �2.

4. Identify the coordinates of the vertex and focus, the 4.equations of the axis of symmetry and directrix, and the direction of opening of the parabola with equation y � �2x2 � 16x � 27.

5. Graph the parabola x � 6y2 � 24y � 25 and find the length 5.of the latus rectum. y

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NAME DATE PERIOD

SCORE


For Questions 1 and 2, write an equation for the circle that satisfies each set of conditions.

1. center (�7, 2), radius 9 units 1.

2. endpoints of a diameter at (�1, 1) and (7, 1) 2.

3. Find the center and radius of the circle with equation 3.x2 � y2 � 2x � 2y � 7. Then graph the circle.

4. Write an equation for an ellipse if the endpoints of the major axis are at (5, 1) and (�5, 1) and the endpoints of the minor axis are at (0, 5) and (0, �3). 4.

5. Find the coordinates of the center and foci and the lengths 5.of the major and minor axes for the ellipse with equation

�(x �

163)2� � �

y42� � 1.

y

xO

NAME DATE PERIOD

SCORE 88

Ass

essm

ent


1. Write an equation for the hyperbola whose graph is shown. 1.

2. Write an equation for the hyperbola 2.with vertices (2, �5) and (2, 3),foci (2, �6) and (2, 4)

3. Find the coordinates of the vertices 3.and foci and the equations of the

asymptotes for the hyperbola �x42� � �3

y62� � 1. Then graph the

hyperbola.

4. Write 2x2 � 12x � y � 5 in standard form. Then state whether the graph of the equation is a parabola, circle,ellipse, or hyperbola. 4.

5. State whether the graph of x2 � 2x � 4y2 � 24y � 37 � 0 is 5.a parabola, circle, ellipse, or hyperbola. State the values used to identify the conic section without writing the equation in standard form.

y

xO

Chapter 8 Quiz (Lesson 8–7)

Graph each system of equations. Use the graph to solve the system.

1. y � (x � 2)2 � 3 1.

�(x �

12)2� � �

(y �4

2)2� � 1

For Questions 3 and 4, find the exact solution(s) of each system of equations.

2. x2 � y2 � � 8 2.y � 2x � 1

3. 2x2 � 5y2 � 22 3.y2 � 3x2 � 1

4. Solve the system of inequalities by graphing. 4.

�x92� � �

y42� � 1

�2x52� � �

y92� � 1

y

xO

2

2

y

xO

NAME DATE PERIOD

SCORE


88

NAME DATE PERIOD

SCORE

88

y

xO

(1, 2)

Chapter 8 Mid-Chapter Test (Lessons 8–1 through 8–4)



1. What is the midpoint of the line segment with endpoints at (�6, 3) and (�10, 7)?A. (�8, 5) B. (�16, 10) C. (2, �2) D. (4, �4) 1.

2. Find the distance between A(�3, 1) and B(5, �5).A. 100 B. 32 C. 4 D. 10 2.

3. Write an equation for the parabola with vertex (1, 2) and directrix x � �34�.

A. y � (x � 2)2 � 1 B. x � (y � 2)2 � 1C. y � (x � 2)2 � 1 D. x � (y � 1)2 � 2 3.

4. Which equation is graphed?A. y � 4x2 � 8x � 4 B. x � 4y2 � 1

C. y � �14�x2 � �

12�x � �

14� D. x � �

14�y2 � �

12�y � �

14� 4.

5. Write an equation of the circle with center (�2, 7) that is tangent to the y-axis.A. (x � 2)2 � (y � 7)2 � 4 B. (x � 2)2 � (y � 7)2 � 49C. (x � 2)2 � (y � 7)2 � 4 D. (x � 2)2 � (y � 7)2 � 49 5.

6. Graph x2 � y2 � 4x � 12. 6.

7. Write an equation of the ellipse centered at (4, 1) if its minor 7.axis is 8 units long and its major axis is 10 units long and parallel to the x-axis.

8. Write the equation of the parabola y � �3x2 � 18x � 5 in 8.standard form.

9. Write an equation for a circle if the endpoints of a diameter 9.are at (�2, �1) and (8, 9).

y

xO

Part II

Part I

NAME DATE PERIOD

SCORE 88

Ass

essm

ent

y

xO


Chapter 8 Cumulative Review (Chapters 1–8)

1. Evaluate �3� 5a � b � if a � �3.5 and b � 10. 1.(Lesson 1-4)

2. Write an equation in slope-intercept form for the line that 2.has a slope of 4 and passes through (2, 5). (Lesson 2-4)

3. Solve the system of equations by using substitution. 3.y � 6x � 52x � 3y � 1. (Lesson 3-2)

4. Perform the indicated matrix operation. If the matrix does 4.

not exist, write impossible. � � � � � (Lesson 4-2)

5. Simplify �22xx

2

2��

77xx

��

43�. Assume that the denominator is not 5.

equal to 0. (Lesson 5-4)

6. Simplify (3 � 4i) � (2 � 5i). (Lesson 5-9) 6.

7. Find the exact solutions to 2x2 � 7x � 5 � 0 by using the 7.Quadratic Formula. (Lesson 6-5)

8. Solve the inequality � 2x � 3 � x2 algebraically. 8.(Lesson 6-7)

9. Determine whether the graph 9.represents an odd degree or an even degree polynomial function. Then state the number of real zeros. (Lesson 7-1)

10. One factor of 2x3 � 7x2 � 2x � 3 is x � 3. 10.Find the remaining factors. (Lesson 7-4)

11. If g(x) � 4x and h(x) � 3x � 5, find [gh](x). (Lesson 7-7) 11.

12. Find the midpoint of the line segment with end points at 12.(�10, 8) and (2, �3). (Lesson 8-1)

13. Write an equation for the parabola with focus (�4, 0) and 13.directrix x � 6. (Lesson 8-2)

14. Find the coordinates of the center and foci and the lengths 14.of the major and minor axes for the ellipse with equation 9x2 � y2 � 9. Then graph the ellipse. (Lesson 8-4)

15. Write the equation 4x2 � 9y2 � 24x � 18y � 9 � 0 in 15.standard form. Then state whether the graph of the equation is a parabola, circle, ellipse, or hyperbola.(Lesson 8-6)

4 3 7�2 9 5

3 6 9�2 �1 0

NAME DATE PERIOD

88

xO

f(x )

Standardized Test Practice (Chapters 1–8)


1. Which of the following is the sum of two consecutive prime numbers? A. 9 B. 11 C. 17 D. 24 1.

2. If (r � 2)(r � 1) � (r � 2)(r � 6), which of the following is true?E. r � 14 F. r � �14 G. r � 14 H. r � 19 2.

3. What is the value of 10m � 3 if 2m � 9?A. 2 B. 8 C. 42 D. 48 3.

4. If 10 pears cost c cents, how many pears will d dollars buy?

E. �100

c0d� F. �1

d0c�

G. �10

cd

� H. �1d0c� 4.

5. What is the value of �xy� is 2.5x � �

131�y and y 0?

A. �565� B. �

23� C. �

2125�

D. �32� 5.

6. The sum of five integers is what percent of the average of the same five integers?E. 5 F. 50 G. 500 H. 5000 6.

7. Which of the following are always true statements?I. x2 � 0 II. x2 � x III. x � 1 � x IV. x � �x

A. I and II only B. I and IV onlyC. III and IV only D. III only 7.

8. The table shows the distribution of quiz scores for a group of students.No student scored less than 50 or greater than 90. What is the mean of the scores?E. 70 F. 72.5G. 75 H. 74.5 8.

9. Square RSTU is inscribed in circle O.If the circumference of circle O is 16�,find the area of triangle ROU.A. 32 B. 32�

C. 64 D. 16� 9.

10. What is the value of t if r � �1 and t � (r � 1)(r � 2)(r � 3)?E. 0 F. 2 G. 6 H. 24 10. HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

NAME DATE PERIOD

88

Ass

essm

ent

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

Score Number ofstudents

90 2

80 6

70 8

60 3

50 1

R S

O

U T


Standardized Test Practice (continued)

11. A jar contains 2 white marbles, 5 red marbles, 11. 12.and 13 blue marbles. How many white marbles must be added to the jar to make the probability of randomly selecting a white

marble �14�?

12. In the figure shown, what is the length of C�D�?

13. If the sales tax on a $22.00 purchase is 13. 14.$1.32, what is the total cost of an item priced at $8.50?

14. Evaluate 7 � 3 � 5 � 22.

Column A Column B

15. 2 � d � e � 2 15.

16. The average of a, b, and c is x. 16.

17. 17.

18. � �a �

cb

� for all real numbers a, b, and c 18.

10 15

10

2 3

4

DCBA

a b

c

DCBA�4

162��3

64�

3a3x

DCBA

d � ee � d

DCBA

Part 3: Quantitative Comparison

Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater;if the quantity in column B is greater;if the quantities are equal; or if the relationship cannot be determined from the information given.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

NAME DATE PERIOD

88

NAME DATE PERIOD

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

A

D

C

B

A E D

CB

15

560˚

Standardized Test PracticeStudent Record Sheet (Use with pages 468–469 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 2

NAME DATE PERIOD

88

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7 9

2 5 8 10

3 6

Solve the problem and write your answer in the blank.

Also enter your answer by writing each number or symbol in a box. Then fill inthe corresponding oval for that number or symbol.

11 13 15 17

12 14 16

Select the best answer from the choices given and fill in the corresponding oval.

18 20

19 21 DCBADCBA

DCBADCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

DCBADCBA

DCBADCBADCBADCBA

DCBADCBADCBADCBA

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison


Answers (Lesson 8-1)

Stu

dy G

uid

e a

nd I

nte

rven

tion

Mid

po

int

and

Dis

tan

ce F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1

©G

lenc

oe/M

cGra

w-H

ill45

5G

lenc

oe A

lgeb

ra 2

Lesson 8-1

The

Mid

po

int

Form

ula

Mid

po

int

Fo

rmu

laT

he m

idpo

int

Mof

a s

egm

ent

with

end

poin

ts (

x 1,

y 1)

and

(x2,

y2)

is �

, �.

y 1�

y 2�

2x 1

�x 2

�2

Fin

d t

he

mid

poi

nt

of t

he

lin

e se

gmen

t w

ith

en

dp

oin

ts a

t (4

,�7)

an

d (

�2,

3).

�,

��

,�

��

,�o

r (1

,�2)

Th

e m

idpo

int

of t

he

segm

ent

is (

1,�

2).

�4

�2

2 � 2

�7

�3

�2

4 �

(�2)

�� 2

y 1�

y 2�

2x 1

�x 2

�2

A d

iam

eter

A�B�

of a

cir

cle

has

en

dp

oin

ts A

(5,�

11)

and

B(�

7,6)

.W

hat

are

th

e co

ord

inat

es o

f th

e ce

nte

rof

th

e ci

rcle

?

Th

e ce

nte

r of

th

e ci

rcle

is

the

mid

poin

t of

all

of i

ts d

iam

eter

s.

�,

��

,�

��

,�o

r ��

1,�

2�

Th

e ci

rcle

has

cen

ter ��

1,�

2�.

1 � 2

1 � 2�

5�2

�2

�2

�11

�6

�� 2

5 �

(�7)

�� 2

y 1�

y 2�

2x 1

�x 2

�2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d t

he

mid

poi

nt

of e

ach

lin

e se

gmen

t w

ith

en

dp

oin

ts a

t th

e gi

ven

coo

rdin

ates

.

1.(1

2,7)

an

d (�

2,11

)2.

(�8,

�3)

an

d (1

0,9)

3.(4

,15)

an

d (1

0,1)

(5,9

)(1

,3)

(7,8

)

4.(�

3,�

3) a

nd

(3,3

)5.

(15,

6) a

nd

(12,

14)

6.(2

2,�

8) a

nd

(�10

,6)

(0,0

)(1

3.5,

10)

(6,�

1)

7.(3

,5)

and

(�6,

11)

8.(8

,�15

) an

d (�

7,13

)9.

(2.5

,�6.

1) a

nd

(7.9

,13.

7)

��,8

��

,�1 �

(5.2

,3.8

)

10.(

�7,

�6)

an

d (�

1,24

) 11

.(3,

�10

) an

d (3

0,�

20)

12.(

�9,

1.7)

an

d (�

11,1

.3)

(�4,

9)�

,�15

�(�

10,1

.5)

13.S

egm

ent

M�N�

has

mid

poin

t P

.If

Mh

as c

oord

inat

es (

14,�

3) a

nd

Ph

as c

oord

inat

es

(�8,

6),w

hat

are

th

e co

ordi

nat

es o

f N

?(�

30,1

5)

14.C

ircl

e R

has

a d

iam

eter

S�T�

.If

Rh

as c

oord

inat

es (

�4,

�8)

an

d S

has

coo

rdin

ates

(1,

4),

wh

at a

re t

he

coor

din

ates

of

T?

(�9,

�20

)

15.S

egm

ent

A�D�

has

mid

poin

t B

,an

d B�

D�h

as m

idpo

int

C.I

f A

has

coo

rdin

ates

(�

5,4)

an

d C

has

coo

rdin

ates

(10

,11)

,wh

at a

re t

he

coor

din

ates

of

Ban

d D

?

Bis

�5,8

�,Dis

�15,

13�.

1 � 32 � 3

33 � 21 � 23 � 2

©G

lenc

oe/M

cGra

w-H

ill45

6G

lenc

oe A

lgeb

ra 2

The

Dis

tan

ce F

orm

ula

Dis

tan

ce F

orm

ula

The

dis

tanc

e be

twee

n tw

o po

ints

(x 1

, y 1

) an

d (x

2, y

2) is

giv

en b

y

d�

�(x

2�

�x 1

)2�

�(y

2�

�y 1

)2�

.

Wh

at i

s th

e d

ista

nce

bet

wee

n (

8,�

2) a

nd

(�

6,�

8)?

d�

�(x

2�

�x 1

)2�

�(y

2�

�y 1

)2�

Dis

tanc

e F

orm

ula

��

(�6

��

8)2

��

[�8

��

(�2)

]�

2 �Le

t (x

1, y

1) �

(8,

�2)

and

(x 2

, y 2

) �

(�6,

�8)

.

��

(�14

)�

2�

(��

6)2

�S

ubtr

act.

��

196

��

36�or

�23

2�

Sim

plify

.

Th

e di

stan

ce b

etw

een

th

e po

ints

is

�23

2�

or a

bou

t 15

.2 u

nit

s.

Fin

d t

he

per

imet

er a

nd

are

a of

sq

uar

e P

QR

Sw

ith

ver

tice

s P

(�4,

1),

Q(�

2,7)

,R(4

,5),

and

S(2

,�1)

.

Fin

d th

e le

ngt

h o

f on

e si

de t

o fi

nd

the

peri

met

er a

nd

the

area

.Ch

oose

P�Q�

.

d�

�(x

2�

�x 1

)2�

�(y

2�

�y 1

)2�

Dis

tanc

e F

orm

ula

��

[�4

��

(�2)

]�

2�

(1�

� 7

)2�

Let

(x1,

y1)

�(�

4, 1

) an

d (x

2, y

2) �

(�2,

7).

��

(�2)

2�

�(�

6�

)2 �S

ubtr

act.

��

40�or

2�

10�S

impl

ify.

Sin

ce o

ne

side

of

the

squ

are

is 2

�10�

,th

e pe

rim

eter

is

8�10�

un

its.

Th

e ar

ea i

s (2

�10�

)2 ,or

40 u

nit

s2.

Fin

d t

he

dis

tan

ce b

etw

een

eac

h p

air

of p

oin

ts w

ith

th

e gi

ven

coo

rdin

ates

.

1.(3

,7)

and

(�1,

4)

2.(�

2,�

10)

and

(10,

�5)

3.

(6,�

6) a

nd

(�2,

0)

5 u

nit

s13

un

its

10 u

nit

s

4.(7

,2)

and

(4,�

1)

5.(�

5,�

2) a

nd

(3,4

) 6.

(11,

5) a

nd

(16,

9)

3�2�

un

its

10 u

nit

s�

41�u

nit

s

7.(�

3,4)

an

d (6

,�11

) 8.

(13,

9) a

nd

(11,

15)

9.(�

15,�

7) a

nd

(2,1

2)

3�34�

un

its

2�10�

un

its

5�26�

un

its

10.R

ecta

ngl

e A

BC

Dh

as v

erti

ces

A(1

,4),

B(3

,1),

C(�

3,�

2),a

nd

D(�

5,1)

.Fin

d th

epe

rim

eter

an

d ar

ea o

f A

BC

D.

2 �13 �

�6 �

5 �u

nit

s;3 �

65 �u

nit

s2

11.C

ircl

e R

has

dia

met

er S�

T�w

ith

en

dpoi

nts

S(4

,5)

and

T(�

2,�

3).W

hat

are

th

eci

rcu

mfe

ren

ce a

nd

area

of

the

circ

le?

(Exp

ress

you

r an

swer

in

ter

ms

of �

.)10

�u

nit

s;25

�u

nit

s2

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Mid

po

int

and

Dis

tan

ce F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Mid

po

int

and

Dis

tan

ce F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1

©G

lenc

oe/M

cGra

w-H

ill45

7G

lenc

oe A

lgeb

ra 2

Lesson 8-1

Fin

d t

he

mid

poi

nt

of e

ach

lin

e se

gmen

t w

ith

en

dp

oin

ts a

t th

e gi

ven

coo

rdin

ates

.

1.(4

,�1)

,(�

4,1)

(0,0

)2.

(�1,

4),(

5,2)

(2,3

)

3.(3

,4),

(5,4

)(4

,4)

4.(6

,2),

(2,�

1)�4,

�

5.(3

,9),

(�2,

�3)

�,3

�6.

(�3,

5),(

�3,

�8)

��3,

��

7.(3

,2),

(�5,

0)(�

1,1)

8.(3

,�4)

,(5,

2)(4

,�1)

9.(�

5,�

9),(

5,4)

�0,�

�10

.(�

11,1

4),(

0,4)

��,9

�

11.(

3,�

6),(

�8,

�3)

��,�

�12

.(0,

10),

(�2,

�5)

��1,

�

Fin

d t

he

dis

tan

ce b

etw

een

eac

h p

air

of p

oin

ts w

ith

th

e gi

ven

coo

rdin

ates

.

13.(

4,12

),(�

1,0)

13 u

nit

s14

.(7,

7),(

�5,

�2)

15 u

nit

s

15.(

�1,

4),(

1,4)

2 u

nit

s16

.(11

,11)

,(8,

15)

5 u

nit

s

17.(

1,�

6),(

7,2)

10 u

nit

s18

.(3,

�5)

,(3,

4)9

un

its

19.(

2,3)

,(3,

5)�

5�u

nit

s20

.(�

4,3)

,(�

1,7)

5 u

nit

s

21.(

�5,

�5)

,(3,

10)

17 u

nit

s22

.(3,

9),(

�2,

�3)

13 u

nit

s

23.(

6,�

2),(

�1,

3)�

74�u

nit

s24

.(�

4,1)

,(2,

�4)

�61�

un

its

25.(

0,�

3),(

4,1)

4�2�

un

its

26.(

�5,

�6)

,(2,

0)�

85�u

nit

s

5 � 29 � 2

5 � 2

11 � 25 � 2

3 � 21 � 2

1 � 2

©G

lenc

oe/M

cGra

w-H

ill45

8G

lenc

oe A

lgeb

ra 2

Fin

d t

he

mid

poi

nt

of e

ach

lin

e se

gmen

t w

ith

en

dp

oin

ts a

t th

e gi

ven

coo

rdin

ates

.

1.(8

,�3)

,(�

6,�

11)

(1,�

7)2.

(�14

,5),

(10,

6)��

2,�

3.(�

7,�

6),(

1,�

2)(�

3,�

4)4.

(8,�

2),(

8,�

8)(8

,�5)

5.(9

,�4)

,(1,

�1)

�5,�

�6.

(3,3

),(4

,9)�

,6�

7.(4

,�2)

,(3,

�7)

�,�

�8.

(6,7

),(4

,4)�5,

�9.

(�4,

�2)

,(�

8,2)

(�6,

0)10

.(5,

�2)

,(3,

7)�4,

�11

.(�

6,3)

,(�

5,�

7)��

,�2 �

12.(

�9,

�8)

,(8,

3)��

,��

13.(

2.6,

�4.

7),(

8.4,

2.5)

(5.5

,�1.

1)14

. ��,6

�, �,4

� �,5

�15

.(�

2.5,

�4.

2),(

8.1,

4.2)

(2.8

,0)

16. �

,�, �

�,�

� ��

,0�

Fin

d t

he

dis

tan

ce b

etw

een

eac

h p

air

of p

oin

ts w

ith

th

e gi

ven

coo

rdin

ates

.

17.(

5,2)

,(2,

�2)

5 u

nit

s18

.(�

2,�

4),(

4,4)

10 u

nit

s

19.(

�3,

8),(

�1,

�5)

�17

3�

un

its

20.(

0,1)

,(9,

�6)

�13

0�

un

its

21.(

�5,

6),(

�6,

6)1

un

it22

.(�

3,5)

,(12

,�3)

17 u

nit

s

23.(

�2,

�3)

,(9,

3)�

157

�u

nit

s24

.(�

9,�

8),(

�7,

8)2�

65�u

nit

s

25.(

9,3)

,(9,

�2)

5 u

nit

s26

.(�

1,�

7),(

0,6)

�17

0�

un

its

27.(

10,�

3),(

�2,

�8)

13 u

nit

s28

.(�

0.5,

�6)

,(1.

5,0)

2�10�

un

its

29. �

,�, �

1,�1

un

it30

.(�

4�2�,

��

5�),(

�5�

2�,4�

5�)�

127

�u

nit

s

31.G

EOM

ETRY

Cir

cle

Oh

as a

dia

met

er A�

B�.I

f A

is a

t (�

6,�

2) a

nd

Bis

at

(�3,

4),f

ind

the

cen

ter

of t

he

circ

le a

nd

the

len

gth

of

its

diam

eter

.��

,1�;3

�5�

un

its

32.G

EOM

ETRY

Fin

d th

e pe

rim

eter

of

a tr

iang

le w

ith

vert

ices

at

(1,�

3),(

�4,

9),a

nd (

�2,

1).

18 �

2�17�

un

its

9 � 2

7 � 53 � 5

2 � 5

1 � 41 � 2

5 � 81 � 2

1 � 8

1 � 62 � 3

1 � 3

5 � 21 � 2

11 � 2

5 � 2

11 � 29 � 2

7 � 2

7 � 25 � 2

11 � 2

Pra

ctic

e (

Ave

rag

e)

Mid

po

int

and

Dis

tan

ce F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1



Readin

g t

o L

earn

Math

em

ati

csM

idp

oin

t an

d D

ista

nce

Fo

rmu

las

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1

©G

lenc

oe/M

cGra

w-H

ill45

9G

lenc

oe A

lgeb

ra 2

Lesson 8-1

Pre-

Act

ivit

yH

ow a

re t

he

Mid

poi

nt

and

Dis

tan

ce F

orm

ula

s u

sed

in

em

erge

ncy

med

icin

e?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-1

at

the

top

of p

age

412

in y

our

text

book

.

How

do

you

fin

d di

stan

ces

on a

roa

d m

ap?

Sam

ple

an

swer

:U

se t

he

scal

e o

f m

iles

on

th

e m

ap.Y

ou

mig

ht

also

use

a r

ule

r.

Rea

din

g t

he

Less

on

1.a.

Wri

te t

he

coor

din

ates

of

the

mid

poin

t of

a s

egm

ent

wit

h e

ndp

oin

ts (

x 1,y

1) a

nd

(x2,

y 2).

�,

�b

.E

xpla

in h

ow t

o fi

nd

the

mid

poin

t of

a s

egm

ent

if y

ou k

now

th

e co

ordi

nat

es o

f th

een

dpoi

nts

.Do

not

use

su

bscr

ipts

in

you

r ex

plan

atio

n.

Sam

ple

an

swer

:To

fin

d t

he

x-co

ord

inat

e o

f th

e m

idp

oin

t,ad

d t

he

x-co

ord

inat

es o

f th

e en

dp

oin

ts a

nd

div

ide

by t

wo

.To

fin

d t

he

y-co

ord

inat

e o

f th

e m

idp

oin

t,d

o t

he

sam

e w

ith

th

e y-

coo

rdin

ates

of

the

end

po

ints

.

2.a.

Wri

te a

n e

xpre

ssio

n f

or t

he

dist

ance

bet

wee

n t

wo

poin

ts w

ith

coo

rdin

ates

(x 1

,y1)

an

d(x

2,y 2

).�

(x2

��

x 1)2

��

(y2

��

y 1)2

�b

.E

xpla

in h

ow t

o fi

nd

the

dist

ance

bet

wee

n t

wo

poin

ts.D

o n

ot u

se s

ubs

crip

ts i

n y

our

expl

anat

ion

.

Sam

ple

an

swer

:F

ind

th

e d

iffe

ren

ce b

etw

een

th

e x-

coo

rdin

ates

an

d s

qu

are

it.F

ind

th

e d

iffe

ren

ce b

etw

een

th

e y-

coo

rdin

ates

an

d s

qu

are

it.A

dd

th

e sq

uar

es.T

hen

fin

d t

he

squ

are

roo

t o

f th

e su

m.

3.C

onsi

der

the

segm

ent

con

nec

tin

g th

e po

ints

(�

3,5)

an

d (9

,11)

.

a.F

ind

the

mid

poin

t of

th

is s

egm

ent.

(3,8

)

b.

Fin

d th

e le

ngt

h o

f th

e se

gmen

t.W

rite

you

r an

swer

in

sim

plif

ied

radi

cal

form

.6�

5�

Hel

pin

g Y

ou

Rem

emb

er

4.H

ow c

an t

he

“mid

”in

mid

poin

t h

elp

you

rem

embe

r th

e m

idpo

int

form

ula

?

Sam

ple

an

swer

:Th

e m

idp

oin

tis

th

e p

oin

t in

th

e m

idd

leo

f a

seg

men

t.It

is h

alfw

ay b

etw

een

th

e en

dp

oin

ts.T

he

coo

rdin

ates

of

the

mid

po

int

are

fou

nd

by

fin

din

g t

he

aver

age

of

the

two

x-c

oo

rdin

ates

(ad

d t

hem

an

dd

ivid

e by

2)

and

th

e av

erag

e o

f th

e tw

o y

-co

ord

inat

es.

y 1�

y 2�

2x 1

�x 2

�2

©G

lenc

oe/M

cGra

w-H

ill46

0G

lenc

oe A

lgeb

ra 2

Qu

adra

tic

Fo

rmC

onsi

der

two

met

hod

s fo

r so

lvin

g th

e fo

llow

ing

equ

atio

n.

(y�

2)2

�5(

y�

2) �

6�

0

On

e w

ay t

o so

lve

the

equ

atio

n i

s to

sim

plif

y fi

rst,

then

use

fac

tori

ng.

y2�

4y�

4 �

5y�

10 �

6�

0y2

�9y

�20

�0

(y�

4)(y

�5)

�0

Th

us,

the

solu

tion

set

is

{4,5

}.

An

oth

er w

ay t

o so

lve

the

equ

atio

n i

s fi

rst

to r

epla

ce y

�2

by a

sin

gle

vari

able

.T

his

wil

l pr

odu

ce a

n e

quat

ion

th

at i

s ea

sier

to

solv

e th

an t

he

orig

inal

equ

atio

n.

Let

t�

y�

2 an

d th

en s

olve

th

e n

ew e

quat

ion

.

(y�

2)2

�5(

y�

2) �

6�

0t2

�5t

�6

�0

(t�

2)(t

�3)

�0

Th

us,

tis

2 o

r 3.

Sin

ce t

�y

�2,

the

solu

tion

set

of

the

orig

inal

equ

atio

n i

s {4

,5}.

Sol

ve e

ach

eq

uat

ion

usi

ng

two

dif

fere

nt

met

hod

s.

1.(z

�2)

2�

8(z

�2)

�7

�0

2.(3

x�

1)2

�(3

x�

1) �

20 �

0

{�3,

�9}

{2,�

1}

3.(2

t�

1)2

�4(

2t�

1) �

3 �

04.

(y2

�1)

2�

(y2

�1)

�2

�0

{0,1

}�0,

��

3��

5.(a

2�

2)2

�2(

a2�

2) �

3 �

06.

(1 �

�c�)

2�

(1 �

�c�)

�6

�0

��1,

��

5��{1

}

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Par

abo

las

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-2

8-2

©G

lenc

oe/M

cGra

w-H

ill46

1G

lenc

oe A

lgeb

ra 2

Lesson 8-2

Equ

atio

ns

of

Para

bo

las

A p

arab

ola

is a

cu

rve

con

sist

ing

of a

ll p

oin

ts i

n t

he

coor

din

ate

plan

e th

at a

re t

he

sam

e di

stan

ce f

rom

a g

iven

poi

nt

(th

e fo

cus)

an

d a

give

n l

ine

(th

e d

irec

trix

).T

he

foll

owin

g ch

art

sum

mar

izes

im

port

ant

info

rmat

ion

abo

ut

para

bola

s.

Sta

nd

ard

Fo

rm o

f E

qu

atio

ny

�a

(x�

h)2

�k

x�

a(y

�k)

2�

h

Axi

s o

f S

ymm

etry

x�

hy

�k

Ver

tex

(h,

k)

(h,

k)

Fo

cus

�h, k

��

�h�

, k �

Dir

ectr

ixy

�k

�x

�h

�

Dir

ecti

on

of

Op

enin

gup

war

d if

a�

0, d

ownw

ard

if a

�0

right

if a

�0,

left

if a

�0

Len

gth

of

Lat

us

Rec

tum

un

its

units

Iden

tify

th

e co

ord

inat

es o

f th

e ve

rtex

an

d f

ocu

s,th

e eq

uat

ion

s of

the

axis

of

sym

met

ry a

nd

dir

ectr

ix,a

nd

th

e d

irec

tion

of

open

ing

of t

he

par

abol

aw

ith

eq

uat

ion

y�

2x2

�12

x�

25.

y�

2x2

�12

x�

25O

rigin

al e

quat

ion

y�

2(x2

�6x

) �

25F

acto

r 2

from

the

x-t

erm

s.

y�

2(x2

�6x

�■

) �

25 �

2(■

)C

ompl

ete

the

squa

re o

n th

e rig

ht s

ide.

y�

2(x2

�6x

�9)

�25

�2(

9)T

he 9

add

ed t

o co

mpl

ete

the

squa

re is

mul

tiplie

d by

2.

y�

2(x

�3)

2�

43W

rite

in s

tand

ard

form

.

Th

e ve

rtex

of

this

par

abol

a is

loc

ated

at

(3,�

43),

the

focu

s is

loc

ated

at �3,

�42

�,th

e

equ

atio

n o

f th

e ax

is o

f sy

mm

etry

is

x�

3,an

d th

e eq

uat

ion

of

the

dire

ctri

x is

y�

�43

.T

he

para

bola

ope

ns

upw

ard.

Iden

tify

th

e co

ord

inat

es o

f th

e ve

rtex

an

d f

ocu

s,th

e eq

uat

ion

s of

th

e ax

is o

fsy

mm

etry

an

d d

irec

trix

,an

d t

he

dir

ecti

on o

f op

enin

g of

th

e p

arab

ola

wit

h t

he

give

n e

qu

atio

n.

1.y

�x2

�6x

�4

2.y

�8x

�2x

2�

103.

x�

y2�

8y�

6

(�3,

�13

),(2

,18)

, �2,1

7�,

(�10

,4),

��9

,4�,

��3,

�12

�,x�

�3,

x�

2,y

�18

,y

�4,

x�

�10

,

y�

�13

,up

do

wn

rig

ht

Wri

te a

n e

qu

atio

n o

f ea

ch p

arab

ola

des

crib

ed b

elow

.

4.fo

cus

(�2,

3),d

irec

trix

x�

�2

5.ve

rtex

(5,

1),f

ocu

s �4

,1�

x�

6(y

�3)

2�

2x

��

3(y

�1)

2�

51 � 24

11 � 121 � 12

1 � 4

1 � 41 � 8

3 � 4

3 � 41 � 8

1 � 8

7 � 8

1 � a1 � a

1 � 4a1 � 4a

1 � 4a1 � 4a

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill46

2G

lenc

oe A

lgeb

ra 2

Gra

ph

Par

abo

las

To

grap

h a

n e

quat

ion

for

a p

arab

ola,

firs

t pu

t th

e gi

ven

equ

atio

n i

nst

anda

rd f

orm

.

y�

a(x

�h

)2�

kfo

r a

para

bola

ope

nin

g u

p or

dow

n,o

rx

�a(

y�

k)2

�h

for

a pa

rabo

la o

pen

ing

to t

he

left

or

righ

t

Use

th

e va

lues

of

a,h

,an

d k

to d

eter

min

e th

e ve

rtex

,foc

us,

axis

of

sym

met

ry,a

nd

len

gth

of

the

latu

s re

ctu

m.T

he

vert

ex a

nd

the

endp

oin

ts o

f th

e la

tus

rect

um

giv

e th

ree

poin

ts o

n t

he

para

bola

.If

you

nee

d m

ore

poin

ts t

o pl

ot a

n a

ccu

rate

gra

ph,s

ubs

titu

te v

alu

es f

or p

oin

tsn

ear

the

vert

ex. G

rap

h y

�(x

�1)

2�

2.

In t

he

equ

atio

n,a

�,h

�1,

k�

2.

Th

e pa

rabo

la o

pen

s u

p,si

nce

a�

0.ve

rtex

:(1,

2)ax

is o

f sy

mm

etry

:x�

1

focu

s:�1,

2 �

�or �1

,2�

len

gth

of

latu

s re

ctu

m:

or 3

un

its

endp

oin

ts o

f la

tus

rect

um

: �2,2

�, ��

,2�

Th

e co

ord

inat

es o

f th

e fo

cus

and

th

e eq

uat

ion

of

the

dir

ectr

ix o

f a

par

abol

a ar

egi

ven

.Wri

te a

n e

qu

atio

n f

or e

ach

par

abol

a an

d d

raw

its

gra

ph

.

1.(3

,5),

y�

12.

(4,�

4),y

��

63.

(5,�

1),x

�3

y�

(x�

3)2

�3

y�

(x�

4)2

�5

x�

(y�

1)2

�4

1 � 41 � 4

1 � 8

x

y

Ox

y

O

x

y

O

3 � 41 � 2

3 � 41 � 2

1 � �1 3�

3 � 41

� 4 ��1 3� �x

y

O

1 � 3

1 � 3

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Par

abo

las

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-2

8-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Par

abo

las

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-2

8-2

©G

lenc

oe/M

cGra

w-H

ill46

3G

lenc

oe A

lgeb

ra 2

Lesson 8-2

Wri

te e

ach

eq

uat

ion

in

sta

nd

ard

for

m.

1.y

�x2

�2x

�2

2.y

�x2

�2x

�4

3.y

�x2

�4x

�1

y�

[x�

(�1)

]2�

1y

�(x

�1)

2�

3y

�[x

�(�

2)]2

�(�

3)

Iden

tify

th

e co

ord

inat

es o

f th

e ve

rtex

an

d f

ocu

s,th

e eq

uat

ion

s of

th

e ax

is o

fsy

mm

etry

an

d d

irec

trix

,an

d t

he

dir

ecti

on o

f op

enin

g of

th

e p

arab

ola

wit

h t

he

give

n e

qu

atio

n.T

hen

fin

d t

he

len

gth

of

the

latu

s re

ctu

m a

nd

gra

ph

th

e p

arab

ola.

4.y

�(x

�2)

25.

x�

(y�

2)2

�3

6.y

��

(x�

3)2

�4

vert

ex:

(2,0

);ve

rtex

:(3

,2);

vert

ex:

(�3,

4);

focu

s:�2,

�;fo

cus:

�3,2

�;fo

cus:

��3,

3�;

axis

of

sym

met

ry:

axis

of

sym

met

ry:

axis

of

sym

met

ry:

x�

2;y

�2;

x�

�3;

dir

ectr

ix:

y�

�;

dir

ectr

ix:

x�

2;

dir

ectr

ix:

y�

4;

op

ens

up

;o

pen

s ri

gh

t;o

pen

s d

ow

n;

latu

s re

ctu

m:

1 u

nit

latu

s re

ctu

m:

1 u

nit

latu

s re

ctu

m:

1 u

nit

Wri

te a

n e

qu

atio

n f

or e

ach

par

abol

a d

escr

ibed

bel

ow.T

hen

dra

w t

he

grap

h.

7.ve

rtex

(0,

0),

8.ve

rtex

(5,

1),

9.ve

rtex

(1,

3),

focu

s �0,

��

focu

s �5,

�di

rect

rix

x�

y�

�3x

2y

�(x

�5)

2�

1x

�2(

y�

3)2

�1 x

y

Ox

y

O

x

y

O

7 � 85 � 4

1 � 12

1 � 43 � 4

1 � 4

3 � 41 � 4

1 � 4

x

y

Ox

y

O

x

y

O

©G

lenc

oe/M

cGra

w-H

ill46

4G

lenc

oe A

lgeb

ra 2

Wri

te e

ach

eq

uat

ion

in

sta

nd

ard

for

m.

1.y

�2x

2�

12x

�19

2.y

�x2

�3x

�3.

y�

�3x

2�

12x

�7

y�

2(x

�3)

2�

1y

�[x

�(�

3)]2

�(�

4)y

��

3[x

�(�

2)]2

�5

Iden

tify

th

e co

ord

inat

es o

f th

e ve

rtex

an

d f

ocu

s,th

e eq

uat

ion

s of

th

e ax

is o

fsy

mm

etry

an

d d

irec

trix

,an

d t

he

dir

ecti

on o

f op

enin

g of

th

e p

arab

ola

wit

h t

he

give

n e

qu

atio

n.T

hen

fin

d t

he

len

gth

of

the

latu

s re

ctu

m a

nd

gra

ph

th

e p

arab

ola.

4.y

�(x

�4)

2�

35.

x�

�y2

�1

6.x

�3(

y�

1)2

�3

vert

ex:

(4,3

);ve

rtex

:(1

,0);

vert

ex:

(�3,

�1)

;

focu

s:�4,

3�;

focu

s:�

,0�;

focu

s:��

2,�

1 �;ax

is:

x�

4;ax

is:

y�

0;ax

is:

y�

�1;

dir

ectr

ix:

y�

2;

dir

ectr

ix:

x�

1;

dir

ectr

ix:

x�

�3

;

op

ens

up

;o

pen

s le

ft;

op

ens

rig

ht;

latu

s re

ctu

m:

1 u

nit

latu

s re

ctu

m:

3 u

nit

sla

tus

rect

um

:u

nit

Wri

te a

n e

qu

atio

n f

or e

ach

par

abol

a d

escr

ibed

bel

ow.T

hen

dra

w t

he

grap

h.

7.ve

rtex

(0,

�4)

,8.

vert

ex (

�2,

1),

9.ve

rtex

(1,

3),

focu

s �0,

�3

�di

rect

rix

x�

�3

axis

of

sym

met

ry x

�1,

latu

s re

ctu

m:2

un

its,

a�

0

y�

2x2

�4

x �

(y �

1)2

�2

y �

�(x

�1)

2�

3

10.T

ELEV

ISIO

NW

rite

the

equ

atio

n in

the

for

m y

�ax

2fo

r a

sate

llit

e di

sh.A

ssum

e th

at t

hebo

ttom

of

the

upw

ard-

faci

ng

dish

pas

ses

thro

ugh

(0,

0) a

nd

that

th

e di

stan

ce f

rom

th

ebo

ttom

to

the

focu

s po

int

is 8

in

ches

.y

�x

21 � 32

x

y

Ox

y

Ox

y

O

1 � 21 � 4

7 � 8

1 � 3

1 � 123 � 4

3 � 4

11 � 121 � 4

1 � 4

x

y

Ox

y

O

x

y

O

1 � 3

1 � 2

1 � 21 � 2

Pra

ctic

e (

Ave

rag

e)

Par

abo

las

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-2

8-2


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csP

arab

ola

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-2

8-2

©G

lenc

oe/M

cGra

w-H

ill46

5G

lenc

oe A

lgeb

ra 2

Lesson 8-2

Pre-

Act

ivit

yH

ow a

re p

arab

olas

use

d i

n m

anu

fact

uri

ng?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-2

at

the

top

of p

age

419

in y

our

text

book

.

Nam

e at

lea

st t

wo

refl

ecti

ve o

bjec

ts t

hat

mig

ht

hav

e th

e sh

ape

of a

para

bola

.

Sam

ple

an

swer

:te

lesc

op

e m

irro

r,sa

telli

te d

ish

Rea

din

g t

he

Less

on

1.In

th

e pa

rabo

la s

how

n i

n t

he

grap

h,t

he

poin

t (2

,�2)

is

call

ed

the

and

the

poin

t (2

,0)

is c

alle

d th

e

.Th

e li

ne

y�

�4

is c

alle

d th

e

,an

d th

e li

ne

x�

2 is

cal

led

the

.

2.a.

Wri

te t

he

stan

dard

for

m o

f th

e eq

uat

ion

of

a pa

rabo

la t

hat

ope

ns

upw

ard

ordo

wn

war

d.y

�a

(x�

h)2

�k

b.T

he

para

bola

ope

ns

dow

nw

ard

if

and

open

s u

pwar

d if

.T

he

equ

atio

n o

f th

e ax

is o

f sy

mm

etry

is

,an

d th

e co

ordi

nat

es o

f th

e ve

rtex

are

.

3.A

par

abol

a h

as e

quat

ion

x�

�(y

�2)

2�

4.T

his

par

abol

a op

ens

to t

he

.

It h

as v

erte

x an

d fo

cus

.Th

e di

rect

rix

is

.Th

e le

ngt

h

of t

he

latu

s re

ctu

m i

s u

nit

s.

Hel

pin

g Y

ou

Rem

emb

er

4.H

ow c

an t

he

way

in

wh

ich

you

plo

t po

ints

in

a r

ecta

ngu

lar

coor

din

ate

syst

em h

elp

you

to

rem

embe

r w

hat

th

e si

gn o

f a

tell

s yo

u a

bou

t th

e di

rect

ion

in

wh

ich

a p

arab

ola

open

s?S

amp

le a

nsw

er:

In p

lott

ing

po

ints

,a p

osi

tive

x-c

oo

rdin

ate

tells

yo

u t

om

ove

to t

he

rig

ht

and

a n

egat

ive

x-co

ord

inat

e te

lls y

ou

to

mov

e to

th

ele

ft.T

his

is li

ke a

par

abo

la w

ho

se e

qu

atio

n is

of

the

form

“x

�…

”;it

op

ens

to t

he

rig

ht

if a

�0

and

to

th

e le

ftif

a�

0.L

ikew

ise,

a p

osi

tive

y-

coo

rdin

ate

tells

yo

u t

o m

ove

up

and

a n

egat

ive

y-co

ord

inat

e te

lls y

ou

to m

ove

do

wn

.Th

is is

like

a p

arab

ola

wh

ose

eq

uat

ion

is o

f th

e fo

rm

“y�

…”;

it o

pen

s u

pw

ard

if a

�0

and

do

wn

war

dif

a�

0.

8

x�

6(2

,2)

(4,2

)

left

1 � 8

(h,k

)

x�

h

a�

0a

�0

axis

of

sym

met

ry

dir

ectr

ix

focu

s

vert

ex

x

y O

( 2, –

2)

( 2, 0

)

y �

–4

©G

lenc

oe/M

cGra

w-H

ill46

6G

lenc

oe A

lgeb

ra 2

Tan

gen

ts t

o P

arab

ola

sA

lin

e th

at i

nte

rsec

ts a

par

abol

a in

exa

ctly

on

e po

int

wit

hou

t cr

ossi

ng

the

curv

e is

a t

ange

nt

to t

he

para

bola

.Th

e po

int

wh

ere

a ta

nge

nt

lin

e to

uch

es

a pa

rabo

la i

s th

e p

oin

t of

tan

gen

cy.T

he

lin

e pe

rpen

dicu

lar

to a

tan

gent

to

a pa

rabo

la a

t th

e po

int

of t

ange

ncy

is c

alle

d th

e n

orm

alto

th

e pa

rabo

la a

t th

at p

oin

t.In

th

e di

agra

m,l

ine

�is

tan

gen

t to

th

e

para

bola

th

at i

s th

e gr

aph

of

y�

x2at

��3 2� ,�9 4� �.

Th

e

x-ax

is i

s ta

nge

nt

to t

he

para

bola

at

O,a

nd

the

y-ax

is

is t

he

nor

mal

to

the

para

bola

at

O.

Sol

ve e

ach

pro

ble

m.

1.F

ind

an e

quat

ion

for

line

�in

the

dia

gram

.Hin

t:A

non

vert

ical

line

wit

h a

neq

uat

ion

of

the

form

y�

mx

�b

wil

l be

tan

gen

t to

th

e gr

aph

of

y�

x2at

��3 2� ,�9 4� �i

f an

d on

ly i

f ��3 2� ,

�9 4� �is

the

only

pai

r of

nu

mbe

rs t

hat

sat

isfi

es b

oth

y�

x2an

d y

�m

x�

b.

m�

3,b

��

�9 4� ,y

�3

x�

�9 4�

2.If

ais

an

y re

al n

um

ber,

then

(a,

a2)

belo

ngs

to

the

grap

h o

f y

�x2

.Exp

ress

m

and

bin

ter

ms

of a

to f

ind

an e

quat

ion

of

the

form

y�

mx

�b

for

the

lin

eth

at i

s ta

nge

nt

to t

he

grap

h o

f y

�x2

at (

a,a2

).

m�

2a

,b�

a2,y

�(2

a)x

�(�

a2)

or

y�

2a

x�

a2

3.F

ind

an e

quat

ion

for

th

e n

orm

al t

o th

e gr

aph

of

y�

x2at

��3 2� ,�9 4� �.

y�

��1 3� x

��1 41 �

4.If

ais

a n

onze

ro r

eal

nu

mbe

r,fi

nd

an e

quat

ion

for

th

e n

orm

al t

o th

e gr

aph

of

y�

x2at

(a,

a2).

y�

�� 21 a�

�x�

�a2�

�1 2� �

x

y

O

�

y �

x2

1–1

–2–3

2

6 5 4 3 2 1

3

�3 – 2, 9 – 4�

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-2

8-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Cir

cles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3

©G

lenc

oe/M

cGra

w-H

ill46

7G

lenc

oe A

lgeb

ra 2

Lesson 8-3

Equ

atio

ns

of

Cir

cles

Th

e eq

uat

ion

of

a ci

rcle

wit

h c

ente

r (h

,k)

and

radi

us

ru

nit

s is

(x

�h

)2�

(y�

k)2

�r2

.

Wri

te a

n e

qu

atio

n f

or a

cir

cle

if t

he

end

poi

nts

of

a d

iam

eter

are

at

(�4,

5) a

nd

(6,

�3)

.

Use

th

e m

idpo

int

form

ula

to

fin

d th

e ce

nte

r of

th

e ci

rcle

.

(h,k

) �

�,

�M

idpo

int

form

ula

��

,�

(x1,

y1)

�(�

4, 5

), (

x 2,

y 2)

�(6

, �

3)

��

,�o

r (1

,1)

Sim

plify

.

Use

th

e co

ordi

nat

es o

f th

e ce

nte

r an

d on

e en

dpoi

nt

of t

he

diam

eter

to

fin

d th

e ra

diu

s.

r�

�(x

2�

x�

1)2

��

(y2

��

y 1)2

�D

ista

nce

form

ula

r�

�(�

4 �

�1)

2�

�(5

��

1)2

�(x

1, y

1) �

(1,

1),

(x2,

y2)

�(�

4, 5

)

��

(�5)

2�

�42

��

�41�

Sim

plify

.

Th

e ra

diu

s of

th

e ci

rcle

is

�41�

,so

r2�

41.

An

equ

atio

n o

f th

e ci

rcle

is

(x�

1)2

�(y

�1)

2�

41.

Wri

te a

n e

qu

atio

n f

or t

he

circ

le t

hat

sat

isfi

es e

ach

set

of

con

dit

ion

s.

1.ce

nte

r (8

,�3)

,rad

ius

6(x

�8)

2�

(y�

3)2

�36

2.ce

nte

r (5

,�6)

,rad

ius

4(x

�5)

2�

(y�

6)2

�16

3.ce

nte

r (�

5,2)

,pas

ses

thro

ugh

(�

9,6)

(x�

5)2

�(y

�2)

2�

32

4.en

dpoi

nts

of

a di

amet

er a

t (6

,6)

and

(10,

12)

(x�

8)2

�(y

�9)

2�

13

5.ce

nte

r (3

,6),

tan

gen

t to

th

e x-

axis

(x�

3)2

�(y

�6)

2�

36

6.ce

nte

r (�

4,�

7),t

ange

nt

to x

�2

(x�

4)2

�(y

�7)

2�

36

7.ce

nte

r at

(�

2,8)

,tan

gen

t to

y�

�4

(x�

2)2

�(y

�8)

2�

144

8.ce

nte

r (7

,7),

pass

es t

hro

ugh

(12

,9)

(x�

7)2

�(y

�7)

2�

29

9.en

dpoi

nts

of

a di

amet

er a

re (

�4,

�2)

an

d (8

,4)

(x�

2)2

�(y

�1)

2�

45

10.e

ndp

oin

ts o

f a

diam

eter

are

(�

4,3)

an

d (6

,�8)

(x�

1)2

�(y

�2.

5)2

�55

.25

2 � 22 � 2

5 �

(�3)

�� 2

�4

�6

�2

y 1�

y 2�

2x 1

�x 2

�2

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill46

8G

lenc

oe A

lgeb

ra 2

Gra

ph

Cir

cles

To

grap

h a

cir

cle,

wri

te t

he

give

n e

quat

ion

in

th

e st

anda

rd f

orm

of

the

equ

atio

n o

f a

circ

le,(

x�

h)2

�(y

�k)

2�

r2.

Plo

t th

e ce

nte

r (h

,k)

of t

he

circ

le.T

hen

use

rto

cal

cula

te a

nd

plot

th

e fo

ur

poin

ts (

h�

r,k)

,(h

�r,

k),(

h,k

�r)

,an

d (h

,k�

r),w

hic

h a

re a

ll p

oin

ts o

n t

he

circ

le.S

ketc

h t

he

circ

le t

hat

goes

th

rou

gh t

hos

e fo

ur

poin

ts.

Fin

d t

he

cen

ter

and

rad

ius

of t

he

circ

le

wh

ose

equ

atio

n i

s x2

�2x

�y2

�4y

�11

.Th

en g

rap

h

the

circ

le. x2

�2x

�y2

�4y

�11

x2�

2x�

■�

y2�

4y�

■�

11 �

■

x2�

2x�

1 �

y2�

4y�

4 �

11 �

1 �

4(x

�1)

2�

(y�

2)2

�16

Th

eref

ore,

the

circ

le h

as i

ts c

ente

r at

(�

1,�

2) a

nd

a ra

diu

s of

�

16��

4.F

our

poin

ts o

n t

he

circ

le a

re (

3,�

2),(

�5,

�2)

,(�

1,2)

,an

d (�

1,�

6).

Fin

d t

he

cen

ter

and

rad

ius

of t

he

circ

le w

ith

th

e gi

ven

eq

uat

ion

.Th

en g

rap

h t

he

circ

le.

1.(x

�3)

2�

y2�

92.

x2�

(y�

5)2

�4

3.(x

�1)

2�

(y�

3)2

�9

(3,0

),r

�3

(0,�

5),r

�2

(1,�

3),r

�3

4.(x

�2)

2�

(y�

4)2

�16

5.x2

�y2

�10

x�

8y�

16 �

06.

x2�

y2�

4x�

6y�

12

(2,�

4),r

�4

(5,�

4),r

�5

(2,�

3),r

�5

x

y

Ox

y

Ox

y

O

x

y

Ox

y

O

x

y

O

x

y

O

x2 �

2x

� y

2 �

4y

� 1

1

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Cir

cles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Cir

cles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3

©G

lenc

oe/M

cGra

w-H

ill46

9G

lenc

oe A

lgeb

ra 2

Lesson 8-3

Wri

te a

n e

qu

atio

n f

or t

he

circ

le t

hat

sat

isfi

es e

ach

set

of

con

dit

ion

s.

1.ce

nte

r (0

,5),

radi

us

1 u

nit

2.ce

nte

r (5

,12)

,rad

ius

8 u

nit

sx

2�

(y�

5)2

�1

(x�

5)2

�(y

�12

)2�

64

3.ce

nte

r (4

,0),

radi

us

2 u

nit

s4.

cen

ter

(2,2

),ra

diu

s 3

un

its

(x�

4)2

�y

2�

4(x

�2)

2�

(y�

2)2

�9

5.ce

nte

r (4

,�4)

,rad

ius

4 u

nit

s6.

cen

ter

(�6,

4),r

adiu

s 5

un

its

(x�

4)2

�(y

�4)

2�

16(x

�6)

2�

(y�

4)2

�25

7.en

dpoi

nts

of

a di

amet

er a

t (�

12,0

) an

d (1

2,0)

x2

�y

2�

144

8.en

dpoi

nts

of

a di

amet

er a

t (�

4,0)

an

d (�

4,�

6)(x

�4)

2�

(y�

3)2

�9

9.ce

nte

r at

(7,

�3)

,pas

ses

thro

ugh

th

e or

igin

(x�

7)2

�(y

�3)

2�

58

10.c

ente

r at

(�

4,4)

,pas

ses

thro

ugh

(�

4,1)

(x�

4)2

�(y

�4)

2�

9

11.c

ente

r at

(�

6,�

5),t

ange

nt

to y

-axi

s(x

�6)

2�

(y�

5)2

�36

12.c

ente

r at

(5,

1),t

ange

nt

to x

-axi

s(x

�5)

2�

(y�

1)2

�1

Fin

d t

he

cen

ter

and

rad

ius

of t

he

circ

le w

ith

th

e gi

ven

eq

uat

ion

.Th

en g

rap

h t

he

circ

le.

13.x

2�

y2�

914

.(x

�1)

2�

(y�

2)2

�4

15.(

x�

1)2

�y2

�16

(0,0

),3

un

its

(1,2

),2

un

its

(�1,

0),4

un

its

16.x

2�

(y�

3)2

�81

17.(

x�

5)2

�(y

�8)

2�

4918

.x2

�y2

�4y

�32

�0

(0,�

3),9

un

its

(5,�

8),7

un

its

(0,2

),6

un

its

x

y

O4

8

8 4 –4 –8

–4–8

x

y

O4

812

–4 –8 –12

x

y

O6

12

12 6 –6 –12

–6–1

2

x

y

Ox

y

Ox

y

O

©G

lenc

oe/M

cGra

w-H

ill47

0G

lenc

oe A

lgeb

ra 2

Wri

te a

n e

qu

atio

n f

or t

he

circ

le t

hat

sat

isfi

es e

ach

set

of

con

dit

ion

s.

1.ce

nte

r (�

4,2)

,rad

ius

8 u

nit

s2.

cen

ter

(0,0

),ra

diu

s 4

un

its

(x�

4)2

�(y

�2)

2�

64x2

�y2

�16

3.ce

nte

r��

,��

3� �,ra

diu

s 5�

2�u

nit

s4.

cen

ter

(2.5

,4.2

),ra

diu

s 0.

9 u

nit

�x�

�2�

( y�

�3�)

2�

50(x

�2.

5)2

�(y

�4.

2)2

�0.

81

5.en

dpoi

nts

of

a di

amet

er a

t (�

2,�

9) a

nd

(0,�

5)(x

�1)

2�

(y�

7)2

�5

6.ce

nte

r at

(�

9,�

12),

pass

es t

hro

ugh

(�

4,�

5)(x

�9)

2�

(y�

12)2

�74

7.ce

nte

r at

(�

6,5)

,tan

gen

t to

x-a

xis

(x�

6)2

�(y

�5)

2�

25

Fin

d t

he

cen

ter

and

rad

ius

of t

he

circ

le w

ith

th

e gi

ven

eq

uat

ion

.Th

en g

rap

h t

he

circ

le.

8.(x

�3)

2�

y2�

169.

3x2

�3y

2�

1210

.x2

�y2

�2x

�6y

�26

(�3,

0),4

un

its

(0,0

),2

un

its

(�1,

�3)

,6 u

nit

s

11.(

x �

1)2

�y2

�4y

�12

12.x

2�

6x�

y2�

013

.x2

�y2

�2x

�6y

��

1(1

,�2)

,4 u

nit

s(3

,0),

3 u

nit

s(�

1,�

3),3

un

its

WEA

THER

For

Exe

rcis

es 1

4 an

d 1

5,u

se t

he

foll

owin

g in

form

atio

n.

On

aver

age,

the

circ

ular

eye

of

a hu

rric

ane

is a

bout

15

mil

es i

n di

amet

er.G

ale

win

ds c

anaf

fect

an

area

up

to 3

00 m

iles

fro

m t

he s

torm

’s c

ente

r.In

199

2,H

urri

cane

And

rew

dev

asta

ted

sout

hern

Flo

rida

.A s

atel

lite

pho

to o

f And

rew

’s l

andf

all

show

ed t

he c

ente

r of

its

eye

on

one

coor

dina

te s

yste

m c

ould

be

appr

oxim

ated

by

the

poin

t (8

0,26

).

14.W

rite

an

equ

atio

n t

o re

pres

ent

a po

ssib

le b

oun

dary

of A

ndr

ew’s

eye

.(x

�80

)2�

(y�

26)2

�56

.25

15.W

rite

an

equ

atio

n t

o re

pres

ent

a po

ssib

le b

oun

dary

of

the

area

aff

ecte

d by

gal

e w

inds

.(x

�80

)2�

(y�

26)2

�90

,000

x

y

O

x

y

O

x

y

O

x

y

O4

8

4 –4 –8

–4–8

x

y

Ox

y

O

1 � 4

1 � 4

Pra

ctic

e (

Ave

rag

e)

Cir

cles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3



Readin

g t

o L

earn

Math

em

ati

csC

ircl

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3

©G

lenc

oe/M

cGra

w-H

ill47

1G

lenc

oe A

lgeb

ra 2

Lesson 8-3

Pre-

Act

ivit

yW

hy

are

circ

les

imp

orta

nt

in a

ir t

raff

ic c

ontr

ol?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-3

at

the

top

of p

age

426

in y

our

text

book

.

A l

arge

hom

e im

prov

emen

t ch

ain

is

plan

nin

g to

en

ter

a n

ew m

etro

poli

tan

area

an

d n

eeds

to

sele

ct l

ocat

ion

s fo

r it

s st

ores

.Mar

ket

rese

arch

has

sh

own

that

pot

enti

al c

ust

omer

s ar

e w

illi

ng

to t

rave

l u

p to

12

mil

es t

o sh

op a

t on

eof

th

eir

stor

es.H

ow c

an c

ircl

es h

elp

the

man

ager

s de

cide

wh

ere

to p

lace

thei

r st

ore?

Sam

ple

an

swer

:A

sto

re w

ill d

raw

cu

sto

mer

s w

ho

live

insi

de

aci

rcle

wit

h c

ente

r at

th

e st

ore

an

d a

rad

ius

of

12 m

iles.

Th

e m

anag

emen

t sh

ou

ld s

elec

t lo

cati

on

s fo

r w

hic

has

man

y p

eop

le a

s p

oss

ible

live

wit

hin

a c

ircl

e o

f ra

diu

s 12

mile

s ar

ou

nd

on

e o

f th

e st

ore

s.

Rea

din

g t

he

Less

on

1.a.

Wri

te t

he

equ

atio

n o

f th

e ci

rcle

wit

h c

ente

r (h

,k)

and

radi

us

r.(x

�h

)2�

(y�

k)2

�r2

b.

Wri

te t

he

equ

atio

n o

f th

e ci

rcle

wit

h c

ente

r (4

,�3)

an

d ra

diu

s 5.

(x�

4)2

�(y

�3)

2�

25

c.T

he

circ

le w

ith

equ

atio

n (

x �

8)2

�y2

�12

1 h

as c

ente

r an

d ra

diu

s

.

d.

Th

e ci

rcle

wit

h e

quat

ion

(x

�10

)2�

(y�

10)2

�1

has

cen

ter

and

radi

us

.

2.a.

In o

rder

to

find

cen

ter

and

radi

us o

f th

e ci

rcle

wit

h eq

uati

on x

2�

y2�

4x�

6y�

3 �

0,

it i

s n

eces

sary

to

.Fil

l in

th

e m

issi

ng

part

s of

th

ispr

oces

s.

x2�

y2�

4x�

6y�

3 �

0

x2�

y2�

4x�

6y�

x2�

4x�

�y2

�6y

��

��

(x�

)2�

(y�

)2�

b.

Th

is c

ircl

e h

as r

adiu

s 4

and

cen

ter

at

.

Hel

pin

g Y

ou

Rem

emb

er

3.H

ow c

an t

he

dist

ance

for

mu

la h

elp

you

to

rem

embe

r th

e eq

uat

ion

of

a ci

rcle

?S

amp

le a

nsw

er:W

rite

th

e d

ista

nce

fo

rmu

la.R

epla

ce (

x 1,y

1) w

ith

(h

,k)

and

(x

2,y 2

) w

ith

(x,

y).

Rep

lace

dw

ith

r.S

qu

are

bo

th s

ides

.No

w y

ou

hav

e th

e eq

uat

ion

of

a ci

rcle

.

(�2,

3)

163

29

43

94

3

com

ple

te t

he

squ

are

1(1

0,�

10)

11(�

8,0)

©G

lenc

oe/M

cGra

w-H

ill47

2G

lenc

oe A

lgeb

ra 2

Tan

gen

ts t

o C

ircl

esA

lin

e th

at i

nte

rsec

ts a

cir

cle

in e

xact

ly o

ne

poin

t is

a

tan

gen

tto

th

e ci

rcle

.In

th

e di

agra

m,l

ine

�is

ta

nge

nt

to t

he

circ

le w

ith

equ

atio

n x

2�

y2�

25 a

t th

e po

int

wh

ose

coor

din

ates

are

(3,

4).

A l

ine

is t

ange

nt

to a

cir

cle

at a

poi

nt

Pon

th

e ci

rcle

if

an

d on

ly i

f th

e li

ne

is p

erpe

ndi

cula

r to

th

e ra

diu

s fr

om t

he

cen

ter

of t

he

circ

le t

o po

int

P.T

his

fac

t en

able

s yo

u t

o fi

nd

an e

quat

ion

of

the

tan

gen

t to

a

circ

le a

t a

poin

t P

if y

ou k

now

an

equ

atio

n f

or t

he

circ

le a

nd

the

coor

din

ates

of

P.

Use

th

e d

iagr

am a

bov

e to

sol

ve e

ach

pro

ble

m.

1.W

hat

is

the

slop

e of

th

e ra

diu

s to

th

e po

int

wit

h c

oord

inat

es (

3,4)

? W

hat

is

the

slop

e of

th

e ta

nge

nt

to t

hat

poi

nt?

�4 3� ,�

�3 4�

2.F

ind

an e

quat

ion

of

the

lin

e �

that

is

tan

gen

t to

th

e ci

rcle

at

(3,4

).

y�

��3 4� x

��2 45 �

3.If

kis

a r

eal

nu

mbe

r be

twee

n �

5 an

d 5,

how

man

y po

ints

on

th

e ci

rcle

hav

e x-

coor

din

ate

k? S

tate

th

e co

ordi

nat

es o

f th

ese

poin

ts i

n t

erm

s of

k.

two

,(k,

��

25 �

�k

2 �)

4.D

escr

ibe

how

you

can

fin

d eq

uat

ion

s fo

r th

e ta

nge

nts

to

the

poin

ts y

ou n

amed

for

Exe

rcis

e 3.

Use

th

e co

ord

inat

es o

f (0

,0)

and

of

on

e o

f th

e g

iven

po

ints

.Fin

d t

he

slo

pe

of

the

rad

ius

to t

hat

po

int.

Use

th

e sl

op

e o

f th

e ra

diu

s to

fin

d w

hat

the

slo

pe

of

the

tan

gen

t m

ust

be.

Use

th

e sl

op

e o

f th

e ta

ng

ent

and

th

eco

ord

inat

es o

f th

e p

oin

t o

n t

he

circ

le t

o f

ind

an

eq

uat

ion

fo

r th

e ta

ng

ent.

5.F

ind

an e

quat

ion

for

th

e ta

nge

nt

at (

�3,

4).

y�

�3 4� x�

�2 x5 �

5

–5

–5

5

(3, 4

)

y

xO

�x2

� y

2 �

25

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Elli

pse

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4

©G

lenc

oe/M

cGra

w-H

ill47

3G

lenc

oe A

lgeb

ra 2

Lesson 8-4

Equ

atio

ns

of

Ellip

ses

An

ell

ipse

is t

he

set

of a

ll p

oin

ts i

n a

pla

ne

such

th

at t

he

sum

of t

he

dist

ance

s fr

om t

wo

give

n p

oin

ts i

n t

he

plan

e,ca

lled

th

e fo

ci,i

s co

nst

ant.

An

ell

ipse

has

tw

o ax

es o

f sy

mm

etry

wh

ich

con

tain

th

e m

ajor

and

min

or a

xes.

In t

he

tabl

e,th

ele

ngt

hs

a,b,

and

car

e re

late

d by

th

e fo

rmu

la c

2�

a2�

b2 .

Sta

nd

ard

Fo

rm o

f E

qu

atio

n�

�1

��

1

Cen

ter

(h,

k)(h

, k)

Dir

ecti

on

of

Maj

or

Axi

sH

oriz

onta

lV

ertic

al

Fo

ci(h

�c,

k),

(h

�c,

k)

(h,

k�

c),

(h,

k�

c)

Len

gth

of

Maj

or

Axi

s2a

units

2aun

its

Len

gth

of

Min

or

Axi

s2b

units

2bun

its

Wri

te a

n e

qu

atio

n f

or t

he

elli

pse

sh

own

.

Th

e le

ngt

h o

f th

e m

ajor

axi

s is

th

e di

stan

ce b

etw

een

(�

2,�

2)

and

(�2,

8).T

his

dis

tan

ce i

s 10

un

its.

2a�

10,s

o a

�5

Th

e fo

ci a

re l

ocat

ed a

t (�

2,6)

an

d (�

2,0)

,so

c�

3.b2

�a2

�c2

�25

�9

�16

Th

e ce

nte

r of

th

e el

lips

e is

at

(�2,

3),s

o h

��

2,k

�3,

a2�

25,a

nd

b2�

16.T

he

maj

or a

xis

is v

erti

cal.

An

equ

atio

n o

f th

e el

lips

e is

�

�1.

Wri

te a

n e

qu

atio

n f

or t

he

elli

pse

th

at s

atis

fies

eac

h s

et o

f co

nd

itio

ns.

1.en

dpoi

nts

of m

ajor

axi

s at

(�

7,2)

and

(5,

2),e

ndpo

ints

of

min

or a

xis

at (

�1,

0) a

nd (

�1,

4)

��

1

2.m

ajor

axi

s 8

unit

s lo

ng a

nd p

aral

lel t

o th

e x-

axis

,min

or a

xis

2 un

its

long

,cen

ter

at (

�2,

�5)

�(y

�5)

2�

1

3.en

dpoi

nts

of

maj

or a

xis

at (

�8,

4) a

nd

(4,4

),fo

ci a

t (�

3,4)

an

d (�

1,4)

��

1

4.en

dpoi

nts

of m

ajor

axi

s at

(3,2

) and

(3,�

14),

endp

oint

s of

min

or a

xis

at (�

1,�

6) a

nd (7

,�6)

��

1

5.m

inor

axi

s 6

unit

s lo

ng a

nd p

aral

lel

to t

he x

-axi

s,m

ajor

axi

s 12

uni

ts l

ong,

cent

er a

t (6

,1)

��

1(x

�6)

2�

9(y

�1)

2�

36

(x�

3)2

�16

(y�

6)2

�64

(y�

4)2

�35

(x�

2)2

�36

(x�

2)2

�16

(y�

2)2

�4

(x�

1)2

�36

(x�

2)2

�16

(y�

3)2

�25

x

F 1 F 2O

y

(x�

h)2

�b

2

(y�

k)2

�a

2(y

�k)

2�

b2

(x�

h)2

�a

2

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill47

4G

lenc

oe A

lgeb

ra 2

Gra

ph

Elli

pse

sT

o gr

aph

an

ell

ipse

,if

nec

essa

ry,w

rite

th

e gi

ven

equ

atio

n i

n t

he

stan

dard

for

m o

f an

equ

atio

n f

or a

n e

llip

se.

��

1 (f

or e

llip

se w

ith

maj

or a

xis

hor

izon

tal)

or

��

1 (f

or e

llip

se w

ith

maj

or a

xis

vert

ical

)

Use

th

e ce

nte

r (h

,k)

and

the

endp

oin

ts o

f th

e ax

es t

o pl

ot f

our

poin

ts o

f th

e el

lips

e.T

o m

ake

a m

ore

accu

rate

gra

ph,u

se a

cal

cula

tor

to f

ind

som

e ap

prox

imat

e va

lues

for

xan

d y

that

sati

sfy

the

equ

atio

n.

Gra

ph

th

e el

lip

se 4

x2

�6y

2�

8x�

36y

��

34.

4x2

�6y

2�

8x�

36y

��

344x

2�

8x�

6y2

�36

y�

�34

4(x2

�2x

�■

) �

6(y2

�6y

�■

) �

�34

�■

4(x2

�2x

�1)

�6(

y2�

6y�

9) �

�34

�58

4(x

�1)

2�

6(y

�3)

2�

24

��

1

Th

e ce

nte

r of

th

e el

lips

e is

(�

1,3)

.Sin

ce a

2�

6,a

��

6�.S

ince

b2

�4,

b�

2.T

he

len

gth

of

the

maj

or a

xis

is 2

�6�,

and

the

len

gth

of

the

min

or a

xis

is 4

.Sin

ce t

he

x-te

rmh

as t

he

grea

ter

den

omin

ator

,th

e m

ajor

axi

s is

hor

izon

tal.

Plo

t th

e en

dpoi

nts

of

the

axes

.T

hen

gra

ph t

he

elli

pse.

Fin

d t

he

coor

din

ates

of

the

cen

ter

and

th

e le

ngt

hs

of t

he

maj

or a

nd

min

or a

xes

for

the

elli

pse

wit

h t

he

give

n e

qu

atio

n.T

hen

gra

ph

th

e el

lip

se.

1.�

�1

(0,0

),4�

3�,6

2.�

�1

(0,0

),10

,4

3.x2

�4y

2�

24y

��

32(0

,�3)

,4,2

4.9x

2�

6y2

�36

x�

12y

�12

(2,�

1),6

,2�

6�

x

y

Ox

y

O

x

y

Ox

y

O

y2� 4

x2� 25

x2� 9

y2� 12

(y�

3)2

�4

(x�

1)2

�6

xO

y

4x2

� 6

y2 �

8x

� 3

6y �

�34

(x�

h)2

�b2

(y�

k)2

�a2

(y�

k)2

�b2

(x�

h)2

�a2

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Elli

pse

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Elli

pse

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4

©G

lenc

oe/M

cGra

w-H

ill47

5G

lenc

oe A

lgeb

ra 2

Lesson 8-4

Wri

te a

n e

qu

atio

n f

or e

ach

ell

ipse

.

1.2.

3.

��

1�

�1

��

1

Wri

te a

n e

qu

atio

n f

or t

he

elli

pse

th

at s

atis

fies

eac

h s

et o

f co

nd

itio

ns.

4.en

dpoi

nts

of

maj

or a

xis

5.

endp

oin

ts o

f m

ajor

axi

s6.

endp

oin

ts o

f m

ajor

axi

s at

(0,

6) a

nd

(0,�

6),

at (

2,6)

an

d (8

,6),

at (

7,3)

an

d (7

,9),

endp

oin

ts o

f m

inor

axi

s en

dpoi

nts

of

min

or a

xis

endp

oin

ts o

f m

inor

axi

s

at (

�3,

0) a

nd

(3,0

)at

(5,

4) a

nd

(5,8

)at

(5,

6) a

nd

(9,6

)

��

1�

�1

��

1

7.m

ajor

axi

s 12

un

its

lon

g 8.

endp

oin

ts o

f m

ajor

axi

s 9.

endp

oin

ts o

f m

ajor

axi

s at

and

para

llel

to

x-ax

is,

at (

�6,

0) a

nd

(6,0

),fo

ci

(0,1

2) a

nd

(0,�

12),

foci

at

min

or a

xis

4 u

nit

s lo

ng,

at ( �

�32�

,0) a

nd

(�32�

,0)

( 0,�

23�) a

nd

( 0,�

�23�

)ce

nte

r at

(0,

0)

��

1�

�1

��

1

Fin

d t

he

coor

din

ates

of

the

cen

ter

and

foc

i an

d t

he

len

gth

s of

th

e m

ajor

an

dm

inor

axe

s fo

r th

e el

lip

se w

ith

th

e gi

ven

eq

uat

ion

.Th

en g

rap

h t

he

elli

pse

.

10.

��

111

.�

�1

12.

��

1

(0,0

);( 0

,��

19�) ;

(0,0

);( �

6�2�,

0);

(0,0

),( 0

,�2�

6�);

20;

1818

;6

14;

10

x

y

O4

8

8 4 –4 –8

–4–8

x

y

O4

8

8 4 –4 –8

–4–8

x

y

O4

8

8 4 –4 –8

–4–8

x2� 25

y2� 49

y2� 9

x2� 81

x2� 81

y2� 10

0

x2

� 121

y2

� 144

y2

� 4x

2� 36

y2

� 4x

2� 36

(x�

7)2

�4

(y�

6)2

�9

(y �

6)2

�4

(x �

5)2

�9

x2

� 9y

2� 36

(y�

2)2

�9

x2

� 16x

2� 16

y2

� 25y

2� 4

x2

� 9

xO

y( 0

, 5)

( 0, –

1)

( –4,

2)

( 4, 2

)

xO

y ( 0, 3

)

( 0, –

3)

( 0, –

5)

( 0, 5

)

xO

y ( 0, 2

)

( 0, –

2)

( –3,

0)

( 3, 0

)

©G

lenc

oe/M

cGra

w-H

ill47

6G

lenc

oe A

lgeb

ra 2

Wri

te a

n e

qu

atio

n f

or e

ach

ell

ipse

.

1.2.

3.

��

1�

�1

��

1

Wri

te a

n e

qu

atio

n f

or t

he

elli

pse

th

at s

atis

fies

eac

h s

et o

f co

nd

itio

ns.

4.en

dpoi

nts

of

maj

or a

xis

5.

endp

oint

s of

maj

or a

xis

6.

maj

or a

xis

20 u

nit

s lo

ng

at (

�9,

0) a

nd

(9,0

),at

(4,

2) a

nd

(4,�

8),

and

para

llel

to

x-ax

is,

endp

oin

ts o

f m

inor

axi

s

endp

oint

s of

min

or a

xis

m

inor

axi

s 10

un

its

lon

g,at

(0,

3) a

nd

(0,�

3)at

(1,

�3)

and

(7,

�3)

cen

ter

at (

2,1)

��

1�

�1

��

1

7.m

ajor

axi

s 10

un

its

lon

g,8.

maj

or a

xis

16 u

nit

s lo

ng,

9.en

dpoi

nts

of

min

or a

xis

min

or a

xis

6 u

nit

s lo

ng

ce

nte

r at

(0,

0),f

oci

at

at (

0,2)

an

d (0

,�2)

,foc

i an

d pa

rall

el t

o x-

axis

,( 0

,2�

15�) a

nd

( 0,�

2�15�

)at

(�

4,0)

an

d (4

,0)

cen

ter

at (

2,�

4)

��

1�

�1

��

1

Fin

d t

he

coor

din

ates

of

the

cen

ter

and

foc

i an

d t

he

len

gth

s of

th

e m

ajor

an

dm

inor

axe

s fo

r th

e el

lip

se w

ith

th

e gi

ven

eq

uat

ion

.Th

en g

rap

h t

he

elli

pse

.

10.

��

111

.�

�1

12.

��

1

(0,0

);( 0

,��

7�);

8;6

(3,1

);( 3

,1 �

�35�

) ;(�

4,�

3);

12;

2 ( �

4 �

2�6�,

�3)

;14

;10

13.S

PORT

SA

n ic

e sk

ater

tra

ces

two

cong

ruen

t el

lipse

s to

for

m a

fig

ure

eigh

t.A

ssum

e th

at t

hece

nter

of

the

firs

t lo

op i

s at

the

ori

gin,

wit

h th

e se

cond

loo

p to

its

rig

ht.W

rite

an

equa

tion

to m

odel

th

e fi

rst

loop

if

its

maj

or a

xis

(alo

ng

the

x-ax

is)

is 1

2 fe

et l

ong

and

its

min

orax

is i

s 6

feet

lon

g.W

rite

an

oth

er e

quat

ion

to

mod

el t

he

seco

nd

loop

.

��

1;�

�1

y2

� 9(x

�12

)2�

� 36y

2� 9

x2

� 36

4

4 –4 –8 –12

–4–8

x

y

O

x

y

O4

8

8 4 –4 –8

–4–8

x

y

O

(y�

3)2

�25

(x�

4)2

�49

(x�

3)2

�1

(y�

1)2

�36

x2� 9

y2� 16

y2

� 4x

2� 20

x2

� 4y

2� 64

(x�

2)2

�9

(y�

4)2

�25

(y �

1)2

�25

(x �

2)2

�10

0(x

�4)

2�

9(y

�3)

2�

25y

2� 9

x2

� 81

(y�

3)2

�9

(x �

1)2

�25

x2

� 4(y

�2)

2�

9y

2� 9

x2

� 121

xO

y

( –5,

3)

( –6,

3)

( 3, 3

)

( 4, 3

)

xO

y

( 0, 2

� �

�5)

( 0, 2

� �

�5)

( 0, –

1)

( 0, 5

)

xO

y( 0

, 3)

( 0, –

3)

( –11

, 0)

( 11,

0)

612

2 –2

–6–1

2

Pra

ctic

e (

Ave

rag

e)

Elli

pse

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csE

llip

ses

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4

©G

lenc

oe/M

cGra

w-H

ill47

7G

lenc

oe A

lgeb

ra 2

Lesson 8-4

Pre-

Act

ivit

yW

hy

are

elli

pse

s im

por

tan

t in

th

e st

ud

y of

th

e so

lar

syst

em?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-4

at

the

top

of p

age

433

in y

our

text

book

.

Is t

he

Ear

th a

lway

s th

e sa

me

dist

ance

fro

m t

he

Su

n?

Exp

lain

you

r an

swer

usi

ng

the

wor

ds c

ircl

ean

d el

lips

e.N

o;

if t

he

Ear

th’s

orb

it w

ere

aci

rcle

,it

wo

uld

alw

ays

be

the

sam

e d

ista

nce

fro

m t

he

Su

nb

ecau

se e

very

po

int

on

a c

ircl

e is

th

e sa

me

dis

tan

ce f

rom

th

ece

nte

r.H

ow

ever

,th

e E

arth

’s o

rbit

is a

n e

llip

se,a

nd

th

e p

oin

tso

n a

n e

llip

se a

re n

ot

all t

he

sam

e d

ista

nce

fro

m t

he

cen

ter.

Rea

din

g t

he

Less

on

1.A

n e

llip

se i

s th

e se

t of

all

poi

nts

in

a p

lan

e su

ch t

hat

th

e of

th

e

dist

ance

s fr

om t

wo

fixe

d po

ints

is

.Th

e tw

o fi

xed

poin

ts a

re c

alle

d th

e

of t

he

elli

pse.

2.C

onsi

der

the

elli

pse

wit

h e

quat

ion

�

�1.

a.F

or t

his

equ

atio

n,a

�an

d b

�.

b.

Wri

te a

n e

quat

ion

th

at r

elat

es t

he

valu

es o

f a,

b,an

d c.

c2

�a

2�

b2

c.F

ind

the

valu

e of

cfo

r th

is e

llip

se.

�5�

3.C

onsi

der

the

elli

pses

wit

h e

quat

ion

s �

�1

and

��

1.C

ompl

ete

the

foll

owin

g ta

ble

to d

escr

ibe

char

acte

rist

ics

of t

hei

r gr

aph

s.

Sta

nd

ard

Fo

rm o

f E

qu

atio

n�

�1

��

1

Dir

ecti

on

of

Maj

or

Axi

sve

rtic

alh

ori

zon

tal

Dir

ecti

on

of

Min

or

Axi

sh

ori

zon

tal

vert

ical

Fo

ci(0

,3),

(0,�

3)( �

5�,0)

,(�

�5�,

0)

Len

gth

of

Maj

or

Axi

s10

un

its

6 u

nit

s

Len

gth

of

Min

or

Axi

s8

un

its

4 u

nit

s

Hel

pin

g Y

ou

Rem

emb

er4.

Som

e st

ude

nts

hav

e tr

oubl

e re

mem

beri

ng

the

two

stan

dard

for

ms

for

the

equ

atio

n o

f an

elli

pse.

How

can

you

rem

embe

r w

hic

h t

erm

com

es f

irst

an

d w

her

e to

pla

ce a

an

d b

inth

ese

equ

atio

ns?

Th

e x-

axis

is h

ori

zon

tal.

If t

he

maj

or

axis

is h

ori

zon

tal,

the

firs

t te

rm is

.T

he

y-ax

is is

ver

tica

l.If

th

e m

ajo

r ax

is is

ver

tica

l,th

e

firs

t te

rm is

.a

is a

lway

s th

e la

rger

of

the

nu

mb

ers

aan

d b

.y

2� a

2

x2

� a2

y2

� 4x2� 9

x2� 16

y2

� 25

y2� 4

x2� 9

x2� 16

y2� 25

23

y2� 4

x2� 9

foci

con

stan

tsu

m

©G

lenc

oe/M

cGra

w-H

ill47

8G

lenc

oe A

lgeb

ra 2

Ecc

entr

icit

y In

an

ell

ipse

,th

e ra

tio

� dc �is

cal

led

the

ecce

ntr

icit

yan

d is

den

oted

by

the

lett

er e

.Ecc

entr

icit

y m

easu

res

the

elon

gati

on o

f an

ell

ipse

.The

clo

ser

eis

to

0,th

e m

ore

an e

llip

se l

ooks

lik

e a

circ

le.T

he

clos

er e

is t

o 1,

the

mor

e el

onga

ted

it i

s.R

ecal

l th

at t

he

equ

atio

n o

f an

ell

ipse

is

� ax2 2��

� by2 2��

1 or

� bx2 2��

� ay2 2��

1

wh

ere

ais

th

e le

ngt

h o

f th

e m

ajor

axi

s,an

d th

at c

��

a2�

b�

2 �.

Fin

d t

he

ecce

ntr

icit

y of

eac

h e

llip

se r

oun

ded

to

the

nea

rest

hu

nd

red

th.

1.�x 92 �

�� 3y 62 �

�1

2.� 8x 12 �

��y 92 �

�1

3.�x 42 �

��y 92 �

�1

0.87

0.94

0.75

4.� 1x 62 �

��y 92 �

�1

5.� 3x 62 �

�� 1y 62 �

�1

6.�x 42 �

�� 3y 62 �

�1

0.66

0.75

0.94

7.Is

a c

ircl

e an

ell

ipse

? E

xpla

in y

our

reas

onin

g.

Yes;

it is

an

elli

pse

wit

h e

ccen

tric

ity

0.

8.T

he

cen

ter

of t

he

sun

is

one

focu

s of

Ear

th's

orb

it a

rou

nd

the

sun

.Th

ele

ngt

h o

f th

e m

ajor

axi

s is

186

,000

,000

mil

es,a

nd

the

foci

are

3,2

00,0

00m

iles

apa

rt.F

ind

the

ecce

ntr

icit

y of

Ear

th's

orb

it.

app

roxi

mat

ely

0.17

9.A

n a

rtif

icia

l sa

tell

ite

orbi

tin

g th

e ea

rth

tra

vels

at

an a

ltit

ude

th

at v

arie

sbe

twee

n 1

32 m

iles

an

d 58

3 m

iles

abo

ve t

he

surf

ace

of t

he

eart

h.I

f th

ece

nte

r of

th

e ea

rth

is

one

focu

s of

its

ell

ipti

cal

orbi

t an

d th

e ra

diu

s of

th

eea

rth

is

3950

mil

es,w

hat

is

the

ecce

ntr

icit

y of

th

e or

bit?

app

roxi

mat

ely

0.05

2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

Hyp

erb

ola

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5

©G

lenc

oe/M

cGra

w-H

ill47

9G

lenc

oe A

lgeb

ra 2

Lesson 8-5

Equ

atio

ns

of

Hyp

erb

ola

sA

hyp

erb

ola

is t

he

set

of a

ll p

oin

ts i

n a

pla

ne

such

th

atth

e ab

solu

te v

alu

e of

th

e d

iffe

ren

ceof

th

e di

stan

ces

from

an

y po

int

on t

he

hyp

erbo

la t

o an

ytw

o gi

ven

poi

nts

in

th

e pl

ane,

call

ed t

he

foci

,is

con

stan

t.

In t

he

tabl

e,th

e le

ngt

hs

a,b,

and

car

e re

late

d by

th

e fo

rmu

la c

2�

a2�

b2.

Sta

nd

ard

Fo

rm o

f E

qu

atio

n�

�1

��

1

Eq

uat

ion

s o

f th

e A

sym

pto

tes

y �

k�

�(x

�h)

y �

k�

�(x

�h)

Tran

sver

se A

xis

Hor

izon

tal

Ver

tical

Fo

ci(h

�c,

k),

(h

�c,

k)

(h,

k�

c),

(h,

k�

c)

Ver

tice

s(h

�a,

k),

(h

�a,

k)

(h,

k�

a),

(h,

k�

a)

Wri

te a

n e

qu

atio

n f

or t

he

hyp

erb

ola

wit

h v

erti

ces

(�2,

1) a

nd

(6,

1)an

d f

oci

(�4,

1) a

nd

(8,

1).

Use

a s

ketc

h t

o or

ien

t th

e h

yper

bola

cor

rect

ly.T

he

cen

ter

of

the

hyp

erbo

la i

s th

e m

idpo

int

of t

he

segm

ent

join

ing

the

two

vert

ices

.Th

e ce

nte

r is

(,1

),or

(2,

1).T

he

valu

e of

ais

th

e

dist

ance

fro

m t

he

cen

ter

to a

ver

tex,

so a

�4.

Th

e va

lue

of c

is

the

dist

ance

fro

m t

he

cen

ter

to a

foc

us,

so c

�6.

c2�

a2�

b2

62�

42�

b2

b2�

36 �

16 �

20

Use

h,k

,a2 ,

and

b2to

wri

te a

n e

quat

ion

of

the

hyp

erbo

la.

��

1

Wri

te a

n e

qu

atio

n f

or t

he

hyp

erb

ola

that

sat

isfi

es e

ach

set

of

con

dit

ion

s.

1.ve

rtic

es (

�7,

0) a

nd

(7,0

),co

nju

gate

axi

s of

len

gth

10

��

1

2.ve

rtic

es (

�2,

�3)

an

d (4

,�3)

,foc

i (�

5,�

3) a

nd

(7,�

3)�

�1

3.ve

rtic

es (

4,3)

an

d (4

,�5)

,con

juga

te a

xis

of l

engt

h 4

��

1

4.ve

rtic

es (

�8,

0) a

nd

(8,0

),eq

uat

ion

of

asym

ptot

es y

��

x�

�1

5.ve

rtic

es (

�4,

6) a

nd

(�4,

�2)

,foc

i (�

4,10

) an

d (�

4,�

6)�

�1

(x�

4)2

�48

(y�

2)2

�16

9y2

� 16x

2� 64

1 � 6

(x�

4)2

�4

(y�

1)2

�16

(y�

3)2

�27

(x�

1)2

�9

y2

� 25x

2� 49

(y�

1)2

�20

(x�

2)2

�16

�2

�6

�2

x

y

O

a � bb � a

(x�

h)2

�b2

(y�

k)2

�a2

(y�

k)2

�b

2(x

�h)

2�

a2

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill48

0G

lenc

oe A

lgeb

ra 2

Gra

ph

Hyp

erb

ola

sT

o gr

aph

a h

yper

bola

,wri

te t

he

give

n e

quat

ion

in

th

e st

anda

rdfo

rm o

f an

equ

atio

n f

or a

hyp

erbo

la

��

1 if

th

e br

anch

es o

f th

e h

yper

bola

ope

n l

eft

and

righ

t,or

��

1 if

th

e br

anch

es o

f th

e h

yper

bola

ope

n u

p an

d do

wn

Gra

ph t

he

poin

t (h

,k),

wh

ich

is

the

cen

ter

of t

he

hyp

erbo

la.D

raw

a r

ecta

ngl

e w

ith

dim

ensi

ons

2aan

d 2b

and

cen

ter

(h,k

).If

th

e h

yper

bola

ope

ns

left

an

d ri

ght,

the

vert

ices

are

(h�

a,k)

an

d (h

�a,

k).I

f th

e h

yper

bola

ope

ns

up

and

dow

n,t

he

vert

ices

are

(h

,k�

a)an

d (h

,k�

a).

Dra

w t

he

grap

h o

f 6y

2�

4x2

�36

y�

8x�

�26

.

Com

plet

e th

e sq

uar

es t

o ge

t th

e eq

uat

ion

in

sta

nda

rd f

orm

.6y

2�

4x2

�36

y�

8x�

�26

6(y2

�6y

�■

) �

4(x2

�2x

�■

) �

�26

�■

6(y2

�6y

�9)

�4(

x2�

2x�

1) �

�26

�50

6(y

�3)

2�

4(x

�1)

2�

24

��

1

Th

e ce

nte

r of

th

e h

yper

bola

is

(�1,

3).

Acc

ordi

ng

to t

he

equ

atio

n,a

2�

4 an

d b2

�6,

so a

�2

and

b�

�6�.

Th

e tr

ansv

erse

axi

s is

ver

tica

l,so

th

e ve

rtic

es a

re (

�1,

5) a

nd

(�1,

1).D

raw

a r

ecta

ngl

e w

ith

vert

ical

dim

ensi

on 4

an

d h

oriz

onta

l di

men

sion

2�

6��

4.9.

Th

e di

agon

als

of t

his

rec

tan

gle

are

the

asym

ptot

es.T

he

bran

ches

of

the

hyp

erbo

la o

pen

up

and

dow

n.U

se t

he

vert

ices

an

dth

e as

ympt

otes

to

sket

ch t

he

hyp

erbo

la.

Fin

d t

he

coor

din

ates

of

the

vert

ices

an

d f

oci

and

th

e eq

uat

ion

s of

th

e as

ymp

tote

sfo

r th

e h

yper

bol

a w

ith

th

e gi

ven

eq

uat

ion

.Th

en g

rap

h t

he

hyp

erb

ola.

1.�

�1

2.(y

�3)

2�

�1

3.�

�1

(2,0

),(�

2,0)

;(�

2,4)

,(�

2,2)

;(0

,4),

(0,�

4);

( 2�

5�,0)

,(�

2�5�,

0);

( �2,

3 �

�10�

) ,(0

,5),

(0,�

5);

y�

�2x

( �2,

3 �

�10�

) ;y

��

x

y �

x �

3,

y �

�x

�2

1 � 31 � 3

2 � 31 � 3

xO

y

4 � 3

x2� 9

y2� 16

(x�

2)2

�9

y2� 16

x2� 4

(x�

1)2

�6

(y�

3)2

�4

xO

y

(x�

h)2

�b2

(y�

k)2

�a2

(y�

k)2

�� b2

(x�

h)2

�a2

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Hyp

erb

ola

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises

xO

y

xO

y


An

swer

s


Skil

ls P

ract

ice

Hyp

erb

ola

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5

©G

lenc

oe/M

cGra

w-H

ill48

1G

lenc

oe A

lgeb

ra 2

Lesson 8-5

Wri

te a

n e

qu

atio

n f

or e

ach

hyp

erb

ola.

1.2.

3.

��

1�

�1

��

1

Wri

te a

n e

qu

atio

n f

or t

he

hyp

erb

ola

that

sat

isfi

es e

ach

set

of

con

dit

ion

s.

4.ve

rtic

es (

�4,

0) a

nd

(4,0

),co

nju

gate

axi

s of

len

gth

8�

�1

5.ve

rtic

es (

0,6)

an

d (0

,�6)

,con

juga

te a

xis

of l

engt

h 1

4�

�1

6.ve

rtic

es (

0,3)

an

d (0

,�3)

,con

juga

te a

xis

of l

engt

h 1

0�

�1

7.ve

rtic

es (

�2,

0) a

nd

(2,0

),co

nju

gate

axi

s of

len

gth

4�

�1

8.ve

rtic

es (

�3,

0) a

nd

(3,0

),fo

ci (

�5,

0)�

�1

9.ve

rtic

es (

0,2)

an

d (0

,�2)

,foc

i (0

,�3)

��

1

10.v

erti

ces

(0,�

2) a

nd

(6,�

2),f

oci

(3 �

�13�

,�2)

��

1

Fin

d t

he

coor

din

ates

of

the

vert

ices

an

d f

oci

and

th

e eq

uat

ion

s of

th

e as

ymp

tote

sfo

r th

e h

yper

bol

a w

ith

th

e gi

ven

eq

uat

ion

.Th

en g

rap

h t

he

hyp

erb

ola.

11.

��

112

.�

�1

13.

��

1

(�3,

0);

( �3�

5�,0)

;(0

,�7)

;( 0

,��

58�) ;

(�4,

0);

( ��

17�,0

) ;y

��

2xy

��

xy

��

x

xO

y

48

8 4 –4 –8

–4–8

xO

y

48

8 4 –4 –8

–4–8

xO

y

1 � 47 � 3

y2� 1

x2� 16

x2� 9

y2� 49

y2� 36

x2� 9

(y�

2)2

�4

(x�

3)2

�9

x2� 5

y2

� 4

y2

� 16x2� 9

y2

� 4x

2� 4

x2

� 25y

2� 9

x2

� 49y

2� 36

y2

� 16x

2� 16

y2

� 25x

2� 4

x2

� 25y

2� 36

y2

� 16x

2� 25

x

y

O

( ��29

, 0)

( –�

�29, 0

)

( 2, 0

)(–

2, 0

)

48

8 4 –4 –8

–4–8

x

y

O

( 0, �

�61)

( 0, –

��61

)

( 0, 6

)

(0, –

6)48

8 4 –4 –8

–4–8

x

y

O

( ��41

, 0)

( –�

�41, 0

)

( 5, 0

)

(–5,

0)

48

8 4 –4 –8

–4–8

©G

lenc

oe/M

cGra

w-H

ill48

2G

lenc

oe A

lgeb

ra 2

Wri

te a

n e

qu

atio

n f

or e

ach

hyp

erb

ola.

1.2.

3.

��

1�

�1

��

1

Wri

te a

n e

qu

atio

n f

or t

he

hyp

erb

ola

that

sat

isfi

es e

ach

set

of

con

dit

ion

s.

4.ve

rtic

es (

0,7)

an

d (0

,�7)

,con

juga

te a

xis

of l

engt

h 1

8 u

nit

s�

�1

5.ve

rtic

es (

1,�

1) a

nd

(1,�

9),c

onju

gate

axi

s of

len

gth

6 u

nit

s�

�1

6.ve

rtic

es (

�5,

0) a

nd

(5,0

),fo

ci (�

�26�

,0)

��

1

7.ve

rtic

es (

1,1)

an

d (1

,�3)

,foc

i (1

,�1

��

5�)�

�1

Fin

d t

he

coor

din

ates

of

the

vert

ices

an

d f

oci

and

th

e eq

uat

ion

s of

th

e as

ymp

tote

sfo

r th

e h

yper

bol

a w

ith

th

e gi

ven

eq

uat

ion

.Th

en g

rap

h t

he

hyp

erb

ola.

8.�

�1

9.�

�1

10.

��

1

(0,�

4);

( 0,�

2�5�)

;(1

,3),

(1,1

);(3

,0),

(3,�

4);

y�

�2x

( 1,2

��

5�);

( 3,�

2 �

2�2�)

;

y�

2 �

�(x

�1)

y�

2 �

�(x

�3)

11.A

STR

ON

OM

YA

stro

nom

ers

use

spe

cial

X-r

ay t

eles

cope

s to

obs

erve

th

e so

urc

es o

fce

lest

ial

X r

ays.

Som

e X

-ray

tel

esco

pes

are

fitt

ed w

ith

a m

etal

mir

ror

in t

he

shap

e of

ah

yper

bola

,wh

ich

ref

lect

s th

e X

ray

s to

a f

ocu

s.S

upp

ose

the

vert

ices

of

such

a m

irro

r ar

elo

cate

d at

(�

3,0)

an

d (3

,0),

and

one

focu

s is

loc

ated

at

(5,0

).W

rite

an

equ

atio

n t

hat

mod

els

the

hyp

erbo

la f

orm

ed b

y th

e m

irro

r.�

�1

y2

� 16x

2� 9

xO

y

xO

y

xO

y

48

8 4 –4 –8

–4–8

1 � 2

(x�

3)2

�4

(y�

2)2

�4

(x �

1)2

�4

(y�

2)2

�1

x2� 4

y2� 16

(x�

1)2

�1

(y�

1)2

�4y

2� 1

x2

� 25

(x�

1)2

�9

(y�

5)2

�16

x2

� 81y

2� 49

(y�

2)2

�16

(x �

1)2

�4

(x �

3)2

�25

(y �

2)2

�9

x2� 36

y2

� 9

x

y

O(–

1, –

2)

(1, –

2)

(3, –

2)x

y O

( –3,

2 �

��34

)

( –3,

2 �

��34

)

( –3,

–1)

(–3,

5)

4

8 4 –4

–4–8

x

y

O

( 0, 3

��5)

( 0, –

3��5)

( 0, 3

)

(0, –

3)48

8 4 –4 –8

–4–8

Pra

ctic

e (

Ave

rag

e)

Hyp

erb

ola

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5



Readin

g t

o L

earn

Math

em

ati

csH

yper

bo

las

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5

©G

lenc

oe/M

cGra

w-H

ill48

3G

lenc

oe A

lgeb

ra 2

Lesson 8-5

Pre-

Act

ivit

yH

ow a

re h

yper

bol

as d

iffe

ren

t fr

om p

arab

olas

?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-5

at

the

top

of p

age

441

in y

our

text

book

.

Loo

k at

th

e sk

etch

of

a h

yper

bola

in

th

e in

trod

uct

ion

to

this

les

son

.Lis

tth

ree

way

s in

wh

ich

hyp

erbo

las

are

diff

eren

t fr

om p

arab

olas

.S

amp

le a

nsw

er:

A h

yper

bo

la h

as t

wo

bra

nch

es,w

hile

ap

arab

ola

is o

ne

con

tin

uo

us

curv

e.A

hyp

erb

ola

has

tw

o f

oci

,w

hile

a p

arab

ola

has

on

e fo

cus.

A h

yper

bo

la h

as t

wo

ver

tice

s,w

hile

a p

arab

ola

has

on

e ve

rtex

.

Rea

din

g t

he

Less

on

1.T

he

grap

h a

t th

e ri

ght

show

s th

e h

yper

bola

wh

ose

equ

atio

n i

n s

tan

dard

for

m i

s �

�1.

Th

e po

int

(0,0

) is

th

e of

th

e h

yper

bola

.

Th

e po

ints

(4,

0) a

nd

(�4,

0) a

re t

he

of t

he

hyp

erbo

la.

Th

e po

ints

(5,

0) a

nd

(�5,

0) a

re t

he

of t

he

hyp

erbo

la.

Th

e se

gmen

t co

nn

ecti

ng

(4,0

) an

d (�

4,0)

is

call

ed t

he

axis

.

Th

e se

gmen

t co

nn

ecti

ng

(0,3

) an

d (0

,�3)

is

call

ed t

he

axis

.

Th

e li

nes

y�

xan

d y

��

xar

e ca

lled

th

e .

2.S

tudy

th

e h

yper

bola

gra

phed

at

the

righ

t.

Th

e ce

nte

r is

.

Th

e va

lue

of a

is

.

Th

e va

lue

of c

is

.

To

fin

d b2

,sol

ve t

he

equ

atio

n

��

.

Th

e eq

uat

ion

in

sta

nda

rd f

orm

for

th

is h

yper

bola

is

.

Hel

pin

g Y

ou

Rem

emb

er

3.W

hat

is

an e

asy

way

to

rem

embe

r th

e eq

uat

ion

rel

atin

g th

e va

lues

of

a,b,

and

cfo

r a

hyp

erbo

la?

Th

is e

qu

atio

n lo

oks

just

like

th

e P

yth

ago

rean

Th

eore

m,

alth

ou

gh

th

e va

riab

les

rep

rese

nt

dif

fere

nt

len

gth

s in

a h

yper

bo

la t

han

ina

rig

ht

tria

ng

le.

�x 42 ��

� 1y 22 ��

1

b2

a2

c2

42

(0,0

)

x

y

O

asym

pto

tes

3 � 43 � 4

con

jug

ate

tran

sver

se

foci

vert

ices

cen

tery2� 9

x2� 16

x

y

O( –

4, 0

)( 4

, 0)

( –5,

0)

( 5, 0

)

y �

3 4xy

� –

3 4x

©G

lenc

oe/M

cGra

w-H

ill48

4G

lenc

oe A

lgeb

ra 2

Rec

tan

gu

lar

Hyp

erb

ola

s A

rec

tan

gula

r h

yper

bol

ais

a h

yper

bola

wit

h p

erpe

ndi

cula

r as

ympt

otes

.F

or e

xam

ple,

the

grap

h o

f x2

�y2

�1

is a

rec

tan

gula

r h

yper

bola

.A h

yper

bola

wit

h a

sym

ptot

es t

hat

are

not

per

pen

dicu

lar

is c

alle

d a

non

rect

angu

lar

hyp

erb

ola.

Th

e gr

aph

s of

equ

atio

ns

of t

he

form

xy

�c,

wh

ere

cis

a c

onst

ant,

are

rect

angu

lar

hyp

erbo

las.

Mak

e a

tab

le o

f va

lues

an

d p

lot

poi

nts

to

grap

h e

ach

rec

tan

gula

rh

yper

bol

a b

elow

.Be

sure

to

con

sid

er n

egat

ive

valu

es f

or t

he

vari

able

s.S

ee s

tud

ents

’tab

les.

1.xy

��

42.

xy�

3

3.xy

��

14.

xy�

8

5.M

ake

a co

nje

ctu

re a

bou

t th

e as

ympt

otes

of

rect

angu

lar

hyp

erbo

las.

Th

e co

ord

inat

e ax

es a

re t

he

asym

pto

tes.

x

y

Ox

y

O

x

y

Ox

y

O

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Co

nic

Sec

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

©G

lenc

oe/M

cGra

w-H

ill48

5G

lenc

oe A

lgeb

ra 2

Lesson 8-6

Stan

dar

d F

orm

An

y co

nic

sec

tion

in

th

e co

ordi

nat

e pl

ane

can

be

desc

ribe

d by

an

equ

atio

n o

f th

e fo

rm

Ax2

�B

xy�

Cy2

�D

x�

Ey

�F

�0,

wh

ere

A,B

,an

d C

are

not

all

zer

o.O

ne

way

to

tell

wh

at k

ind

of c

onic

sec

tion

an

equ

atio

n r

epre

sen

ts i

s to

rea

rran

ge t

erm

s an

dco

mpl

ete

the

squ

are,

if n

eces

sary

,to

get

one

of t

he

stan

dard

for

ms

from

an

ear

lier

les

son

.T

his

met

hod

is

espe

cial

ly u

sefu

l if

you

are

goi

ng

to g

raph

th

e eq

uat

ion

.

Wri

te t

he

equ

atio

n 3

x2�

4y2

�30

x�

8y

�59

�0

in s

tan

dar

d f

orm

.S

tate

wh

eth

er t

he

grap

h o

f th

e eq

uat

ion

is

a p

ara

bola

,cir

cle,

elli

pse

,or

hyp

erbo

la.

3x2

�4y

2�

30x

� 8

y�

59�

0O

rigin

al e

quat

ion

3x2

�30

x�

4y2

�8y

��

59Is

olat

e te

rms.

3(x2

�10

x�

■)

�4(

y2�

2y�

■)

��

59 �

■�

■F

acto

r ou

t co

mm

on m

ultip

les.

3(x2

�10

x�

25)

�4(

y2�

2y�

1)�

�59

�3(

25)

� (

�4)

(1)

Com

plet

e th

e sq

uare

s.

3(x

�5)

2�

4(y

�1)

2�

12S

impl

ify.

��

1D

ivid

e ea

ch s

ide

by 1

2.

Th

e gr

aph

of

the

equ

atio

n i

s a

hyp

erbo

la w

ith

its

cen

ter

at (

5,�

1).T

he

len

gth

of

the

tran

sver

se a

xis

is 4

un

its

and

the

len

gth

of

the

con

juga

te a

xis

is 2

�3�

un

its.

Wri

te e

ach

eq

uat

ion

in

sta

nd

ard

for

m.S

tate

wh

eth

er t

he

grap

h o

f th

e eq

uat

ion

is

a p

ara

bola

,cir

cle,

elli

pse

,or

hyp

erbo

la.

1.x2

�y2

�6x

�4y

�3

�0

2.x2

�2y

2�

6x�

20y

�53

�0

(x�

3)2

�(y

�2)

2�

10;

circ

le�

�1;

ellip

se

3.6x

2�

60x

�y

�16

1 �

04.

x2�

y2�

4x�

14y

�29

�0

y�

6(x

�5)

2�

11;

par

abo

la(x

�2)

2�

(y�

7)2

�24

;cir

cle

5.6x

2�

5y2

�24

x�

20y

�56

�0

6.3y

2�

x�

24y

�46

�0

��

1;hy

per

bo

lax

��

3(y

�4)

2�

2;p

arab

ola

7.x2

�4y

2�

16x

�24

y�

36 �

08.

x2�

2y2

�8x

�4y

�2

�0

��

1;hy

per

bo

la�

�1;

ellip

se

9.4x

2�

48x

�y

�15

8 �

010

.3x2

�y2

�48

x�

4y�

184

�0

y�

�4(

x�

6)2

�14

;p

arab

ola

��

1;el

lipse

11.�

3x2

�2y

2�

18x

�20

y�

5 �

012

.x2

�y2

�8x

�2y

�8

�0

��

1;hy

per

bo

la(x

�4)

2�

(y�

1)2

�9;

circ

le(x

�3)

2�

6(y

�5)

2�

9

(y�

2)2

�12

(x�

8)2

�4

(y�

1)2

�8

(x�

4)2

�16

(y�

3)2

�16

(x�

8)2

�64

(y�

2)2

�12

(x�

2)2

�10

(y�

5)2

�3

(x�

3)2

�6

(y�

1)2

�3

(x�

5)2

�4

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill48

6G

lenc

oe A

lgeb

ra 2

Iden

tify

Co

nic

Sec

tio

ns

If y

ou a

re g

iven

an

equ

atio

n o

f th

e fo

rmA

x2�

Bxy

�C

y2�

Dx

�E

y�

F�

0,w

ith

B�

0,yo

u c

an d

eter

min

e th

e ty

pe o

f co

nic

sec

tion

just

by

con

side

rin

g th

e va

lues

of

Aan

d C

.Ref

erto

th

e fo

llow

ing

char

t.

Rel

atio

nsh

ip o

f A

and

CTy

pe

of

Co

nic

Sec

tio

n

A�

0 or

C�

0, b

ut n

ot b

oth.

para

bola

A �

Cci

rcle

Aan

d C

have

the

sam

e si

gn,

but

A

C.

ellip

se

Aan

d C

have

opp

osite

sig

ns.

hype

rbol

a

Wit

hou

t w

riti

ng

the

equ

atio

n i

n s

tan

dar

d f

orm

,sta

te w

het

her

th

egr

aph

of

each

eq

uat

ion

is

a p

ara

bola

,cir

cle,

elli

pse

,or

hyp

erbo

la.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Co

nic

Sec

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

Exam

ple

Exam

ple

a.3x

2�

3y2

�5x

�12

�0

A�

3 an

d C

��

3 h

ave

oppo

site

sig

ns,

soth

e gr

aph

of

the

equ

atio

n i

s a

hyp

erbo

la.

b.y

2�

7y�

2x�

13A

�0,

so t

he

grap

h o

f th

e eq

uat

ion

is

a pa

rabo

la.

Exer

cises

Exer

cises

Wit

hou

t w

riti

ng

the

equ

atio

n i

n s

tan

dar

d f

orm

,sta

te w

het

her

th

e gr

aph

of

each

equ

atio

n i

s a

pa

rabo

la,c

ircl

e,el

lip

se,o

r h

yper

bola

.

1.x2

�17

x�

5y�

82.

2x2

�2y

2�

3x�

4y�

5p

arab

ola

circ

le3.

4x2

�8x

�4y

2�

6y�

104.

8(x

�x2

) �

4(2y

2�

y) �

100

hyp

erb

ola

circ

le5.

6y2

�18

�24

�4x

26.

y�

27x

�y2

ellip

sep

arab

ola

7.x2

�4(

y�

y2)

�2x

�1

8.10

x�

x2�

2y2

�5y

ellip

seel

lipse

9.x

�y2

�5y

�x2

�5

10.1

1x2

�7y

2�

77ci

rcle

hyp

erb

ola

11.3

x2�

4y2

�50

�y2

12.y

2�

8x�

11ci

rcle

par

abo

la

13.9

y2�

99y

�3(

3x�

3x2 )

14.6

x2�

4 �

5y2

�3

circ

lehy

per

bo

la

15.1

11 �

11x2

�10

y216

.120

x2�

119y

2�

118x

�11

7y�

0el

lipse

hyp

erb

ola

17.3

x2�

4y2

�12

18.1

50 �

x2�

120

�y

hyp

erb

ola

par

abo

la



Skil

ls P

ract

ice

Co

nic

Sec

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

©G

lenc

oe/M

cGra

w-H

ill48

7G

lenc

oe A

lgeb

ra 2

Lesson 8-6

Wri

te e

ach

eq

uat

ion

in

sta

nd

ard

for

m.S

tate

wh

eth

er t

he

grap

h o

f th

e eq

uat

ion

is

a p

ara

bola

,cir

cle,

elli

pse

,or

hyp

erbo

la.T

hen

gra

ph

th

e eq

uat

ion

.

1.x2

�25

y2�

25hy

per

bo

la2.

9x2

�4y

2�

36el

lipse

3.x2

�y2

�16

�0

circ

le�

�1

��

1x

2�

y2

�16

4.x2

�8x

�y2

�9

circ

le5.

x2�

2x�

15 �

yp

arab

ola

6.10

0x2

�25

y2�

400 ellip

se(x

�4)

2�

y2

�25

y�

(x�

1)2

�16

��

1

Wit

hou

t w

riti

ng

the

equ

atio

n i

n s

tan

dar

d f

orm

,sta

te w

het

her

th

e gr

aph

of

each

equ

atio

n i

s a

pa

rabo

la,c

ircl

e,el

lip

se,o

r h

yper

bola

.

7.9x

2�

4y2

�36

ellip

se8.

x2�

y2�

25ci

rcle

9.y

�x2

�2x

par

abo

la10

.y�

2x2

�4x

�4

par

abo

la

11.4

y2�

25x2

�10

0hy

per

bo

la12

.16x

2�

y2�

16el

lipse

13.1

6x2

�4y

2�

64hy

per

bo

la14

.5x2

�5y

2�

25ci

rcle

15.2

5y2

�9x

2�

225

ellip

se16

.36y

2�

4x2

�14

4hy

per

bo

la

17.y

�4x

2�

36x

�14

4p

arab

ola

18.x

2�

y2�

144

�0

circ

le

19.(

x�

3)2

�(y

�1)

2�

4ci

rcle

20.2

5y2

�50

y�

4x2

�75

ellip

se

21.x

2�

6y2

�9

�0

hyp

erb

ola

22.x

�y2

�5y

�6

par

abo

la

23.(

x�

5)2

�y2

�10

circ

le24

.25x

2�

10y2

�25

0 �

0el

lipse

x

y

O

xy

O4

8

–4 –8 –12

–16

–4–8

x

y

O4

8

8 4 –4 –8

–4–8

y2

� 16x

2� 4

x

y

Ox

y

Ox

O

y

48

4 2 –2 –4

–4–8

y2� 9

x2� 4

y2

� 1x

2� 25

©G

lenc

oe/M

cGra

w-H

ill48

8G

lenc

oe A

lgeb

ra 2

Wri

te e

ach

eq

uat

ion

in

sta

nd

ard

for

m.S

tate

wh

eth

er t

he

grap

h o

f th

e eq

uat

ion

is

a p

ara

bola

,cir

cle,

elli

pse

,or

hyp

erbo

la.T

hen

gra

ph

th

e eq

uat

ion

.

1.y2

��

3x2.

x2�

y2�

6x�

73.

5x2

�6y

2�

30x

�12

y�

�9

par

abo

laci

rcle

hyp

erb

ola

x�

�y

2(x

�3)

2�

y2�

16�

�1

4.19

6y2

�12

25 �

100x

25.

3x2

�9

�3y

2�

6y6.

9x2

�y2

�54

x�

6y�

�81

ellip

seci

rcle

ellip

se

��

1x

2�

(y�

1)2

�4

��

1

Wit

hou

t w

riti

ng

the

equ

atio

n i

n s

tan

dar

d f

orm

,sta

te w

het

her

th

e gr

aph

of

each

equ

atio

n i

s a

pa

rabo

la,c

ircl

e,el

lip

se,o

r h

yper

bola

.

7.6x

2�

6y2

�36

8.4x

2�

y2�

169.

9x2

�16

y2�

64y

�80

�0

circ

lehy

per

bo

lael

lipse

10.5

x2�

5y2

�45

�0

11.x

2�

2x�

y12

.4y2

�36

x2�

4x �

144

�0

circ

lep

arab

ola

hyp

erb

ola

13.A

STR

ON

OM

YA

sat

elli

te t

rave

ls i

n a

n h

yper

boli

c or

bit.

It r

each

es t

he

vert

ex o

f it

s or

bit

at (

5,0)

an

d th

en t

rave

ls a

lon

g a

path

th

at g

ets

clos

er a

nd

clos

er t

o th

e li

ne

y�

x.

Wri

te a

n e

quat

ion

th

at d

escr

ibes

th

e pa

th o

f th

e sa

tell

ite

if t

he

cen

ter

of i

ts h

yper

boli

cor

bit

is a

t (0

,0).

��

1y

2� 4

x2

� 25

2 � 5x

y

O

x

y

Ox

y

O

(y�

3)2

�9

(x�

3)2

�1

y2

� 6.25

x2

� 12.2

5

xO

y

x

y

Ox

y

O

(y �

1)2

�5

(x �

3)2

�6

1 � 3Pra

ctic

e (

Ave

rag

e)

Co

nic

Sec

tio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csC

on

ic S

ecti

on

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

©G

lenc

oe/M

cGra

w-H

ill48

9G

lenc

oe A

lgeb

ra 2

Lesson 8-6

Pre-

Act

ivit

yH

ow c

an y

ou u

se a

fla

shli

ght

to m

ake

con

ic s

ecti

ons?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-6

at

the

top

of p

age

449

in y

our

text

book

.

Th

e fi

gure

s in

th

e in

trod

uct

ion

sh

ow h

ow a

pla

ne

can

sli

ce a

dou

ble

con

e to

form

th

e co

nic

sec

tion

s.N

ame

the

con

ic s

ecti

on t

hat

is

form

ed i

f th

e pl

ane

slic

es t

he

dou

ble

con

e in

eac

h o

f th

e fo

llow

ing

way

s:

•T

he

plan

e is

par

alle

l to

th

e ba

se o

f th

e do

ubl

e co

ne

and

slic

es t

hro

ugh

one

of t

he

con

es t

hat

for

m t

he

dou

ble

con

e.ci

rcle

•T

he

plan

e is

per

pen

dicu

lar

to t

he

base

of

the

dou

ble

con

e an

d sl

ices

thro

ugh

bot

h o

f th

e co

nes

th

at f

orm

th

e do

ubl

e co

ne.

hyp

erb

ola

Rea

din

g t

he

Less

on

1.N

ame

the

con

ic s

ecti

on t

hat

is

the

grap

h o

f ea

ch o

f th

e fo

llow

ing

equ

atio

ns.

Giv

e th

eco

ordi

nat

es o

f th

e ve

rtex

if

the

con

ic s

ecti

on i

s a

para

bola

an

d of

th

e ce

nte

r if

it

is a

circ

le,a

n e

llip

se,o

r a

hyp

erbo

la.

a.�

�1

ellip

se;

(3,�

5)

b.

x�

�2(

y�

1)2

�7

par

abo

la;

(7,�

1)

c.(x

�5)

2�

(y�

5)2

�1

hyp

erb

ola

;(5

,�5)

d.

(x�

6)2

�(y

�2)

2�

1ci

rcle

;(�

6,2)

2.E

ach

of

the

foll

owin

g is

th

e eq

uat

ion

of

a co

nic

sec

tion

.For

eac

h e

quat

ion

,ide

nti

fy t

he

valu

es o

f A

and

C.T

hen

,wit

hou

t w

riti

ng

the

equ

atio

n i

n s

tan

dard

for

m,s

tate

wh

eth

erth

e gr

aph

of

each

equ

atio

n i

s a

para

bola

,cir

cle,

elli

pse,

or h

yper

bola

.

a.2x

2�

y2�

6x�

8y�

12 �

0A

�;C

�;t

ype

of g

raph

:

b.

2x2

�3x

�2y

�5

�0

A�

;C�

;typ

e of

gra

ph:

c.5x

2�

10x

�5y

2�

20y

�1

�0

A�

;C�

;typ

e of

gra

ph:

d.

x2�

y2�

4x�

2y�

5 �

0A

�;C

�;t

ype

of g

raph

:

Hel

pin

g Y

ou

Rem

emb

er

3.W

hat

is

an e

asy

way

to

reco

gniz

e th

at a

n e

quat

ion

rep

rese

nts

a p

arab

ola

rath

er t

han

one

of t

he

oth

er c

onic

sec

tion

s?

If t

he

equ

atio

n h

as a

n x

2te

rm a

nd

yte

rm b

ut

no

y2

term

,th

en t

he

gra

ph

is a

par

abo

la.L

ikew

ise,

if t

he

equ

atio

n h

as a

y2

term

an

d x

term

bu

t n

ox

2te

rm,t

hen

th

e g

rap

h is

a p

arab

ola

.

hyp

erb

ola

�1

1

circ

le5

5

par

abo

la0

2

ellip

se1

2

(y�

5)2

�15

(x�

3)2

�36

©G

lenc

oe/M

cGra

w-H

ill49

0G

lenc

oe A

lgeb

ra 2

Lo

ciA

loc

us

(plu

ral,

loci

) is

th

e se

t of

all

poi

nts

,an

d on

ly t

hos

e po

ints

,th

at s

atis

fya

give

n s

et o

f co

ndi

tion

s.In

geo

met

ry,f

igu

res

ofte

n a

re d

efin

ed a

s lo

ci.F

orex

ampl

e,a

circ

le is

the

locu

s of

poi

nts

of a

pla

ne t

hat

are

a gi

ven

dist

ance

from

a g

iven

poi

nt.T

he

defi

nit

ion

lea

ds n

atu

rall

y to

an

equ

atio

n w

hos

e gr

aph

is t

he

curv

e de

scri

bed.

Wri

te a

n e

qu

atio

n o

f th

e lo

cus

of p

oin

ts t

hat

are

th

esa

me

dis

tan

ce f

rom

(3,

4) a

nd

y�

�4.

Rec

ogni

zing

tha

t th

e lo

cus

is a

par

abol

a w

ith

focu

s (3

,4)

and

dire

ctri

x y

��

4,yo

u ca

n fi

nd t

hat

h�

3,k

�0,

and

a�

4 w

here

(h,

k) is

the

ver

tex

and

4 un

its

is t

he

dist

ance

fro

m t

he

vert

ex t

o bo

th t

he

focu

s an

d di

rect

rix.

Th

us,

an e

quat

ion

for

th

e pa

rabo

la i

s y

�� 11 6�

(x�

3)2 .

Th

e pr

oble

m a

lso

may

be

appr

oach

ed a

nal

ytic

ally

as

foll

ows:

Let

(x,

y) b

e a

poin

t of

th

e lo

cus.

Th

e di

stan

ce f

rom

(3,

4) t

o (x

,y)

�th

e di

stan

ce f

rom

y�

�4

to (

x,y)

.

�(x

�3

�)2

�(

�y

�4)

�2 ��

�(x

�x

�)2

�(

�y

�(�

�4)

)2�

(x�

3)2

�y2

�8y

�16

�y2

�8y

�16

(x�

3)2

�16

y

� 11 6�(x

�3)

2�

y

Des

crib

e ea

ch l

ocu

s as

a g

eom

etri

c fi

gure

.Th

en w

rite

an

eq

uat

ion

for

the

locu

s.

1.A

ll p

oin

ts t

hat

are

th

e sa

me

dist

ance

fro

m (

0,5)

an

d (4

,5).

line,

x�

2

2.A

ll p

oin

ts t

hat

are

4 u

nit

s fr

om t

he

orig

in.

circ

le,x

2�

y2

�4

3.A

ll p

oin

ts t

hat

are

th

e sa

me

dist

ance

fro

m (

�2,

�1)

an

d x

�2.

par

abo

la,x

�� 81 �

(y2

�2y

�1)

4.T

he

locu

s of

poi

nts

su

ch t

hat

th

e su

m o

f th

e di

stan

ces

from

(�

2,0)

an

d (2

,0)

is 6

.

ellip

se,�

x 92 ��

�y 52 ��

1

5.T

he

locu

s of

poi

nts

su

ch t

hat

th

e ab

solu

te v

alu

e of

th

e d

iffe

ren

ce o

f th

e di

stan

ces

from

(�

3,0)

an

d (3

,0)

is 2

.

hyp

erb

ola

,�x 12 �

��y 82 �

�1

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

Exam

ple

Exam

ple



Stu

dy G

uid

e a

nd I

nte

rven

tion

So

lvin

g Q

uad

rati

c S

yste

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

©G

lenc

oe/M

cGra

w-H

ill49

1G

lenc

oe A

lgeb

ra 2

Lesson 8-7

Syst

ems

of

Qu

adra

tic

Equ

atio

ns

Lik

e sy

stem

s of

lin

ear

equ

atio

ns,

syst

ems

ofqu

adra

tic

equ

atio

ns

can

be

solv

ed b

y su

bsti

tuti

on a

nd

elim

inat

ion

.If

the

grap

hs

are

a co

nic

sect

ion

an

d a

lin

e,th

e sy

stem

wil

l h

ave

0,1,

or 2

sol

uti

ons.

If t

he

grap

hs

are

two

con

icse

ctio

ns,

the

syst

em w

ill

hav

e 0,

1,2,

3,or

4 s

olu

tion

s.

Sol

ve t

he

syst

em o

f eq

uat

ion

s.y

�x

2�

2x�

15x

�y

��

3

Rew

rite

th

e se

con

d eq

uat

ion

as

y�

�x

�3

and

subs

titu

te i

nto

th

e fi

rst

equ

atio

n.

�x

�3

�x2

�2x

�15

0 �

x2�

x�

12A

dd x

�3

to e

ach

side

.

0 �

(x�

4)(x

�3)

Fac

tor.

Use

th

e Z

ero

Pro

duct

pro

pert

y to

get

x�

4 or

x�

�3.

Su

bsti

tute

th

ese

valu

es f

or x

in x

�y

��

3:

4 �

y�

�3

or�

3 �

y�

�3

y�

�7

y�

0

Th

e so

luti

ons

are

(4,�

7) a

nd

(�3,

0).

Fin

d t

he

exac

t so

luti

on(s

) of

eac

h s

yste

m o

f eq

uat

ion

s.

1.y�

x2�

52.

x2�

(y�

5)2

�25

y�x

�3

y�

�x2

(2,�

1),(

�1,

�4)

(0,0

)

3.x2

�(y

�5)

2�

254.

x2�

y2�

9y

�x2

x2�

y�

3

(0,0

),(3

,9),

(�3,

9)(0

,3),

(�5�,

�2)

,(�

�5�,

�2)

5.x2

�y2

�1

6.y

�x

�3

x2�

y2�

16x

�y2

�4

�,

�, �,�

�,�

,�,

��,

�, ��

,��

�,

�1

��

29��

� 27

��

29��

� 2�

30��

2�

34��

2�

30��

2�

34��

2

1 �

�29�

�� 2

7 �

�29�

�� 2

�30�

�2

�34�

�2

�30�

�2

�34�

�2

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill49

2G

lenc

oe A

lgeb

ra 2

Syst

ems

of

Qu

adra

tic

Ineq

ual

itie

sS

yste

ms

of q

uad

rati

c in

equ

alit

ies

can

be

solv

edby

gra

phin

g.

Sol

ve t

he

syst

em o

f in

equ

alit

ies

by

grap

hin

g.x

2�

y2

25

�x�

�2�

y2

Th

e gr

aph

of

x2�

y2

25 c

onsi

sts

of a

ll p

oin

ts o

n o

r in

side

th

e ci

rcle

wit

h c

ente

r (0

,0)

and

radi

us

5.T

he

grap

h o

f

�x�

�2�

y2�

con

sist

s of

all

poi

nts

on

or

outs

ide

the

circ

le w

ith

cen

ter �

,0�a

nd

radi

us

.Th

e so

luti

on o

f th

e

syst

em i

s th

e se

t of

poi

nts

in

bot

h r

egio

ns.

Sol

ve t

he

syst

em o

f in

equ

alit

ies

by

grap

hin

g.x2

�y2

25

��

1

Th

e gr

aph

of

x2�

y2

25 c

onsi

sts

of a

ll p

oin

ts o

n o

r in

side

th

e ci

rcle

wit

h c

ente

r (0

,0)

and

radi

us

5.T

he

grap

h o

f

��

1 ar

e th

e po

ints

“in

side

”bu

t n

ot o

n t

he

bran

ches

of

the

hyp

erbo

la s

how

n.T

he

solu

tion

of

the

syst

em i

s th

e se

t of

poin

ts i

n b

oth

reg

ion

s.

Sol

ve e

ach

sys

tem

of

ineq

ual

itie

s b

elow

by

grap

hin

g.

1.�

1

2.x2

�y2

16

93.

y�

(x�

2)2

y�

x�

2x2

�9y

2�

225

(x�

1)2

�(y

�1)

2

16 x

y

Ox

y

O6

12

12 6 –6 –12

–6–1

2x

y

O

1 � 2

y2� 4

x2� 16

x2� 9

y2� 4

x2 � 9y2 � 4

x

y

O

5 � 25 � 2

25 � 45 � 2

25 � 45 � 2

x

y

O

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

So

lvin

g Q

uad

rati

c S

yste

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

So

lvin

g Q

uad

rati

c S

yste

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

©G

lenc

oe/M

cGra

w-H

ill49

3G

lenc

oe A

lgeb

ra 2

Lesson 8-7

Fin

d t

he

exac

t so

luti

on(s

) of

eac

h s

yste

m o

f eq

uat

ion

s.

1.y

�x

�2

(0,�

2),(

1,�

1)2.

y�

x�

3(�

1,2)

,3.

y�

3x(0

,0)

y�

x2�

2y

�2x

2(1

.5,4

.5)

x�

y2

4.y

�x

( �2�,

�2�)

,5.

x�

�5

(�5,

0)6.

y�

7n

o s

olu

tio

nx2

�y2

�4

( ��

2�,�

�2�)

x2�

y2�

25x2

�y2

�9

7.y

��

2x�

2(2

,�2)

,8.

x�

y�

1 �

0(1

,2)

9.y

�2

�x

(0,2

),(3

,�1)

y2�

2x�

,1�

y2�

4xy

�x2

�4x

�2

10.y

�x

�1

no

so

luti

on

11.y

�3x

2(0

,0)

12.y

�x2

�1

(�1,

2),

y�

x2y

��

3x2

y�

�x2

�3

(1,2

)

13.y

�4x

(�1,

�4)

,(1,

4)14

.y�

�1

(0,�

1)15

.4x2

�9y

2�

36(�

3,0)

,4x

2�

y2�

204x

2�

y2�

1x2

�9y

2�

9(3

,0)

16.3

(y�

2)2

�4(

x�

3)2

�12

17.x

2�

4y2

�4

(�2,

0),

18.y

2�

4x2

�4

no

y

��

2x�

2(0

,2),

(3,�

4)x2

�y2

�4

(2,0

)y

�2x

solu

tio

n

Sol

ve e

ach

sys

tem

of

ineq

ual

itie

s b

y gr

aph

ing.

19.y

3x

�2

20.y

x

21.4

y2�

9x2

�14

4x2

�y2

�16

y�

�2x

2�

4x2

�8y

2�

16

22.G

AR

DEN

ING

An

ell

ipti

cal

gard

en b

ed h

as a

pat

h f

rom

poi

nt

Ato

po

int

B.I

f th

e be

d ca

n b

e m

odel

ed b

y th

e eq

uat

ion

x2

�3y

2�

12

and

the

path

can

be

mod

eled

by

the

lin

e y

��

x,w

hat

are

th

e

coor

din

ates

of

poin

ts A

and

B?

(�3,

1) a

nd

(3,

�1)

1 � 3x

y

B

A

O

x

y

O4

8

8 4 –4 –8

–4–8

x

y

Ox

y

O

1 � 2

©G

lenc

oe/M

cGra

w-H

ill49

4G

lenc

oe A

lgeb

ra 2

Fin

d t

he

exac

t so

luti

on(s

) of

eac

h s

yste

m o

f eq

uat

ion

s.

1.(x

�2)

2�

y2�

52.

x�

2(y

�1)

2�

63.

y2�

3x2

�6

4.x2

�2y

2�

1x

�y

�1

x�

y�

3y

�2x

�1

y�

�x

�1

(0,�

1),(

3,2)

(2,1

),(6

.5,�

3.5)

(�1,

�3)

,(5,

9)(1

,0),

�,

�5.

4y2

�9x

2�

366.

y�

x2�

37.

x2�

y2�

258.

y2�

10 �

6x2

4x2

�9y

2�

36x2

�y2

�9

4y�

3x4y

2�

40 �

2x2

no

so

luti

on

(0,�

3),(

��

5�,2)

(4,3

),(�

4,�

3)( 0

,��

10�)

9.x2

�y2

�25

10.4

x2�

9y2

�36

11.x

��

(y�

3)2

�2

12.

��

1x

�3y

�5

2x2

�9y

2�

18x

�(y

�3)

2�

3x2

�y2

�9

(�5,

0),(

4,3)

(�3,

0)n

o s

olu

tio

n(�

3,0)

13.2

5x2

�4y

2�

100

14.x

2�

y2�

415

.x2

�y2

�3

x�

��

�1

y2�

x2�

3

no

so

luti

on

(�2,

0)n

o s

olu

tio

n

16.

��

117

.x�

2y�

318

.x2

�y2

�64

3x2

�y2

�9

x2�

y2�

9x2

�y2

�8

( �2,

��

3�)(3

,0),

��,

�( �

6,�

2�7�)

Sol

ve e

ach

sys

tem

of

ineq

ual

itie

s b

y gr

aph

ing.

19.y

�x2

20.x

2�

y2�

3621

.�

1

y�

�x

�2

x2�

y2�

16(x

�1)

2�

(y�

2)2

4

22.G

EOM

ETRY

Th

e to

p of

an

iro

n g

ate

is s

hap

ed l

ike

hal

f an

el

lips

e w

ith

tw

o co

ngr

uen

t se

gmen

ts f

rom

th

e ce

nte

r of

th

eel

lips

e to

th

e el

lips

e as

sh

own

.Ass

um

e th

at t

he

cen

ter

ofth

e el

lips

e is

at

(0,0

).If

th

e el

lips

e ca

n b

e m

odel

ed b

y th

eeq

uat

ion

x2

�4y

2�

4 fo

r y

�0

and

the

two

con

gru

ent

segm

ents

can

be

mod

eled

by

y�

xan

d y

��

x,

wh

at a

re t

he

coor

din

ates

of

poin

ts A

and

B?

�3�

�2

�3�

�2

BA

(0, 0

)

x

y

O

x

y

O4

8

8 4 –4 –8

–4–8

x

y

O

(x�

2)2

�4

(y�

3)2

�16

12 � 59 � 5

y2� 7

x2� 7

y2� 8

x2� 4

5 � 2

y2� 16

x2� 9

2 � 31 � 3

Pra

ctic

e (

Ave

rag

e)

So

lvin

g Q

uad

rati

c S

yste

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

��1,

�an

d �1

,�

�3�

�2

�3�

�2



Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Qu

adra

tic

Sys

tem

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

©G

lenc

oe/M

cGra

w-H

ill49

5G

lenc

oe A

lgeb

ra 2

Lesson 8-7

Pre-

Act

ivit

yH

ow d

o sy

stem

s of

eq

uat

ion

s ap

ply

to

vid

eo g

ames

?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-7

at

the

top

of p

age

455

in y

our

text

book

.

Th

e fi

gure

in

you

r te

xtbo

ok s

how

s th

at t

he

spac

esh

ip h

its

the

circ

ula

r fo

rce

fiel

d in

tw

o po

ints

.Is

it p

ossi

ble

for

the

spac

esh

ip t

o h

it t

he

forc

e fi

eld

inei

ther

few

er o

r m

ore

than

tw

o po

ints

? S

tate

all

pos

sibi

liti

es a

nd

expl

ain

how

th

ese

cou

ld h

appe

n.

Sam

ple

an

swer

:Th

e sp

aces

hip

co

uld

hit

the

forc

e fi

eld

in z

ero

po

ints

if t

he

spac

esh

ip m

isse

d t

he

forc

efi

eld

all

tog

eth

er.T

he

spac

esh

ip c

ou

ld a

lso

hit

th

e fo

rce

fiel

din

on

e p

oin

t if

th

e sp

aces

hip

just

to

uch

ed t

he

edg

e o

f th

efo

rce

fiel

d.

Rea

din

g t

he

Less

on

1.D

raw

a s

ketc

h t

o il

lust

rate

eac

h o

f th

e fo

llow

ing

poss

ibil

itie

s.

a.a

para

bola

an

d a

lin

e b

.an

ell

ipse

an

d a

circ

le

c.a

hyp

erbo

la a

nd

ath

at i

nte

rsec

t in

th

at i

nte

rsec

t in

li

ne

that

in

ters

ect

in2

poin

ts4

poin

ts1

poin

t

2.C

onsi

der

the

foll

owin

g sy

stem

of

equ

atio

ns.

x2�

3 �

y2

2x2

�3y

2�

11

a.W

hat

kin

d of

con

ic s

ecti

on i

s th

e gr

aph

of

the

firs

t eq

uat

ion

?hy

per

bo

la

b.

Wh

at k

ind

of c

onic

sec

tion

is

the

grap

h o

f th

e se

con

d eq

uat

ion

?el

lipse

c.B

ased

on

you

r an

swer

s to

par

ts a

an

d b,

wh

at a

re t

he

poss

ible

nu

mbe

rs o

f so

luti

ons

that

th

is s

yste

m c

ould

hav

e?0,

1,2,

3,o

r 4

Hel

pin

g Y

ou

Rem

emb

er

3.S

upp

ose

that

th

e gr

aph

of

a qu

adra

tic

ineq

ual

ity

is a

reg

ion

wh

ose

bou

nda

ry i

s a

circ

le.

How

can

you

rem

embe

r w

het

her

to

shad

e th

e in

teri

or o

r th

e ex

teri

or o

f th

e ci

rcle

?S

amp

le a

nsw

er:T

he

solu

tio

ns

of

an in

equ

alit

y o

f th

e fo

rm x

2�

y2

�r2

are

all p

oin

ts t

hat

are

less

th

an r

un

its

fro

m t

he

ori

gin

,so

th

e g

rap

h is

the

inte

rio

ro

f th

e ci

rcle

.Th

e so

luti

on

s o

f an

ineq

ual

ity

of

the

form

x

2�

y2

�r2

are

the

po

ints

th

at a

re m

ore

th

an r

un

its

fro

m t

he

ori

gin

,so

the

gra

ph

is t

he

exte

rio

ro

f th

e ci

rcle

.

x

y

Ox

y

Ox

y O

©G

lenc

oe/M

cGra

w-H

ill49

6G

lenc

oe A

lgeb

ra 2

Gra

ph

ing

Qu

adra

tic

Eq

uat

ion

s w

ith

xy-

Term

sYo

u c

an u

se a

gra

phin

g ca

lcu

lato

r to

exa

min

e gr

aph

s of

qu

adra

tic

equ

atio

ns

that

con

tain

xy-

term

s.

Use

a g

rap

hin

g ca

lcu

lato

r to

dis

pla

y th

e gr

aph

of

x2�

xy�

y2�

4.

Sol

ve t

he

equ

atio

n f

or y

in t

erm

s of

xby

usi

ng

the

quad

rati

c fo

rmu

la.

y2�

xy�

(x2

�4)

�0

To

use

th

e fo

rmu

la,l

et a

�1,

b�

x,an

d c

�(x

2�

4).

y�

y�

To

grap

h t

he

equ

atio

n o

n t

he

grap

hin

g ca

lcu

lato

r,en

ter

the

two

equ

atio

ns:

y�

and

y�

Use

a g

rap

hin

g ca

lcu

lato

r to

gra

ph

eac

h e

qu

atio

n.S

tate

th

e ty

pe

of c

urv

e ea

ch g

rap

h r

epre

sen

ts.

1.y2

�xy

�8

2.x2

�y2

�2x

y�

x�

0

hyp

erb

ola

par

abo

la

3.x2

�xy

�y2

�15

4.x2

�xy

�y2

��

9

ellip

seg

rap

h is

�

5.2x

2�

2xy

�y2

�4x

�20

6.x2

�xy

�2y

2�

2x�

5y�

3 �

0

hyp

erb

ola

two

inte

rsec

tin

g li

nes

�x

��

16 �

�3x

2�

��

�2

�x

��

16 �

�3x

2�

��

�2

�x

��

16 �

�3x

2�

��

�2

�x

��

x2�

4�

(1)(

x2�

�4)

��

��

2

x

y

O1

–1–2

2

2 1 –1 –2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

Exam

ple

Exam

ple


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9. C

D

B

B

B

A

C

B

D

a. 4; b. any positivenumber except 4; c. any negativenumber; d. 0

A

C

D

A

C

D

B

B

A

D

A

D

C

C

B

A

D

C

A

B

An

swer

s

(continued on the next page)

Chapter 8 Assessment Answer Key Form 1 Form 2APage 497 Page 498 Page 499


10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: no solutions

A

C

D

D

C

B

B

A

B

C

A

A

C

D

B

C

D

C

A

C

no solutions

C

D

B

D

B

D

C

A

D

A

D

Chapter 8 Assessment Answer Key Form 2A (continued) Form 2BPage 500 Page 501 Page 502


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: (x � 3)2 � (y � 1)2 � 38

y

xO

y

xO(4, 0)

(1, �3)

(4, 0), (1, �3)

hyperbola; A � 4,C � �4

parabola; C � 0

�(x �

93)2� � �

(y �4

1)2� � 1;

ellipse

(x � 1)2 � (y � 1)2 � 25;circle

(5, �1), (1, �1); (3 � 2�2�, �1);

y � 1 � �(x � 3)

�(y �

362)2� � �

(x �3

1)2� � 1

�8x1

2� � �

2y5

2� � 1

(�2�, �2), (��2�, �2)

�(y �

365)2� � �

(x �16

2)2� � 1

�(x

1�00

1)2� � �

(y �25

3)2� � 1

y

xO

y

xO

(x � 4)2 � (y � 2)2 � 16

(2, 4); ��94

�, 4�;y � 4; x � �

74

�; right

y � 3(x � 1)2 � 1

x � �112�(y � 4)2 � 1

�61� units

�6, �32

��

An

swer

s

Chapter 8 Assessment Answer Key Form 2CPage 503 Page 504


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: (x � 5)2 � (y � 2)2 � 66

y

xO

y

xO

no solution

ellipse; A � 10; C � 3

�(y

1�00

3)2� � �

(x �25

1)2� � 1;

hyperbola

y � 4(x � 2)2 � 5;parabola

(1, 3), (�3, 3); (�1 � 2�2�, 3);

y � 3 � �(x � 1)

�(x �

493)2� � �

(y �21

1)2� � 1

�1y4

2

4� � �

1x6

2� � 1

(�5�, �3), (��5�, �3)

�(x �

641)2� � �

(y �9

4)2� � 1

�(y �

492)2� � �

(x �25

2)2� � 1

y

xO

y

xO

�x � �12

��2� (y � 2)2 � 4

(1, 3); �1, �285��,

x � 1; y � �283�;

upward

y � �116�(x � 1)2 � 3

�202� units

��32

�, 1�

Chapter 8 Assessment Answer Key Form 2DPage 505 Page 506


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15.

16.

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18.

19.

20.

B: (�2.5, 0.725)

y

xO

y

xO

(5, 3)(–1, 3)

(2, 0)

(�1, 3), (5, 3), (2, 0)

no solution

parabola; A � 0, C � 4

circle; A � C � �32

�

x � �8�y � �12

��2� 94;

parabola

2(x � 1)2 � 3�y � �43

��2� 67

�3

;

ellipse

�(y �

92)2� � �

(x �16

4)2� � 1

(2, 3); (2, 2), (2, 4); 2�6�; 2�5�

�(y �

362)2� � �

(x �16

3)2� � 1

y

xO

y

xO

(x � 3)2 � (y � 5)2 � 25

(10, �1); ��349�, �1�,

y � �1; x � �441�;

y � ��1160�(x � 80)2 � 40

x � ��16

�(y � 1)2 � 5

x � �(y � 3)2 � 2

�166� units

(�3.45, 4.15)

An

swer

s

Chapter 8 Assessment Answer Key Form 3Page 507 Page 508

�(x �

495)2� � � 1

�y � �12

��2

�25


Chapter 8 Assessment Answer KeyPage 509, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts ofidentifying, graphing, and writing equations of conicsections, and solving systems of quadratic equations andinequalities.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of identifying,graphing, and writing equations of conic sections, andsolving systems of quadratic equations and inequalities.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts ofidentifying, graphing, and writing equations of conicsections, and solving systems of quadratic equations andinequalities.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work is shown to substantiate

the final computation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the concepts ofidentifying, graphing, and writing equations of conicsections, and solving systems of quadratic equations andinequalities.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Does not satisfy requirements of problems.• No answer may be given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations


1a. The coefficients of the quadratic termsare the same number and have thesame sign (A � C � 1).

1b. The radius would be the square root of anegative number (�5).

1c. Student responses should indicate thatthe constant in the original equationshould be changed to a number lessthan 25, or that the constant obtainedwhen the equation is written instandard form should be changed to apositive number, or that one or both ofthe coefficients of the linear terms, xand y, must be changed to a numbersufficiently large to result in a positivenumber on the right side of thestandard form of the equation. Sampleanswer: Change the constant in theoriginal equation to 24. The center ofthe circle is (�4, 3) and the radius is 1unit.

2. The graphs of circles, ellipses,hyperbolas, and parabolas that open tothe left and right never representrelations that are functions. Of all theconic sections studied in this chapter,only parabolas that open upward ordownward have graphs which pass thevertical line test and are thereforefunctions.

3. The parabolas share the same vertex.Sample answer: The graph ofy � (x � 2)2 � 1 opens upward while thegraph of x � (y � 1)2 � 2 opens to theright.

4a. (x � 4)2 � (y � 3)2 � 4y � (x � 4)2 � 3 Region 2 is the intersection of the regioninside the circle, including its boundary(�) and the region above the parabola,not including its boundary (�).

4b. (x � 4)2 � (y � 3)2 � 4y � (x � 4)2 � 3Region 3 is the intersection of the regionoutside the circle, including itsboundary (�) and the region below theparabola, not including its boundary(�).

4c. Region 1 is the intersection of the regionoutside the circle, including itsboundary (�) and the region above theparabola, not including its boundary(�).

5a. Students must select both values suchthat �5 � k � 1 so that the graph of thehorizontal line y � k will intersect thegraph of the ellipse twice.

5b. Students may select only k � 1 andk � �5, the equations of the only twohorizontal lines that are tangent to theellipse, each intersecting the ellipse inexactly one point.

5c. Students must select both values suchthat k � 1 or k � �5 so that the graphof the horizontal line y � k will notintersect the graph of the ellipse.

An

swer

s

Chapter 8 Assessment Answer Key Page 509, Open-Ended Assessment

Sample Answers

In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.


1. parabola; focus;directrix

2. ellipse; foci of the ellipse

3. hyperbola

4. minor axis;major axis

5. transverse axis

6. tangent

7. latus rectum

8. asymptote

9. conjugate axis

10. distance formula

11. Sample answer: A circle is the set ofall points in a planethat are the samedistance from agiven point, whichis the center.

12. Sample answer: A vertex of ahyperbola is thepoint on a branch of the hyperbolathat is closest to the center of thehyperbola.

1.

2.

3.

4.

5.

Quiz (Lessons 8–3 and 8–4)

Page 511

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3.

4.

5.

Quiz (Lesson 8–7)

Page 512

1.

2.

3.

4. y

xO

2

2

(1, 2), (1, �2), (�1, 2), (�1, �2)

��73

�, ��131��, (1, 3)

y

xO

(1, �2) (3, �2)

(1, �2), (3, �2)

ellipse; A � 1, C � 4

y � 2(x � 3)2 � 23;parabola

y

xO

(�2, 0); (�2�10�, 0);y � �3x

�(y �

161)2� � �

(x �9

2)2� � 1

�(x �

41)2� � �

(y �4

2)2� � 1

(3, 0); (3 � 2�3�, 0); 8; 4

�2x5

2� � �

(y �16

1)2� � 1

y

xO

(1, �1); 3 units

(x � 3)2 � (y � 1)2 � 16

(x � 7)2 � (y � 2)2 � 81

yxO

�16

� unit

(�4, 5); ��4, �389��;

x � �4; y � �481�;

downward

y � �112�(x � 1)2 � 1

A

��1, �72

��

Chapter 8 Assessment Answer Key Vocabulary Test/Review Quiz (Lessons 8–1 and 8–2) Quiz (Lessons 8–5 and 8–6)

Page 510 Page 511 Page 512


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9.

10.

11.

12.

13.

14.

15.�(x �

93)2� � �

(y �4

1)2� � 1;

ellipse

(0, 0); (0, �2�2�); 6; 2

x � ��210�y2 � 1

��4, �52

��

even; 4

{x � x � �3 or x � 1}

1, �52

�

�xx

��

43

�

(�1, �1)

�22.5

(x � 3)2 � (y � 4)2 � 50

y � �3(x � 3)2 � 32

�(x �

254)2� � �

(y �16

1)2� � 1

y

xO

A

C

B

D

A

An

swer

s

Chapter 8 Assessment Answer Key Mid-Chapter Test Cumulative ReviewPage 513 Page 514

� �7 9 16�4 8 5


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13. 14.

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18. DCBA

DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

4 3 / 4

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

9 . 0 1

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 2 . 5

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

4

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

Chapter 8 Assessment Answer KeyStandardized Test Practice

Page 515 Page 516

Documents

Chapter 8 Resource Masters - ktlmathclass.weebly.com · ©Glencoe/McGraw-Hill iv Glencoe Algebra 2 Teacher’s Guide to Using the Chapter 8 Resource Masters The Fast FileChapter Resource