Chapter 6 Resource Masters - Glencoe/McGraw-Hill v Glencoe Algebra 2 Assessment Options The assessment masters in the Chapter 6 Resource Mastersoffer a wide range of assessment tools

Chapter 6Resource 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 6 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: 0-07-828009-5 Algebra 2Chapter 6 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 6-1Study Guide and Intervention . . . . . . . . 313–314Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 315Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 316Reading to Learn Mathematics . . . . . . . . . . 317Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 318







Chapter 6 AssessmentChapter 6 Test, Form 1 . . . . . . . . . . . . 355–356Chapter 6 Test, Form 2A . . . . . . . . . . . 357–358Chapter 6 Test, Form 2B . . . . . . . . . . . 359–360Chapter 6 Test, Form 2C . . . . . . . . . . . 361–362Chapter 6 Test, Form 2D . . . . . . . . . . . 363–364Chapter 6 Test, Form 3 . . . . . . . . . . . . 365–366Chapter 6 Open-Ended Assessment . . . . . . 367Chapter 6 Vocabulary Test/Review . . . . . . . 368Chapter 6 Quizzes 1 & 2 . . . . . . . . . . . . . . . 369Chapter 6 Quizzes 3 & 4 . . . . . . . . . . . . . . . 370Chapter 6 Mid-Chapter Test . . . . . . . . . . . . 371Chapter 6 Cumulative Review . . . . . . . . . . . 372Chapter 6 Standardized Test Practice . . 373–374

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

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

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 6 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 6 Resource Masters includes the core materials neededfor Chapter 6. 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 6-1.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 6Resource 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 342–343. 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 _____

66

© 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 6.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

axis of symmetry

completing the square

constant term

discriminant

dihs·KRIH·muh·nuhnt

linear term

maximum value

minimum value

parabola

puh·RA·buh·luh

quadratic equation

kwah·DRA·tihk

Quadratic Formula

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

quadratic function

quadratic inequality

quadratic term

roots

Square Root Property

vertex

vertex form

Zero Product Property

zeros

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

66

Study Guide and InterventionGraphing Quadratic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

© Glencoe/McGraw-Hill 313 Glencoe Algebra 2

Less

on

6-1

Graph Quadratic Functions

Quadratic Function A function defined by an equation of the form f (x) � ax2 � bx � c, where a � 0

Graph of a Quadratic A parabola with these characteristics: y intercept: c ; axis of symmetry: x � ;Function x-coordinate of vertex:

Find the y-intercept, the equation of the axis of symmetry, and thex-coordinate of the vertex for the graph of f(x) � x2 � 3x � 5. Use this informationto graph the function.

a � 1, b � �3, and c � 5, so the y-intercept is 5. The equation of the axis of symmetry is

x � or . The x-coordinate of the vertex is .

Next make a table of values for x near .

x x2 � 3x � 5 f(x ) (x, f(x ))

0 02 � 3(0) � 5 5 (0, 5)

1 12 �3(1) � 5 3 (1, 3)

� �2� 3� � � 5 � , �

2 22 � 3(2) � 5 3 (2, 3)

3 32 � 3(3) � 5 5 (3, 5)

For Exercises 1–3, complete parts a–c for each quadratic function.a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate

of the vertex.b. Make a table of values that includes the vertex.c. Use this information to graph the function.1. f(x) � x2 � 6x � 8 2. f(x) � �x2 �2x � 2 3. f(x) � 2x2 � 4x � 3

8, x � �3, �3 2, x � �1, �1 3, x � 1, 1

x

f(x)

O

12

8

4

4 8–4

x

f(x)

O

4

–4

–8

4 8–8 –4

x

(x)

O 4–4

4

8

–8

12

–4

x 1 0 2 3

f (x) 1 3 3 9

x �1 0 �2 1

f (x) 3 2 2 �1

x �3 �2 �1 �4

f (x) �1 0 3 0

11�4

3�2

11�4

3�2

3�2

3�2

x

f(x)

O

3�2

3�2

3�2

�(�3)�2(1)

�b�2a

�b�2a

ExampleExample

ExercisesExercises


Maximum and Minimum Values The y-coordinate of the vertex of a quadraticfunction is the maximum or minimum value of the function.

Maximum or Minimum Value The graph of f(x ) � ax2 � bx � c, where a � 0, opens up and has a minimumof a Quadratic Function when a � 0. The graph opens down and has a maximum when a � 0.

Determine whether each function has a maximum or minimumvalue. Then find the maximum or minimum value of each function.

Study Guide and Intervention (continued)

Graphing Quadratic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

ExampleExample

a. f(x) � 3x2 � 6x � 7For this function, a � 3 and b � �6.Since a � 0, the graph opens up, and thefunction has a minimum value.The minimum value is the y-coordinateof the vertex. The x-coordinate of the vertex is � � � 1.

Evaluate the function at x � 1 to find theminimum value.f(1) � 3(1)2 � 6(1) � 7 � 4, so theminimum value of the function is 4.

�6�2(3)

�b�2a

b. f(x) � 100 � 2x � x2

For this function, a � �1 and b � �2.Since a � 0, the graph opens down, andthe function has a maximum value.The maximum value is the y-coordinate ofthe vertex. The x-coordinate of the vertex is � � � �1.

Evaluate the function at x � �1 to findthe maximum value.f(�1) � 100 � 2(�1) � (�1)2 � 101, sothe minimum value of the function is 101.

�2�2(�1)

�b�2a

ExercisesExercises

Determine whether each function has a maximum or minimum value. Then findthe maximum or minimum value of each function.

1. f(x) � 2x2 � x � 10 2. f(x) � x2 � 4x � 7 3. f(x) � 3x2 � 3x � 1

min., 9 min., �11 min.,

4. f(x) � 16 � 4x �x2 5. f(x) � x2 � 7x � 11 6. f(x) � �x2 � 6x � 4

max., 20 min., � max., 5

7. f(x) � x2 � 5x � 2 8. f(x) � 20 � 6x � x2 9. f(x) � 4x2 � x � 3

min., � max., 29 min., 2

10. f(x) � �x2 � 4x � 10 11. f(x) � x2 � 10x � 5 12. f(x) � �6x2 � 12x � 21

max., 14 min., �20 max., 27

13. f(x) � 25x2 � 100x � 350 14. f(x) � 0.5x2 � 0.3x � 1.4 15. f(x) � � � 8

min., 250 min., �1.445 max., �7 31�

x�4

�x2�2

15�

17�

5�

1�

7�

Skills PracticeGraphing Quadratic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1


Less

on

6-1

For each quadratic function, find the y-intercept, the equation of the axis ofsymmetry, and the x-coordinate of the vertex.

1. f(x) � 3x2 2. f(x) � x2 � 1 3. f(x) � �x2 � 6x � 150; x � 0; 0 1; x � 0; 0 �15; x � 3; 3

4. f(x) � 2x2 � 11 5. f(x) � x2 � 10x � 5 6. f(x) � �2x2 � 8x � 7�11; x � 0; 0 5; x � 5; 5 7; x � 2; 2

Complete parts a–c for each quadratic function.a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate

of the vertex.b. Make a table of values that includes the vertex.c. Use this information to graph the function.

7. f(x) � �2x2 8. f(x) � x2 � 4x � 4 9. f(x) � x2 � 6x � 80; x � 0; 0 4; x � 2; 2 8; x � 3; 3

Determine whether each function has a maximum or a minimum value. Then findthe maximum or minimum value of each function.

10. f(x) � 6x2 11. f(x) � �8x2 12. f(x) � x2 � 2xmin.; 0 max.; 0 min.; �1

13. f(x) � x2 � 2x � 15 14. f(x) � �x2 � 4x � 1 15. f(x) � x2 � 2x � 3min.; 14 max.; 3 min.; �4

16. f(x) � �2x2 � 4x � 3 17. f(x) � 3x2 � 12x � 3 18. f(x) � 2x2 � 4x � 1max.; �1 min.; �9 min.; �1

x

f(x)

Ox

f(x)

O

16

12

8

4

2–2 4 6

x

f(x)

O

x 0 2 3 4 6

f (x) 8 0 �1 0 8

x �2 0 2 4 6

f (x) 16 4 0 4 16

x �2 �1 0 1 2

f (x) �8 �2 0 �2 �8


Complete parts a–c for each quadratic function.a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate

of the vertex.b. Make a table of values that includes the vertex.c. Use this information to graph the function.

1. f(x) � x2 � 8x � 15 2. f(x) � �x2 � 4x � 12 3. f(x) � 2x2 � 2x � 115; x � 4; 4 12; x � �2; �2 1; x � 0.5; 0.5

Determine whether each function has a maximum or a minimum value. Then findthe maximum or minimum value of each function.

4. f(x) � x2 � 2x � 8 5. f(x) � x2 � 6x � 14 6. v(x) � �x2 � 14x � 57min.; �9 min.; 5 max.; �8

7. f(x) � 2x2 � 4x � 6 8. f(x) � �x2 � 4x � 1 9. f(x) � ��23�x2 � 8x � 24

min.; �8 max.; 3 max.; 0

10. GRAVITATION From 4 feet above a swimming pool, Susan throws a ball upward with avelocity of 32 feet per second. The height h(t) of the ball t seconds after Susan throws itis given by h(t) � �16t2 � 32t � 4. Find the maximum height reached by the ball andthe time that this height is reached. 20 ft; 1 s

11. HEALTH CLUBS Last year, the SportsTime Athletic Club charged $20 to participate inan aerobics class. Seventy people attended the classes. The club wants to increase theclass price this year. They expect to lose one customer for each $1 increase in the price.

a. What price should the club charge to maximize the income from the aerobics classes?$45

b. What is the maximum income the SportsTime Athletic Club can expect to make?$2025

16

12

8

4

x

f(x)

O 2–2–4–6x

f(x)

O

16

12

8

4

2 4 6 8

x �1 0 0.5 1 2

f (x) 5 1 0.5 1 5

x �6 �4 �2 0 2

f (x) 0 12 16 12 0

x 0 2 4 6 8

f (x) 15 3 �1 3 15

Practice (Average)

Graphing Quadratic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

Reading to Learn MathematicsGraphing Quadratic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1


Less

on

6-1

Pre-Activity How can income from a rock concert be maximized?

Read the introduction to Lesson 6-1 at the top of page 286 in your textbook.

• Based on the graph in your textbook, for what ticket price is the incomethe greatest? $40

• Use the graph to estimate the maximum income. about $72,000

Reading the Lesson1. a. For the quadratic function f(x) � 2x2 � 5x � 3, 2x2 is the term,

5x is the term, and 3 is the term.

b. For the quadratic function f(x) � �4 � x � 3x2, a � , b � , and

c � .

2. Consider the quadratic function f(x) � ax2 � bx � c, where a � 0.

a. The graph of this function is a .

b. The y-intercept is .

c. The axis of symmetry is the line .

d. If a � 0, then the graph opens and the function has a

value.

e. If a � 0, then the graph opens and the function has a

value.

3. Refer to the graph at the right as you complete the following sentences.

a. The curve is called a .

b. The line x � �2 is called the .

c. The point (�2, 4) is called the .

d. Because the graph contains the point (0, �1), �1 is

the .

Helping You Remember4. How can you remember the way to use the x2 term of a quadratic function to tell

whether the function has a maximum or a minimum value? Sample answer:Remember that the graph of f(x) � x2 (with a � 0) is a U-shaped curvethat opens up and has a minimum. The graph of g(x) � �x2 (with a � 0)is just the opposite. It opens down and has a maximum.

y-intercept

vertex

axis of symmetry

parabola

x

f(x)

O(0, –1)

(–2, 4)

maximumdownward

minimumupward

x � ��2ba�

c

parabola

�41�3

constantlinearquadratic


Finding the Axis of Symmetry of a ParabolaAs you know, if f(x) � ax2 � bx � c is a quadratic function, the values of x

that make f(x) equal to zero are and .

The average of these two number values is ��2ba�.

The function f(x) has its maximum or minimum

value when x � ��2ba�. Since the axis of symmetry

of the graph of f (x) passes through the point where the maximum or minimum occurs, the axis of

symmetry has the equation x � ��2ba�.

Find the vertex and axis of symmetry for f(x) � 5x2 � 10x � 7.

Use x � ��2ba�.

x � ��21(05)� � �1 The x-coordinate of the vertex is �1.

Substitute x � �1 in f(x) � 5x2 � 10x � 7.f(�1) � 5(�1)2 � 10(�1) � 7 � �12The vertex is (�1,�12).The axis of symmetry is x � ��2

ba�, or x � �1.

Find the vertex and axis of symmetry for the graph of each function using x � ��2

ba�.

1. f(x) � x2 � 4x � 8 2. g(x) � �4x2 � 8x � 3

3. y � �x2 � 8x � 3 4. f(x) � 2x2 � 6x � 5

5. A(x) � x2 � 12x � 36 6. k(x) � �2x2 � 2x � 6

O

f(x)

x

– –, f( ( (( b––2a

b––2a

b––2ax = –

f(x) = ax2 + bx + c

�b � �b2 � 4�ac��2a

�b � �b2 � 4�ac��2a

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

ExampleExample

Study Guide and InterventionSolving Quadratic Equations by Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2


Less

on

6-2

Solve Quadratic Equations

Quadratic Equation A quadratic equation has the form ax2 � bx � c � 0, where a � 0.

Roots of a Quadratic Equation solution(s) of the equation, or the zero(s) of the related quadratic function

The zeros of a quadratic function are the x-intercepts of its graph. Therefore, finding the x-intercepts is one way of solving the related quadratic equation.

Solve x2 � x � 6 � 0 by graphing.

Graph the related function f(x) � x2 � x � 6.

The x-coordinate of the vertex is � � , and the equation of the

axis of symmetry is x � � .

Make a table of values using x-values around � .

x �1 � 0 1 2

f(x) �6 �6 �6 �4 0

From the table and the graph, we can see that the zeros of the function are 2 and �3.

Solve each equation by graphing.

1. x2 � 2x � 8 � 0 2, �4 2. x2 � 4x � 5 � 0 5, �1 3. x2 � 5x � 4 � 0 1, 4

4. x2 � 10x � 21 � 0 5. x2 � 4x � 6 � 0 6. 4x2 � 4x � 1 � 0

3, 7 no real solutions � 1�

x

f(x)

Ox

f(x)

O

x

f(x)

O

x

f(x)

O

x

f(x)

Ox

f(x)

O

1�4

1�2

1�2

1�2

1�2

�b�2a x

f(x)

O

ExampleExample

ExercisesExercises


Estimate Solutions Often, you may not be able to find exact solutions to quadraticequations by graphing. But you can use the graph to estimate solutions.

Solve x2 � 2x � 2 � 0 by graphing. If exact roots cannot be found,state the consecutive integers between which the roots are located.

The equation of the axis of symmetry of the related function is

x � � � 1, so the vertex has x-coordinate 1. Make a table of values.

x �1 0 1 2 3

f (x) 1 �2 �3 �2 1

The x-intercepts of the graph are between 2 and 3 and between 0 and�1. So one solution is between 2 and 3, and the other solution isbetween 0 and �1.

Solve the equations by graphing. If exact roots cannot be found, state theconsecutive integers between which the roots are located.

1. x2 � 4x � 2 � 0 2. x2 � 6x � 6 � 0 3. x2 � 4x � 2� 0

between 0 and 1; between �2 and �1; between �1 and 0;between 3 and 4 between �5 and �4 between �4 and �3

4. �x2 � 2x � 4 � 0 5. 2x2 � 12x � 17 � 0 6. � x2 � x � � 0

between 3 and 4; between 2 and 3; between �2 and �1;between �2 and �1 between 3 and 4 between 3 and 4

x

f(x)

O

x

f(x)

Ox

f(x)

O

5�2

1�2

x

f(x)

Ox

f(x)

Ox

f(x)

O

�2�2(1)

x

f(x)

O


Solving Quadratic Equations by Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2

ExampleExample

ExercisesExercises

Skills PracticeSolving Quadratic Equations By Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2


Less

on

6-2

Use the related graph of each equation to determine its solutions.

1. x2 � 2x � 3 � 0 2. �x2 � 6x � 9 � 0 3. 3x2 � 4x � 3 � 0

�3, 1 �3 no real solutions

Solve each equation by graphing. If exact roots cannot be found, state theconsecutive integers between which the roots are located.

4. x2 � 6x � 5 � 0 5. �x2 � 2x � 4 � 0 6. x2 � 6x � 4 � 01, 5 no real solutions between 0 and 1;

between 5 and 6

Use a quadratic equation to find two real numbers that satisfy each situation, orshow that no such numbers exist.

7. Their sum is �4, and their product is 0. 8. Their sum is 0, and their product is �36.

�x2 � 4x � 0; 0, �4 �x2 � 36 � 0; �6, 6

x

f(x)

O 6–6 12–12

36

24

12

x

f(x)

O

x

f(x)

O

x

f(x)

O

x

f(x)

O

x

f(x)

O

f(x) � 3x2 � 4x � 3

x

f(x)

O

f(x) � �x2 � 6x � 9

x

f(x)

O

f(x) � x2 � 2x � 3


Use the related graph of each equation to determine its solutions.

1. �3x2 � 3 � 0 2. 3x2 � x � 3 � 0 3. x2 � 3x � 2 � 0

�1, 1 no real solutions 1, 2Solve each equation by graphing. If exact roots cannot be found, state theconsecutive integers between which the roots are located.

4. �2x2 � 6x � 5 � 0 5. x2 � 10x � 24 � 0 6. 2x2 � x � 6 � 0between 0 and 1; �6, �4 between �2 and �1, between �4 and �3 2

Use a quadratic equation to find two real numbers that satisfy each situation, orshow that no such numbers exist.

7. Their sum is 1, and their product is �6. 8. Their sum is 5, and their product is 8.

For Exercises 9 and 10, use the formula h(t) � v0t � 16t2, where h(t) is the heightof an object in feet, v0 is the object’s initial velocity in feet per second, and t is thetime in seconds.

9. BASEBALL Marta throws a baseball with an initial upward velocity of 60 feet per second.Ignoring Marta’s height, how long after she releases the ball will it hit the ground? 3.75 s

10. VOLCANOES A volcanic eruption blasts a boulder upward with an initial velocity of240 feet per second. How long will it take the boulder to hit the ground if it lands at thesame elevation from which it was ejected? 15 s

�x2 � 5x � 8 � 0;no such realnumbers exist

�x2 � x � 6 � 0;3, �2

x

f(x)

O

x

f(x)

O

x

f(x)

O–4 –2–6

12

8

4

x

f(x)

O

f(x) � x2 � 3x � 2

x

f(x)

O

f(x) � 3x2 � x � 3

x

f(x)

O

f(x) � �3x2 � 3

Practice (Average)

Solving Quadratic Equations By Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2

Reading to Learn MathematicsSolving Quadratic Equations by Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2


Less

on

6-2

Pre-Activity How does a quadratic function model a free-fall ride?


Write a quadratic function that describes the height of a ball t seconds afterit is dropped from a height of 125 feet. h(t) � �16t 2 � 125

Reading the Lesson

1. The graph of the quadratic function f(x) � �x2 � x � 6 is shown at the right. Use the graph to find the solutions of thequadratic equation �x2 � x � 6 � 0. �2 and 3

2. Sketch a graph to illustrate each situation.

a. A parabola that opens b. A parabola that opens c. A parabola that opensdownward and represents a upward and represents a downward and quadratic function with two quadratic function with represents a real zeros, both of which are exactly one real zero. The quadratic function negative numbers. zero is a positive number. with no real zeros.

Helping You Remember

3. Think of a memory aid that can help you recall what is meant by the zeros of a quadraticfunction.

Sample answer: The basic facts about a subject are sometimes calledthe ABCs. In the case of zeros, the ABCs are the XYZs, because thezeros are the x-values that make the y-values equal to zero.

x

y

Ox

y

Ox

y

O

x

y

O


Graphing Absolute Value Equations You can solve absolute value equations in much the same way you solved quadratic equations. Graph the related absolute value function for each equation using a graphing calculator. Then use the ZERO feature in the CALC menu to find its real solutions, if any. Recall that solutions are points where the graph intersects the x-axis.

For each equation, make a sketch of the related graph and find the solutions rounded to the nearest hundredth.

1. |x � 5| � 0 2. |4x � 3| � 5 � 0 3. |x � 7| � 0

5 No solutions 7

4. |x � 3| � 8 � 0 5. �|x � 3| � 6 � 0 6. |x � 2| � 3 � 0

�11, 5 �9, 3 �1, 5

7. |3x � 4| � 2 8. |x � 12| � 10 9. |x | � 3 � 0

�2, ��23

� �22, �2 �3, 3

10. Explain how solving absolute value equations algebraically and finding zeros of absolute value functions graphically are related.Sample answer: values of x when solving algebraically are the x-intercepts (or zeros) of the function when graphed.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2

Study Guide and InterventionSolving Quadratic Equations by Factoring

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3


Less

on

6-3

Solve Equations by Factoring When you use factoring to solve a quadratic equation,you use the following property.

Zero Product Property For any real numbers a and b, if ab � 0, then either a � 0 or b �0, or both a and b � 0.

Solve each equation by factoring.ExampleExamplea. 3x2 � 15x

3x2 � 15x Original equation

3x2 � 15x � 0 Subtract 15x from both sides.

3x(x � 5) � 0 Factor the binomial.

3x � 0 or x � 5 � 0 Zero Product Property

x � 0 or x � 5 Solve each equation.

The solution set is {0, 5}.

b. 4x2 � 5x � 214x2 � 5x � 21 Original equation

4x2 � 5x � 21 � 0 Subtract 21 from both sides.

(4x � 7)(x � 3) � 0 Factor the trinomial.

4x � 7 � 0 or x � 3 � 0 Zero Product Property

x � � or x � 3 Solve each equation.

The solution set is �� , 3�.7�4

7�4

ExercisesExercises

Solve each equation by factoring.

1. 6x2 � 2x � 0 2. x2 � 7x 3. 20x2 � �25x

�0, � {0, 7} �0, � �4. 6x2 � 7x 5. 6x2 � 27x � 0 6. 12x2 � 8x � 0

�0, � �0, � �0, �7. x2 � x � 30 � 0 8. 2x2 � x � 3 � 0 9. x2 � 14x � 33 � 0

{5, �6} � , �1� {�11, �3}

10. 4x2 � 27x � 7 � 0 11. 3x2 � 29x � 10 � 0 12. 6x2 � 5x � 4 � 0

� , �7� ��10, � �� , �13. 12x2 � 8x � 1 � 0 14. 5x2 � 28x � 12 � 0 15. 2x2 � 250x � 5000 � 0

� , � � , �6� {100, 25}

16. 2x2 � 11x � 40 � 0 17. 2x2 � 21x � 11 � 0 18. 3x2 � 2x � 21 � 0

�8, � � ��11, � � , �3�19. 8x2 � 14x � 3 � 0 20. 6x2 � 11x � 2 � 0 21. 5x2 � 17x � 12 � 0

� , � ��2, � � , �4�22. 12x2 � 25x � 12 � 0 23. 12x2 � 18x � 6 � 0 24. 7x2 � 36x � 5 � 0

�� , � � �� , �1� � , 5�1�

1�

3�

4�

3�

1�

1�

3�

7�

1�

5�

2�

1�

1�

4�

1�

1�

1�

3�

2�

9�

7�

5�

1�


Write Quadratic Equations To write a quadratic equation with roots p and q, let(x � p)(x � q) � 0. Then multiply using FOIL.

Write a quadratic equation with the given roots. Write the equationin the form ax2 � bx � c � 0.


Solving Quadratic Equations by Factoring

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3

ExampleExample

a. 3, �5(x � p)(x � q) � 0 Write the pattern.

(x � 3)[x � (�5)] � 0 Replace p with 3, q with �5.

(x � 3)(x � 5) � 0 Simplify.

x2 � 2x � 15 � 0 Use FOIL.

The equation x2 � 2x � 15 � 0 has roots 3 and �5.

b. � ,

(x � p)(x � q) � 0

�x � �� x � � � 0

�x � ��x � � � 0

� � 0

� 24 � 0

24x2 � 13x � 7 � 0

The equation 24x2 � 13x � 7 � 0 has

roots � and .1�3

7�8

24 � (8x � 7)(3x � 1)��24

(3x � 1)�3

(8x � 7)�8

1�3

7�8

1�3

7�8

1�3

7�8

ExercisesExercises

Write a quadratic equation with the given roots. Write the equation in the formax2 � bx � c � 0.

1. 3, �4 2. �8, �2 3. 1, 9x2 � x � 12 � 0 x2 � 10x � 16 � 0 x2 � 10x � 9 � 0

4. �5 5. 10, 7 6. �2, 15x2 � 10x � 25 � 0 x2 � 17x � 70 � 0 x2 � 13x � 30 � 0

7. � , 5 8. 2, 9. �7,

3x2 � 14x � 5 � 0 3x2 � 8x � 4 � 0 4x2 � 25x � 21 � 0

10. 3, 11. � , �1 12. 9,

5x2 � 17x � 6 � 0 9x2 � 13x � 4 � 0 6x2 � 55x � 9 � 0

13. , � 14. , � 15. ,

9x2 � 4 � 0 8x2 � 6x � 5 � 0 35x2 � 22x � 3 � 0

16. � , 17. , 18. ,

16x2 � 42x � 49 8x2 � 10x � 3 � 0 48x2 � 14x � 1 � 0

1�6

1�8

3�4

1�2

7�2

7�8

1�5

3�7

1�2

5�4

2�3

2�3

1�6

4�9

2�5

3�4

2�3

1�3

Skills PracticeSolving Quadratic Equations by Factoring

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3


Less

on

6-3


1. x2 � 64 {�8, 8} 2. x2 � 100 � 0 {10, �10}

3. x2 � 3x � 2 � 0 {1, 2} 4. x2 � 4x � 3 � 0 {1, 3}

5. x2 � 2x � 3 � 0 {1, �3} 6. x2 � 3x � 10 � 0 {5, �2}

7. x2 � 6x � 5 � 0 {1, 5} 8. x2 � 9x � 0 {0, 9}

9. �x2 � 6x � 0 {0, 6} 10. x2 � 6x � 8 � 0 {�2, �4}

11. x2 � �5x {0, �5} 12. x2 � 14x � 49 � 0 {7}

13. x2 � 6 � 5x {2, 3} 14. x2 � 18x � �81 {�9}

15. x2 � 4x � 21 {�3, 7} 16. 2x2 � 5x � 3 � 0 � , �3�

17. 4x2 � 5x � 6 � 0 � , �2� 18. 3x2 � 13x � 10 � 0 �� , 5�

Write a quadratic equation with the given roots. Write the equation in the formax2 � bx � c � 0, where a, b, and c are integers.

19. 1, 4 x2 � 5x � 4 � 0 20. 6, �9 x2 � 3x � 54 � 0

21. �2, �5 x2 � 7x � 10 � 0 22. 0, 7 x2 � 7x � 0

23. � , �3 3x2 �10x � 3 � 0 24. � , 8x2 � 2x � 3 � 0

25. Find two consecutive integers whose product is 272. 16, 17

3�4

1�2

1�3

2�

3�

1�



1. x2 � 4x � 12 � 0 {6, �2} 2. x2 � 16x � 64 � 0 {8} 3. x2 � 20x � 100 � 0 {10}

4. x2 � 6x � 8 � 0 {2, 4} 5. x2 � 3x � 2 � 0 {�2, �1} 6. x2 � 9x � 14 � 0 {2, 7}

7. x2 � 4x � 0 {0, 4} 8. 7x2 � 4x �0, � 9. x2 � 25 � 10x {5}

10. 10x2 � 9x �0, � 11. x2 � 2x � 99 {�9, 11}

12. x2 � 12x � �36 {�6} 13. 5x2 � 35x � 60 � 0 {3, 4}

14. 36x2 � 25 � , � � 15. 2x2 � 8x � 90 � 0 {9, �5}

16. 3x2 � 2x � 1 � 0 � , �1� 17. 6x2 � 9x �0, �18. 3x2 � 24x � 45 � 0 {�5, �3} 19. 15x2 � 19x � 6 � 0 �� , � �20. 3x2 � 8x � �4 �2, � 21. 6x2 � 5x � 6 � , � �Write a quadratic equation with the given roots. Write the equation in the formax2 � bx � c � 0, where a, b, and c are integers.

22. 7, 2 23. 0, 3 24. �5, 8x2 � 9x � 14 � 0 x2 � 3x � 0 x2 � 3x � 40 � 0

25. �7, �8 26. �6, �3 27. 3, �4x2 � 15x � 56 � 0 x2 � 9x � 18 � 0 x2 � x � 12 � 0

28. 1, 29. , 2 30. 0, �

2x2 � 3x � 1 � 0 3x2 � 7x � 2 � 0 2x2 � 7x � 0

31. , �3 32. 4, 33. � , �

3x2 � 8x � 3 � 0 3x2 � 13x � 4 � 0 15x2 � 22x � 8 � 0

34. NUMBER THEORY Find two consecutive even positive integers whose product is 624.24, 26

35. NUMBER THEORY Find two consecutive odd positive integers whose product is 323.17, 19

36. GEOMETRY The length of a rectangle is 2 feet more than its width. Find thedimensions of the rectangle if its area is 63 square feet. 7 ft by 9 ft

37. PHOTOGRAPHY The length and width of a 6-inch by 8-inch photograph are reduced bythe same amount to make a new photograph whose area is half that of the original. Byhow many inches will the dimensions of the photograph have to be reduced? 2 in.

4�5

2�3

1�3

1�3

7�2

1�3

1�2

2�

3�

2�

2�

3�

3�

1�

5�

5�

9�

4�

Practice (Average)

Solving Quadratic Equations by Factoring

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3

Reading to Learn MathematicsSolving Quadratic Equations by Factoring

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3


Less

on

6-3

Pre-Activity How is the Zero Product Property used in geometry?


What does the expression x(x � 5) mean in this situation?

It represents the area of the rectangle, since the area is theproduct of the width and length.

Reading the Lesson

1. The solution of a quadratic equation by factoring is shown below. Give the reason foreach step of the solution.

x2 � 10x � �21 Original equation

x2 � 10x � 21 � 0 Add 21 to each side.

(x � 3)(x � 7) � 0 Factor the trinomial.

x � 3 � 0 or x � 7 � 0 Zero Product Property

x � 3 x � 7 Solve each equation.

The solution set is .

2. On an algebra quiz, students were asked to write a quadratic equation with �7 and 5 asits roots. The work that three students in the class wrote on their papers is shown below.

Marla Rosa Larry(x �7)(x � 5) � 0 (x � 7)(x � 5) � 0 (x � 7)(x � 5) � 0x2 � 2x � 35 � 0 x2 � 2x � 35 � 0 x2 � 2x � 35 � 0

Who is correct? RosaExplain the errors in the other two students’ work.

Sample answer: Marla used the wrong factors. Larry used the correctfactors but multiplied them incorrectly.


3. A good way to remember a concept is to represent it in more than one way. Describe analgebraic way and a graphical way to recognize a quadratic equation that has a doubleroot.

Sample answer: Algebraic: Write the equation in the standard form ax2 � bx � c � 0 and examine the trinomial. If it is a perfect squaretrinomial, the quadratic function has a double root. Graphical: Graph therelated quadratic function. If the parabola has exactly one x-intercept,then the equation has a double root.

{3, 7}


Euler’s Formula for Prime NumbersMany mathematicians have searched for a formula that would generate prime numbers. One such formula was proposed by Euler and uses a quadratic polynomial, x2 � x � 41.

Find the values of x2 � x � 41 for the given values of x. State whether each value of the polynomial is or is not a prime number.

1. x � 0 2. x � 1 3. x � 2

4. x � 3 5. x � 4 6. x � 5

7. x � 6 8. x � 17 9. x � 28

10. x � 29 11. x � 30 12. x � 35

13. Does the formula produce all prime numbers greater than 40? Give examples in your answer.

14. Euler’s formula produces primes for many values of x, but it does not work for all of them. Find the first value of x for which the formula fails.(Hint: Try multiples of ten.)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3

Study Guide and InterventionCompleting the Square

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4


Less

on

6-4

Square Root Property Use the following property to solve a quadratic equation that isin the form “perfect square trinomial � constant.”

Square Root Property For any real number x if x2 � n, then x � n.

Solve each equation by using the Square Root Property.ExampleExamplea. x2 � 8x � 16 � 25

x2 � 8x � 16 � 25(x � 4)2 � 25

x � 4 � �25� or x � 4 � ��25�x � 5 � 4 � 9 or x � �5 � 4 � �1

The solution set is {9, �1}.

b. 4x2 � 20x � 25 � 324x2 � 20x � 25 � 32

(2x � 5)2 � 322x � 5 � �32� or 2x � 5 � ��32�2x � 5 � 4�2� or 2x � 5 � �4�2�

x �

The solution set is � �.5 4�2��2

5 4�2��2

ExercisesExercises

Solve each equation by using the Square Root Property.

1. x2 � 18x � 81 � 49 2. x2 � 20x � 100 � 64 3. 4x2 � 4x � 1 � 16

{2, 16} {�2, �18} � , � �

4. 36x2 � 12x � 1 � 18 5. 9x2 � 12x � 4 � 4 6. 25x2 � 40x � 16 � 28

� � �0, � � �

7. 4x2 � 28x � 49 � 64 8. 16x2 � 24x � 9 � 81 9. 100x2 � 60x � 9 � 121

� , � � � , �3� {�0.8, 1.4}

10. 25x2 � 20x � 4 � 75 11. 36x2 � 48x � 16 � 12 12. 25x2 � 30x � 9 � 96

� � � � � �3 � 4�6��

�2 � �3��

�2 � 5�3��

3�

1�

15�

�4 � 2�7��

4��1 � 3�2�

��

5�

3�


Complete the Square To complete the square for a quadratic expression of the form x2 � bx, follow these steps.

1. Find . ➞ 2. Square . ➞ 3. Add � �2to x2 � bx.b

�2b�2

b�2


Completing the Square

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4

Find the value ofc that makes x2 � 22x � c aperfect square trinomial. Thenwrite the trinomial as thesquare of a binomial.

Step 1 b � 22; � 11

Step 2 112 � 121Step 3 c � 121

The trinomial is x2 � 22x � 121,which can be written as (x � 11)2.

b�2

Solve 2x2 � 8x � 24 � 0 bycompleting the square.

2x2 � 8x � 24 � 0 Original equation

� Divide each side by 2.

x2 � 4x � 12 � 0 x2 � 4x � 12 is not a perfect square.

x2 � 4x � 12 Add 12 to each side.

x2 � 4x � 4 � 12 � 4 Since �� 2

� 4, add 4 to each side.

(x � 2)2 � 16 Factor the square.

x � 2 � 4 Square Root Property

x � 6 or x � � 2 Solve each equation.

The solution set is {6, �2}.

4�2

0�2

2x2 � 8x � 24��2

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the value of c that makes each trinomial a perfect square. Then write thetrinomial as a perfect square.

1. x2 � 10x � c 2. x2 � 60x � c 3. x2 � 3x � c

25; (x � 5)2 900; (x � 30)2 ; �x � �2

4. x2 � 3.2x � c 5. x2 � x � c 6. x2 � 2.5x � c

2.56; (x � 1.6)2 ; �x � �2 1.5625; (x � 1.25)2

Solve each equation by completing the square.

7. y2 � 4y � 5 � 0 8. x2 � 8x � 65 � 0 9. s2 � 10s � 21 � 0�1, 5 �5, 13 3, 7

10. 2x2 � 3x � 1 � 0 11. 2x2 � 13x � 7 � 0 12. 25x2 � 40x � 9 � 0

1, � , 7 , �

13. x2 � 4x � 1 � 0 14. y2 � 12y � 4 � 0 15. t2 � 3t � 8 � 0

�2 � �3� �6 � 4�2� �3 � �41��2

9�

1�

1�

1�

1�

1�

1�2

3�

9�

Skills PracticeCompleting the Square

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4


Less

on

6-4


1. x2 � 8x � 16 � 1 3, 5 2. x2 � 4x � 4 � 1 �1, �3

3. x2 � 12x � 36 � 25 �1, �11 4. 4x2 � 4x � 1 � 9 �1, 2

5. x2 � 4x � 4 � 2 �2 � �2� 6. x2 � 2x � 1 � 5 1 � �5�

7. x2 � 6x � 9 � 7 3 � �7� 8. x2 � 16x � 64 � 15 �8 � �15�


9. x2 � 10x � c 25; (x � 5)2 10. x2 � 14x � c 49; (x � 7)2

11. x2 � 24x � c 144; (x � 12)2 12. x2 � 5x � c ; �x � �2

13. x2 � 9x � c ; �x � �2 14. x2 � x � c ; �x � �2


15. x2 � 13x � 36 � 0 4, 9 16. x2 � 3x � 0 0, �3

17. x2 � x � 6 � 0 2, �3 18. x2 � 4x � 13 � 0 2 � �17�

19. 2x2 � 7x � 4 � 0 �4, 20. 3x2 � 2x � 1 � 0 , �1

21. x2 � 3x � 6 � 0 22. x2 � x � 3 � 0

23. x2 � �11 �i �11� 24. x2 � 2x � 4 � 0 1 � i �3�

1 � �13��2

�3 � �33��2

1�

1�

1�

1�

9�

81�

5�

25�



1. x2 � 8x � 16 � 1 2. x2 � 6x � 9 � 1 3. x2 � 10x � 25 � 16

�5, �3 �4, �2 �9, �1

4. x2 � 14x � 49 � 9 5. 4x2 � 12x � 9 � 4 6. x2 � 8x � 16 � 8

4, 10 � , � 4 � 2�2�

7. x2 � 6x � 9 � 5 8. x2 � 2x � 1 � 2 9. 9x2 � 6x � 1 � 2

3 � �5� 1 � �2�


10. x2 � 12x � c 11. x2 � 20x � c 12. x2 � 11x � c

36; (x � 6)2 100; (x � 10)2 ; �x � �2

13. x2 � 0.8x � c 14. x2 � 2.2x � c 15. x2 � 0.36x � c

0.16; (x � 0.4)2 1.21; (x � 1.1)2 0.0324; (x � 0.18)2

16. x2 � x � c 17. x2 � x � c 18. x2 � x � c

; �x � �2 ; �x � �2 ; �x � �2


19. x2 � 6x � 8 � 0 �4, �2 20. 3x2 � x � 2 � 0 , �1 21. 3x2 � 5x � 2 � 0 1,

22. x2 � 18 � 9x 23. x2 � 14x � 19 � 0 24. x2 � 16x � 7 � 06, 3 7 � �30� �8 � �71�

25. 2x2 � 8x � 3 � 0 26. x2 � x � 5 � 0 27. 2x2 � 10x � 5 � 0

28. x2 � 3x � 6 � 0 29. 2x2 � 5x � 6 � 0 30. 7x2 � 6x � 2 � 0

31. GEOMETRY When the dimensions of a cube are reduced by 4 inches on each side, thesurface area of the new cube is 864 square inches. What were the dimensions of theoriginal cube? 16 in. by 16 in. by 16 in.

32. INVESTMENTS The amount of money A in an account in which P dollars is invested for2 years is given by the formula A � P(1 � r)2, where r is the interest rate compoundedannually. If an investment of $800 in the account grows to $882 in two years, at whatinterest rate was it invested? 5%

�3 � i�5��7

�5 � i�23��4

�3 � i �15��2

5 � �15��2

�1 � �21��2

�4 � �22��2

2�

2�

5�

25�

1�

1�

5�

25�

5�3

1�4

5�6

11�

121�

1 � �2��3

5�

1�

Practice (Average)

Completing the Square

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4

Reading to Learn MathematicsCompleting the Square

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4


Less

on

6-4

Pre-Activity How can you find the time it takes an accelerating race car toreach the finish line?


Explain what it means to say that the driver accelerates at a constant rateof 8 feet per second square.

If the driver is traveling at a certain speed at a particularmoment, then one second later, the driver is traveling 8 feetper second faster.

Reading the Lesson

1. Give the reason for each step in the following solution of an equation by using theSquare Root Property.

x2 � 12x � 36 � 81 Original equation

(x � 6)2 � 81 Factor the perfect square trinomial.

x � 6 � �81� Square Root Property

x � 6 � 9 81 � 9

x � 6 � 9 or x � 6 � �9 Rewrite as two equations.

x � 15 x � �3 Solve each equation.

2. Explain how to find the constant that must be added to make a binomial into a perfectsquare trinomial.

Sample answer: Find half of the coefficient of the linear term and squareit.

3. a. What is the first step in solving the equation 3x2 � 6x � 5 by completing the square?Divide the equation by 3.

b. What is the first step in solving the equation x2 � 5x � 12 � 0 by completing thesquare? Add 12 to each side.


4. How can you use the rules for squaring a binomial to help you remember the procedurefor changing a binomial into a perfect square trinomial?

One of the rules for squaring a binomial is (x � y)2 � x2 � 2xy � y2. Incompleting the square, you are starting with x2 � bx and need to find y2.

This shows you that b � 2y, so y � . That is why you must take half of

the coefficient and square it to get the constant that must be added tocomplete the square.

b�


The Golden Quadratic EquationsA golden rectangle has the property that its length can be written as a � b, where a is the width of the

rectangle and �a �a

b� � �

ab�. Any golden rectangle can be

divided into a square and a smaller golden rectangle,as shown.

The proportion used to define golden rectangles can be used to derive two quadratic equations. These aresometimes called golden quadratic equations.

Solve each problem.

1. In the proportion for the golden rectangle, let a equal 1. Write the resulting quadratic equation and solve for b.

2. In the proportion, let b equal 1. Write the resulting quadratic equation and solve for a.

3. Describe the difference between the two golden quadratic equations you found in exercises 1 and 2.

4. Show that the positive solutions of the two equations in exercises 1 and 2 are reciprocals.

5. Use the Pythagorean Theorem to find a radical expression for the diagonal of a golden rectangle when a � 1.

6. Find a radical expression for the diagonal of a golden rectangle when b � 1.

a

a

a

b

b

a

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4

Study Guide and InterventionThe Quadratic Formula and the Discriminant

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5


Less

on

6-5

Quadratic Formula The Quadratic Formula can be used to solve any quadraticequation once it is written in the form ax2 � bx � c � 0.

Quadratic Formula The solutions of ax 2 � bx � c � 0, with a � 0, are given by x � .

Solve x2 � 5x � 14 by using the Quadratic Formula.

Rewrite the equation as x2 � 5x � 14 � 0.

x � Quadratic Formula

� Replace a with 1, b with �5, and c with �14.

� Simplify.

�

� 7 or �2

The solutions are �2 and 7.

Solve each equation by using the Quadratic Formula.

1. x2 � 2x � 35 � 0 2. x2 � 10x � 24 � 0 3. x2 � 11x � 24 � 0

5, �7 �4, �6 3, 8

4. 4x2 � 19x � 5 � 0 5. 14x2 � 9x � 1 � 0 6. 2x2 � x � 15 � 0

, �5 � , � 3, �

7. 3x2 � 5x � 2 8. 2y2 � y � 15 � 0 9. 3x2 � 16x � 16 � 0

�2, , �3 4,

10. 8x2 � 6x � 9 � 0 11. r2 � � � 0 12. x2 � 10x � 50 � 0

� , , 5 � 5�3�

13. x2 � 6x � 23 � 0 14. 4x2 � 12x � 63 � 0 15. x2 � 6x � 21 � 0

�3 � 4�2� 3 � 2i�3�3 � 6�2��

1�

2�

3�

3�

2�25

3r�5

4�

5�

1�

5�

1�

1�

1�

5 9�2

5 �81��2

�(�5) �(�5)2�� 4(1�)(�14�)��2(1)

�b �b2 � 4�ac��2a

�b �b2 ��4ac��

2a

ExampleExample

ExercisesExercises


Roots and the Discriminant

DiscriminantThe expression under the radical sign, b2 � 4ac, in the Quadratic Formula is called the discriminant.

Roots of a Quadratic Equation

Discriminant Type and Number of Roots

b2 � 4ac � 0 and a perfect square 2 rational roots

b2 � 4ac � 0, but not a perfect square 2 irrational roots

b2 � 4ac � 0 1 rational root

b2 � 4ac � 0 2 complex roots

Find the value of the discriminant for each equation. Then describethe number and types of roots for the equation.


The Quadratic Formula and the Discriminant

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5

ExampleExample

a. 2x2 � 5x � 3The discriminant is b2 � 4ac � 52 � 4(2)(3) or 1.The discriminant is a perfect square, sothe equation has 2 rational roots.

b. 3x2 � 2x � 5The discriminant is b2 � 4ac � (�2)2 � 4(3)(5) or �56.The discriminant is negative, so theequation has 2 complex roots.

ExercisesExercises

For Exercises 1�12, complete parts a�c for each quadratic equation.a. Find the value of the discriminant.b. Describe the number and type of roots.c. Find the exact solutions by using the Quadratic Formula.

1. p2 � 12p � �4 128; 2. 9x2 � 6x � 1 � 0 0; 3. 2x2 � 7x � 4 � 0 81; two irrational roots; one rational root; 2 rational roots; � ,4�6 � 4�2�

4. x2 � 4x � 4 � 0 32; 5. 5x2 � 36x � 7 � 0 1156; 6. 4x2 � 4x � 11 � 0

2 irrational roots; 2 rational roots; �160; 2 complexroots; �2 � 2�2� , 7

7. x2 � 7x � 6 � 0 25; 8. m2 � 8m � �14 8; 9. 25x2 � 40x � �16 0; 2 rational roots; 2 irrational roots; 1 rational root; 1, 6 4 � �2�

10. 4x2 � 20x � 29 � 0 �64; 11. 6x2 � 26x � 8 � 0 484; 12. 4x2 � 4x � 11 � 0 192; 2 complex roots; 2 rational roots; 2 irrational roots;

4�

1 � i �10��

1�

1�

1�

Skills PracticeThe Quadratic Formula and the Discriminant

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5


Less

on

6-5

Complete parts a�c for each quadratic equation.a. Find the value of the discriminant.b. Describe the number and type of roots.c. Find the exact solutions by using the Quadratic Formula.

1. x2 � 8x � 16 � 0 2. x2 � 11x � 26 � 0

0; 1 rational root; 4 225; 2 rational roots; �2, 13

3. 3x2 � 2x � 0 4. 20x2 � 7x � 3 � 0

4; 2 rational roots; 0, 289; 2 rational roots; � ,

5. 5x2 � 6 � 0 6. x2 � 6 � 0

120; 2 irrational roots; � 24; 2 irrational roots; ��6�

7. x2 � 8x � 13 � 0 8. 5x2 � x � 1 � 0

12; 2 irrational roots; �4 � �3� 21; 2 irrational roots;

9. x2 � 2x � 17 � 0 10. x2 � 49 � 0

72; 2 irrational roots; 1 � 3�2� �196; 2 complex roots; �7i

11. x2 � x � 1 � 0 12. 2x2 � 3x � �2

�3; 2 complex roots; �7; 2 complex roots;

Solve each equation by using the method of your choice. Find exact solutions.

13. x2 � 64 �8 14. x2 � 30 � 0 ��30�

15. x2 � x � 30 �5, 6 16. 16x2 � 24x � 27 � 0 , �

17. x2 � 4x � 11 � 0 2 � �15� 18. x2 � 8x � 17 � 0 4 � �33�

19. x2 � 25 � 0 �5i 20. 3x2 � 36 � 0 �2i �3�

21. 2x2 � 10x � 11 � 0 22. 2x2 � 7x � 4 � 0

23. 8x2 � 1 � 4x 24. 2x2 � 2x � 3 � 0

25. PARACHUTING Ignoring wind resistance, the distance d(t) in feet that a parachutistfalls in t seconds can be estimated using the formula d(t) � 16t2. If a parachutist jumpsfrom an airplane and falls for 1100 feet before opening her parachute, how many secondspass before she opens the parachute? about 8.3 s

�1 � i�5��2

1 � i�

4

7 � �17��4

�5 � �3��2

3�

9�

3 � i �7��4

1 � i �3��2

1 � �21��10

�30��5

1�

3�

2�


Complete parts a�c for each quadratic equation.a. Find the value of the discriminant.b. Describe the number and type of roots.c. Find the exact solutions by using the Quadratic Formula.

1. x2 � 16x � 64 � 0 2. x2 � 3x 3. 9x2 � 24x � 16 � 0

0; 1 rational; 8 9; 2 rational; 0, 3 0; 1 rational;

4. x2 � 3x � 40 5. 3x2 � 9x � 2 � 0 105; 6. 2x2 � 7x � 0

169; 2 rational; �5, 8 2 irrational; 49; 2 rational; 0, �

7. 5x2 � 2x � 4 � 0 �76; 8. 12x2 � x � 6 � 0 289; 9. 7x2 � 6x � 2 � 0 �20;

2 complex; 2 rational; , � 2 complex;

10. 12x2 � 2x � 4 � 0 196; 11. 6x2 � 2x � 1 � 0 28; 12. x2 � 3x � 6 � 0 �15;

2 rational; , � 2 irrational; 2 complex;

13. 4x2 � 3x2 � 6 � 0 105; 14. 16x2 � 8x � 1 � 0 15. 2x2 � 5x � 6 � 0 73;

2 irrational; 0; 1 rational; 2 irrational;

Solve each equation by using the method of your choice. Find exact solutions.

16. 7x2 � 5x � 0 0, 17. 4x2 � 9 � 0 �

18. 3x2 � 8x � 3 , �3 19. x2 � 21 � 4x �3, 7

20. 3x2 � 13x � 4 � 0 , 4 21. 15x2 � 22x � �8 � , �

22. x2 � 6x � 3 � 0 3 � �6� 23. x2 � 14x � 53 � 0 7 � 2i

24. 3x2 � �54 �3i �2� 25. 25x2 � 20x � 6 � 0

26. 4x2 � 4x � 17 � 0 27. 8x � 1 � 4x2

28. x2 � 4x � 15 2 � i �11� 29. 4x2 � 12x � 7 � 0

30. GRAVITATION The height h(t) in feet of an object t seconds after it is propelled straight upfrom the ground with an initial velocity of 60 feet per second is modeled by the equationh(t) � �16t2 � 60t. At what times will the object be at a height of 56 feet? 1.75 s, 2 s

31. STOPPING DISTANCE The formula d � 0.05s2 � 1.1s estimates the minimum stoppingdistance d in feet for a car traveling s miles per hour. If a car stops in 200 feet, what is thefastest it could have been traveling when the driver applied the brakes? about 53.2 mi/h

3 � �2��2

2 � �3��2

1 � 4i�

2 � �10��5

4�

2�

1�

1�

3�

5�

5 � �73��4

1�

3 � �105��8

�3 � i �15��1 � �7��

62�

1�

�3 � i �5��7

2�

3�

1 � i �19��5

7�

�9 � �105��6

4�

Practice (Average)

The Quadratic Formula and the Discriminant

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5

Reading to Learn MathematicsThe Quadratic Formula and the Discriminant

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5


Less

on

6-5

Pre-Activity How is blood pressure related to age?


Describe how you would calculate your normal blood pressure using one ofthe formulas in your textbook.

Sample answer: Substitute your age for A in the appropriateformula (for females or males) and evaluate the expression.

Reading the Lesson

1. a. Write the Quadratic Formula. x �

b. Identify the values of a, b, and c that you would use to solve 2x2 � 5x � �7, but donot actually solve the equation.

a � b � c �

2. Suppose that you are solving four quadratic equations with rational coefficients andhave found the value of the discriminant for each equation. In each case, give thenumber of roots and describe the type of roots that the equation will have.

Value of Discriminant Number of Roots Type of Roots

64 2 real, rational

�8 2 complex

21 2 real, irrational

0 1 real, rational


3. How can looking at the Quadratic Formula help you remember the relationshipsbetween the value of the discriminant and the number of roots of a quadratic equationand whether the roots are real or complex?

Sample answer: The discriminant is the expression under the radical inthe Quadratic Formula. Look at the Quadratic Formula and consider whathappens when you take the principal square root of b2 � 4ac and apply� in front of the result. If b2 � 4ac is positive, its principal square rootwill be a positive number and applying � will give two different realsolutions, which may be rational or irrational. If b2 � 4ac � 0, itsprincipal square root is 0, so applying � in the Quadratic Formula willonly lead to one solution, which will be rational (assuming a, b, and c areintegers). If b2 � 4ac is negative, since the square roots of negativenumbers are not real numbers, you will get two complex roots,corresponding to the � and � in the � symbol.

7�52

�b � �b2 �4�ac��2a


Sum and Product of Roots Sometimes you may know the roots of a quadratic equation without knowing the equationitself. Using your knowledge of factoring to solve an equation, you can work backward tofind the quadratic equation. The rule for finding the sum and product of roots is as follows:

Sum and Product of RootsIf the roots of ax2 � bx � c � 0, with a ≠ 0, are s1 and s2,

then s1 � s2 � ��ba

� and s1 � s2 � �ac

�.

A road with an initial gradient, or slope, of 3% can be represented bythe formula y � ax2 � 0. 03x � c, where y is the elevation and x is the distance alongthe curve. Suppose the elevation of the road is 1105 feet at points 200 feet and 1000feet along the curve. You can find the equation of the transition curve. Equationsof transition curves are used by civil engineers to design smooth and safe roads.

The roots are x � 3 and x � �8.

3 � (�8) � �5 Add the roots.

3(�8) � �24 Multiply the roots.

Equation: x2 � 5x � 24 � 0

Write a quadratic equation that has the given roots.

1. 6, �9 2. 5, �1 3. 6, 6

x2 � 3x � 54 � 0 x2 � 4x � 5 � 0 x2 � 12x � 36 � 0

4. 4 �3� 6. ��25�, �

27� 6.

x2 � 8x � 13 � 0 35x2 � 4x � 4 � 0 49x2 � 42x � 205 � 0

Find k such that the number given is a root of the equation.

7. 7; 2x2 � kx � 21 � 0 8. �2; x2 � 13x � k � 0 �11 �30

�2 3�5��7

x

y

O

(–5–2, –301–

4)

10

–10

–20

–30

2 4–2–4–6–8

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5

ExampleExample

Study Guide and InterventionAnalyzing Graphs of Quadratic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6


Less

on

6-6

Analyze Quadratic Functions

The graph of y � a (x � h)2 � k has the following characteristics:• Vertex: (h, k )

Vertex Form • Axis of symmetry: x � hof a Quadratic • Opens up if a � 0Function • Opens down if a � 0

• Narrower than the graph of y � x2 if a � 1• Wider than the graph of y � x2 if a � 1

Identify the vertex, axis of symmetry, and direction of opening ofeach graph.

a. y � 2(x � 4)2 � 11The vertex is at (h, k) or (�4, �11), and the axis of symmetry is x � �4. The graph opensup, and is narrower than the graph of y � x2.

a. y � � (x � 2)2 � 10

The vertex is at (h, k) or (2, 10), and the axis of symmetry is x � 2. The graph opensdown, and is wider than the graph of y � x2.

Each quadratic function is given in vertex form. Identify the vertex, axis ofsymmetry, and direction of opening of the graph.

1. y � (x � 2)2 � 16 2. y � 4(x � 3)2 � 7 3. y � (x � 5)2 � 3

(2, 16); x � 2; up (�3, �7); x � �3; up (5, 3); x � 5; up

4. y � �7(x � 1)2 � 9 5. y � (x � 4)2 � 12 6. y � 6(x � 6)2 � 6

(�1, �9); x � �1; down (4, �12); x � 4; up (�6, 6); x � �6; up

7. y � (x � 9)2 � 12 8. y � 8(x � 3)2 � 2 9. y � �3(x � 1)2 � 2

(9, 12); x � 9; up (3, �2); x � 3; up (1, �2); x � 1; down

10. y � � (x � 5)2 � 12 11. y � (x � 7)2 � 22 12. y � 16(x � 4)2 � 1

(�5, 12); x � �5; down (7, 22); x � 7; up (4, 1); x � 4; up

13. y � 3(x � 1.2)2 � 2.7 14. y � �0.4(x � 0.6)2 � 0.2 15. y � 1.2(x � 0.8)2 � 6.5

(1.2, 2.7); x � 1.2; up (0.6, �0.2); x � 0.6; (�0.8, 6.5); x � �0.8;down up

4�3

5�2

2�5

1�5

1�2

1�4

ExampleExample

ExercisesExercises


Write Quadratic Functions in Vertex Form A quadratic function is easier tograph when it is in vertex form. You can write a quadratic function of the form y � ax2 � bx � c in vertex from by completing the square.

Write y � 2x2 � 12x � 25 in vertex form. Then graph the function.

y � 2x2 � 12x � 25y � 2(x2 � 6x) � 25y � 2(x2 � 6x � 9) � 25 � 18y � 2(x � 3)2 � 7

The vertex form of the equation is y � 2(x � 3)2 � 7.

Write each quadratic function in vertex form. Then graph the function.

1. y � x2 � 10x � 32 2. y � x2 � 6x 3. y � x2 � 8x � 6y � (x � 5)2 � 7 y � (x � 3)2 � 9 y � (x � 4)2 � 10

4. y � �4x2 � 16x � 11 5. y � 3x2 � 12x � 5 6. y � 5x2 � 10x � 9y � �4(x � 2)2 � 5 y � 3(x � 2)2 � 7 y � 5(x� 1)2 � 4

x

y

O

x

y

O

x

y

O

x

y

O 4–4 8

8

4

–4

–8

–12

x

y

O

x

y

O

x

y

O


Analyzing Graphs of Quadratic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6

ExampleExample

ExercisesExercises

Skills PracticeAnalyzing Graphs of Quadratic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6


Less

on

6-6

Write each quadratic function in vertex form, if not already in that form. Thenidentify the vertex, axis of symmetry, and direction of opening.

1. y � (x � 2)2 2. y � �x2 � 4 3. y � x2 � 6y � (x � 2)2 � 0; y � �(x � 0)2 � 4; y � (x � 0)2 � 6;(2, 0); x � 2; up (0, 4); x � 0; down (0, �6); x � 0; up

4. y � �3(x � 5)2 5. y � �5x2 � 9 6. y � (x � 2)2 � 18y � �3(x � 5)2 � 0; y � �5(x � 0)2 � 9; y � (x � 2)2 � 18; (�5, 0); x � �5; down (0, 9); x � 0; down (2, �18); x � 2; up

7. y � x2 � 2x � 5 8. y � x2 � 6x � 2 9. y � �3x2 � 24xy � (x � 1)2 � 6; y � (x � 3)2 � 7; y � �3(x � 4)2 � 48; (1, �6); x � 1; up (�3, �7); x � �3; up (4, 48); x � 4; down

Graph each function.

10. y � (x � 3)2 � 1 11. y � (x � 1)2 � 2 12. y � �(x � 4)2 � 4

13. y � � (x � 2)2 14. y � �3x2 � 4 15. y � x2 � 6x � 4

Write an equation for the parabola with the given vertex that passes through thegiven point.

16. vertex: (4, �36) 17. vertex: (3, �1) 18. vertex: (�2, 2)point: (0, �20) point: (2, 0) point: (�1, 3)y � (x � 4)2 � 36 y � (x � 3)2 � 1 y � (x � 2)2 � 2

x

y

Ox

y

O

x

y

O

1�2

x

y

O

x

y

Ox

y

O


Write each quadratic function in vertex form, if not already in that form. Thenidentify the vertex, axis of symmetry, and direction of opening.

1. y � �6(x � 2)2 � 1 2. y � 2x2 � 2 3. y � �4x2 � 8xy � �6(x � 2)2 � 1; y � 2(x � 0)2 � 2; y � �4(x � 1)2 � 4;(�2, �1); x � �2; down (0, 2); x � 0; up (1, 4); x � 1; down

4. y � x2 � 10x � 20 5. y � 2x2 � 12x � 18 6. y � 3x2 � 6x � 5y � (x � 5)2 � 5; y � 2(x � 3)2; (�3, 0); y � 3(x � 1)2 � 2; (�5, �5); x � �5; up x � �3; up (1, 2); x � 1; up

7. y � �2x2 � 16x � 32 8. y � �3x2 � 18x � 21 9. y � 2x2 � 16x � 29y � �2(x � 4)2; y � �3(x � 3)2 � 6; y � 2(x � 4)2 � 3; (�4, 0); x � �4; down (3, 6); x � 3; down (�4, �3); x � �4; up

Graph each function.

10. y � (x � 3)2 � 1 11. y � �x2 � 6x � 5 12. y � 2x2 � 2x � 1

Write an equation for the parabola with the given vertex that passes through thegiven point.

13. vertex: (1, 3) 14. vertex: (�3, 0) 15. vertex: (10, �4)point: (�2, �15) point: (3, 18) point: (5, 6)y � �2(x � 1)2 � 3 y � (x � 3)2 y � (x � 10)2 � 4

16. Write an equation for a parabola with vertex at (4, 4) and x-intercept 6.y � �(x � 4)2 � 4

17. Write an equation for a parabola with vertex at (�3, �1) and y-intercept 2.y � (x � 3)2 � 1

18. BASEBALL The height h of a baseball t seconds after being hit is given by h(t) � �16t2 � 80t � 3. What is the maximum height that the baseball reaches, andwhen does this occur? 103 ft; 2.5 s

19. SCULPTURE A modern sculpture in a park contains a parabolic arc thatstarts at the ground and reaches a maximum height of 10 feet after ahorizontal distance of 4 feet. Write a quadratic function in vertex formthat describes the shape of the outside of the arc, where y is the heightof a point on the arc and x is its horizontal distance from the left-handstarting point of the arc. y � � (x � 4)2 � 105

�

10 ft

4 ft

1�

2�

1�

x

y

O

x

y

O

Practice (Average)

Analyzing Graphs of Quadratic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6

Reading to Learn MathematicsAnalyzing Graphs of Quadratic Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6


Less

on

6-6

Pre-Activity How can the graph of y � x2 be used to graph any quadraticfunction?


• What does adding a positive number to x2 do to the graph of y � x2?It moves the graph up.

• What does subtracting a positive number to x before squaring do to thegraph of y � x2? It moves the graph to the right.

Reading the Lesson

1. Complete the following information about the graph of y � a(x � h)2 � k.

a. What are the coordinates of the vertex? (h, k)

b. What is the equation of the axis of symmetry? x � h

c. In which direction does the graph open if a � 0? If a � 0? up; down

d. What do you know about the graph if a � 1?It is wider than the graph of y � x2.

If a � 1? It is narrower than the graph of y � x2.

2. Match each graph with the description of the constants in the equation in vertex form.

a. a � 0, h � 0, k � 0 iii b. a � 0, h � 0, k � 0 iv

c. a � 0, h � 0, k � 0 ii d. a � 0, h � 0, k � 0 i

i. ii. iii. iv.


3. When graphing quadratic functions such as y � (x � 4)2 and y � (x � 5)2, many studentshave trouble remembering which represents a translation of the graph of y � x2 to the leftand which represents a translation to the right. What is an easy way to remember this?

Sample answer: In functions like y � (x � 4)2, the plus sign puts thegraph “ahead” so that the vertex comes “sooner” than the origin and thetranslation is to the left. In functions like y � (x � 5)2, the minus puts thegraph “behind” so that the vertex comes “later” than the origin and thetranslation is to the right.

x

y

Ox

y

Ox

y

Ox

y

O


Patterns with Differences and Sums of SquaresSome whole numbers can be written as the difference of two squares andsome cannot. Formulas can be developed to describe the sets of numbersalgebraically.

If possible, write each number as the difference of two squares.Look for patterns.

1. 0 02 � 02 2. 1 12 � 02 3. 2 cannot 4. 3 22 � 12

5. 4 22 � 02 6. 5 32 � 22 7. 6 cannot 8. 7 42 � 32

9. 8 32 � 12 10. 9 32 � 02 11. 10 cannot 12. 11 62 � 52

13. 12 42 � 22 14. 13 72 � 62 15. 14 cannot 16. 15 42 � 12

Even numbers can be written as 2n, where n is one of the numbers 0, 1, 2, 3, and so on. Odd numbers can be written 2n � 1. Use these expressions for these problems.

17. Show that any odd number can be written as the difference of two squares.2n � 1 � (n � 1)2 � n2

18. Show that the even numbers can be divided into two sets: those that can be written in the form 4n and those that can be written in the form 2 � 4n.Find 4n for n � 0, 1, 2, and so on. You get {0, 4, 8, 12, …}. For 2 � 4n,you get {2, 6, 10, 12, …}. Together these sets include all even numbers.

19. Describe the even numbers that cannot be written as the difference of two squares. 2 � 4n, for n � 0, 1, 2, 3, …

20. Show that the other even numbers can be written as the difference of two squares. 4n � (n � 1)2 � (n � 1)2

Every whole number can be written as the sum of squares. It is never necessary to use more than four squares. Show that this is true for the whole numbers from 0 through 15 by writing each one as the sum of the least number of squares.

21. 0 02 22. 1 12 23. 2 12 � 12

24. 3 12 � 12 � 12 25. 4 22 26. 5 12 � 22

27. 6 12 � 12 � 22 28. 7 12 � 12 � 12 � 22 29. 8 22 � 22

30. 9 32 31. 10 12 � 32 32. 11 12 � 12 � 32

33. 12 12 � 12 � 12 � 32 34. 13 22 � 32 35. 14 12 � 22 � 32

36. 15 12 � 12 � 22 � 32

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6

Study Guide and InterventionGraphing and Solving Quadratic Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

6-76-7


Less

on

6-7

Graph Quadratic Inequalities To graph a quadratic inequality in two variables, usethe following steps:

1. Graph the related quadratic equation, y � ax2 � bx � c.Use a dashed line for � or �; use a solid line for or �.

2. Test a point inside the parabola.If it satisfies the inequality, shade the region inside the parabola;otherwise, shade the region outside the parabola.

Graph the inequality y � x2 � 6x � 7.

First graph the equation y � x2 � 6x � 7. By completing the square, you get the vertex form of the equation y � (x � 3)2 � 2,so the vertex is (�3, �2). Make a table of values around x � �3,and graph. Since the inequality includes �, use a dashed line.Test the point (�3, 0), which is inside the parabola. Since (�3)2 � 6(�3) � 7 � �2, and 0 � �2, (�3, 0) satisfies the inequality. Therefore, shade the region inside the parabola.

Graph each inequality.

1. y � x2 � 8x � 17 2. y x2 � 6x � 4 3. y � x2 � 2x � 2

4. y � �x2 � 4x � 6 5. y � 2x2 � 4x 6. y � �2x2 � 4x � 2

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

ExampleExample

ExercisesExercises


Solve Quadratic Inequalities Quadratic inequalities in one variable can be solvedgraphically or algebraically.

To solve ax2 � bx � c � 0:First graph y � ax2 � bx � c. The solution consists of the x-values

Graphical Methodfor which the graph is below the x-axis.

To solve ax2 � bx � c � 0:First graph y � ax2 � bx � c. The solution consists the x-values for which the graph is above the x-axis.

Find the roots of the related quadratic equation by factoring,

Algebraic Methodcompleting the square, or using the Quadratic Formula.2 roots divide the number line into 3 intervals.Test a value in each interval to see which intervals are solutions.

If the inequality involves or �, the roots of the related equation are included in thesolution set.

Solve the inequality x2 � x � 6 � 0.

First find the roots of the related equation x2 � x � 6 � 0. Theequation factors as (x � 3)(x � 2) � 0, so the roots are 3 and �2.The graph opens up with x-intercepts 3 and �2, so it must be on or below the x-axis for �2 x 3. Therefore the solution set is {x �2 x 3}.

Solve each inequality.

1. x2 � 2x � 0 2. x2 � 16 � 0 3. 0 � 6x � x2 � 5

{x�2 � x � 0} {x�4 � x � 4} {x1 � x � 5}

4. c2 4 5. 2m2 � m � 1 6. y2 � �8

{c�2 � c � 2} �m� � m � 1�

7. x2 � 4x � 12 � 0 8. x2 � 9x � 14 � 0 9. �x2 � 7x � 10 � 0

{x�2 � x � 6} {xx � �7 or x � �2} {x2 � x � 5}

10. 2x2 � 5x� 3 0 11. 4x2 � 23x � 15 � 0 12. �6x2 � 11x � 2 � 0

�x�3 � x � � �xx � or x � 5� �xx � �2 or x � �13. 2x2 � 11x � 12 � 0 14. x2 � 4x � 5 � 0 15. 3x2 � 16x � 5 � 0

�xx � or x � 4� �x � x � 5�1�

3�

1�

3�

1�

1�

x

y

O


Graphing and Solving Quadratic Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

6-76-7

ExampleExample

ExercisesExercises

Skills PracticeGraphing and Solving Quadratic Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

6-76-7


Less

on

6-7


1. y � x2 � 4x � 4 2. y x2 � 4 3. y � x2 � 2x � 5

Use the graph of its related function to write the solutions of each inequality.

4. x2 � 6x � 9 0 5. �x2 � 4x � 32 � 0 6. x2 � x � 20 � 0

3 �8 � x � 4 x � �5 or x � 4

Solve each inequality algebraically.

7. x2 � 3x � 10 � 0 8. x2 � 2x � 35 � 0{x�2 � x � 5} {xx � �7 or x 5}

9. x2 � 18x � 81 0 10. x2 36{xx � 9} {x�6 � x � 6}

11. x2 � 7x � 0 12. x2 � 7x � 6 � 0{xx � 0 or x � 7} {x�6 � x � �1}

13. x2 � x � 12 � 0 14. x2 � 9x � 18 0{xx � �4 or x � 3} {x�6 � x � �3}

15. x2 � 10x � 25 � 0 16. �x2 � 2x � 15 � 0all reals {x�5 � x � 3}

17. x2 � 3x � 0 18. 2x2 � 2x � 4{xx � �3 or x � 0} {xx � �2 or x � 1}

19. �x2 � 64 �16x 20. 9x2 � 12x � 9 � 0all reals

x

y

O 2

5

x

y

O 2

6

x

y

O

x

y

O

x

y

O

x

y

O



1. y x2 � 4 2. y � x2 � 6x � 6 3. y � 2x2 � 4x � 2

Use the graph of its related function to write the solutions of each inequality.

4. x2 � 8x � 0 5. �x2 � 2x � 3 � 0 6. x2 � 9x � 14 0

x � 0 or x � 8 �3 � x � 1 2 � x � 7

Solve each inequality algebraically.

7. x2 � x � 20 � 0 8. x2 � 10x � 16 � 0 9. x2 � 4x � 5 0

{xx � �4 or x � 5} {x2 � x � 8}

10. x2 � 14x � 49 � 0 11. x2 � 5x � 14 12. �x2 � 15 � 8x

all reals {xx � �2 or x � 7} {x�5 � x � �3}

13. �x2 � 5x � 7 0 14. 9x2 � 36x � 36 0 15. 9x 12x2

all reals {xx � �2} �xx � 0 or x �16. 4x2 � 4x � 1 � 0 17. 5x2 � 10 � 27x 18. 9x2 � 31x � 12 0

�xx � � � �xx � or x 5� �x�3 � x � � �19. FENCING Vanessa has 180 feet of fencing that she intends to use to build a rectangular

play area for her dog. She wants the play area to enclose at least 1800 square feet. Whatare the possible widths of the play area? 30 ft to 60 ft

20. BUSINESS A bicycle maker sold 300 bicycles last year at a profit of $300 each. The makerwants to increase the profit margin this year, but predicts that each $20 increase inprofit will reduce the number of bicycles sold by 10. How many $20 increases in profit canthe maker add in and expect to make a total profit of at least $100,000? from 5 to 10

4�

2�

1�

3�

x

y

O

x

y

Ox

y

O 2 4 6

6

–6

–12

8

x

y

O

x

y

O

Practice (Average)

Graphing and Solving Quadratic Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

6-76-7

Reading to Learn MathematicsGraphing and Solving Quadratic Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

6-76-7


Less

on

6-7

Pre-Activity How can you find the time a trampolinist spends above a certainheight?


• How far above the ground is the trampoline surface? 3.75 feet• Using the quadratic function given in the introduction, write a quadratic

inequality that describes the times at which the trampolinist is morethan 20 feet above the ground. �16t 2 � 42t � 3.75 � 20

Reading the Lesson

1. Answer the following questions about how you would graph the inequality y � x2 � x � 6.

a. What is the related quadratic equation? y � x2 � x � 6

b. Should the parabola be solid or dashed? How do you know?solid; The inequality symbol is .

c. The point (0, 2) is inside the parabola. To use this as a test point, substitute

for x and for y in the quadratic inequality.

d. Is the statement 2 � 02 � 0 � 6 true or false? true

e. Should the region inside or outside the parabola be shaded? inside

2. The graph of y � �x2 � 4x is shown at the right. Match each of the following related inequalities with its solution set.

a. �x2 � 4x � 0 ii i. {x x � 0 or x � 4}

b. �x2 � 4x 0 iii ii. {x 0 � x � 4}

c. �x2 � 4x � 0 iv iii. {x x 0 or x � 4}

d. �x2 � 4x � 0 i iv. {x 0 x 4}


3. A quadratic inequality in two variables may have the form y � ax2 � bx � c,y � ax2 � bx � c, y � ax2 � bx � c, or y ax2 � bx � c. Describe a way to rememberwhich region to shade by looking at the inequality symbol and without using a test point.Sample answer: If the symbol is � or , shade the region above theparabola. If the symbol is � or �, shade the region below the parabola.

x

y

O(0, 0) (4, 0)

(2, 4)

20


Graphing Absolute Value Inequalities You can solve absolute value inequalities by graphing in much the same manner you graphed quadratic inequalities. Graph the related absolute function for each inequality by using a graphing calculator. For � and �, identify the x-values, if any, for which the graph lies below the x-axis. For � and , identify the x values, if any, for which the graph lies above the x-axis.

For each inequality, make a sketch of the related graph and find the solutions rounded to the nearest hundredth.

1. |x � 3| � 0 2. |x| � 6 � 0 3. �|x � 4| � 8 � 0

�6 � x � 6 �12 � x � 4

4. 2|x � 6| � 2 � 0 5. |3x � 3| � 0 6. |x � 7| � 5

x � �7 or x �5 all real numbers 2 � x � 12

7. |7x � 1| � 13 8. |x � 3.6| 4.2 9. |2x � 5| 7

x � �1.71 or x � 2 �0.6 � x � 7.8 �6 � x � 1

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-76-7

Chapter 6 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. Find the y-intercept for f(x) � �(x � 1)2.A. 1 B. �1 C. x D. 0 1.

2. What is the equation of the axis of symmetry of y � �3(x � 6)2 � 12?A. x � 2 B. x � �6 C. x � 6 D. x � �18 2.

3. Find the minimum value of f(x) � x2 � 6x.A. 3 B. �6 C. �9 D. 27 3.

4. The graph of f(x) � �2x2 � x opens _____ and has a _____ value.A. down; maximum B. down; minimumC. up; maximum D. up; minimum 4.

5. The related graph of a quadratic equation is shown at the right.Use the graph to determine the solutions of the equation.A. �2, 3 B. �3, 2C. 0, �6 D. 0, 2 5.

6. The quadratic function f(x) � x2 has _____.A. no zeros B. exactly one zeroC. exactly two zeros D. more than two zeros 6.

For Questions 7 and 8, solve each equation by factoring.

7. x2 � 3x � 10 � 0A. {�5, 2} B. (�2, 5) C. {�2, 5} D. {�10, 1} 7.

8. 2x2 � 6x � 0A. {�3, 0} B. {0, 3} C. {0, 6} D. {�3, 3} 8.

9. Which quadratic equation has roots �2 and 3?A. x2 � x � 6 � 0 B. x2 � x � 6 � 0C. x2 � 6x � 1 � 0 D. x2 � x � 6 � 0 9.

10. To solve x2 � 8x � 16 � 25 by using the Square Root Property, you would first rewrite the equation as _____.A. (x � 4)2 � 25 B. x2 � 8x � 9 � 0C. (x � 4)2 � 5 D. x2 � 8x � 9 10.

11. Find the value of c that makes x2 � 10x � c a perfect square.A. 100 B. 25 C. 10 D. 50 11.

66

xO

f(x )


Chapter 6 Test, Form 1 (continued)

12. The quadratic equation x2 � 6x � 1 is to be solved by completing the square.Which equation would be the first step in that solution?A. x2 � 6x � 1 � 0 B. x2 � 6x � 36 � 1 � 36C. x(x � 6) � 1 D. x2 � 6x � 9 � 1 � 9 12.

13. Find the exact solutions to x2 � 3x � 1 � 0 by using the Quadratic Formula.

A. ��3 �

2�5�

� B. �3 �

2�13�� C. ��3 �

2�13�� D. �

3 �2�5�� 13.

For Questions 14 and 15, use the value of the discriminant to determine the number and type of roots for each equation.

14. x2 � 3x � 7 � 0A. 2 complex roots B. 2 real, irrational rootsC. 2 real, rational roots D. 1 real, rational root 14.

15. x2 � 4x � 4A. 2 real, rational roots B. 2 real, irrational rootsC. 1 real, rational root D. no real roots 15.

16. What is the vertex of y � 2(x � 3)2 � 6?A. (�3, �6) B. (3, �6) C. (�3, 6) D. (3, 6) 16.

17. What is the equation of the axis of symmetry of y � �3(x � 6)2 � 1?A. x � 2 B. x � �6 C. x � �3 D. x � 6 17.

18. Which quadratic function has its vertex at (2, 3) and passes through (1, 0)?A. y � 2(x � 2)2 � 3 B. y � �3(x � 2)2 � 3C. y � �3(x � 2)2 � 3 D. y � 2(x � 2)2 � 3 18.

19. Which quadratic inequality is graphed at the right?A. y � (x � 1)2 � 4B. y � �(x � 1)2 � 4C. y � �(x � 1)2 � 4D. y � �(x � 1)2 � 4 19.

20. Solve (x � 4)(x � 2) � 0.A. {x � x � �2 or x � 4} B. {x � �4 � x � 2}C. {x � �2 � x � 4} D. {x � x � �2 or x � 4} 20.

Bonus Find the x-intercepts and the y-intercept of the graph B:of y � 2(x � 4)2 � 18.

NAME DATE PERIOD

66

y

xO

Chapter 6 Test, Form 2A

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. Identify the y-intercept and the axis of symmetry for the graph of f(x) � 10x2 � 40x � 42.A. 42; x � 4 B. 0; x � �4 C. 42; x � �2 D. �42; x � 2 1.

2. Identify the quadratic function graphed at the right.A. f(x) � �x2 � 2xB. f(x) � �x2 � 2xC. f(x) � x2 � 2xD. f(x) � �(x � 2)2 2.

3. Determine whether f(x) � 4x2 � 16x � 6 has a maximum or a minimum value and find that value.A. minimum; �10 B. minimum; 2 C. maximum; �10 D. maximum; 2 3.

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.

4. �x2 � 4xA. 4, 0 B. �4, 0C. between �4 and 4 D. �2, 4 4.

5. x2 � 2x � 5A. between �4 and �3; B. between �2 and �1;

between 1 and 2 between 3 and 4C. no real solutions D. �1, �6 5.


6. x2 � 3x � 18A. {6} B. {�6, 3} C. {�9, 2} D. {�3, 6} 6.

7. 3x2 � 20 � 7x

A. {�10, 2} B. ��5, �43�� C. ��4, �

53�� D. ��20, �

13�� 7.

8. Which quadratic equation has roots �2 and �15�?

A. x2 � 4x � 4 � 0 B. 5x2 � 9x � 2 � 0C. 5x2 � 9x � 2 � 0 D. 5x2 � 11x � 2 � 0 8.

9. To solve 9x2 � 12x � 4 � 49 by using the Square Root Property, you would first rewrite the equation as _____.A. 9x2 � 12x � 45 � 0 B. (3x � 2)2 � �49C. (3x � 2)2 � 7 D. (3x � 2)2 � 49 9.

10. Find the value of c that makes x2 � 9x � c a perfect square.

A. �841� B. �

92� C. ��

841� D. 81 10.

66

xO

f(x )


Chapter 6 Test, Form 2A (continued)

11. The quadratic equation x2 � 8x � �20 is to be solved by completing the square. Which equation would be a step in that solution?A. (x � 4)2 � 4 B. x � 4 � �2iC. x2 � 8x � 20 � 0 D. x2 � 8x � 16 � �20 11.

12. Find the exact solutions to 3x2 � 5x � 1 by using the Quadratic Formula.

A. ��5 �6

�13�� B. �

5 �3�13�� C. �

5 �6�37�� D. �

5 �6�13�� 12.


13. 2x2 � 7x � 9 � 0A. 2 real, rational B. 2 real, irrationalC. 2 complex D. 1 real, rational 13.

14. x2 � 20 � 12x � 16A. 1 real, irrational B. 2 real, rationalC. no real D. 1 real, rational 14.

15. Identify the vertex, axis of symmetry, and direction of opening for

y � �12�(x � 8)2 � 2.

A. (�8, 2); x � �8; up B. (�8, �2); x � �8; downC. (8, �2); x � 8; up D. (8, 2); x � 8; up 15.

16. Which quadratic function has its vertex at (�2, 7) and opens down?A. y � �3(x � 2)2 � 7 B. y � (x � 2)2 � 7C. y � �12(x � 2)2 � 7 D. y � �2(x � 2)2 � 7 16.

17. Write y � x2 � 4x � 1 in vertex form.A. y � (x � 2)2 � 5 B. y � (x � 2)2 � 5C. y � (x � 2)2 � 1 D. y � (x � 2)2 � 3 17.

18. Write an equation for the parabola whose vertex is at (�8, 4) and passes through (�6, �2).

A. y � ��32�(x � 8)2 � 4 B. y � ��

14�(x � 8)2 � 4

C. y � �32�(x � 6)2 � 2 D. y � ��

32�(x � 8)2 � 4 18.

19. Which quadratic inequality is graphed at the right?A. y � (x � 2)(x � 3) B. y � (x � 2)(x � 3)C. y � (x � 2)(x � 3) D. y (x � 2)(x � 3) 19.

20. Solve x2 � 2x � 24.A. {x � �4 � x � 6} B. {x � �6 � x � 4}C. {x � x � �6 or x � 4} D. {x � x � �4 or x � 6} 20.

Bonus Write a quadratic equation with roots ��i�

43�

�. B:

NAME DATE PERIOD

66

yxO

Chapter 6 Test, Form 2B

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. Identify the y-intercept and the axis of symmetry for the graph of f(x) � �3x2 � 6x � 12.A. 2; x � �12 B. 12; x � 1 C. �2; x � 0 D. �12; x � �1 1.

2. Identify the quadratic function graphed at the right.A. f(x) � x2 � 4xB. f(x) � �x2 � 4xC. f(x) � �x2 � 4xD. f(x) � �(x � 4)2 2.

3. Determine whether f(x) � �5x2 � 10x � 6 has a maximum or a minimum value and find that value.A. minimum; �1 B. maximum; 11 C. maximum; �1 D. minimum; 11 3.

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.

4. x2 � 4xA. �4, 0 B. between �4 and 4C. 2, �4 D. 0, 4 4.

5. x2 � 2x � �2A. between �3 and �2; B. between �1 and 0;

between 0 and 1 between 2 and 3C. no real solutions D. �1, 1 5.


6. x2 � 3x � 28A. {�4, 7} B. {�14, 2} C. {�7, 4} D. {�2, 14} 6.

7. 5x2 � 4 � 19x

A. ��2, �25�� B. ��

25�, 2� C. ��

15�, 4� D. ��4, �

15�� 7.

8. Which quadratic equation has roots 7 and ��23�?

A. 2x2 � 11x � 21 � 0 B. 3x2 � 19x � 14 � 0C. 3x2 � 23x � 14 � 0 D. 2x2 � 11x � 21 � 0 8.

9. To solve 4x2 � 28x � 49 � 25 by using the Square Root Property, you would first rewrite the equation as _____.A. (2x � 7)2 � 25 B. (2x � 7)2 � 5C. (2x � 7)2 � �5 D. 4x2 � 28x � 24 � 0 9.

10. Find the value of c that makes x2 � 5x � c a perfect square trinomial.

A. �2156�

B. �54� C. �

245� D. �

52� 10.

66

xO

f(x )


Chapter 6 Test, Form 2B (continued)

11. The quadratic equation x2 � 18x � �106 is to be solved by completing the square. Which equation would be a step in that solution?A. (x � 9)2 � 25 B. x2 � 18x � 106 � 0C. x � 9 � �5i D. x2 � 18x � 81 � �106 11.

12. Find the exact solutions to 2x2 � 5x � 1 by using the Quadratic Formula.

A. ��5 �4

�17�� B. �

5 �4�17�� C. �

5 �4�33�� D. �

5 �2�17�� 12.


13. 3x2 � x � 12 � 0A. 2 complex roots B. 1 real, rational rootC. 2 real, rational roots D. 2 real, irrational roots 13.

14. x2 � 10 � 3x � 3A. 2 complex roots B. 2 real, irrational rootsC. 1 real, rational root D. 2 real, rational roots 14.

15. Identify the vertex, axis of symmetry, and direction of opening for y � �8(x � 2)2.A. (�8, �2); x � �8 up B. (�2, 0); x � �2; downC. (2, 0); x � 2; down D. (�2, �8); x � �2; down 15.

16. Which quadratic function has its vertex at (�3, 5) and opens down?A. y � (x � 3)2 � 5 B. y � (x � 3)2 � 5C. y � �(x � 3)2 � 5 D. y � �(x � 3)2 � 5 16.

17. Write y � x2 � 18x � 52 in vertex form.A. y � (x � 9)2 � 113 B. y � (x � 9)2 � 29C. y � (x � 9)2 � 52 D. y � (x � 9)2 � 29 17.

18. Write an equation for the parabola whose vertex is at (�5, 7) and passes through (�3, �1).

A. y � ��111�

(x � 5)2 � 7 B. y � �2(x � 5)2 � 7

C. y � ��12�(x � 5)2 � 7 D. y � ��

12�(x � 5)2 � 7 18.

19. Which quadratic inequality is graphed at the right?A. y � (x � 3)(x � 1) B. y � (x � 3)(x � 1)C. y � (x � 3)(x � 1) D. y (x � 3)(x � 1) 19.

20. Solve 2x � 3 � x2.A. {x � �1 � x � 3} B. {x � �3 � x � 1}C. {x � x � �1 or x � 3} D. {x � x � �3 or x � 1} 20.

Bonus Write a quadratic equation with roots ��i�

32�

�. B:

NAME DATE PERIOD

66

y

xO

Chapter 6 Test, Form 2C


1. Graph f(x) � �5x2 � 10x, labeling the y-intercept, vertex, 1.and axis of symmetry.

2. Determine whether f(x) � �3x2 � 6x � 1 has a maximum 2.or a minimum value and find that value.

For Questions 3 and 4, solve each equation by graphing.If exact roots cannot be found, state the consecutive integers between which the roots are located.

3. x2 � 6x � 83.

4. x2 � x � 5 � 0 4.

5. Solve 5x2 � 13x � 6 by factoring. 5.

6. GEOMETRY The length of a rectangle is 7 inches longer 6.than its width. If the area of the rectangle is 144 square inches, what are its dimensions?

7. Write a quadratic equation with �6 and �34� as its roots. 7.

Write the equation in the form ax2 � bx � c � 0, where a, b, and c are integers.


8. x2 � 6x � 9 � 25 8.

9. 4x2 � 20x � 25 � 7 9.

y

xO

y

xO

xO

f(x )

NAME DATE PERIOD

SCORE 66

Ass

essm

ent


Chapter 6 Test, Form 2C (continued)

For Questions 10 and 11, solve each equation by completing the square.

10. x2 � 4x � 9 � 0 10.

11. 2x2 � 3x � 2 � 0 11.

12. Find the exact solutions to 5x2 � 3x � 2 by using the Quadratic Formula. 12.

For Questions 13 and 14, find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation.

13. 9x2 � 12x � 4 � 0 13.

14. 4x2 � 1 � 9x � 2 14.

15. Identify the vertex, axis of symmetry, and direction of 15.

opening for y � ��23�(x � 5)2 � 7.

16. Write an equation for the parabola with vertex at (2, �1) 16.and y-intercept 5.

17. Write y � x2 � 6x � 8 in vertex form. 17.

18. PHYSICS The height h (in feet) of a certain rocket 18.t seconds after it leaves the ground is modeled by h(t) � �16t2 � 48t � 15. Write the function in vertex form and find the maximum height reached by the rocket.

19. Graph y x2 � 6x � 9. 19.

20. Solve 2x2 � 5x � 3 � 0 algebraically. 20.

Bonus Write a quadratic equation with roots ��37�

�. Write the B:

equation in the form ax2 � bx � c � 0, where a, b, and c are integers.

y

xO

NAME DATE PERIOD

66

Chapter 6 Test, Form 2D


1. Graph f(x) � x2 � 4x � 3, labeling the y-intercept, vertex, 1.and axis of symmetry.

2. Determine whether f(x) � 5x2 � 20x � 3 has a maximum or 2.a minimum value and find that value.

For Questions 3 and 4, solve each equation by graphing.If exact roots cannot be found, state the consecutive integers between which the roots are located.

3. x2 � 2x � 3 � 0 3.

4. 2x2 � 2x � 3 � 0 4.

5. Solve 3x2 � x � 4 by factoring. 5.

6. GEOMETRY The length of a rectangle is 10 inches longer 6.than its width. If the area of the rectangle is 144 square inches, what are its dimensions?

7. Write a quadratic equation with �4 and �32� as its roots. 7.


y

xO

y

xO

xO

f(x )

NAME DATE PERIOD

SCORE 66

Ass

essm

ent


Chapter 6 Test, Form 2D (continued)


8. x2 � 14x � 49 � 16 8.

9. 9x2 � 12x � 4 � 6 9.


10. x2 � 8x � 14 � 0 10.

11. 3x2 � x � 2 � 0 11.

12. Find the exact solutions to 2x2 � 9x � 5 by using the 12.Quadratic Formula.

For Questions 13 and 14, find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation.

13. 25x2 � 20x � 4 � 0 13.

14. 2x2 � 10x � 9 � 2x 14.

15. Identify the vertex, axis of symmetry, and direction of 15.opening for y � �(x � 6)2 � 5.

16. Write an equation for the parabola with vertex at (�4, 2) 16.and y-intercept �2.

17. Write y � x2 � 4x � 8 in vertex form. 17.

18. PHYSICS The height h (in feet) of a certain rocket 18.t seconds after it leaves the ground is modeled by h(t) � �16t2 � 64t � 12. Write the function in vertex form and find the maximum height reached by the rocket.

19. Graph y � x2 � 4x � 4.19.

20. Solve 2x2 � 7x � 15 0 algebraically. 20.

Bonus Write a quadratic equation with roots ��45�

�. B:

Write the equation in the form ax2 � bx � c � 0,where a, b, and c are integers.

y

xO

NAME DATE PERIOD

66

Chapter 6 Test, Form 3


1. Graph f(x) � 3 � 3x2 � 2x, labeling the y-intercept, vertex, 1.and axis of symmetry.

2. Determine whether f(x) � 1 � �35�x � �

34�x2

has a maximum or a minimum value and find that value.

3. BUSINESS Khalid charges $10 for a one-year subscription to his on-line newsletter. Khalid currently has 600 subscribers and he estimates that for each $1 decrease in the subscription price, he would gain 100 new subscribers. What subscription price will maximize Khalid’s 2.income? If he charges this price, how much income should Khalid expect? 3.

For Questions 4–6, solve each equation by graphing. 4.If exact roots cannot be found, state the consecutive integers between which the roots are located.

4. 0.5x2 � 9 � 4.5x

5. �23�x � 3 � �

13�x2 5.

6. 4x(x � 3) � �9 6.

7. Solve 18x2 � 15 � 39x by factoring. 7.

8. Write a quadratic equation with ��23� and 1.75 as its roots. 8.


9. If the roots of an equation are �5 and 3, what is the 9.equation of the axis of symmetry?

y

xO

y

xO

2

2

y

xO

2

2

NAME DATE PERIOD

SCORE 66

Ass

essm

ent

xO

f(x )


Chapter 6 Test, Form 3 (continued)

10. Solve 4x2 � 2x � 0.25 � 1.44 by using the 10.Square Root Property.

For Questions 11 and 12, solve each equation by completing the square.

11. 2x2 � �52�x � 2 � 0 11.

12. x2 � 2.5x � 3 � 0.5 12.

13. Find the exact solutions to �14�x2 � 3x � 1 � 0 by using the 13.

Quadratic Formula.

14. Find the value of the discriminant for 14.3x(0.2x � 0.4) � 1 � 0.9. Then describe the number and type of roots for the equation.

15. Find all values of k such that x2 � kx � 1 � 0 has two 15.complex roots.

16. Write an equation of the parabola with equation 16.

y � ��35��x � �

12��

2� �

52�, translated 4 units left and 2 units up.

Then identify the vertex, axis of symmetry, and direction of opening of your function.

17. PHYSICS The height h (in feet) of a certain aircraft 17.t seconds after it leaves the ground is modeled by h(t) � �9.1t2 � 591.5t � 20,388.125. Write the function in vertex form and find the maximum height reached by the aircraft.

18. Write an equation for the parabola that has the same 18.

vertex as y � �13�x2 � 6x � �

823� and x-intercept 1.

19. Graph y �(x2 � 2x) � 5.25. 19.

20. Solve �x � �72��(x � 1)2 � 0. 20.

Bonus Write a quadratic equation with roots ��3 �42i�5��. B:

Write the equation in the form ax2 � bx � c � 0,where a, b, and c are integers.

y

xO

NAME DATE PERIOD

66

Chapter 6 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. Mr. Moseley asked the students in his Algebra class to work ingroups to solve (x � 3)2 � 25, stating that each student in thefirst group to solve the equation correctly would earn five bonuspoints on the next quiz. Mi-Ling’s group solved the equationusing the Square Root Property. Emilia’s group used theQuadratic Formula to find the solutions. In which group wouldyou prefer to be? Explain your reasoning.

2. The next day, Mr. Moseley had his students work in pairs toreview for their chapter exam. He asked each student to write apractice problem for his or her partner. Len wrote the followingproblem for his partner, Jocelyn: Write an equation for theparabola whose vertex is (�3, �4), that passes through (�1, 0),and that opens downward.a. Jocelyn had trouble solving Len’s problem. Explain why.b. How would you change Len’s problem?c. Make the change you suggested in part b and complete the

problem.

3. a. Write a quadratic function in vertex form whose maximumvalue is 8.

b. Write a quadratic function that transforms the graph of yourfunction from part a so that it is shifted horizontally. Explainthe change you made and describe the transformation thatresults from this change.

4. When asked to write f(x) � 2x2 � 12x � 5 in vertex form, Josephwrote:

f(x) � 2x2 � 12x � 5Step 1 f(x) � 2(x2 � 6x) � 5Step 2 f(x) � 2(x2 � 6x � 9) � 5 � 9Step 3 f(x) � 2(x � 3)2 � 4Is Joseph’s answer correct? Explain your reasoning.

5. The graph of y � x2 � 4x � 4 is shown. Susan used this graph to solve three quadratic inequalities. Her three solutions are given below. Replace each ● with an inequality symbol (, �, �, �) so that each solution is correct. Explain your reasoning for each.a. The solution of x2 � 4x � 4 ● 0 is

{x � x �2 or x � �2}.b. The solution of x2 � 4x � 4 ● 0 is .c. The solution of x2 � 4x � 4 ● 0 is all real numbers.

NAME DATE PERIOD

SCORE 66

Ass

essm

ent

y

xO


Chapter 6 Vocabulary Test/Review

Write whether each sentence is true or false. If false, replace the underlined word or words to make a true sentence.

1. The Square Root Property is used when a quadratic 1.equation is solved by factoring.

2. In f(x) � 3x2 � 2x � 5, the linear term is 5. 2.

3. 2x2 � 3x � 4 � 0 is an example of a quadratic equation. 3.

4. The solutions of a quadratic equation are called its zeros. 4.

5. The quadratic function y � 2(x � 3)2 � 1 is written in 5.vertex form.

6. If a parabola opens upward, the y-coordinate of the vertex 6.is the maximum value.

7. In f(x) � �x2 � 2x � 1, the constant term is �x2. 7.

8. It is necessary to identify the values of a, b, and c in order 8.to solve a quadratic equation by completing the square.

9. The highest or lowest point on a parabola is called the 9.vertex.

10. In the Quadratic Formula, the expression b2 � 4ac is 10.called the quadratic term.

In your own words—Define each term.

11. parabola

12. axis of symmetry

axis of symmetrycompleting the squareconstant termdiscriminantlinear term

maximum valueminimum valueparabolaquadratic equationQuadratic Formula

quadratic functionquadratic inequalityquadratic termrootsSquare Root Property

vertexvertex formZero Product Propertyzeros

NAME DATE PERIOD

SCORE 66

Chapter 6 Quiz (Lessons 6–1 and 6–2)

66


For Questions 1 and 2, consider f(x) � x2 � 2x � 3.

1. Find the y-intercept, the equation of the axis of symmetry, 1.and the x-coordinate of the vertex.

2. Graph the function, labeling the y-intercept, vertex, and 2.axis of symmetry.

3. Determine whether f(x) � 2x2 � 8x � 9 has a maximum or 3.a minimum value and find that value.

Solve each equation. If exact roots cannot be found, state the consecutive integers between which the roots are located.

4. x2 � 2x � 3 4.

5. x2 � 4x � 7 � 0 5.

NAME DATE PERIOD

SCORE


For Questions 1 and 2, solve each equation by factoring.1. 3x2 � 10 � 13x 2. x2 � 4x � 45

3. STANDARDIZED TEST PRACTICE What is the integer solution of 6x2 � 9 � 21x?

A. �3 B. 3 C. �12� D. 2

Write a quadratic equation with the given roots. Write the equation in the form ax2 � bx � c � 0, where a, b,and c are integers.

4. �6 and 2 5. �23� and �4

Solve each equation by using the Square Root Property.6. x2 � 8x � 16 � 36 7. x2 � 2x � 1 � 45

8. 25x2 � 20x � 4 � 3

Solve each equation by completing the square.9. x2 � 10x � 11 10. x2 � 4x � 29 � 11

NAME DATE PERIOD

SCORE 66

Ass

essm

ent

xO

f(x )

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.


1. Solve x2 � 4x � 1 by using the Quadratic Formula. 1.Find exact solutions.

2. Find the value of the discriminant for 3x2 � 6x � 11. Then 2.describe the number and type of roots for the equation.

3. Graph y � �(x � 2)2 � 1. Show and label the vertex and 3.axis of symmetry.

4. Write y � �3x2 � 12x � 6 in vertex form. 4.

5. Write an equation for the parabola whose vertex is at 5.(�5, 0) and passes through (0, 50).

xO

y

Chapter 6 Quiz (Lesson 6–7)

1. Graph y � ��13�(x � 2)2 � 3. 1.

2. Use the graph of its related function to write the 2.solutions of �x2 � 6x � 5 � 0.

3. Solve 0 � x2 � 4x � 3 by graphing. 3.

4. Solve 4x2 � 1 � 4x algebraically. 4.

y

xO

y

xO

NAME DATE PERIOD

SCORE


66

NAME DATE PERIOD

SCORE

66

Chapter 6 Mid-Chapter Test (Lessons 6–1 through 6–4)



1. Which function is graphed?A. f(x) � x2 � 2x � 3B. f(x) � x2 � 2x � 3C. f(x) � x2 � x � 3D. f(x) � (x � 3)2 1.

2. By the Zero Product Property, if (2x � 1)(x � 5) � 0, then _____.

A. x � 1 or x � 5 B. x � �12� or x � 5

C. x � ��12� or x � �5 D. x � �1 or x � �5 2.

3. Write a quadratic equation with 7 and �25� as its roots.

Write the equation in the form ax2 � bx � c � 0, where a, b, and c are integers.A. y � 5x2 � 37x � 14 B. y � 2x2 � 9x � 35C. y � 5x2 � 37x � 14 D. y � 2x2 � 9x � 35 3.

4. The quadratic equation x2 � 4x � 16 is to be solved by completing the square. Which equation would be a step in that solution?A. (x � 2)2 � �20 B. x2 � 4x � 16 � 0C. (x � 2)2 � 20 D. (x � 2)2 � 4 4.

5. Solve x2 � 6x � �6. If exact roots cannot be found, state the consecutive integers between which the roots are located.A. �2, �3 B. between �4 and �3; between �2 and �1C. �3 D. between �5 and �4; between �2 and �1 5.

6. Solve x2 � 4x � 3 � 0 by graphing. 6.

7. Determine whether f(x) � �12�x2 � x � 9 7.

has a maximum or a minimum value and find that value.


8. x2 � 7x � 18 9. 4x2 � x 9.

10. Solve 9x2 � 6x � 1 � 5 by using the Square Root Property. 10.

y

xO

Part II

Part I

NAME DATE PERIOD

SCORE 66

Ass

essm

ent

xO

f(x )

8.


Chapter 6 Cumulative Review (Chapters 1–6)

1. Find the value of 12 � 36 � 4 � (5 � 7)2. (Lesson 1-1) 1.

2. Find the slope of the line that is parallel to the line with 2.equation 3x � 4y � 10. (Lesson 2-3)

3. Describe the system 2x � 3y � 21 and y � 5 � �23�x as 3.

consistent and independent, consistent and dependent, or inconsistent. (Lesson 3-1)

4. Find the coordinates of the vertices of the figure formed 4.by the system of inequalities. (Lesson 3-3)

x � �2 x � y � 6y � �2 x � y � �2

5. Find the value of � �. (Lesson 4-5) 5.

6. Solve � � � � � by using inverse matrices. 6.

(Lesson 4-8)

7. Use synthetic division to find 7.

(2x4 � 5x3 � x2 � 10x � 4) � (x � 3). (Lesson 5-3)

8. Use a calculator to approximate �4

983� to three 8.decimal places. (Lesson 5-5)

9. Solve �x � 2� � 1 � 8. (Lesson 5-8) 9.

10. PHYSICS An object is thrown straight up from the top of 10.a 100-foot platform at a velocity of 48 feet per second. The

height h(t) of the object t seconds after being thrown is given by h(t) � �16t2 � 48t � 100. Find the maximum height reached by the object and the time it takes to achieve this height. (Lesson 6-1)

11. Solve x2 � 2x � 3 by graphing. (Lesson 6-2)

11.

12. Solve 4x2 � 4x � 24 by factoring. (Lesson 6-3) 12.

13. Find the value of the discriminant for 7x2 � 5x � 1 � 0. 13.Then describe the number and type of roots for the equation. (Lesson 6-5)

14. Write y � x2 � 7x � 5 in vertex form. (Lesson 6-6) 14.

y

xO

11�13

ab

�13

4�2

124

5�6

NAME DATE PERIOD

66

Standardized Test Practice (Chapters 1–6)


1. If �ab� � �

32�, then 8a equals which of the following?

A. 16b B. 12b C. �32b� D. �

83�b 1.

2. 20% of 3 yards is how many fifths of 9 feet?E. 1 F. 6 G. 10 H. 15 2.

3. If u � v and t � 0, which of the following are true?I. ut � vt II. u � t � v � t III. u � t � v � tA. I only B. III onlyC. I and II only D. I, II, and III 3.

4. Which of the following is the greatest?

E. �23� F. �

79� G. �

1105�

H. �181�

4.

5. If 2a � 3b represents the perimeter of a rectangle and a � 2b represents its width, the length is ______.

A. 7b B. b C. �72b� D. 14b 5.

6. In the figure, what is the area of the shaded region?E. 30 F. 36G. 54 H. 27 6.

7. Mr. Salazár rented a car for d days. The rental agency charged x dollars per day plus c cents per mile for the model he selected.When Mr. Salazár returned the car, he paid a total of T dollars.In terms of d, x, c, and T, how many miles did he drive?

A. T � (xd � c) B. T � �xcd� C. �xd

T� c� D. �

T �c

xd� 7.

8. If P(3, 2) and Q(7, 10) are the endpoints of the diameter of a circle, what is the area of the circle?E. 2�5� F. 80 G. 4�5� H. 20 8.

9. If (x � y)2 � 100 and xy � 20, what is the value of x2 � y2?A. 120 B. 140 C. 80 D. 60 9.

10. The tenth term in the sequence 7, 12, 19, 28, … is ______.E. 124 F. 103 G. 57 H. 147 10. HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

NAME DATE PERIOD

66

Ass

essm

ent

6

3

15

Part 1: Multiple Choice

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


Standardized Test Practice (continued)

11. If t2 � 6t � �9, what is the value of �t � �12��

2? 11. 12.

12. All four walls of a rectangular room that is 14 feet wide, 20 feet long, and 8 feet high, are to be painted. What is the minimum cost of paint if one gallon covers at most 130 square feet and the paint costs $22 per gallon?

13. The bar graph shows the distribution of votes among the 13. 14.candidates for senior class president. If 220 seniors voted in all, how many students voted for either Theo or Pam?

14. Find the median of x, 2x � 1, �2x

� � 13, 45, and

x � 22 if the mean of this set of numbers is 83.

Column A Column B

15. 15.

16. 1 a c 16.

17. x2 � 25 17.y2 � 16

yx

DCBA

�ac

��ac

�

DCBA

zx

DCBA

y˚ y˚ z˚x˚

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

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.

NAME DATE PERIOD

66

NAME DATE PERIOD

JoeyAnaPamTheo

20

30

10

0

40

Perc

ent

of

vote

s re

ceiv

ed 50

Candidates

A

D

C

B

Standardized Test PracticeStudent Record Sheet (Use with pages 342–343 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 2

NAME DATE PERIOD

66

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.

For Questions 14–20, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.

11 15 17 19

12

13

14 16 18 20

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

21 23 25 27

22 24 26 28 DCBADCBADCBADCBA

DCBADCBADCBADCBA

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 6-1)

Stu

dy G

uid

e a

nd I

nte

rven

tion

Gra

ph

ing

Qu

adra

tic

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-1

6-1

©G

lenc

oe/M

cGra

w-H

ill31

3G

lenc

oe A

lgeb

ra 2

Lesson 6-1

Gra

ph

Qu

adra

tic

Fun

ctio

ns

Qu

adra

tic

Fu

nct

ion

Afu

nctio

n de

fined

by

an e

quat

ion

of t

he f

orm

f(x

) �

ax2

�bx

�c,

whe

re a

�0

Gra

ph

of

a Q

uad

rati

cA

par

abo

law

ith t

hese

cha

ract

eris

tics:

yin

terc

ept:

c;

axis

of

sym

met

ry:

x�

;F

un

ctio

nx-

coor

dina

te o

f ve

rtex

:

Fin

d t

he

y-in

terc

ept,

the

equ

atio

n o

f th

e ax

is o

f sy

mm

etry

,an

d t

he

x-co

ord

inat

e of

th

e ve

rtex

for

th

e gr

aph

of

f(x)

�x2

�3x

�5.

Use

th

is i

nfo

rmat

ion

to g

rap

h t

he

fun

ctio

n.

a�

1,b

��

3,an

d c

�5,

so t

he

y-in

terc

ept

is 5

.Th

e eq

uat

ion

of

the

axis

of

sym

met

ry i

s

x�

or

.Th

e x-

coor

din

ate

of t

he

vert

ex i

s .

Nex

t m

ake

a ta

ble

of v

alu

es f

or x

nea

r .

xx

2�

3x�

5f(

x)

(x,f

(x))

002

�3(

0) �

55

(0,

5)

112

�3(

1) �

53

(1,

3)

��2

�3 �

��5

�,

�2

22�

3(2)

�5

3(2

, 3)

332

�3(

3) �

55

(3,

5)

For

Exe

rcis

es 1

–3,c

omp

lete

par

ts a

–c f

or e

ach

qu

adra

tic

fun

ctio

n.

a.F

ind

th

e y-

inte

rcep

t,th

e eq

uat

ion

of

the

axis

of

sym

met

ry,a

nd

th

e x-

coor

din

ate

of t

he

vert

ex.

b.

Mak

e a

tab

le o

f va

lues

th

at i

ncl

ud

es t

he

vert

ex.

c.U

se t

his

in

form

atio

n t

o gr

aph

th

e fu

nct

ion

.1.

f(x)

�x2

�6x

�8

2.f(

x) �

�x2

�2x

�2

3.f(

x) �

2x2

�4x

�3

8,x

��

3,�

32,

x�

�1,

�1

3,x

�1,

1 ( 1, 1

)x

f(x)

O12 8 4

48

–4

( –1,

3)

x

f(x)

O4 –4 –8

48

–8–4

( –3,

–1)

x

f(x)

O4

–4

48

–8

12 –4

x1

02

3

f(x

)1

33

9

x�

10

�2

1

f(x

)3

22

�1

x�

3�

2�

1�

4

f(x

)�

10

30

11 � 43 � 2

11 � 43 � 2

3 � 23 � 2

x

f (x)

O

3 � 2

3 � 23 � 2

�(�

3)�

2(1)

�b

� 2a

�b

� 2a

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill31

4G

lenc

oe A

lgeb

ra 2

Max

imu

m a

nd

Min

imu

m V

alu

esT

he

y-co

ordi

nat

e of

th

e ve

rtex

of

a qu

adra

tic

fun

ctio

n i

s th

e m

axim

um

or

min

imu

m v

alu

e of

th

e fu

nct

ion

.

Max

imu

m o

r M

inim

um

Val

ue

The

gra

ph o

f f(

x)

�ax

2�

bx�

c, w

here

a�

0, o

pens

up

and

has

a m

inim

umo

f a

Qu

adra

tic

Fu

nct

ion

whe

n a

�0.

The

gra

ph o

pens

dow

n an

d ha

s a

max

imum

whe

n a

�0.

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

has

a m

axim

um

or

min

imu

mva

lue.

Th

en f

ind

th

e m

axim

um

or

min

imu

m v

alu

e of

eac

h f

un

ctio

n.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Gra

ph

ing

Qu

adra

tic

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-1

6-1

Exam

ple

Exam

ple

a.f(

x) �

3x2

�6x

�7

For

th

is f

un

ctio

n,a

�3

and

b�

�6.

Sin

ce a

�0,

the

grap

h o

pen

s u

p,an

d th

efu

nct

ion

has

a m

inim

um

val

ue.

Th

e m

inim

um

val

ue

is t

he

y-co

ordi

nat

eof

th

e ve

rtex

.Th

e x-

coor

din

ate

of t

he

vert

ex i

s �

��

1.

Eva

luat

e th

e fu

nct

ion

at

x�

1 to

fin

d th

em

inim

um

val

ue.

f(1)

�3(

1)2

�6(

1) �

7 �

4,so

th

em

inim

um

val

ue

of t

he

fun

ctio

n i

s 4.

�6

� 2(3)

�b

� 2a

b.f

(x)

�10

0 �

2x�

x2

For

th

is f

un

ctio

n,a

��

1 an

d b

��

2.S

ince

a�

0,th

e gr

aph

ope

ns

dow

n,a

nd

the

fun

ctio

n h

as a

max

imu

m v

alu

e.T

he

max

imu

m v

alu

e is

th

e y-

coor

din

ate

ofth

e ve

rtex

.Th

e x-

coor

din

ate

of t

he

vert

ex

is

��

��

1.

Eva

luat

e th

e fu

nct

ion

at

x�

�1

to f

ind

the

max

imu

m v

alu

e.f(

�1)

�10

0 �

2(�

1) �

(�1)

2�

101,

soth

e m

inim

um

val

ue

of t

he

fun

ctio

n i

s 10

1.

�2

� 2(�

1)�

b� 2a

Exer

cises

Exer

cises

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

has

a m

axim

um

or

min

imu

m v

alu

e.T

hen

fin

dth

e m

axim

um

or

min

imu

m v

alu

e of

eac

h f

un

ctio

n.

1.f(

x) �

2x2

�x

�10

2.f(

x) �

x2�

4x�

73.

f(x)

�3x

2�

3x�

1

min

.,9

min

.,�

11m

in.,

4.f(

x) �

16 �

4x�

x25.

f(x)

�x2

�7x

�11

6.f(

x) �

�x2

�6x

�4

max

.,20

min

.,�

max

.,5

7.f(

x) �

x2�

5x�

28.

f(x)

�20

�6x

�x2

9.f(

x) �

4x2

�x

�3

min

.,�

max

.,29

min

.,2

10.f

(x)

��

x2�

4x�

1011

.f(x

) �

x2�

10x

�5

12.f

(x)

��

6x2

�12

x�

21

max

.,14

min

.,�

20m

ax.,

27

13.f

(x)

�25

x2�

100x

�35

014

.f(x

) �

0.5x

2�

0.3x

�1.

415

.f(x

) �

��

8

min

.,25

0m

in.,

�1.

445

max

.,�

731 � 32

x � 4�

x2�

215 � 1617 � 4

5 � 4

1 � 47 � 8


An

swer

s


Skil

ls P

ract

ice

Gra

ph

ing

Qu

adra

tic

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-1

6-1

©G

lenc

oe/M

cGra

w-H

ill31

5G

lenc

oe A

lgeb

ra 2

Lesson 6-1

For

eac

h q

uad

rati

c fu

nct

ion

,fin

d t

he

y-in

terc

ept,

the

equ

atio

n o

f th

e ax

is o

fsy

mm

etry

,an

d t

he

x-co

ord

inat

e of

th

e ve

rtex

.

1.f(

x) �

3x2

2.f(

x) �

x2�

13.

f(x)

��

x2�

6x�

150;

x�

0;0

1;x

�0;

0�

15;

x�

3;3

4.f(

x) �

2x2

�11

5.f(

x) �

x2�

10x

�5

6.f(

x) �

�2x

2�

8x�

7�

11;

x�

0;0

5;x

�5;

57;

x�

2;2

Com

ple

te p

arts

a–c

for

eac

h q

uad

rati

c fu

nct

ion

.a.

Fin

d t

he

y-in

terc

ept,

the

equ

atio

n o

f th

e ax

is o

f sy

mm

etry

,an

d t

he

x-co

ord

inat

eof

th

e ve

rtex

.b

.M

ake

a ta

ble

of

valu

es t

hat

in

clu

des

th

e ve

rtex

.c.

Use

th

is i

nfo

rmat

ion

to

grap

h t

he

fun

ctio

n.

7.f(

x) �

�2x

28.

f(x)

�x2

�4x

�4

9.f(

x) �

x2�

6x�

80;

x�

0;0

4;x

�2;

28;

x�

3;3

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

has

a m

axim

um

or

a m

inim

um

val

ue.

Th

en f

ind

the

max

imu

m o

r m

inim

um

val

ue

of e

ach

fu

nct

ion

.

10.f

(x)

�6x

211

.f(x

) �

�8x

212

.f(x

) �

x2�

2xm

in.;

0m

ax.;

0m

in.;

�1

13.f

(x)

�x2

�2x

�15

14.f

(x)

��

x2�

4x�

115

.f(x

) �

x2�

2x�

3m

in.;

14m

ax.;

3m

in.;

�4

16.f

(x)

��

2x2

�4x

�3

17.f

(x)

�3x

2�

12x

�3

18.f

(x)

�2x

2�

4x�

1m

ax.;

�1

min

.;�

9m

in.;

�1

( 3, –

1)x

f (x)

O( 2

, 0)

x

f(x)

O16 12 8 4

2–2

46

( 0, 0

)x

f(x)

O

x0

23

46

f(x

)8

0�

10

8

x�

20

24

6

f(x

)16

40

416

x�

2�

10

12

f(x

)�

8�

20

�2

�8

©G

lenc

oe/M

cGra

w-H

ill31

6G

lenc

oe A

lgeb

ra 2

Com

ple

te p

arts

a–c

for

eac

h q

uad

rati

c fu

nct

ion

.a.

Fin

d t

he

y-in

terc

ept,

the

equ

atio

n o

f th

e ax

is o

f sy

mm

etry

,an

d t

he

x-co

ord

inat

eof

th

e ve

rtex

.b

.M

ake

a ta

ble

of

valu

es t

hat

in

clu

des

th

e ve

rtex

.c.

Use

th

is i

nfo

rmat

ion

to

grap

h t

he

fun

ctio

n.

1.f(

x) �

x2�

8x�

152.

f(x)

��

x2�

4x�

123.

f(x)

�2x

2�

2x�

115

;x

�4;

412

;x

��

2;�

21;

x�

0.5;

0.5

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

has

a m

axim

um

or

a m

inim

um

val

ue.

Th

en f

ind

the

max

imu

m o

r m

inim

um

val

ue

of e

ach

fu

nct

ion

.

4.f(

x) �

x2�

2x�

85.

f(x)

�x2

�6x

�14

6.v(

x) �

�x2

�14

x�

57m

in.;

�9

min

.;5

max

.;�

8

7.f(

x) �

2x2

�4x

�6

8.f(

x) �

�x2

�4x

�1

9.f(

x) �

��2 3� x

2�

8x�

24m

in.;

�8

max

.;3

max

.;0

10.G

RA

VIT

ATI

ON

Fro

m 4

fee

t ab

ove

a sw

imm

ing

pool

,Su

san

th

row

s a

ball

upw

ard

wit

h a

velo

city

of

32 f

eet

per

seco

nd.

Th

e h

eigh

t h

(t)

of t

he

ball

tse

con

ds a

fter

Su

san

th

row

s it

is g

iven

by

h(t

) �

�16

t2�

32t

�4.

Fin

d th

e m

axim

um

hei

ght

reac

hed

by

the

ball

an

dth

e ti

me

that

th

is h

eigh

t is

rea

ched

.20

ft;

1 s

11.H

EALT

H C

LUB

SL

ast

year

,th

e S

port

sTim

e A

thle

tic

Clu

b ch

arge

d $2

0 to

par

tici

pate

in

an a

erob

ics

clas

s.S

even

ty p

eopl

e at

ten

ded

the

clas

ses.

Th

e cl

ub

wan

ts t

o in

crea

se t

he

clas

s pr

ice

this

yea

r.T

hey

exp

ect

to l

ose

one

cust

omer

for

eac

h $

1 in

crea

se i

n t

he

pric

e.

a.W

hat

pri

ce s

hou

ld t

he

clu

b ch

arge

to

max

imiz

e th

e in

com

e fr

om t

he

aero

bics

cla

sses

?$4

5b

.W

hat

is

the

max

imu

m i

nco

me

the

Spo

rtsT

ime

Ath

leti

c C

lub

can

exp

ect

to m

ake?

$202

5

f(x)

( 0.5

, 0.5

)x

O

16 12 8 4

( –2,

16)

x

f (x)

O2

–2–4

–6( 4

, –1)

x

f (x)

O16 12 8 4

24

68

x�

10

0.5

12

f(x

)5

10.

51

5

x�

6�

4�

20

2

f(x

)0

1216

120

x0

24

68

f(x

)15

3�

13

15

Pra

ctic

e (

Ave

rag

e)

Gra

ph

ing

Qu

adra

tic

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-1

6-1



Readin

g t

o L

earn

Math

em

ati

csG

rap

hin

g Q

uad

rati

c F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-1

6-1

©G

lenc

oe/M

cGra

w-H

ill31

7G

lenc

oe A

lgeb

ra 2

Lesson 6-1

Pre-

Act

ivit

yH

ow c

an i

nco

me

from

a r

ock

con

cert

be

max

imiz

ed?

Rea

d th

e in

trod

uct

ion

to

Les

son

6-1

at

the

top

of p

age

286

in y

our

text

book

.

•B

ased

on

th

e gr

aph

in

you

r te

xtbo

ok,f

or w

hat

tic

ket

pric

e is

th

e in

com

eth

e gr

eate

st?

$40

•U

se t

he

grap

h t

o es

tim

ate

the

max

imu

m i

nco

me.

abo

ut

$72,

000

Rea

din

g t

he

Less

on

1.a.

For

th

e qu

adra

tic

fun

ctio

n f

(x)

�2x

2�

5x�

3,2x

2is

th

e te

rm,

5xis

th

e te

rm,a

nd

3 is

th

e te

rm.

b.

For

th

e qu

adra

tic

fun

ctio

n f

(x)

��

4 �

x�

3x2 ,

a�

,b�

,an

d

c�

.

2.C

onsi

der

the

quad

rati

c fu

nct

ion

f(x

) �

ax2

�bx

�c,

wh

ere

a�

0.

a.T

he

grap

h o

f th

is f

un

ctio

n i

s a

.

b.

Th

e y-

inte

rcep

t is

.

c.T

he

axis

of

sym

met

ry i

s th

e li

ne

.

d.

If a

�0,

then

th

e gr

aph

ope

ns

and

the

fun

ctio

n h

as a

valu

e.

e.If

a�

0,th

en t

he

grap

h o

pen

s an

d th

e fu

nct

ion

has

a

valu

e.

3.R

efer

to

the

grap

h a

t th

e ri

ght

as y

ou c

ompl

ete

the

foll

owin

g se

nte

nce

s.

a.T

he

curv

e is

cal

led

a .

b.

Th

e li

ne

x�

�2

is c

alle

d th

e .

c.T

he

poin

t (�

2,4)

is

call

ed t

he

.

d.

Bec

ause

th

e gr

aph

con

tain

s th

e po

int

(0,�

1),�

1 is

the

.

Hel

pin

g Y

ou

Rem

emb

er4.

How

can

you

rem

embe

r th

e w

ay t

o u

se t

he

x2te

rm o

f a

quad

rati

c fu

nct

ion

to

tell

wh

eth

er t

he

fun

ctio

n h

as a

max

imu

m o

r a

min

imu

m v

alu

e?S

amp

le a

nsw

er:

Rem

emb

er t

hat

th

e g

rap

h o

f f(

x) �

x2

(wit

h a

�0)

is a

U-s

hap

ed c

urv

eth

at o

pen

s u

p a

nd

has

a m

inim

um

.Th

e g

rap

h o

f g

(x)

��

x2

(wit

h a

�0)

is ju

st t

he

op

po

site

.It

op

ens

do

wn

an

d h

as a

max

imu

m.

y-in

terc

ept

vert

ex

axis

of

sym

met

ry

par

abo

la

x

f(x)

O ( 0, –

1)

( –2,

4)

max

imu

md

ow

nw

ard

min

imu

mu

pw

ard

x�

�� 2b a�

c

par

abo

la

�4

1�

3

con

stan

tlin

ear

qu

adra

tic

©G

lenc

oe/M

cGra

w-H

ill31

8G

lenc

oe A

lgeb

ra 2

Fin

din

g t

he

Axi

s o

f S

ymm

etry

of

a P

arab

ola

As

you

kn

ow,i

f f(

x) �

ax2

�bx

�c

is a

qu

adra

tic

fun

ctio

n,t

he

valu

es o

f x

that

mak

e f(

x) e

qual

to

zero

are

an

d .

Th

e av

erag

e of

th

ese

two

nu

mbe

r va

lues

is

�� 2b a�

.

Th

e fu

nct

ion

f(x

) h

as i

ts m

axim

um

or

min

imu

m

valu

e w

hen

x�

�� 2b a�

.Sin

ce t

he

axis

of

sym

met

ry

of t

he

grap

h o

f f(

x) p

asse

s th

rou

gh t

he

poin

t w

her

e th

e m

axim

um

or

min

imu

m o

ccu

rs,t

he

axis

of

sym

met

ry h

as t

he

equ

atio

n x

��

� 2b a�.

Fin

d t

he

vert

ex a

nd

axi

s of

sym

met

ry f

or f

(x)

�5x

2�

10x

�7.

Use

x�

�� 2b a�

.

x�

�� 21 (0 5)�

��

1T

he

x-co

ordi

nat

e of

th

e ve

rtex

is

�1.

Su

bsti

tute

x�

�1

in f

(x)

�5x

2�

10x

�7.

f(�

1) �

5(�

1)2

�10

(�1)

�7

��

12T

he

vert

ex i

s (�

1,�

12).

Th

e ax

is o

f sy

mm

etry

is

x�

�� 2b a�

,or

x�

�1.

Fin

d t

he

vert

ex a

nd

axi

s of

sym

met

ry f

or t

he

grap

h o

f ea

ch f

un

ctio

n

usi

ng

x�

�� 2b a�

.

1.f(

x) �

x2�

4x�

8(2

,�12

);x

�2

2.g(

x) �

�4x

2�

8x�

3(�

1,7)

;x

��

1

3.y

��

x2�

8x�

3(4

,19)

;x

�4

4.f(

x) �

2x2

�6x

�5

��3 2� ,

�1 2� �;x

��

�3 2�

5.A

(x)

�x2

�12

x�

36(�

6,0)

;x

��

66.

k(x)

��

2x2

�2x

�6

��1 2� ,�

5�1 2� �;

x�

�1 2�

O

f(x)

x

––

,f

((

((

b –– 2a b –– 2a

b –– 2ax

= –

f(x

) =

ax

2 +

bx

+ c

�b

��

b2�

4�

ac��

��

2a�

b�

�b2

�4

�ac�

��

�2a

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-1

6-1

Exam

ple

Exam

ple


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

So

lvin

g Q

uad

rati

c E

qu

atio

ns

by G

rap

hin

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-2

6-2

©G

lenc

oe/M

cGra

w-H

ill31

9G

lenc

oe A

lgeb

ra 2

Lesson 6-2

Solv

e Q

uad

rati

c Eq

uat

ion

s

Qu

adra

tic

Eq

uat

ion

Aqu

adra

tic e

quat

ion

has

the

form

ax

2�

bx�

c�

0, w

here

a�

0.

Ro

ots

of

a Q

uad

rati

c E

qu

atio

nso

lutio

n(s)

of

the

equa

tion,

or

the

zero

(s)

of t

he r

elat

ed q

uadr

atic

fun

ctio

n

Th

e ze

ros

of a

qu

adra

tic

fun

ctio

n a

re t

he

x-in

terc

epts

of

its

grap

h.T

her

efor

e,fi

ndi

ng

the

x-in

terc

epts

is

one

way

of

solv

ing

the

rela

ted

quad

rati

c eq

uat

ion

.

Sol

ve x

2�

x �

6 �

0 b

y gr

aph

ing.

Gra

ph t

he

rela

ted

fun

ctio

n f

(x)

�x2

�x

�6.

Th

e x-

coor

din

ate

of t

he

vert

ex i

s �

�,a

nd

the

equ

atio

n o

f th

e

axis

of

sym

met

ry i

s x

��

.

Mak

e a

tabl

e of

val

ues

usi

ng

x-va

lues

aro

un

d �

.

x�

1�

01

2

f(x

)�

6�

6�

6�

40

Fro

m t

he

tabl

e an

d th

e gr

aph

,we

can

see

th

at t

he

zero

s of

th

e fu

nct

ion

are

2 a

nd

�3.

Sol

ve e

ach

eq

uat

ion

by

grap

hin

g.

1.x2

�2x

�8

�0

2,�

42.

x2�

4x�

5 �

05,

�1

3.x2

�5x

�4

�0

1,4

4.x2

�10

x�

21 �

05.

x2�

4x�

6 �

06.

4x2

�4x

�1

�0

3,7

no

rea

l so

luti

on

s�

1 � 2

x

f(x)

Ox

f (x)

O

x

f(x)

O

x

f(x)

O

x

f(x)

Ox

f(x)

O

1 � 41 � 2

1 � 2

1 � 2

1 � 2�

b� 2a

x

f (x)

O

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill32

0G

lenc

oe A

lgeb

ra 2

Esti

mat

e So

luti

on

sO

ften

,you

may

not

be

able

to

fin

d ex

act

solu

tion

s to

qu

adra

tic

equ

atio

ns

by g

raph

ing.

Bu

t yo

u c

an u

se t

he

grap

h t

o es

tim

ate

solu

tion

s.

Sol

ve x

2�

2x�

2 �

0 b

y gr

aph

ing.

If e

xact

roo

ts c

ann

ot b

e fo

un

d,

stat

e th

e co

nse

cuti

ve i

nte

gers

bet

wee

n w

hic

h t

he

root

s ar

e lo

cate

d.

Th

e eq

uat

ion

of

the

axis

of

sym

met

ry o

f th

e re

late

d fu

nct

ion

is

x�

��

1,so

the

ver

tex

has

x-co

ordi

nate

1.M

ake

a ta

ble

of v

alue

s.

x�

10

12

3

f(x

)1

�2

�3

�2

1

Th

e x-

inte

rcep

ts o

f th

e gr

aph

are

bet

wee

n 2

an

d 3

and

betw

een

0 a

nd

�1.

So

one

solu

tion

is

betw

een

2 a

nd

3,an

d th

e ot

her

sol

uti

on i

sbe

twee

n 0

an

d �

1.

Sol

ve t

he

equ

atio

ns

by

grap

hin

g.If

exa

ct r

oots

can

not

be

fou

nd

,sta

te t

he

con

secu

tive

in

tege

rs b

etw

een

wh

ich

th

e ro

ots

are

loca

ted

.

1.x2

�4x

�2

�0

2.x2

�6x

�6

�0

3.x2

�4x

�2�

0

bet

wee

n 0

an

d 1

;b

etw

een

�2

and

�1;

bet

wee

n �

1 an

d 0

;b

etw

een

3 a

nd

4b

etw

een

�5

and

�4

bet

wee

n �

4 an

d �

3

4.�

x2�

2x�

4 �

05.

2x2

�12

x�

17 �

06.

�x2

�x

��

0

bet

wee

n 3

an

d 4

;b

etw

een

2 a

nd

3;

bet

wee

n �

2 an

d �

1;b

etw

een

�2

and

�1

bet

wee

n 3

an

d 4

bet

wee

n 3

an

d 4 x

f(x)

O

x

f(x)

Ox

f(x)

O

5 � 21 � 2

x

f(x)

Ox

f (x)

Ox

f(x)

O

�2

� 2(1)

x

f (x)

O

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

So

lvin

g Q

uad

rati

c E

qu

atio

ns

by G

rap

hin

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-2

6-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

So

lvin

g Q

uad

rati

c E

qu

atio

ns

By

Gra

ph

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-2

6-2

©G

lenc

oe/M

cGra

w-H

ill32

1G

lenc

oe A

lgeb

ra 2

Lesson 6-2

Use

th

e re

late

d g

rap

h o

f ea

ch e

qu

atio

n t

o d

eter

min

e it

s so

luti

ons.

1.x2

�2x

�3

�0

2.�

x2�

6x�

9 �

03.

3x2

�4x

�3

�0

�3,

1�

3n

o r

eal s

olu

tio

ns

Sol

ve e

ach

eq

uat

ion

by

grap

hin

g.If

exa

ct r

oots

can

not

be

fou

nd

,sta

te t

he

con

secu

tive

in

tege

rs b

etw

een

wh

ich

th

e ro

ots

are

loca

ted

.

4.x2

�6x

�5

�0

5.�

x2�

2x�

4 �

06.

x2�

6x�

4 �

01,

5n

o r

eal s

olu

tio

ns

bet

wee

n 0

an

d 1

;b

etw

een

5 a

nd

6

Use

a q

uad

rati

c eq

uat

ion

to

fin

d t

wo

real

nu

mb

ers

that

sat

isfy

eac

h s

itu

atio

n,o

rsh

ow t

hat

no

such

nu

mb

ers

exis

t.

7.T

hei

r su

m i

s �

4,an

d th

eir

prod

uct

is

0.8.

Th

eir

sum

is

0,an

d th

eir

prod

uct

is

�36

.

�x

2�

4x�

0;0,

�4

�x

2�

36 �

0;�

6,6

f(x) �

�x2

� 3

6

x

f (x)

O6

–612

–12

36 24 12

f(x) �

�x2

� 4

x

x

f (x)

O

f(x) �

x2

� 6

x �

4

x

f (x)

Of(x

) � �

x2 �

2x

� 4x

f (x)

O

f(x) �

x2

� 6

x �

5

x

f (x)

O

x

f (x) O

f(x) �

3x2

� 4

x �

3

x

f(x)

O

f(x) �

�x2

� 6

x �

9

x

f (x)

O

f(x) �

x2

� 2

x �

3

©G

lenc

oe/M

cGra

w-H

ill32

2G

lenc

oe A

lgeb

ra 2

Use

th

e re

late

d g

rap

h o

f ea

ch e

qu

atio

n t

o d

eter

min

e it

s so

luti

ons.

1.�

3x2

�3

�0

2.3x

2�

x�

3 �

03.

x2�

3x�

2 �

0

�1,

1n

o r

eal s

olu

tio

ns

1,2

Sol

ve e

ach

eq

uat

ion

by

grap

hin

g.If

exa

ct r

oots

can

not

be

fou

nd

,sta

te t

he

con

secu

tive

in

tege

rs b

etw

een

wh

ich

th

e ro

ots

are

loca

ted

.

4.�

2x2

�6x

�5

�0

5.x2

�10

x�

24 �

06.

2x2

�x

�6

�0

bet

wee

n 0

an

d 1

;�

6,�

4b

etw

een

�2

and

�1,

bet

wee

n �

4 an

d �

32

Use

a q

uad

rati

c eq

uat

ion

to

fin

d t

wo

real

nu

mb

ers

that

sat

isfy

eac

h s

itu

atio

n,o

rsh

ow t

hat

no

such

nu

mb

ers

exis

t.

7.T

hei

r su

m i

s 1,

and

thei

r pr

odu

ct i

s �

6.8.

Th

eir

sum

is

5,an

d th

eir

prod

uct

is

8.

For

Exe

rcis

es 9

an

d 1

0,u

se t

he

form

ula

h(t

) �

v 0t

�16

t2,w

her

e h

(t)

is t

he

hei

ght

of a

n o

bje

ct i

n f

eet,

v 0is

th

e ob

ject

’s i

nit

ial

velo

city

in

fee

t p

er s

econ

d,a

nd

tis

th

eti

me

in s

econ

ds.

9.B

ASE

BA

LLM

arta

thr

ows

a ba

seba

ll w

ith

an in

itia

l upw

ard

velo

city

of

60 f

eet

per

seco

nd.

Igno

ring

Mar

ta’s

hei

ght,

how

long

aft

er s

he r

elea

ses

the

ball

will

it h

it t

he g

roun

d?3.

75 s

10.V

OLC

AN

OES

A v

olca

nic

eru

ptio

n b

last

s a

bou

lder

upw

ard

wit

h a

n i

nit

ial

velo

city

of

240

feet

per

sec

ond.

How

lon

g w

ill

it t

ake

the

bou

lder

to

hit

th

e gr

oun

d if

it

lan

ds a

t th

esa

me

elev

atio

n f

rom

wh

ich

it

was

eje

cted

?15

s

�x2

�5x

�8

�0;

no

su

ch r

eal

nu

mb

ers

exis

tx

f (x)

Of(x

) � �

x2 �

5x

� 8

�x

2�

x�

6 �

0;3,

�2

f(x) �

�x2

� x

� 6 x

f (x)

O

x

f (x)

O

f(x) �

2x2

� x

� 6

f(x) �

x2

� 1

0x �

24

x

f (x)

O

f(x) �

�2x

2 �

6x

� 5

x

f (x)

O–4

–2–6

12 8 4

x

f (x)

O

f(x) �

x2

� 3

x �

2

x

f(x) O

f(x) �

3x2

� x

� 3

x

f (x)

O

f(x) �

�3x

2 �

3

Pra

ctic

e (

Ave

rag

e)

So

lvin

g Q

uad

rati

c E

qu

atio

ns

By

Gra

ph

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-2

6-2


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Qu

adra

tic

Eq

uat

ion

s by

Gra

ph

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-2

6-2

©G

lenc

oe/M

cGra

w-H

ill32

3G

lenc

oe A

lgeb

ra 2

Lesson 6-2

Pre-

Act

ivit

yH

ow d

oes

a q

uad

rati

c fu

nct

ion

mod

el a

fre

e-fa

ll r

ide?

Rea

d th

e in

trod

uct

ion

to

Les

son

6-2

at

the

top

of p

age

294

in y

our

text

book

.

Wri

te a

qu

adra

tic

fun

ctio

n t

hat

des

crib

es t

he

hei

ght

of a

bal

l t

seco

nds

aft

erit

is

drop

ped

from

a h

eigh

t of

125

fee

t.h

(t)

��

16t2

�12

5

Rea

din

g t

he

Less

on

1.T

he

grap

h o

f th

e qu

adra

tic

fun

ctio

n f

(x)

��

x2�

x�

6 is

sh

own

at

the

righ

t.U

se t

he

grap

h t

o fi

nd

the

solu

tion

s of

th

equ

adra

tic

equ

atio

n �

x2�

x�

6 �

0.�

2 an

d 3

2.S

ketc

h a

gra

ph t

o il

lust

rate

eac

h s

itu

atio

n.

a.A

par

abol

a th

at o

pen

s b

.A

par

abol

a th

at o

pen

s c.

A p

arab

ola

that

ope

ns

dow

nw

ard

and

repr

esen

ts a

u

pwar

d an

d re

pres

ents

a

dow

nw

ard

and

qu

adra

tic

fun

ctio

n w

ith

tw

o qu

adra

tic

fun

ctio

n w

ith

re

pres

ents

a

re

al z

eros

,bot

h o

f w

hic

h a

reex

actl

y on

e re

al z

ero.

Th

e

quad

rati

c fu

nct

ion

n

egat

ive

nu

mbe

rs.

zero

is

a po

siti

ve n

um

ber.

wit

h n

o re

al z

eros

.

Hel

pin

g Y

ou

Rem

emb

er

3.T

hin

k of

a m

emor

y ai

d th

at c

an h

elp

you

rec

all

wh

at i

s m

ean

t by

th

e ze

ros

of a

qu

adra

tic

fun

ctio

n.

Sam

ple

an

swer

:Th

e b

asic

fac

ts a

bo

ut

a su

bje

ct a

re s

om

etim

es c

alle

d t

he

AB

Cs.

In t

he

case

of

zero

s,th

e A

BC

s ar

e th

e X

YZ

s,b

ecau

se t

he

zero

sar

e th

e x-

valu

es t

hat

mak

e th

e y-

valu

es e

qu

al t

o z

ero

.

x

y

Ox

y

Ox

y

O

x

y

O

©G

lenc

oe/M

cGra

w-H

ill32

4G

lenc

oe A

lgeb

ra 2

Gra

ph

ing

Ab

solu

te V

alu

e E

qu

atio

ns

You

can

sol

ve a

bsol

ute

val

ue

equ

atio

ns

in m

uch

th

e sa

me

way

you

sol

ved

quad

rati

c eq

uat

ion

s.G

raph

th

e re

late

d ab

solu

te v

alu

e fu

nct

ion

for

eac

h

equ

atio

n u

sin

g a

grap

hin

g ca

lcu

lato

r.T

hen

use

th

e ZE

ROfe

atu

re i

n t

he

CALC

men

u t

o fi

nd

its

real

sol

uti

ons,

if a

ny.

Rec

all

that

sol

uti

ons

are

poin

ts

wh

ere

the

grap

h i

nte

rsec

ts t

he

x-ax

is.

For

eac

h e

qu

atio

n,m

ake

a sk

etch

of

the

rela

ted

gra

ph

an

d f

ind

th

e so

luti

ons

rou

nd

ed t

o th

e n

eare

st h

un

dre

dth

.

1.|x

�5|

�0

2.|4

x�

3| �

5 �

03.

|x�

7| �

0

�5

No

so

luti

on

s7

4.|x

�3|

�8

�0

5.�

|x�

3| �

6 �

06.

|x�

2| �

3 �

0

�11

,5�

9,3

�1,

5

7.|3

x �

4| �

28.

|x �

12| �

109.

|x|�

3 �

0

�2,

��2 3�

�22

,�2

�3,

3

10.E

xpla

in h

ow s

olvi

ng

abso

lute

val

ue

equ

atio

ns

alge

brai

call

y an

d fi

ndi

ng

zero

s of

abs

olu

te v

alu

e fu

nct

ion

s gr

aph

ical

ly a

re r

elat

ed.

Sam

ple

an

swer

:va

lues

of

xw

hen

so

lvin

g a

lgeb

raic

ally

are

th

e x-

inte

rcep

ts (

or

zero

s) o

f th

e fu

nct

ion

wh

en g

rap

hed

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-2

6-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

So

lvin

g Q

uad

rati

c E

qu

atio

ns

by F

acto

rin

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-3

6-3

©G

lenc

oe/M

cGra

w-H

ill32

5G

lenc

oe A

lgeb

ra 2

Lesson 6-3

Solv

e Eq

uat

ion

s b

y Fa

cto

rin

gW

hen

you

use

fac

tori

ng

to s

olve

a q

uad

rati

c eq

uat

ion

,yo

u u

se t

he

foll

owin

g pr

oper

ty.

Zer

o P

rod

uct

Pro

per

tyF

or a

ny r

eal n

umbe

rs a

and

b, if

ab

�0,

the

n ei

ther

a�

0 or

b�

0, o

r bo

th a

and

b�

0.

Sol

ve e

ach

eq

uat

ion

by

fact

orin

g.Ex

ampl

eEx

ampl

ea.

3x2

�15

x3x

2�

15x

Orig

inal

equ

atio

n

3x2

�15

x�

0S

ubtr

act

15x

from

bot

h si

des.

3x(x

�5)

�0

Fac

tor

the

bino

mia

l.

3x �

0or

x�

5 �

0Z

ero

Pro

duct

Pro

pert

y

x�

0or

x�

5S

olve

eac

h eq

uatio

n.

Th

e so

luti

on s

et i

s {0

,5}.

b.4

x2�

5x�

214x

2�

5x�

21O

rigin

al e

quat

ion

4x2

�5x

�21

�0

Sub

trac

t 21

fro

m b

oth

side

s.

(4x

�7)

(x�

3)�

0F

acto

r th

e tr

inom

ial.

4x�

7 �

0or

x�

3 �

0Z

ero

Pro

duct

Pro

pert

y

x�

�or

x

�3

Sol

ve e

ach

equa

tion.

Th

e so

luti

on s

et i

s ��

,3�.

7 � 4

7 � 4

Exer

cises

Exer

cises

Sol

ve e

ach

eq

uat

ion

by

fact

orin

g.

1.6x

2�

2x�

02.

x2�

7x3.

20x2

��

25x

�0,�

{0,7

}�0,

��

4.6x

2�

7x5.

6x2

�27

x�

06.

12x2

�8x

�0

�0,�

�0,�

�0,�

7.x2

�x

�30

�0

8.2x

2�

x�

3 �

09.

x2�

14x

�33

�0

{5,�

6}�

,�1 �

{�11

,�3}

10.4

x2�

27x

�7

�0

11.3

x2�

29x

�10

�0

12.6

x2�

5x�

4 �

0

�,�

7 ��

10,

��

,�

13.1

2x2

�8x

�1

�0

14.5

x2�

28x

�12

�0

15.2

x2�

250x

�50

00 �

0

�,

��

,�6 �

{100

,25}

16.2

x2�

11x

�40

�0

17.2

x2�

21x

�11

�0

18.3

x2�

2x�

21 �

0

�8,�

��

11,

��

,�3 �

19.8

x2�

14x

�3

�0

20.6

x2�

11x

�2

�0

21.5

x2�

17x

�12

�0

�,

��

2,�

�,�

4 �22

.12x

2�

25x

�12

�0

23.1

2x2

�18

x�

6 �

024

.7x2

�36

x�

5 �

0

��,�

��

,�1 �

�,5

�1 � 7

1 � 23 � 4

4 � 3

3 � 51 � 6

1 � 43 � 2

7 � 31 � 2

5 � 2

2 � 51 � 2

1 � 6

4 � 31 � 2

1 � 31 � 4

3 � 2

2 � 39 � 2

7 � 6

5 � 41 � 3

©G

lenc

oe/M

cGra

w-H

ill32

6G

lenc

oe A

lgeb

ra 2

Wri

te Q

uad

rati

c Eq

uat

ion

sT

o w

rite

a q

uad

rati

c eq

uat

ion

wit

h r

oots

pan

d q,

let

(x�

p)(x

�q)

�0.

Th

en m

ult

iply

usi

ng

FO

IL.

Wri

te a

qu

adra

tic

equ

atio

n w

ith

th

e gi

ven

roo

ts.W

rite

th

e eq

uat

ion

in t

he

form

ax2

�bx

�c

�0.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

So

lvin

g Q

uad

rati

c E

qu

atio

ns

by F

acto

rin

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-3

6-3

Exam

ple

Exam

ple

a.3,

�5 (x

�p)

(x�

q) �

0W

rite

the

patte

rn.

(x�

3)[x

�(�

5)]

�0

Rep

lace

pw

ith 3

, q

with

�5.

(x�

3)(x

�5)

�0

Sim

plify

.

x2�

2x�

15 �

0U

se F

OIL

.

Th

e eq

uat

ion

x2

�2x

�15

�0

has

roo

ts

3 an

d �

5.

b.�

,

(x�

p)(x

�q)

�0

�x�

��

x�

��0

�x�

��x�

��0

��

0

�24

�0

24x2

�13

x�

7 �

0

Th

e eq

uat

ion

24x

2�

13x

�7

�0

has

root

s �

and

.1 � 3

7 � 8

24 �

(8x

�7)

(3x

�1)

��

�24

(3x

�1)

�3

(8x

�7)

�8

1 � 37 � 8

1 � 37 � 8

1 � 37 � 8

Exer

cises

Exer

cises

Wri

te a

qu

adra

tic

equ

atio

n w

ith

th

e gi

ven

roo

ts.W

rite

th

e eq

uat

ion

in

th

e fo

rma

x2�

bx�

c�

0.

1.3,

�4

2.�

8,�

23.

1,9

x2

�x

�12

�0

x2

�10

x�

16 �

0x

2�

10x

�9

�0

4.�

55.

10,7

6.�

2,15

x2

�10

x�

25 �

0x

2�

17x

�70

�0

x2

�13

x�

30 �

0

7.�

,58.

2,9.

�7,

3x2

�14

x�

5 �

03x

2�

8x�

4 �

04x

2�

25x

�21

�0

10.3

,11

.�,�

112

.9,

5x2

�17

x�

6 �

09x

2�

13x

�4

�0

6x2

�55

x�

9 �

0

13.

,�14

.,�

15.

,

9x2

�4

�0

8x2

�6x

�5

�0

35x

2�

22x

�3

�0

16.�

,17

.,

18.

,

16x

2�

42x

�49

8x2

�10

x�

3 �

048

x2

�14

x�

1 �

0

1 � 61 � 8

3 � 41 � 2

7 � 27 � 8

1 � 53 � 7

1 � 25 � 4

2 � 32 � 3

1 � 64 � 9

2 � 5

3 � 42 � 3

1 � 3


An

swer

s


Skil

ls P

ract

ice

So

lvin

g Q

uad

rati

c E

qu

atio

ns

by F

acto

rin

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-3

6-3

©G

lenc

oe/M

cGra

w-H

ill32

7G

lenc

oe A

lgeb

ra 2

Lesson 6-3

Sol

ve e

ach

eq

uat

ion

by

fact

orin

g.

1.x2

�64

{�8,

8}2.

x2�

100

�0

{10,

�10

}

3.x2

�3x

�2

�0

{1,2

}4.

x2�

4x�

3 �

0{1

,3}

5.x2

�2x

�3

�0

{1,�

3}6.

x2�

3x�

10 �

0{5

,�2}

7.x2

�6x

�5

�0

{1,5

}8.

x2�

9x�

0{0

,9}

9.�

x2�

6x�

0{0

,6}

10.x

2�

6x�

8 �

0{�

2,�

4}

11.x

2�

�5x

{0,�

5}12

.x2

�14

x�

49 �

0{7

}

13.x

2�

6 �

5x{2

,3}

14.x

2�

18x

��

81{�

9}

15.x

2�

4x�

21{�

3,7}

16.2

x2�

5x�

3 �

0�

,�3 �

17.4

x2�

5x�

6 �

0�

,�2 �

18.3

x2�

13x

�10

�0

��,5

�

Wri

te a

qu

adra

tic

equ

atio

n w

ith

th

e gi

ven

roo

ts.W

rite

th

e eq

uat

ion

in

th

e fo

rma

x2�

bx�

c�

0,w

her

e a

,b,a

nd

car

e in

tege

rs.

19.1

,4x

2�

5x�

4 �

020

.6,�

9x

2�

3x�

54 �

0

21.�

2,�

5x

2�

7x�

10 �

022

.0,7

x2

�7x

�0

23.�

,�3

3x2

�10

x�

3 �

024

.�,

8x2

�2x

�3

�0

25.F

ind

two

con

secu

tive

in

tege

rs w

hos

e pr

odu

ct i

s 27

2.16

,17

3 � 41 � 2

1 � 3

2 � 33 � 4

1 � 2

©G

lenc

oe/M

cGra

w-H

ill32

8G

lenc

oe A

lgeb

ra 2

Sol

ve e

ach

eq

uat

ion

by

fact

orin

g.

1.x2

�4x

�12

�0

{6,�

2}2.

x2�

16x

�64

�0

{8}

3.x2

�20

x�

100

�0

{10}

4.x2

�6x

�8

�0

{2,4

}5.

x2�

3x�

2 �

0{�

2,�

1}6.

x2�

9x�

14 �

0{2

,7}

7.x2

�4x

�0

{0,4

}8.

7x2

�4x

�0,�

9.x2

�25

�10

x{5

}

10.1

0x2

�9x

�0,�

11.x

2�

2x�

99{�

9,11

}

12.x

2�

12x

��

36{�

6}13

.5x2

�35

x�

60 �

0{3

,4}

14.3

6x2

�25

�,�

�15

.2x2

�8x

�90

�0

{9,�

5}

16.3

x2�

2x�

1 �

0�

,�1 �

17.6

x2�

9x�0,

�18

.3x2

�24

x�

45 �

0{�

5,�

3}19

.15x

2�

19x

�6

�0

��,�

�20

.3x2

�8x

��

4�2,

�21

.6x2

�5x

�6

�,�

�W

rite

a q

uad

rati

c eq

uat

ion

wit

h t

he

give

n r

oots

.Wri

te t

he

equ

atio

n i

n t

he

form

ax2

�bx

�c

�0,

wh

ere

a,b

,an

d c

are

inte

gers

.

22.7

,223

.0,3

24

.�5,

8x

2�

9x�

14 �

0x

2�

3x�

0x

2�

3x�

40 �

0

25.�

7,�

826

.�6,

�3

27.3

,�4

x2

�15

x�

56 �

0x

2�

9x�

18 �

0x

2�

x�

12 �

0

28.1

,29

.,2

30.0

,�

2x2

�3x

�1

�0

3x2

�7x

�2

�0

2x2

�7x

�0

31.

,�3

32.4

,33

.�,�

3x2

�8x

�3

�0

3x2

�13

x�

4 �

015

x2

�22

x�

8 �

0

34.N

UM

BER

TH

EORY

Fin

d tw

o co

nse

cuti

ve e

ven

pos

itiv

e in

tege

rs w

hos

e pr

odu

ct i

s 62

4.24

,26

35.N

UM

BER

TH

EORY

Fin

d tw

o co

nse

cuti

ve o

dd p

osit

ive

inte

gers

wh

ose

prod

uct

is

323.

17,1

936

.GEO

MET

RYT

he

len

gth

of

a re

ctan

gle

is 2

fee

t m

ore

than

its

wid

th.F

ind

the

dim

ensi

ons

of t

he

rect

angl

e if

its

are

a is

63

squ

are

feet

.7

ft b

y 9

ft

37.P

HO

TOG

RA

PHY

Th

e le

ngt

h a

nd

wid

th o

f a

6-in

ch b

y 8-

inch

ph

otog

raph

are

red

uce

d by

the

sam

e am

oun

t to

mak

e a

new

ph

otog

raph

wh

ose

area

is

hal

f th

at o

f th

e or

igin

al.B

yh

ow m

any

inch

es w

ill

the

dim

ensi

ons

of t

he

phot

ogra

ph h

ave

to b

e re

duce

d?2

in.

4 � 52 � 3

1 � 31 � 3

7 � 21 � 3

1 � 2

2 � 33 � 2

2 � 3

2 � 33 � 5

3 � 21 � 3

5 � 65 � 6

9 � 10

4 � 7

Pra

ctic

e (

Ave

rag

e)

So

lvin

g Q

uad

rati

c E

qu

atio

ns

by F

acto

rin

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-3

6-3



Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Qu

adra

tic

Eq

uat

ion

s by

Fac

tori

ng

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-3

6-3

©G

lenc

oe/M

cGra

w-H

ill32

9G

lenc

oe A

lgeb

ra 2

Lesson 6-3

Pre-

Act

ivit

yH

ow i

s th

e Z

ero

Pro

du

ct P

rop

erty

use

d i

n g

eom

etry

?

Rea

d th

e in

trod

uct

ion

to

Les

son

6-3

at

the

top

of p

age

301

in y

our

text

book

.

Wh

at d

oes

the

expr

essi

on x

(x�

5) m

ean

in

th

is s

itu

atio

n?

It r

epre

sen

ts t

he

area

of

the

rect

ang

le,s

ince

th

e ar

ea is

th

ep

rod

uct

of

the

wid

th a

nd

len

gth

.

Rea

din

g t

he

Less

on

1.T

he

solu

tion

of

a qu

adra

tic

equ

atio

n b

y fa

ctor

ing

is s

how

n b

elow

.Giv

e th

e re

ason

for

each

ste

p of

th

e so

luti

on.

x2�

10x

��

21O

rigin

al e

quat

ion

x2�

10x

�21

�0

Ad

d 2

1 to

eac

h s

ide.

(x�

3)(x

�7)

�0

Fact

or

the

trin

om

ial.

x�

3 �

0 or

x �

7 �

0Z

ero

Pro

du

ct P

rop

erty

x�

3 x

�7

So

lve

each

eq

uat

ion

.

Th

e so

luti

on s

et i

s .

2.O

n a

n a

lgeb

ra q

uiz

,stu

den

ts w

ere

aske

d to

wri

te a

qu

adra

tic

equ

atio

n w

ith

�7

and

5 as

its

root

s.T

he

wor

k th

at t

hre

e st

ude

nts

in

th

e cl

ass

wro

te o

n t

hei

r pa

pers

is

show

n b

elow

.

Mar

laR

osa

Lar

ry(x

�7)

(x�

5) �

0(x

�7)

(x�

5) �

0(x

�7)

(x�

5) �

0x2

�2x

�35

�0

x2�

2x�

35 �

0x2

�2x

�35

�0

Wh

o is

cor

rect

?R

osa

Exp

lain

th

e er

rors

in

th

e ot

her

tw

o st

ude

nts

’ wor

k.

Sam

ple

an

swer

:M

arla

use

d t

he

wro

ng

fac

tors

.Lar

ry u

sed

th

e co

rrec

tfa

cto

rs b

ut

mu

ltip

lied

th

em in

corr

ectl

y.

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

a co

nce

pt i

s to

rep

rese

nt

it i

n m

ore

than

on

e w

ay.D

escr

ibe

anal

gebr

aic

way

an

d a

grap

hic

al w

ay t

o re

cogn

ize

a qu

adra

tic

equ

atio

n t

hat

has

a d

oubl

ero

ot.

Sam

ple

an

swer

:A

lgeb

raic

:Wri

te t

he

equ

atio

n in

th

e st

and

ard

fo

rm

ax2

�b

x�

c�

0 an

d e

xam

ine

the

trin

om

ial.

If it

is a

per

fect

sq

uar

etr

ino

mia

l,th

e q

uad

rati

c fu

nct

ion

has

a d

ou

ble

ro

ot.

Gra

ph

ical

:G

rap

h t

he

rela

ted

qu

adra

tic

fun

ctio

n.I

f th

e p

arab

ola

has

exa

ctly

on

e x-

inte

rcep

t,th

en t

he

equ

atio

n h

as a

do

ub

le r

oo

t.

{3,7

}

©G

lenc

oe/M

cGra

w-H

ill33

0G

lenc

oe A

lgeb

ra 2

Eu

ler’

s F

orm

ula

fo

r P

rim

e N

um

ber

sM

any

mat

hem

atic

ian

s h

ave

sear

ched

for

a f

orm

ula

th

at w

ould

gen

erat

e pr

ime

nu

mbe

rs.O

ne

such

for

mu

la w

as p

ropo

sed

by E

ule

r an

d u

ses

a qu

adra

tic

poly

nom

ial,

x2�

x�

41.

Fin

d t

he

valu

es o

f x2

�x

�41

for

th

e gi

ven

val

ues

of

x.S

tate

wh

eth

er

each

val

ue

of t

he

pol

ynom

ial

is o

r is

not

a p

rim

e n

um

ber

.

1.x

�0

2.x

�1

3.x

�2

41,p

rim

e43

,pri

me

47,p

rim

e

4.x

�3

5.x

�4

6.x

�5

53,p

rim

e61

,pri

me

71,p

rim

e

7.x

�6

8.x

�17

9.x

�28

83,p

rim

e34

7,p

rim

e85

3,p

rim

e

10.x

�29

11.

x�

3012

.x�

35

911,

pri

me

971,

pri

me

1301

,pri

me

13.D

oes

the

form

ula

prod

uce

all p

rim

e nu

mbe

rs g

reat

er t

han

40?

Giv

e ex

ampl

es

in y

our

answ

er.

No

.Am

on

g t

he

pri

mes

om

itte

d a

re 5

9,67

,73,

79,8

9,10

1,10

3,10

7,10

9,an

d 1

27.

14.E

ule

r’s

form

ula

pro

duce

s pr

imes

for

man

y va

lues

of

x,bu

t it

doe

s n

ot w

ork

for

all

of t

hem

.Fin

d th

e fi

rst

valu

e of

xfo

r w

hic

h t

he

form

ula

fai

ls.

(Hin

t:T

ry m

ult

iple

s of

ten

.)

x�

40 g

ives

168

1,w

hic

h e

qu

als

412 .

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-3

6-3


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Co

mp

leti

ng

th

e S

qu

are

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

6-4

©G

lenc

oe/M

cGra

w-H

ill33

1G

lenc

oe A

lgeb

ra 2

Lesson 6-4

Squ

are

Ro

ot

Pro

per

tyU

se t

he

foll

owin

g pr

oper

ty t

o so

lve

a qu

adra

tic

equ

atio

n t

hat

is

in t

he

form

“pe

rfec

t sq

uar

e tr

inom

ial

�co

nst

ant.

”

Sq

uar

e R

oo

t P

rop

erty

For

any

rea

l num

ber

xif

x2

�n,

the

n x

�

n.

Sol

ve e

ach

eq

uat

ion

by

usi

ng

the

Sq

uar

e R

oot

Pro

per

ty.

Exam

ple

Exam

ple

a.x2

�8x

�16

�25

x2�

8x�

16 �

25(x

�4)

2�

25x

�4

��

25�or

x�

4 �

��

25�x

�5

�4

�9

or

x�

�5

�4

��

1

Th

e so

luti

on s

et i

s {9

,�1}

.

b.4

x2�

20x

�25

�32

4x2

�20

x�

25�

32(2

x�

5)2

�32

2x�

5 �

�32�

or 2

x�

5 �

��

32�2x

�5

�4�

2�or

2x

�5

��

4�2�

x�

Th

e so

luti

on s

et i

s �

�.5

4�

2��

� 2

5

4 �2�

�� 2

Exer

cises

Exer

cises

Sol

ve e

ach

eq

uat

ion

by

usi

ng

the

Sq

uar

e R

oot

Pro

per

ty.

1.x2

�18

x�

81 �

492.

x2�

20x

�10

0 �

643.

4x2

�4x

�1

�16

{2,1

6}{�

2,�

18}

�,�

�

4.36

x2�

12x

�1

�18

5.9x

2�

12x

�4

�4

6.25

x2�

40x

�16

�28

��

�0,�

��

7.4x

2�

28x

�49

�64

8.16

x2�

24x

�9

�81

9.10

0x2

�60

x�

9 �

121

�,�

��

,�3 �

{�0.

8,1.

4}

10.2

5x2

�20

x�

4 �

7511

.36x

2�

48x

�16

�12

12.2

5x2

�30

x�

9 �

96

��

��

��

3 �

4�6�

�� 5

�2

��

3��

� 3�

2 �

5�3�

�� 5

3 � 21 � 2

15 � 2

�4

�2 �

7��

� 54 � 3

�1

�3�

2��

� 6

5 � 23 � 2

©G

lenc

oe/M

cGra

w-H

ill33

2G

lenc

oe A

lgeb

ra 2

Co

mp

lete

th

e Sq

uar

eT

o co

mpl

ete

the

squ

are

for

a qu

adra

tic

expr

essi

on o

f th

e fo

rm

x2�

bx,

foll

ow t

hes

e st

eps.

1.F

ind

.➞

2.S

quar

e .

➞3.

Add

��2

to x

2�

bx.

b � 2b � 2

b � 2

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Co

mp

leti

ng

th

e S

qu

are

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

6-4

Fin

d t

he

valu

e of

cth

at m

akes

x2

�22

x�

ca

per

fect

sq

uar

e tr

inom

ial.

Th

enw

rite

th

e tr

inom

ial

as t

he

squ

are

of a

bin

omia

l.

Ste

p 1

b�

22;

�11

Ste

p 2

112

�12

1S

tep

3c

�12

1

The

tri

nom

ial

is x

2�

22x

�12

1,w

hic

h c

an b

e w

ritt

en a

s (x

�11

)2.

b � 2

Sol

ve 2

x2�

8x�

24 �

0 b

yco

mp

leti

ng

the

squ

are.

2x2

�8x

�24

�0

Orig

inal

equ

atio

n

�D

ivid

e ea

ch s

ide

by 2

.

x2�

4x�

12 �

0x2

�4x

�12

is n

ot a

per

fect

squ

are.

x2�

4x�

12A

dd 1

2 to

eac

h si

de.

x2�

4x�

4 �

12 �

4S

ince

��2

�4,

add

4 t

o ea

ch s

ide.

(x�

2)2

�16

Fac

tor

the

squa

re.

x�

2 �

4

Squ

are

Roo

t P

rope

rty

x�

6 or

x�

�2

Sol

ve e

ach

equa

tion.

Th

e so

luti

on s

et i

s {6

,�2}

.

4 � 2

0 � 22x

2�

8x�

24�

� 2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d t

he

valu

e of

cth

at m

akes

eac

h t

rin

omia

l a

per

fect

sq

uar

e.T

hen

wri

te t

he

trin

omia

l as

a p

erfe

ct s

qu

are.

1.x2

�10

x�

c2.

x2�

60x

�c

3.x2

�3x

�c

25;

(x�

5)2

900;

(x�

30)2

;�x

��2

4.x2

�3.

2x�

c5.

x2�

x�

c6.

x2�

2.5x

�c

2.56

;(x

� 1

.6)2

;�x

��2

1.56

25;

(x�

1.25

)2

Sol

ve e

ach

eq

uat

ion

by

com

ple

tin

g th

e sq

uar

e.

7.y2

�4y

�5

�0

8.x2

�8x

�65

�0

9.s2

�10

s�

21 �

0�

1,5

�5,

133,

7

10.2

x2�

3x�

1 �

011

.2x2

�13

x�

7 �

012

.25x

2�

40x

�9

�0

1,�

,7,�

13.x

2�

4x�

1 �

014

.y2

�12

y�

4 �

015

.t2

�3t

�8

�0

�2

��

3��

6 �

4�2�

�3

��

41 ��

� 29 � 51 � 5

1 � 21 � 2

1 � 41 � 16

1 � 2

3 � 29 � 4



Skil

ls P

ract

ice

Co

mp

leti

ng

th

e S

qu

are

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

6-4

©G

lenc

oe/M

cGra

w-H

ill33

3G

lenc

oe A

lgeb

ra 2

Lesson 6-4

Sol

ve e

ach

eq

uat

ion

by

usi

ng

the

Sq

uar

e R

oot

Pro

per

ty.

1.x2

�8x

�16

�1

3,5

2.x2

�4x

�4

�1

�1,

�3

3.x2

�12

x�

36 �

25�

1,�

114.

4x2

�4x

�1

�9

�1,

2

5.x2

�4x

�4

�2

�2

��

2�6.

x2�

2x�

1 �

51

��

5�

7.x2

�6x

�9

�7

3 �

�7�

8.x2

�16

x�

64 �

15�

8 �

�15�

Fin

d t

he

valu

e of

cth

at m

akes

eac

h t

rin

omia

l a

per

fect

sq

uar

e.T

hen

wri

te t

he

trin

omia

l as

a p

erfe

ct s

qu

are.

9.x2

�10

x�

c25

;(x

�5)

210

.x2

�14

x�

c49

;(x

�7)

2

11.x

2�

24x

�c

144;

(x�

12)2

12.x

2�

5x�

c;�x

��2

13.x

2�

9x�

c;�x

��2

14.x

2�

x�

c;�x

��2

Sol

ve e

ach

eq

uat

ion

by

com

ple

tin

g th

e sq

uar

e.

15.x

2�

13x

�36

�0

4,9

16.x

2�

3x�

00,

�3

17.x

2�

x�

6 �

02,

�3

18.x

2�

4x�

13 �

02

��

17�

19.2

x2�

7x�

4 �

0�

4,20

.3x2

�2x

�1

�0

,�1

21.x

2�

3x�

6 �

022

.x2

�x

�3

�0

23.x

2�

�11

�i�

11�24

.x2

�2x

�4

�0

1 �

i�3�

1 �

�13�

�� 2

�3

��

33��

� 2

1 � 31 � 2

1 � 21 � 4

9 � 281 � 4

5 � 225 � 4

©G

lenc

oe/M

cGra

w-H

ill33

4G

lenc

oe A

lgeb

ra 2

Sol

ve e

ach

eq

uat

ion

by

usi

ng

the

Sq

uar

e R

oot

Pro

per

ty.

1.x2

�8x

�16

�1

2.x2

�6x

�9

�1

3.x2

�10

x�

25 �

16

�5,

�3

�4,

�2

�9,

�1

4.x2

�14

x�

49 �

95.

4x2

�12

x�

9 �

46.

x2�

8x�

16 �

8

4,10

�,�

4 �

2�2�

7.x2

�6x

�9

�5

8.x2

�2x

�1

�2

9.9x

2�

6x�

1 �

2

3 �

�5�

1 �

�2�

Fin

d t

he

valu

e of

cth

at m

akes

eac

h t

rin

omia

l a

per

fect

sq

uar

e.T

hen

wri

te t

he

trin

omia

l as

a p

erfe

ct s

qu

are.

10.x

2�

12x

�c

11.x

2�

20x

�c

12.x

2�

11x

�c

36;

(x�

6)2

100;

(x�

10)2

;�x

��2

13.x

2�

0.8x

�c

14.x

2�

2.2x

�c

15.x

2�

0.36

x�

c

0.16

;(x

�0.

4)2

1.21

;(x

�1.

1)2

0.03

24;

(x�

0.18

)2

16.x

2�

x�

c17

.x2

�x

�c

18.x

2�

x�

c

;�x

��2

;�x

��2

;�x

��2

Sol

ve e

ach

eq

uat

ion

by

com

ple

tin

g th

e sq

uar

e.

19.x

2�

6x�

8 �

0�

4,�

220

.3x2

�x

�2

�0

,�1

21.3

x2�

5x�

2 �

01,

22.x

2�

18 �

9x23

.x2

�14

x�

19 �

024

.x2

�16

x�

7 �

06,

37

��

30��

8 �

�71�

25.2

x2�

8x�

3 �

026

.x2

�x

�5

�0

27.2

x2�

10x

�5

�0

28.x

2�

3x�

6 �

029

.2x2

�5x

�6

�0

30.7

x2�

6x�

2 �

0

31.G

EOM

ETRY

Wh

en t

he

dim

ensi

ons

of a

cu

be a

re r

edu

ced

by 4

in

ches

on

eac

h s

ide,

the

surf

ace

area

of

the

new

cu

be i

s 86

4 sq

uar

e in

ches

.Wh

at w

ere

the

dim

ensi

ons

of t

he

orig

inal

cu

be?

16 in

.by

16 in

.by

16 in

.

32.I

NV

ESTM

ENTS

Th

e am

oun

t of

mon

ey A

in a

n a

ccou

nt

in w

hic

h P

doll

ars

is i

nve

sted

for

2 ye

ars

is g

iven

by

the

form

ula

A�

P(1

�r)

2 ,w

her

e r

is t

he

inte

rest

rat

e co

mpo

un

ded

ann

ual

ly.I

f an

in

vest

men

t of

$80

0 in

th

e ac

cou

nt

grow

s to

$88

2 in

tw

o ye

ars,

at w

hat

inte

rest

rat

e w

as i

t in

vest

ed?

5%

�3

�i�

5��

� 7�

5 �

i�23�

�� 4

�3

�i�

15��

� 2

5 �

�15�

�� 2

�1

��

21��

� 2�

4 �

�22�

�� 2

2 � 32 � 3

5 � 625 � 36

1 � 81 � 64

5 � 1225 � 14

4

5 � 31 � 4

5 � 6

11 � 212

1�

41 �

�2�

�3

5 � 21 � 2

Pra

ctic

e (

Ave

rag

e)

Co

mp

leti

ng

th

e S

qu

are

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

6-4


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csC

om

ple

tin

g t

he

Sq

uar

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

6-4

©G

lenc

oe/M

cGra

w-H

ill33

5G

lenc

oe A

lgeb

ra 2

Lesson 6-4

Pre-

Act

ivit

yH

ow c

an y

ou f

ind

th

e ti

me

it t

akes

an

acc

eler

atin

g ra

ce c

ar t

ore

ach

th

e fi

nis

h l

ine?

Rea

d th

e in

trod

uct

ion

to

Les

son

6-4

at

the

top

of p

age

306

in y

our

text

book

.

Exp

lain

wh

at i

t m

ean

s to

say

th

at t

he

driv

er a

ccel

erat

es a

t a

con

stan

t ra

teof

8 f

eet

per

seco

nd

squ

are.

If t

he

dri

ver

is t

rave

ling

at

a ce

rtai

n s

pee

d a

t a

par

ticu

lar

mo

men

t,th

en o

ne

seco

nd

late

r,th

e d

rive

r is

tra

velin

g 8

fee

tp

er s

eco

nd

fas

ter.

Rea

din

g t

he

Less

on

1.G

ive

the

reas

on f

or e

ach

ste

p in

th

e fo

llow

ing

solu

tion

of

an e

quat

ion

by

usi

ng

the

Squ

are

Roo

t P

rope

rty.

x2�

12x

�36

�81

Orig

inal

equ

atio

n

(x�

6)2

�81

Fact

or

the

per

fect

sq

uar

e tr

ino

mia

l.

x�

6 �

�

81�S

qu

are

Ro

ot

Pro

per

ty

x�

6 �

9

81 �

9

x�

6 �

9 or

x�

6 �

�9

Rew

rite

as

two

eq

uat

ion

s.

x�

15

x�

�3

So

lve

each

eq

uat

ion

.

2.E

xpla

in h

ow t

o fi

nd

the

con

stan

t th

at m

ust

be

adde

d to

mak

e a

bin

omia

l in

to a

per

fect

squ

are

trin

omia

l.

Sam

ple

an

swer

:Fin

d h

alf

of

the

coef

ficie

nt

of

the

linea

r te

rm a

nd

sq

uar

e it.

3.a.

Wh

at i

s th

e fi

rst

step

in

sol

vin

g th

e eq

uat

ion

3x2

�6x

�5

by c

ompl

etin

g th

e sq

uar

e?D

ivid

e th

e eq

uat

ion

by

3.

b.

Wh

at i

s th

e fi

rst

step

in

sol

vin

g th

e eq

uat

ion

x2

�5x

�12

�0

by c

ompl

etin

g th

esq

uar

e?A

dd

12

to e

ach

sid

e.

Hel

pin

g Y

ou

Rem

emb

er

4.H

ow c

an y

ou u

se t

he

rule

s fo

r sq

uar

ing

a bi

nom

ial

to h

elp

you

rem

embe

r th

e pr

oced

ure

for

chan

gin

g a

bin

omia

l in

to a

per

fect

squ

are

trin

omia

l?

On

e o

f th

e ru

les

for

squ

arin

g a

bin

om

ial i

s (x

�y

)2�

x2

�2x

y�

y2 .

Inco

mp

leti

ng

th

e sq

uar

e,yo

u a

re s

tart

ing

wit

h x

2�

bx

and

nee

d t

o f

ind

y2 .

Th

is s

ho

ws

you

th

at b

�2y

,so

y�

.Th

at is

why

yo

u m

ust

tak

e h

alf

of

the

coef

fici

ent

and

sq

uar

e it

to

get

th

e co

nst

ant

that

mu

st b

e ad

ded

to

com

ple

te t

he

squ

are.

b � 2

©G

lenc

oe/M

cGra

w-H

ill33

6G

lenc

oe A

lgeb

ra 2

Th

e G

old

en Q

uad

rati

c E

qu

atio

ns

A g

old

en r

ecta

ngl

eh

as t

he

prop

erty

th

at i

ts l

engt

h

can

be

wri

tten

as

a�

b,w

her

e a

is t

he

wid

th o

f th

e

rect

angl

e an

d �a

� ab

��

�a b� .A

ny

gold

en r

ecta

ngl

e ca

n b

e

divi

ded

into

a s

quar

e an

d a

smal

ler

gold

en r

ecta

ngl

e,as

sh

own

.

Th

e pr

opor

tion

use

d to

def

ine

gold

en r

ecta

ngl

es c

an b

e u

sed

to d

eriv

e tw

o qu

adra

tic

equ

atio

ns.

The

se a

reso

met

imes

call

ed g

old

en q

uad

rati

c eq

uat

ion

s.

Sol

ve e

ach

pro

ble

m.

1.In

th

e pr

opor

tion

for

th

e go

lden

rec

tan

gle,

let

aeq

ual

1.W

rite

th

e re

sult

ing

quad

rati

c eq

uat

ion

an

d so

lve

for

b.

b2

�b

�1

�0

b�

2.In

th

e pr

opor

tion

,let

beq

ual

1.W

rite

th

e re

sult

ing

quad

rati

c eq

uat

ion

an

d so

lve

for

a.

a2

�a

�1

�0

a�

3.D

escr

ibe

the

diff

eren

ce b

etw

een

the

two

gold

en q

uad

rati

c eq

uat

ion

s yo

u

fou

nd

in e

xerc

ises

1 a

nd

2.

Th

e si

gn

s o

f th

e fi

rst-

deg

ree

term

s ar

e o

pp

osi

te.

4.S

how

th

at t

he

posi

tive

sol

uti

ons

of t

he

two

equ

atio

ns

in e

xerc

ises

1 a

nd

2 ar

e re

cipr

ocal

s.

��

��

��1 4�

5�

�1

5.U

se t

he

Pyt

hag

orea

n T

heo

rem

to

fin

d a

radi

cal

expr

essi

on f

or t

he

diag

onal

of

a g

olde

n r

ecta

ngl

e w

hen

a�

1.

d�

6.F

ind

a ra

dica

l ex

pres

sion

for

th

e di

agon

al o

f a

gold

en r

ecta

ngl

e w

hen

b�

1.

d�

�10

�2

��

5��

�� 2

�10

�2

��

5��

�� 2

�( 1

2 )�

( �5�)

2

�� 4

1 �

�5�

�2

�1

��

5��

� 2

1 �

�5�

�2

�1

��

5��

� 2

a

a a

b b

a

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

6-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

Th

e Q

uad

rati

c F

orm

ula

an

d t

he

Dis

crim

inan

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-5

6-5

©G

lenc

oe/M

cGra

w-H

ill33

7G

lenc

oe A

lgeb

ra 2

Lesson 6-5

Qu

adra

tic

Form

ula

Th

e Q

uad

rati

c F

orm

ula

can

be

use

d to

sol

ve a

ny

quad

rati

ceq

uat

ion

on

ce i

t is

wri

tten

in

th

e fo

rm a

x2�

bx�

c�

0.

Qu

adra

tic

Fo

rmu

laT

he s

olut

ions

of

ax2

�bx

�c

�0,

with

a�

0, a

re g

iven

by

x�

.

Sol

ve x

2�

5x�

14 b

y u

sin

g th

e Q

uad

rati

c F

orm

ula

.

Rew

rite

th

e eq

uat

ion

as

x2�

5x�

14 �

0.

x�

Qua

drat

ic F

orm

ula

�R

epla

ce a

with

1,

bw

ith �

5, a

nd c

with

�14

.

�S

impl

ify.

� �7

or �

2

Th

e so

luti

ons

are

�2

and

7.

Sol

ve e

ach

eq

uat

ion

by

usi

ng

the

Qu

adra

tic

For

mu

la.

1.x2

�2x

�35

�0

2.x2

�10

x�

24 �

03.

x2�

11x

�24

�0

5,�

7�

4,�

63,

8

4.4x

2�

19x

�5

�0

5.14

x2�

9x�

1 �

06.

2x2

�x

�15

�0

,�5

�,�

3,�

7.3x

2�

5x�

28.

2y2

�y

�15

�0

9.3x

2�

16x

�16

�0

�2,

,�3

4,

10.8

x2�

6x�

9 �

011

.r2

��

�0

12.x

2�

10x

�50

�0

�,

,5

�5�

3�

13.x

2�

6x�

23 �

014

.4x2

�12

x�

63 �

015

.x2

�6x

�21

�0

�3

�4�

2�3

�2i

�3�

3 �

6 �2 �

�� 21 � 5

2 � 53 � 4

3 � 2

2 � 253r � 5

4 � 35 � 2

1 � 3

5 � 21 � 7

1 � 21 � 45

9

�2

5

�81�

�� 2

�(�

5)

�(�

5)2

��

4(1

�)(

�14

�)�

��

��

2(1)

�b

�

b2�

4�

ac��

��

2a

�b

�

b2

��

4ac

��

��

2a

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill33

8G

lenc

oe A

lgeb

ra 2

Ro

ots

an

d t

he

Dis

crim

inan

t

Dis

crim

inan

tT

he e

xpre

ssio

n un

der

the

radi

cal s

ign,

b2

�4a

c, in

the

Qua

drat

ic F

orm

ula

is c

alle

d th

e d

iscr

imin

ant.

Ro

ots

of

a Q

uad

rati

c Eq

uat

ion

Dis

crim

inan

tTy

pe

and

Nu

mb

er o

f R

oo

ts

b2

�4a

c�

0 an

d a

perf

ect

squa

re2

ratio

nal r

oots

b2

�4a

c�

0, b

ut n

ot

a pe

rfec

t sq

uare

2 irr

atio

nal r

oots

b2

�4a

c�

01

ratio

nal r

oot

b2

�4a

c�

02

com

plex

roo

ts

Fin

d t

he

valu

e of

th

e d

iscr

imin

ant

for

each

eq

uat

ion

.Th

en d

escr

ibe

the

nu

mb

er a

nd

typ

es o

f ro

ots

for

the

equ

atio

n.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Th

e Q

uad

rati

c F

orm

ula

an

d t

he

Dis

crim

inan

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-5

6-5

Exam

ple

Exam

ple

a.2x

2�

5x�

3T

he

disc

rim

inan

t is

b2

�4a

c�

52�

4(2)

(3)

or 1

.T

he

disc

rim

inan

t is

a p

erfe

ct s

quar

e,so

the

equ

atio

n h

as 2

rat

ion

al r

oots

.

b.3

x2�

2x�

5T

he

disc

rim

inan

t is

b2

�4a

c�

(�2)

2�

4(3)

(5)

or �

56.

Th

e di

scri

min

ant

is n

egat

ive,

so t

he

equ

atio

n h

as 2

com

plex

roo

ts.

Exer

cises

Exer

cises

For

Exe

rcis

es 1

�12

,com

ple

te p

arts

a�

c fo

r ea

ch q

uad

rati

c eq

uat

ion

.a.

Fin

d t

he

valu

e of

th

e d

iscr

imin

ant.

b.

Des

crib

e th

e n

um

ber

an

d t

ype

of r

oots

.c.

Fin

d t

he

exac

t so

luti

ons

by

usi

ng

the

Qu

adra

tic

For

mu

la.

1.p2

�12

p�

�4

128;

2.9x

2�

6x�

1 �

00;

3.2x

2�

7x�

4 �

081

;tw

o ir

rati

on

alro

ots

;o

ne

rati

on

al r

oo

t;2

rati

on

al r

oo

ts;

�,4

�6

�4 �

2�

4.x2

�4x

�4

�0

32;

5.5x

2�

36x

�7

�0

1156

;6.

4x2

�4x

�11

�0

2 ir

rati

on

al r

oo

ts;

2 ra

tio

nal

ro

ots

;�

160;

2 co

mp

lex

roo

ts;

�2

�2 �

2�,7

7.x2

�7x

�6

�0

25;

8.m

2�

8m�

�14

8;9.

25x2

�40

x�

�16

0;2

rati

on

al r

oo

ts;

2 ir

rati

on

al r

oo

ts;

1 ra

tio

nal

ro

ot;

1,6

4 �

�2�

10.4

x2�

20x

�29

�0

�64

;11

.6x2

�26

x�

8 �

048

4;12

.4x2

�4x

�11

�0

192;

2 co

mp

lex

roo

ts;

2 ra

tio

nal

ro

ots

;2

irra

tio

nal

ro

ots

;4 � 5

1 �

i�10�

�� 2

1 � 5

1 � 21 � 3


An

swer

s


Skil

ls P

ract

ice

Th

e Q

uad

rati

c F

orm

ula

an

d t

he

Dis

crim

inan

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

6-5

6-5

©G

lenc

oe/M

cGra

w-H

ill33

9G

lenc

oe A

lgeb

ra 2

Lesson 6-5

Com

ple

te p

arts

a�

c fo

r ea

ch q

uad

rati

c eq

uat

ion

.a.

Fin

d t

he

valu

e of

th

e d

iscr

imin

ant.

b.

Des

crib

e th

e n

um

ber

an

d t

ype

of r

oots

.c.

Fin

d t

he

exac

t so

luti

ons

by

usi

ng

the

Qu

adra

tic

For

mu

la.

1.x2

�8x

�16

�0

2.x2

�11

x�

26 �

0

0;1

rati

on

al r

oo

t;4

225;

2 ra

tio

nal

ro

ots

;�

2,13

3.3x

2�

2x�

04.

20x2

�7x

�3

�0

4;2

rati

on

al r

oo

ts;

0,28

9;2

rati

on

al r

oo

ts;

�,

5.5x

2�

6 �

06.

x2�

6 �

0

120;

2 ir

rati

on

al r

oo

ts;

�24

;2

irra

tio

nal

ro

ots

;�

�6�

7.x2

�8x

�13

�0

8.5x

2�

x�

1 �

0

12;

2 ir

rati

on

al r

oo

ts;

�4

��

3�21

;2

irra

tio

nal

ro

ots

;

9.x2

�2x

�17

�0

10.x

2�

49 �

0

72;

2 ir

rati

on

al r

oo

ts;

1 �

3�2�

�19

6;2

com

ple

x ro

ots

;�

7i

11.x

2�

x�

1 �

012

.2x2

�3x

��

2

�3;

2 co

mp

lex

roo

ts;

�7;

2 co

mp

lex

roo

ts;

Sol

ve e

ach

eq

uat

ion

by

usi

ng

the

met

hod

of

you

r ch

oice

.Fin

d e

xact

sol

uti

ons.

13.x

2�

64�

814

.x2

�30

�0

��

30�

15.x

2�

x�

30�

5,6

16.1

6x2

�24

x�

27 �

0,�

17.x

2�

4x�

11 �

02

� �

15�18

.x2

�8x

�17

�0

4 �

�33�

19.x

2�

25 �

0�

5i20

.3x2

�36

�0

�2i

�3�

21.2

x2�

10x

�11

�0

22.2

x2�

7x�

4 �

0

23.8

x2�

1 �

4x24

.2x2

�2x

�3

�0

25.P

AR

AC

HU

TIN

GIg

nor

ing

win

d re

sist

ance

,th

e di

stan

ce d

(t)

in f

eet

that

a p

arac

hu

tist

fall

s in

tse

con

ds c

an b

e es

tim

ated

usi

ng

the

form

ula

d(t

) �

16t2

.If

a pa

rach

uti

st ju

mps

from

an

air

plan

e an

d fa

lls

for

1100

fee

t be

fore

ope

nin

g h

er p

arac

hu

te,h

ow m

any

seco

nds

pass

bef

ore

she

open

s th

e pa

rach

ute

?ab

ou

t 8.

3 s

�1

�i�

5��

� 21

�i

�4

7 �

�17�

�� 4

�5

��

3��

� 2

3 � 49 � 4

3 �

i�7�

�� 4

1 �

i�3�

�� 2

1 �

�21�

�� 10

�30�

�5

1 � 43 � 5

2 � 3

©G

lenc

oe/M

cGra

w-H

ill34

0G

lenc

oe A

lgeb

ra 2

Com

ple

te p

arts

a�

c fo

r ea

ch q

uad

rati

c eq

uat

ion

.a.

Fin

d t

he

valu

e of

th

e d

iscr

imin

ant.

b.

Des

crib

e th

e n

um

ber

an

d t

ype

of r

oots

.c.

Fin

d t

he

exac

t so

luti

ons

by

usi

ng

the

Qu

adra

tic

For

mu

la.

1.x2

�16

x�

64 �

02.

x2�

3x3.

9x2

�24

x�

16 �

0

0;1

rati

on

al;

89;

2 ra

tio

nal

;0,

30;

1 ra

tio

nal

;

4.x2

�3x

�40

5.3x

2�

9x�

2 �

010

5;6.

2x2

�7x

�0

169;

2 ra

tio

nal

;�

5,8

2 ir

rati

on

al;

49;

2 ra

tio

nal

;0,

�

7.5x

2�

2x�

4 �

0�

76;

8.12

x2�

x�

6 �

028

9;9.

7x2

�6x

�2

�0

�20

;

2 co

mp

lex;

2 ra

tio

nal

;,�

2 co

mp

lex;

10.1

2x2

�2x

�4

�0

196;

11.6

x2�

2x�

1 �

028

;12

.x2

�3x

�6

�0

�15

;

2 ra

tio

nal

;,�

2 ir

rati

on

al;

2 co

mp

lex;

13.4

x2�

3x2

�6

�0

105;

14.1

6x2

�8x

�1

�0

15.2

x2�

5x�

6 �

073

;

2 ir

rati

on

al;

0;1

rati

on

al;

2 ir

rati

on

al;

Sol

ve e

ach

eq

uat

ion

by

usi

ng

the

met

hod

of

you

r ch

oice

.Fin

d e

xact

sol

uti

ons.

16.7

x2�

5x�

00,

17.4

x2�

9 �

0�

18.3

x2�

8x�

3,�

319

.x2

�21

�4x

�3,

7

20.3

x2�

13x

�4

�0

,421

.15x

2�

22x

��

8�

,�

22.x

2�

6x�

3 �

03

��

6�23

.x2

�14

x�

53 �

07

�2i

24.3

x2�

�54

�3i

�2�

25.2

5x2

�20

x�

6 �

0

26.4

x2�

4x�

17 �

027

.8x

�1

�4x

2

28.x

2�

4x�

152

�i�

11�29

.4x2

�12

x�

7 �

0

30. G

RA

VIT

ATI

ON

The

hei

ght

h(t)

in f

eet

of a

n ob

ject

tse

cond

s af

ter

it is

pro

pelle

d st

raig

ht u

pfr

om t

he

grou

nd

wit

h a

n i

nit

ial

velo

city

of

60 f

eet

per

seco

nd

is m

odel

ed b

y th

e eq

uat

ion

h(t

) �

�16

t2�

60t.

At

wh

at t

imes

wil

l th

e ob

ject

be

at a

hei

ght

of 5

6 fe

et?

1.75

s,2

s

31.S

TOPP

ING

DIS

TAN

CE

Th

e fo

rmu

la d

�0.

05s2

�1.

1ses

tim

ates

th

e m

inim

um

sto

ppin

gdi

stan

ce d

in f

eet

for

a ca

r tr

avel

ing

sm

iles

per

hou

r.If

a c

ar s

tops

in 2

00 f

eet,

wha

t is

the

fast

est

it c

ould

hav

e be

en t

rave

ling

whe

n th

e dr

iver

app

lied

the

brak

es?

abo

ut

53.2

mi/h

3 �

�2�

�2

2 �

�3�

�2

1 �

4i�

2

2 �

�10�

�� 54 � 5

2 � 31 � 3

1 � 3

3 � 25 � 7

5 �

�73�

�� 4

1 � 43

��

105

��

� 8

�3

�i �

15��

� 21

��

7��

62 � 3

1 � 2

�3

�i�

5��

� 72 � 3

3 � 41

�i�

19��

� 5

7 � 2�

9 �

�10

5�

�� 6

4 � 3

Pra

ctic

e (

Ave

rag

e)

Th

e Q

uad

rati

c F

orm

ula

an

d t

he

Dis

crim

inan

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-5

6-5



Readin

g t

o L

earn

Math

em

ati

csT

he

Qu

adra

tic

Fo

rmu

la a

nd

th

e D

iscr

imin

ant

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-5

6-5

©G

lenc

oe/M

cGra

w-H

ill34

1G

lenc

oe A

lgeb

ra 2

Lesson 6-5

Pre-

Act

ivit

yH

ow i

s b

lood

pre

ssu

re r

elat

ed t

o ag

e?

Rea

d th

e in

trod

uct

ion

to

Les

son

6-5

at

the

top

of p

age

313

in y

our

text

book

.

Des

crib

e h

ow y

ou w

ould

cal

cula

te y

our

nor

mal

blo

od p

ress

ure

usi

ng

one

ofth

e fo

rmu

las

in y

our

text

book

.

Sam

ple

an

swer

:S

ub

stit

ute

yo

ur

age

for

Ain

th

e ap

pro

pri

ate

form

ula

(fo

r fe

mal

es o

r m

ales

) an

d e

valu

ate

the

exp

ress

ion

.

Rea

din

g t

he

Less

on

1.a.

Wri

te t

he

Qu

adra

tic

For

mu

la.

x�

b.

Iden

tify

th

e va

lues

of

a,b,

and

cth

at y

ou w

ould

use

to

solv

e 2x

2�

5x�

�7,

but

don

ot a

ctu

ally

sol

ve t

he

equ

atio

n.

a�

b�

c�

2.S

upp

ose

that

you

are

sol

vin

g fo

ur

quad

rati

c eq

uat

ion

s w

ith

rat

ion

al c

oeff

icie

nts

an

dh

ave

fou

nd

the

valu

e of

th

e di

scri

min

ant

for

each

equ

atio

n.I

n e

ach

cas

e,gi

ve t

he

nu

mbe

r of

roo

ts a

nd

desc

ribe

th

e ty

pe o

f ro

ots

that

th

e eq

uat

ion

wil

l h

ave.

Val

ue

of

Dis

crim

inan

tN

um

ber

of

Ro

ots

Typ

e o

f R

oo

ts

642

real

,rat

ion

al

�8

2co

mp

lex

212

real

,irr

atio

nal

01

real

,rat

ion

al

Hel

pin

g Y

ou

Rem

emb

er

3.H

ow c

an l

ooki

ng

at t

he

Qu

adra

tic

For

mu

la h

elp

you

rem

embe

r th

e re

lati

onsh

ips

betw

een

th

e va

lue

of t

he

disc

rim

inan

t an

d th

e n

um

ber

of r

oots

of

a qu

adra

tic

equ

atio

nan

d w

het

her

th

e ro

ots

are

real

or

com

plex

?

Sam

ple

an

swer

:Th

e d

iscr

imin

ant

is t

he

exp

ress

ion

un

der

th

e ra

dic

al in

the

Qu

adra

tic

Fo

rmu

la.L

oo

k at

th

e Q

uad

rati

c F

orm

ula

an

d c

on

sid

er w

hat

hap

pen

s w

hen

yo

u t

ake

the

pri

nci

pal

sq

uar

e ro

ot

of

b2

�4a

can

d a

pp

ly�

in f

ron

t o

f th

e re

sult

.If

b2

�4a

cis

po

siti

ve,i

ts p

rin

cip

al s

qu

are

roo

tw

ill b

e a

po

siti

ve n

um

ber

an

d a

pp

lyin

g �

will

giv

e tw

o d

iffe

ren

t re

also

luti

on

s,w

hic

h m

ay b

e ra

tio

nal

or

irra

tio

nal

.If

b2

�4a

c�

0,it

sp

rin

cip

al s

qu

are

roo

t is

0,s

o a

pp

lyin

g �

in t

he

Qu

adra

tic

Fo

rmu

la w

illo

nly

lead

to

on

e so

luti

on

,wh

ich

will

be

rati

on

al (

assu

min

g a

,b,a

nd

car

ein

teg

ers)

.If

b2

�4a

cis

neg

ativ

e,si

nce

th

e sq

uar

e ro

ots

of

neg

ativ

en

um

ber

s ar

e n

ot

real

nu

mb

ers,

you

will

get

tw

o c

om

ple

x ro

ots

,co

rres

po

nd

ing

to

th

e �

and

�in

th

e �

sym

bo

l.7�

52

�b

��

b2

�4

�ac �

��

2a

©G

lenc

oe/M

cGra

w-H

ill34

2G

lenc

oe A

lgeb

ra 2

Su

m a

nd

Pro

du

ct o

f R

oo

ts

Som

etim

es y

ou m

ay k

now

th

e ro

ots

of a

qu

adra

tic

equ

atio

n w

ith

out

know

ing

the

equ

atio

nit

self

.Usi

ng

you

r kn

owle

dge

of f

acto

rin

g to

sol

ve a

n e

quat

ion

,you

can

wor

k ba

ckw

ard

tofi

nd

the

quad

rati

c eq

uat

ion

.Th

e ru

le f

or f

indi

ng

the

sum

an

d pr

odu

ct o

f ro

ots

is a

s fo

llow

s:

Su

m a

nd

Pro

du

ct o

f R

oo

tsIf

the

root

s of

ax2

�bx

�c

�0,

with

a≠

0, a

re s

1an

d s 2

,

then

s1

�s 2

��

�b a�an

d s 1

�s 2

�� ac � .

A r

oad

wit

h a

n i

nit

ial

grad

ien

t,or

slo

pe,

of 3

% c

an b

e re

pre

sen

ted

by

the

form

ula

y�

ax2

�0.

03x

�c,

wh

ere

yis

th

e el

evat

ion

an

d x

is t

he

dis

tan

ce a

lon

gth

e cu

rve.

Su

pp

ose

the

elev

atio

n o

f th

e ro

ad i

s 11

05 f

eet

at p

oin

ts 2

00 f

eet

and

100

0fe

et a

lon

g th

e cu

rve.

You

can

fin

d t

he

equ

atio

n o

f th

e tr

ansi

tion

cu

rve.

Eq

uat

ion

sof

tra

nsi

tion

cu

rves

are

use

d b

y ci

vil

engi

nee

rs t

o d

esig

n s

moo

th a

nd

saf

e ro

ads.

Th

e ro

ots

are

x�

3 an

d x

��

8.

3 �

(�8)

��

5A

dd t

he r

oots

.

3(�

8) �

�24

Mul

tiply

the

roo

ts.

Equ

atio

n:x

2�

5x�

24 �

0

Wri

te a

qu

adra

tic

equ

atio

n t

hat

has

th

e gi

ven

roo

ts.

1.6,

�9

2.5,

�1

3.6,

6

x2

�3x

�54

�0

x2

�4x

�5

�0

x2

�12

x�

36 �

0

4.4

�

3�6.

��2 5� ,

�2 7�6.

x2

�8x

�13

�0

35x

2�

4x�

4 �

049

x2

�42

x�

205

�0

Fin

d k

such

th

at t

he

nu

mb

er g

iven

is

a ro

ot o

f th

e eq

uat

ion

.

7.7;

2x2

�kx

�21

�0

8.�

2;x2

�13

x�

k�

0 �

11�

30

�2

3�

5��

� 7

x

y

O

(–5 – 2,

–30

1 – 4)

10 –10

–20

–30

24

–2–4

–6–8

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-5

6-5

Exam

ple

Exam

ple


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

An

alyz

ing

Gra

ph

s o

f Q

uad

rati

c F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-6

6-6

©G

lenc

oe/M

cGra

w-H

ill34

3G

lenc

oe A

lgeb

ra 2

Lesson 6-6

An

alyz

e Q

uad

rati

c Fu

nct

ion

s

The

gra

ph o

f y

�a

(x�

h)2

�k

has

the

follo

win

g ch

arac

teris

tics:

•V

erte

x: (

h, k

)V

erte

x F

orm

•A

xis

of s

ymm

etry

: x

�h

of

a Q

uad

rati

c•

Ope

ns u

p if

a�

0F

un

ctio

n•

Ope

ns d

own

if a

�0

•N

arro

wer

tha

n th

e gr

aph

of y

�x

2if

a

�1

•W

ider

tha

n th

e gr

aph

of y

�x

2if

a

�1

Iden

tify

th

e ve

rtex

,axi

s of

sym

met

ry,a

nd

dir

ecti

on o

f op

enin

g of

each

gra

ph

.

a.y

�2(

x�

4)2

�11

Th

e ve

rtex

is

at (

h,k

) or

(�

4,�

11),

and

the

axis

of

sym

met

ry i

s x

��

4.T

he

grap

h o

pen

su

p,an

d is

nar

row

er t

han

th

e gr

aph

of

y �

x2.

a.y

��

(x�

2)2

�10

Th

e ve

rtex

is

at (

h,k

) or

(2,

10),

and

the

axis

of

sym

met

ry i

s x

�2.

Th

e gr

aph

ope

ns

dow

n,a

nd

is w

ider

th

an t

he

grap

h o

f y

�x2

.

Eac

h q

uad

rati

c fu

nct

ion

is

give

n i

n v

erte

x fo

rm.I

den

tify

th

e ve

rtex

,axi

s of

sym

met

ry,a

nd

dir

ecti

on o

f op

enin

g of

th

e gr

aph

.

1.y

�(x

�2)

2�

162.

y�

4(x

�3)

2�

73.

y�

(x�

5)2

�3

(2,1

6);

x�

2;u

p(�

3,�

7);

x�

�3;

up

(5,3

);x

�5;

up

4.y

��

7(x

�1)

2�

95.

y�

(x�

4)2

�12

6.y

�6(

x�

6)2

�6

(�1,

�9)

;x

��

1;d

ow

n(4

,�12

);x

�4;

up

(�6,

6);

x�

�6;

up

7.y

�(x

�9)

2�

128.

y�

8(x

�3)

2�

29.

y�

�3(

x�

1)2

�2

(9,1

2);

x�

9;u

p(3

,�2)

;x

�3;

up

(1,�

2);

x�

1;d

ow

n

10.y

��

(x�

5)2

�12

11.y

�(x

�7)

2�

2212

.y�

16(x

�4)

2�

1

(�5,

12);

x�

�5;

do

wn

(7,2

2);

x�

7;u

p(4

,1);

x�

4;u

p

13.y

�3(

x�

1.2)

2�

2.7

14.y

��

0.4(

x�

0.6)

2�

0.2

15.y

�1.

2(x

�0.

8)2

�6.

5

(1.2

,2.7

);x

�1.

2;u

p(0

.6,�

0.2)

;x

�0.

6;(�

0.8,

6.5)

;x

��

0.8;

do

wn

up

4 � 35 � 2

2 � 5

1 � 5

1 � 2

1 � 4

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill34

4G

lenc

oe A

lgeb

ra 2

Wri

te Q

uad

rati

c Fu

nct

ion

s in

Ver

tex

Form

A q

uad

rati

c fu

nct

ion

is

easi

er t

ogr

aph

wh

en i

t is

in

ver

tex

form

.You

can

wri

te a

qu

adra

tic

fun

ctio

n o

f th

e fo

rm

y�

ax2

�bx

�c

in v

erte

x fr

om b

y co

mpl

etin

g th

e sq

uar

e.

Wri

te y

�2x

2�

12x

�25

in

ver

tex

form

.Th

en g

rap

h t

he

fun

ctio

n.

y�

2x2

�12

x�

25y

�2(

x2�

6x)

�25

y�

2(x2

�6x

�9)

�25

�18

y�

2(x

�3)

2�

7

Th

e ve

rtex

for

m o

f th

e eq

uat

ion

is

y�

2(x

�3)

2�

7.

Wri

te e

ach

qu

adra

tic

fun

ctio

n i

n v

erte

x fo

rm.T

hen

gra

ph

th

e fu

nct

ion

.

1.y

�x2

�10

x �

322.

y �

x2�

6x3.

y�

x2�

8x�

6y

�(x

�5)

2�

7y

�(x

�3)

2�

9y

�(x

�4)

2�

10

4.y

��

4x2

�16

x�

115.

y�

3x2

�12

x�

56.

y�

5x2

�10

x�

9y

��

4(x

�2)

2�

5y

�3(

x�

2)2

�7

y�

5(x�

1)2

�4 x

y

O

x

y

O

x

y

O

x

y

O4

–48

8 4 –4 –8 –12

x

y

O

x

y

O

x

y

O

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

An

alyz

ing

Gra

ph

s o

f Q

uad

rati

c F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-6

6-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

An

alyz

ing

Gra

ph

s o

f Q

uad

rati

c F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-6

6-6

©G

lenc

oe/M

cGra

w-H

ill34

5G

lenc

oe A

lgeb

ra 2

Lesson 6-6

Wri

te e

ach

qu

adra

tic

fun

ctio

n i

n v

erte

x fo

rm,i

f n

ot a

lrea

dy

in t

hat

for

m.T

hen

iden

tify

th

e ve

rtex

,axi

s of

sym

met

ry,a

nd

dir

ecti

on o

f op

enin

g.

1.y

�(x

�2)

22.

y�

�x2

�4

3.y

�x2

�6

y�

(x�

2)2

�0;

y�

�(x

�0)

2�

4;y

�(x

�0)

2�

6;(2

,0);

x�

2;u

p(0

,4);

x�

0;d

ow

n(0

,�6)

;x

�0;

up

4.y

��

3(x

�5)

25.

y�

�5x

2�

96.

y�

(x�

2)2

�18

y�

�3(

x�

5)2

�0;

y�

�5(

x�

0)2

�9;

y�

(x�

2)2

�18

;(�

5,0)

;x

��

5;d

ow

n(0

,9);

x�

0;d

ow

n(2

,�18

);x

�2;

up

7.y

�x2

�2x

�5

8.y

�x2

�6x

�2

9.y

��

3x2

�24

xy

�(x

�1)

2�

6;y

�(x

�3)

2�

7;y

��

3(x

�4)

2�

48;

(1,�

6);

x�

1;u

p(�

3,�

7);

x�

�3;

up

(4,4

8);

x�

4;d

ow

n

Gra

ph

eac

h f

un

ctio

n.

10.y

�(x

�3)

2�

111

.y�

(x�

1)2

�2

12.y

��

(x�

4)2

�4

13.y

��

(x�

2)2

14.y

��

3x2

�4

15.y

�x2

�6x

�4

Wri

te a

n e

qu

atio

n f

or t

he

par

abol

a w

ith

th

e gi

ven

ver

tex

that

pas

ses

thro

ugh

th

egi

ven

poi

nt.

16.v

erte

x:(4

,�36

)17

.ver

tex:

(3,�

1)18

.ver

tex:

(�2,

2)po

int:

(0,�

20)

poin

t:(2

,0)

poin

t:(�

1,3)

y�

(x�

4)2

�36

y�

(x�

3)2

�1

y�

(x�

2)2

�2x

y

Ox

y

O

x

y

O

1 � 2

x

y

O

x

y

Ox

y

O

©G

lenc

oe/M

cGra

w-H

ill34

6G

lenc

oe A

lgeb

ra 2

Wri

te e

ach

qu

adra

tic

fun

ctio

n i

n v

erte

x fo

rm,i

f n

ot a

lrea

dy

in t

hat

for

m.T

hen

iden

tify

th

e ve

rtex

,axi

s of

sym

met

ry,a

nd

dir

ecti

on o

f op

enin

g.

1.y

��

6(x

�2)

2�

12.

y�

2x2

�2

3.y

��

4x2

�8x

y�

�6(

x�

2)2

�1;

y�

2(x

�0)

2�

2;y

��

4(x

�1)

2�

4;(�

2,�

1);

x�

�2;

do

wn

(0,2

);x

�0;

up

(1,4

);x

�1;

do

wn

4.y

�x2

�10

x�

205.

y�

2x2

�12

x�

186.

y�

3x2

�6x

�5

y�

(x�

5)2

�5;

y�

2(x

�3)

2 ;(�

3,0)

;y

�3(

x�

1)2

�2;

(�5,

�5)

;x

��

5;u

px

��

3;u

p(1

,2);

x�

1;u

p

7.y

��

2x2

�16

x�

328.

y�

�3x

2�

18x

�21

9.y

�2x

2�

16x

�29

y�

�2(

x�

4)2 ;

y�

�3(

x�

3)2

�6;

y�

2(x

�4)

2�

3;(�

4,0)

;x

��

4;d

ow

n(3

,6);

x�

3;d

ow

n(�

4,�

3);

x�

�4;

up

Gra

ph

eac

h f

un

ctio

n.

10.y

�(x

�3)

2�

111

.y�

�x2

�6x

�5

12.y

�2x

2�

2x�

1

Wri

te a

n e

qu

atio

n f

or t

he

par

abol

a w

ith

th

e gi

ven

ver

tex

that

pas

ses

thro

ugh

th

egi

ven

poi

nt.

13.v

erte

x:(1

,3)

14.v

erte

x:(�

3,0)

15

.ver

tex:

(10,

�4)

poin

t:(�

2,�

15)

poin

t:(3

,18)

poin

t:(5

,6)

y�

�2(

x�

1)2

�3

y�

(x�

3)2

y�

(x�

10)2

�4

16.W

rite

an

equ

atio

n f

or a

par

abol

a w

ith

ver

tex

at (

4,4)

an

d x-

inte

rcep

t 6.

y�

�(x

�4)

2�

4

17.W

rite

an

equ

atio

n f

or a

par

abol

a w

ith

ver

tex

at (

�3,

�1)

an

d y-

inte

rcep

t 2.

y�

(x�

3)2

�1

18.B

ASE

BA

LLT

he

hei

ght

hof

a b

aseb

all

tse

con

ds a

fter

bei

ng

hit

is

give

n b

y h

(t)

��

16t2

�80

t�

3.W

hat

is

the

max

imu

m h

eigh

t th

at t

he

base

ball

rea

ches

,an

dw

hen

doe

s th

is o

ccu

r?10

3 ft

;2.

5 s

19.S

CU

LPTU

RE

A m

oder

n sc

ulpt

ure

in a

par

k co

ntai

ns a

par

abol

ic a

rc t

hat

star

ts a

t th

e gr

oun

d an

d re

ach

es a

max

imu

m h

eigh

t of

10

feet

aft

er a

hor

izon

tal

dist

ance

of

4 fe

et.W

rite

a q

uad

rati

c fu

nct

ion

in

ver

tex

form

that

des

crib

es t

he

shap

e of

th

e ou

tsid

e of

th

e ar

c,w

her

e y

is t

he

hei

ght

of a

poi

nt

on t

he

arc

and

xis

its

hor

izon

tal

dist

ance

fro

m t

he

left

-han

dst

arti

ng

poin

t of

th

e ar

c.y

��

(x�

4)2

�10

5 � 8

10 ft

4 ft

1 � 3

2 � 51 � 2

x

y O

x

y

O

x

y

O

Pra

ctic

e (

Ave

rag

e)

An

alyz

ing

Gra

ph

s o

f Q

uad

rati

c F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-6

6-6


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csA

nal

yzin

g G

rap

hs

of

Qu

adra

tic

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-6

6-6

©G

lenc

oe/M

cGra

w-H

ill34

7G

lenc

oe A

lgeb

ra 2

Lesson 6-6

Pre-

Act

ivit

yH

ow c

an t

he

grap

h o

f y

�x2

be

use

d t

o gr

aph

an

y q

uad

rati

cfu

nct

ion

?

Rea

d th

e in

trod

uct

ion

to

Les

son

6-6

at

the

top

of p

age

322

in y

our

text

book

.

•W

hat

doe

s ad

din

g a

posi

tive

nu

mbe

r to

x2

do t

o th

e gr

aph

of

y�

x2?

It m

oves

th

e g

rap

h u

p.

•W

hat

doe

s su

btra

ctin

g a

posi

tive

nu

mbe

r to

xbe

fore

squ

arin

g do

to

the

grap

h o

f y

�x2

?It

mov

es t

he

gra

ph

to

th

e ri

gh

t.

Rea

din

g t

he

Less

on

1.C

ompl

ete

the

foll

owin

g in

form

atio

n a

bou

t th

e gr

aph

of

y�

a(x

�h

)2�

k.

a.W

hat

are

th

e co

ordi

nat

es o

f th

e ve

rtex

?(h

,k)

b.

Wh

at i

s th

e eq

uat

ion

of

the

axis

of

sym

met

ry?

x�

h

c.In

wh

ich

dir

ecti

on d

oes

the

grap

h o

pen

if

a�

0? I

f a

�0?

up

;d

ow

n

d.

Wh

at d

o yo

u k

now

abo

ut

the

grap

h i

f a

�

1?It

is w

ider

th

an t

he

gra

ph

of

y�

x2 .

If

a�

1?It

is n

arro

wer

th

an t

he

gra

ph

of

y�

x2 .

2.M

atch

eac

h g

raph

wit

h t

he

desc

ript

ion

of

the

con

stan

ts i

n t

he

equ

atio

n i

n v

erte

x fo

rm.

a.a

�0,

h�

0,k

�0

iiib

.a�

0,h

�0,

k�

0iv

c.a

�0,

h�

0,k

�0

iid

.a�

0,h

�0,

k�

0i

i.ii

.ii

i.iv

.

Hel

pin

g Y

ou

Rem

emb

er

3.W

hen

grap

hing

qua

drat

ic f

unct

ions

suc

h as

y�

(x�

4)2

and

y�

(x�

5)2 ,

man

y st

uden

tsha

ve t

roub

le r

emem

beri

ng w

hich

rep

rese

nts

a tr

ansl

atio

n of

the

gra

ph o

f y

�x2

to t

he le

ftan

d w

hich

rep

rese

nts

a tr

ansl

atio

n to

the

rig

ht.W

hat

is a

n ea

sy w

ay t

o re

mem

ber

this

?

Sam

ple

an

swer

:In

fu

nct

ion

s lik

e y

�(x

�4)

2 ,th

e p

lus

sig

n p

uts

th

eg

rap

h “

ahea

d”

so t

hat

th

e ve

rtex

co

mes

“so

on

er”

than

th

e o

rig

in a

nd

th

etr

ansl

atio

n is

to

th

e le

ft.I

n f

un

ctio

ns

like

y�

(x�

5)2 ,

the

min

us

pu

ts t

he

gra

ph

“b

ehin

d”

so t

hat

th

e ve

rtex

co

mes

“la

ter”

than

th

e o

rig

in a

nd

th

etr

ansl

atio

n is

to

th

e ri

gh

t.

x

y

Ox

y

Ox

y

Ox

y

O

©G

lenc

oe/M

cGra

w-H

ill34

8G

lenc

oe A

lgeb

ra 2

Pat

tern

s w

ith

Dif

fere

nce

s an

d S

um

s o

f S

qu

ares

Som

e w

hol

e n

um

bers

can

be

wri

tten

as

the

diff

eren

ce o

f tw

o sq

uar

es a

nd

som

e ca

nn

ot.F

orm

ula

s ca

n b

e de

velo

ped

to d

escr

ibe

the

sets

of

nu

mbe

rsal

gebr

aica

lly.

If p

ossi

ble

,wri

te e

ach

nu

mb

er a

s th

e d

iffe

ren

ce o

f tw

o sq

uar

es.

Loo

k f

or p

atte

rns.

1.0

02�

022.

112

�02

3.2

can

no

t4.

322

�12

5.4

22�

026.

532

�22

7.6

can

no

t8.

742

�32

9.8

32�

1210

.932

�02

11.

10ca

nn

ot

12.1

162

�52

13.1

242

�22

14.1

372

�62

15.1

4ca

nn

ot

16.1

542

�12

Eve

n n

um

ber

s ca

n b

e w

ritt

en a

s 2n

,wh

ere

nis

on

e of

th

e n

um

ber

s 0,

1,2,

3,an

d s

o on

.Od

d n

um

ber

s ca

n b

e w

ritt

en 2

n�

1.U

se t

hes

e ex

pre

ssio

ns

for

thes

e p

rob

lem

s.

17.S

how

th

at a

ny

odd

nu

mbe

r ca

n b

e w

ritt

en a

s th

e di

ffer

ence

of

two

squ

ares

.2n

�1

�(n

�1)

2�

n2

18.S

how

th

at t

he

even

nu

mbe

rs c

an b

e di

vide

d in

to t

wo

sets

:th

ose

that

can

be

wri

tten

in

th

e fo

rm 4

nan

d th

ose

that

can

be

wri

tten

in

th

e fo

rm 2

�4n

.F

ind

4n

for

n�

0,1,

2,an

d s

o o

n.Y

ou

get

{0,

4,8,

12,…

}.F

or

2 �

4n,y

ou

get

{2,

6,10

,12,

…}.

Tog

eth

er t

hes

e se

ts in

clu

de

all e

ven

nu

mb

ers.

19.D

escr

ibe

the

even

nu

mbe

rs t

hat

can

not

be

wri

tten

as

the

diff

eren

ce o

f tw

o sq

uar

es.

2 �

4n,f

or

n�

0,1,

2,3,

…

20.S

how

th

at t

he

oth

er e

ven

nu

mbe

rs c

an b

e w

ritt

en a

s th

e di

ffer

ence

of

two

squ

ares

.4n

�(n

�1)

2�

(n�

1)2

Eve

ry w

hol

e n

um

ber

can

be

wri

tten

as

the

sum

of

squ

ares

.It

is n

ever

n

eces

sary

to

use

mor

e th

an f

our

squ

ares

.Sh

ow t

hat

th

is i

s tr

ue

for

the

wh

ole

nu

mb

ers

from

0 t

hro

ugh

15

by

wri

tin

g ea

ch o

ne

as t

he

sum

of

the

leas

t n

um

ber

of

squ

ares

.

21.0

0222

.112

23.2

12�

12

24.3

12�

12�

1225

.422

26.5

12�

22

27.6

12�

12�

2228

.712

�12

�12

�22

29.8

22�

22

30.9

3231

.10

12�

3232

.11

12�

12�

32

33.1

212

�12

�12

�32

34.1

322

�32

35.1

412

�22

�32

36.1

512

�12

�22

�32

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-6

6-6



Stu

dy G

uid

e a

nd I

nte

rven

tion

Gra

ph

ing

an

d S

olv

ing

Qu

adra

tic

Ineq

ual

itie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-7

6-7

©G

lenc

oe/M

cGra

w-H

ill34

9G

lenc

oe A

lgeb

ra 2

Lesson 6-7

Gra

ph

Qu

adra

tic

Ineq

ual

itie

sT

o gr

aph

a q

uad

rati

c in

equ

alit

y in

tw

o va

riab

les,

use

the

foll

owin

g st

eps:

1.G

raph

th

e re

late

d qu

adra

tic

equ

atio

n,y

�ax

2�

bx�

c.U

se a

das

hed

lin

e fo

r �

or �

;use

a s

olid

lin

e fo

r

or �

.

2.T

est

a po

int

insi

de t

he

para

bola

.If

it

sati

sfie

s th

e in

equ

alit

y,sh

ade

the

regi

on i

nsi

de t

he

para

bola

;ot

her

wis

e,sh

ade

the

regi

on o

uts

ide

the

para

bola

.

Gra

ph

th

e in

equ

alit

y y

�x2

�6x

�7.

Fir

st g

raph

th

e eq

uat

ion

y�

x2�

6x�

7.B

y co

mpl

etin

g th

e sq

uar

e,yo

u g

et t

he

vert

ex f

orm

of

the

equ

atio

n y

�(x

�3)

2�

2,so

th

e ve

rtex

is

(�3,

�2)

.Mak

e a

tabl

e of

val

ues

aro

un

d x

��

3,an

d gr

aph

.Sin

ce t

he

ineq

ual

ity

incl

ude

s �

,use

a d

ash

ed l

ine.

Tes

t th

e po

int

(�3,

0),w

hic

h i

s in

side

th

e pa

rabo

la.S

ince

(�

3)2

�6(

�3)

�7

��

2,an

d 0

��

2,(�

3,0)

sat

isfi

es t

he

ineq

ual

ity.

Th

eref

ore,

shad

e th

e re

gion

in

side

th

e pa

rabo

la.

Gra

ph

eac

h i

neq

ual

ity.

1.y

�x2

�8x

�17

2.y

x2

�6x

�4

3.y

�x2

�2x

�2

4.y

��

x2�

4x�

65.

y�

2x2

�4x

6.y

��

2x2

�4x

�2

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill35

0G

lenc

oe A

lgeb

ra 2

Solv

e Q

uad

rati

c In

equ

alit

ies

Qu

adra

tic

ineq

ual

itie

s in

on

e va

riab

le c

an b

e so

lved

grap

hic

ally

or

alge

brai

call

y.

To s

olve

ax

2�

bx�

c�

0:F

irst

grap

h y

�ax

2�

bx�

c. T

he s

olut

ion

cons

ists

of

the

x-va

lues

Gra

ph

ical

Met

ho

dfo

r w

hich

the

gra

ph is

bel

ow

the

x-ax

is.

To s

olve

ax

2�

bx�

c�

0:F

irst

grap

h y

�ax

2�

bx�

c. T

he s

olut

ion

cons

ists

the

x-v

alue

s fo

r w

hich

the

gra

ph is

ab

ove

the

x-ax

is.

Fin

d th

e ro

ots

of t

he r

elat

ed q

uadr

atic

equ

atio

n by

fac

torin

g,

Alg

ebra

ic M

eth

od

com

plet

ing

the

squa

re,

or u

sing

the

Qua

drat

ic F

orm

ula.

2 ro

ots

divi

de t

he n

umbe

r lin

e in

to 3

inte

rval

s.Te

st a

val

ue in

eac

h in

terv

al t

o se

e w

hich

inte

rval

s ar

e so

lutio

ns.

If t

he

ineq

ual

ity

invo

lves

or

�,t

he

root

s of

th

e re

late

d eq

uat

ion

are

in

clu

ded

in t

he

solu

tion

set

.

Sol

ve t

he

ineq

ual

ity

x2�

x�

6 �

0.

Fir

st f

ind

the

root

s of

th

e re

late

d eq

uat

ion

x2

�x

�6

�0.

Th

eeq

uat

ion

fac

tors

as

(x�

3)(x

�2)

�0,

so t

he

root

s ar

e 3

and

�2.

Th

e gr

aph

ope

ns

up

wit

h x

-in

terc

epts

3 a

nd

�2,

so i

t m

ust

be

on

or b

elow

th

e x-

axis

for

�2

x

3.

Th

eref

ore

the

solu

tion

set

is

{x�

2

x

3}.

Sol

ve e

ach

in

equ

alit

y.

1.x2

�2x

�0

2.x2

�16

�0

3.0

�6x

�x2

�5

{x�

2 �

x�

0}{x

�4

�x

�4}

{x1

�x

�5}

4.c2

4

5.2m

2�

m�

16.

y2�

�8

{c�

2 �

c �

2}�m

��

m�

1 �

7.x2

�4x

�12

�0

8.x2

�9x

�14

�0

9.�

x2�

7x�

10 �

0

{x�

2 �

x�

6}{x

x�

�7

or

x�

�2}

{x2

�x

�5}

10.2

x2�

5x�

3

011

.4x2

�23

x�

15 �

012

.�6x

2�

11x

�2

�0

�x�

3 �

x�

��x

x�

or

x�

5 ��x

x�

�2

or

x�

�13

.2x2

�11

x�

12 �

014

.x2

�4x

�5

�0

15.3

x2�

16x

�5

�0

�xx

�o

r x

�4 �

�x

�x

�5 �

1 � 33 � 2

1 � 63 � 4

1 � 2

1 � 2

x

y

O

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Gra

ph

ing

an

d S

olv

ing

Qu

adra

tic

Ineq

ual

itie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-7

6-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Gra

ph

ing

an

d S

olv

ing

Qu

adra

tic

Ineq

ual

itie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-7

6-7

©G

lenc

oe/M

cGra

w-H

ill35

1G

lenc

oe A

lgeb

ra 2

Lesson 6-7

Gra

ph

eac

h i

neq

ual

ity.

1.y

�x2

�4x

�4

2.y

x2

�4

3.y

�x2

�2x

�5

Use

th

e gr

aph

of

its

rela

ted

fu

nct

ion

to

wri

te t

he

solu

tion

s of

eac

h i

neq

ual

ity.

4.x2

�6x

�9

0

5.�

x2�

4x�

32 �

06.

x2�

x�

20 �

0

3�

8 �

x�

4x

��

5 o

r x

�4

Sol

ve e

ach

in

equ

alit

y al

geb

raic

ally

.

7.x2

�3x

�10

�0

8.x2

�2x

�35

�0

{x�

2 �

x�

5}{x

x�

�7

or

x

5}

9.x2

�18

x�

81

010

.x2

36

{xx

�9}

{x�

6 �

x�

6}

11.x

2�

7x�

012

.x2

�7x

�6

�0

{xx

�0

or

x�

7}{x

�6

�x

��

1}

13.x

2�

x�

12 �

014

.x2

�9x

�18

0

{xx

��

4 o

r x

�3}

{x�

6 �

x�

�3}

15.x

2�

10x

�25

�0

16.�

x2�

2x�

15 �

0al

l rea

ls{x

�5

�x

�3}

17.x

2�

3x�

018

.2x2

�2x

�4

{xx

��

3 o

r x

�0}

{xx

��

2 o

r x

�1}

19.�

x2�

64

�16

x20

.9x2

�12

x�

9 �

0al

l rea

ls

x

y O2

5

x

y O2

6

x

y O

x

y

O

x

y

O

x

y

O

©G

lenc

oe/M

cGra

w-H

ill35

2G

lenc

oe A

lgeb

ra 2

Gra

ph

eac

h i

neq

ual

ity.

1.y

x2

�4

2.y

�x2

�6x

�6

3.y

�2x

2�

4x�

2

Use

th

e gr

aph

of

its

rela

ted

fu

nct

ion

to

wri

te t

he

solu

tion

s of

eac

h i

neq

ual

ity.

4.x2

�8x

�0

5.�

x2�

2x�

3 �

06.

x2�

9x�

14

0

x�

0 o

r x

�8

�3

�x

�1

2 �

x�

7

Sol

ve e

ach

in

equ

alit

y al

geb

raic

ally

.

7.x2

�x

�20

�0

8.x2

�10

x�

16 �

09.

x2�

4x�

5

0

{xx

��

4 o

r x

�5}

{x2

�x

�8}

10.x

2�

14x

�49

�0

11.x

2�

5x�

1412

.�x2

�15

�8x

all r

eals

{xx

��

2 o

r x

�7}

{x�

5 �

x�

�3}

13.�

x2�

5x�

7

014

.9x2

�36

x�

36

015

.9x

12

x2

all r

eals

{xx

��

2}�x

x�

0 o

r x

�

16.4

x2�

4x�

1 �

017

.5x2

�10

�27

x18

.9x2

�31

x�

12

0

�xx

��

��x

x�

or

x

5 ��x

�3

�x

��

�19

.FEN

CIN

GV

anes

sa h

as 1

80 f

eet

of f

enci

ng

that

sh

e in

ten

ds t

o u

se t

o bu

ild

a re

ctan

gula

rpl

ay a

rea

for

her

dog

.Sh

e w

ants

th

e pl

ay a

rea

to e

ncl

ose

at l

east

180

0 sq

uar

e fe

et.W

hat

are

the

poss

ible

wid

ths

of t

he

play

are

a?30

ft

to 6

0 ft

20.B

USI

NES

SA

bic

ycle

mak

er s

old

300

bicy

cles

last

yea

r at

a p

rofi

t of

$30

0 ea

ch.T

he m

aker

wan

ts t

o in

crea

se t

he

prof

it m

argi

n t

his

yea

r,bu

t pr

edic

ts t

hat

eac

h $

20 i

ncr

ease

in

prof

it w

ill

redu

ce t

he n

umbe

r of

bic

ycle

s so

ld b

y 10

.How

man

y $2

0 in

crea

ses

in p

rofi

t ca

nth

e m

aker

add

in

an

d ex

pect

to

mak

e a

tota

l pr

ofit

of

at l

east

$10

0,00

0?fr

om

5 t

o 1

0

4 � 92 � 5

1 � 2

3 � 4x

y

O

x

y

Ox

y

O2

46

6 –6 –12

8

x

y Ox

y

O

x

y

OPra

ctic

e (

Ave

rag

e)

Gra

ph

ing

an

d S

olv

ing

Qu

adra

tic

Ineq

ual

itie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-7

6-7



Readin

g t

o L

earn

Math

em

ati

csG

rap

hin

g a

nd

So

lvin

g Q

uad

rati

c In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-7

6-7

©G

lenc

oe/M

cGra

w-H

ill35

3G

lenc

oe A

lgeb

ra 2

Lesson 6-7

Pre-

Act

ivit

yH

ow c

an y

ou f

ind

th

e ti

me

a tr

amp

olin

ist

spen

ds

abov

e a

cert

ain

hei

ght?

Rea

d th

e in

trod

uct

ion

to

Les

son

6-7

at

the

top

of p

age

329

in y

our

text

book

.

•H

ow f

ar a

bove

th

e gr

oun

d is

th

e tr

ampo

lin

e su

rfac

e?3.

75 f

eet

•U

sin

g th

e qu

adra

tic

fun

ctio

n g

iven

in

th

e in

trod

uct

ion

,wri

te a

qu

adra

tic

ineq

ual

ity

that

des

crib

es t

he

tim

es a

t w

hic

h t

he

tram

poli

nis

t is

mor

eth

an 2

0 fe

et a

bove

th

e gr

oun

d.�

16t2

�42

t�

3.75

�20

Rea

din

g t

he

Less

on

1.A

nsw

er t

he

foll

owin

g qu

esti

ons

abou

t h

ow y

ou w

ould

gra

ph t

he

ineq

ual

ity

y�

x2�

x�

6.

a.W

hat

is

the

rela

ted

quad

rati

c eq

uat

ion

?y

�x

2�

x�

6

b.

Sh

ould

th

e pa

rabo

la b

e so

lid

or d

ash

ed?

How

do

you

kn

ow?

solid

;Th

e in

equ

alit

y sy

mb

ol i

s

.

c.T

he

poin

t (0

,2)

is i

nsi

de t

he

para

bola

.To

use

th

is a

s a

test

poi

nt,

subs

titu

te

for

xan

d fo

r y

in t

he

quad

rati

c in

equ

alit

y.

d.

Is t

he

stat

emen

t 2

�02

�0

�6

tru

e or

fal

se?

tru

e

e.S

hou

ld t

he

regi

on i

nsi

de o

r ou

tsid

e th

e pa

rabo

la b

e sh

aded

?in

sid

e

2.T

he

grap

h o

f y

��

x2�

4xis

sh

own

at

the

righ

t.M

atch

eac

h

of t

he

foll

owin

g re

late

d in

equ

alit

ies

wit

h i

ts s

olu

tion

set

.

a.�

x2�

4x�

0ii

i.{x

x�

0 or

x�

4}

b.

�x2

�4x

0

iiiii

.{x

0 �

x�

4}

c.�

x2�

4x�

0iv

iii.

{xx

0

or x

�4}

d.

�x2

�4x

�0

iiv

.{x

0

x

4}

Hel

pin

g Y

ou

Rem

emb

er

3.A

qu

adra

tic

ineq

ual

ity

in t

wo

vari

able

s m

ay h

ave

the

form

y�

ax2

�bx

�c,

y�

ax2

�bx

�c,

y�

ax2

�bx

�c,

or y

ax

2�

bx�

c.D

escr

ibe

a w

ay t

o re

mem

ber

whi

ch r

egio

n to

sha

de b

y lo

okin

g at

the

ine

qual

ity

sym

bol

and

wit

hout

usi

ng a

tes

t po

int.

Sam

ple

an

swer

:If

th

e sy

mb

ol i

s �

or

,s

had

e th

e re

gio

n a

bov

e th

ep

arab

ola

.If

the

sym

bo

l is

�o

r �

,sh

ade

the

reg

ion

bel

ow

th

e p

arab

ola

.x

y

O( 0

, 0)

( 4, 0

)

( 2, 4

)

20

©G

lenc

oe/M

cGra

w-H

ill35

4G

lenc

oe A

lgeb

ra 2

Gra

ph

ing

Ab

solu

te V

alu

e In

equ

alit

ies

You

can

sol

ve a

bsol

ute

val

ue

ineq

ual

itie

s by

gra

phin

g in

mu

ch t

he

sam

e m

ann

er y

ou g

raph

ed q

uad

rati

c in

equ

alit

ies.

Gra

ph t

he

rela

ted

abso

lute

fu

nct

ion

fo

r ea

ch i

neq

ual

ity

by u

sin

g a

grap

hin

g ca

lcu

lato

r.F

or �

and

�,i

den

tify

th

e x-

valu

es,i

f an

y,fo

r w

hic

h t

he

grap

h l

ies

belo

wth

e x-

axis

.For

�an

d

,ide

nti

fy

the

xva

lues

,if

any,

for

wh

ich

th

e gr

aph

lie

s ab

ove

the

x-ax

is.

For

eac

h i

neq

ual

ity,

mak

e a

sket

ch o

f th

e re

late

d g

rap

h a

nd

fin

d t

he

solu

tion

s ro

un

ded

to

the

nea

rest

hu

nd

red

th.

1.|x

�3|

�0

2.|x|

�6

�0

3.�

|x �

4| �

8 �

0

x�

3 o

r x

�3

�6

�x

�6

�12

�x

�4

4.2|x

�6|

�2

�0

5.|3x

�3|

�0

6.|x

�7|

�5

x�

�7

or

x

�5

all r

eal n

um

ber

s2

�x

�12

7.|7x

�1|

�13

8.|x

�3.

6|

4.2

9.|2x

�5|

7

x�

�1.

71 o

r x

�2

�0.

6 �

x�

7.8

�6

�x

�1

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-7

6-7


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.

10. A

D

C

C

D

A

B

A

B

C

1 and 7; 14

C

B

C

B

D

C

A

D

D

B

A

B

B

C

B

B

A

C

B

B

Chapter 6 Assessment Answer KeyForm 1 Form 2APage 355 Page 356 Page 357

An

swer

s

(continued on the next page)


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:

A

A

B

D

C

B

A

D

B

C

C

A

B

D

A

C

D

B

C

B

D

C

A

B

A

D

D

C

D

B

Chapter 6 Assessment Answer KeyForm 2A (continued) Form 2BPage 358 Page 359 Page 360

Sample answer: 16x2 � 3 � 0

Sample answer: 9x2 � 2 � 0


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: 9x2 � 7 � 0

�x � x � ��12

� or x � 3�

y

xO

y � (x � 3)2 � 1

y � �32

�(x � 2)2 � 1

(�5, �7); x � �5; down

33; 2 real, irrational roots

0; 1 real, rational root

�3 �

1i0�31��

��2, �12

��{�2 � �13�}

��5 �2

�7��{�8, 2}

4x2 � 21x � 18 � 0

9 in. by 16 in.

��3, �25

��

y

xO

y

xO

2, 4

maximum; 4

Chapter 6 Assessment Answer KeyForm 2CPage 361 Page 362

An

swer

s

xO

f(x )

(3, 0)

(1, 5)

f (x) � �5x 2 � 10x

x � 1

between �2 and �1;between 1 and 2 h(t ) � �16(t � 1.5)2 �

51; 51 ft


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: 16x2 � 5 � 0

�x � ��32

� � x � 5�

y

xO

y � (x � 2)2 � 4

y � ��14

�(x � 4)2 � 2

(6, �5); x � 6; down

�8; 2 complex roots

0; 1 real, rational root

�9 �

4�41��

��1, �23

��{4 � �2�}

��2 �3

�6��{3, 11}

2x2 � 5x � 12

8 in. by 18 in.

��1, �43

��

y

xO

between �1 and 0;between 1 and 2

y

xO

1, �3

minimum; �17

Chapter 6 Assessment Answer KeyForm 2DPage 363 Page 364

xO

f(x )

(2, �1)

(0, 3)

f (x) � x 2 � 4x � 3

x � 2

h(t) � �16(t � 2)2 �76; 76 ft


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: 16x2 � 24x � 29 � 0

�x � x � ��72

� or x � 1�

y

xO

y � ��22090

�(x � 9)2 � �229�

h(t) � �9.1(t � 32.5)2 �

30,000; 30,000 ft

�2 � k � 2

1.2; two real,irrational roots

6 � 4�2�

{�3.5, 1}

��5 �8i�39��

{�0.35, 0.85}

x � �1

12x2 � 13x � 14 � 0

��12

�, �53

��

y

xO

between 1 and 2

y

xO

2

2

between �3 and �2;between 4 and 5

y

xO

2

2

3, 6

$8.00; $6400

minimum; �2225�

xO

f(x )

(0, 3)

f (x) � 3x 2 � 2x � 3

x � � 13

– , 13

83( )

An

swer

s

Chapter 6 Assessment Answer KeyForm 3Page 365 Page 366

y � ��35

��x � �72

�� 2� �

12

�;

��72

�, ��12

��; x � ��72

�;

down


Chapter 6 Assessment Answer KeyPage 367, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts ofgraphing, analyzing, and finding the maximum andminimum values of quadratic functions; solving quadraticequations; and solving inequalities.

• 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 graphing,analyzing, and finding the maximum and minimum valuesof quadratic functions; solving quadratic equations; andsolving 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 ofgraphing, analyzing, and finding the maximum andminimum values of quadratic functions; solving quadraticequations; and solving inequalities.

• 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 conceptsof graphing, analyzing, and finding the maximum andminimum values of quadratic functions; solving quadraticequations; and solving inequalities.

• 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

Chapter 6 Assessment Answer KeyPage 367, Open-Ended Assessment

Sample Answers


1. Student responses should indicate thatusing the Square Root Property, as Mi-Ling’s group did, would take lesstime than the other two methods sincethe equation is already set up as aperfect square set equal to a constant.To solve using either of the other twomethods, the binomial would need to beexpanded and the constant on the rightbrought to the left side of the equalsign.

2a. Jocelyn had trouble because theproblem is impossible. No suchparabola exists.

2b. Student responses will vary. One of thethree conditions must be omitted ormodified. Sample answer: Delete“...and passes through (�1, 0).”

2c. Answers will vary and depend on theanswer for part b. For example, for thesample answer in part b above, apossible equation is:y � �2(x � 3)2 � 4.

3a. Answer must be of the form y � a(x � h)2 � 8 where h is any realnumber and a � 0.

3b. Answers must be of the form y � a[x � (h � n)]2 � 8 where h and arepresent the same values as in part a.The student choice is for the value ofn. The student should indicate that thegraph will shift to the left n units ifhis or her value of n is negative, butwill shift the graph to the right n unitsif the chosen value of n is positive.

4. Students should indicate that Joseph’sanswer is not correct. In Step 2, whenhe completed the square by inserting�9 inside the parentheses, he actuallyadded 2(9) � 18 to the right side of theequation, so he must subtract 18 fromthe constant on the same side, ratherthan add 9, to keep the statementsequivalent. The correct solution is f(x) � 2(x � 3)2 � 23.

5a. �; The graph is strictly above the x-axis for all values of x other than 2.

5b. �; The graph is never below the x-axis.

5c. �; The graph is always on or above the x-axis.

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.

An

swer

s


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

Page 368 Page 369 Page 370

1. false; Zero ProductProperty

2. false; constant term

3. false; quadraticinequality

4. false; roots

5. true

6. false; minimumvalue

7. false; quadraticterm

8. false; (the)Quadratic Formula

9. true

10. false; discriminant

11. Sample answer: A parabola is asmooth curve thatis the graph of aquadratic function.

12. Sample answer: An axis of symmetryis a line along whichyou can fold a graphand get matchingparts on both sidesof the line.

1.

2.

3.

4.

5.

Quiz (Lessons 6–3 and 6–4)

Page 369

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

1.

2.

3.

4.

5.

Quiz (Lesson 6–7)

Page 370

1.

2.

3.

4. all reals

{x � x � 1 or x � 3}

y

xO

{x � 1 � x � 5}

y

xO

y � 2(x � 5)2

y � �3(x � 2)2 � 6

xO

y

(2, �1)

x � 2

�96; 2 complex roots

2 � �5�

{2 � i�14�}

{�1, 11}

��2 �5

�3��{1 � 3�5�}

{�10, 2}

3x2 � 10x � 8 � 0

x2 � 4x � 12 � 0

B

{�9, 5}

��5, �23

��

between 1 and 2;between �6 and �5

3, �1

minimum; 1

�3; x � �1; �1

xO

f(x )

(�1, �4) (0, �3)

f (x) � x 2 � 2x � 3

x � �1


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.y � �x � �

72

��2� �

249�

�3, 2 complex roots

{�2, 3}

y

xO

�1, 3

136 ft; 1.5 s

51

5.599

(2, �3)

92

(�2, 0), (�2, 8),(0, �2), (8, �2)

inconsistent

��34

�

17

��1 �3

�5��0, �

14

��{�2, 9}

minimum; �9�12

�

y

xO

1, 3

D

C

A

B

B

Chapter 6 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 371 Page 372

An

swer

s

2x3 � x2 � 2x �

4 � �x �

83

�


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. 12.

13. 14.

15.

16.

17. DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

8 0

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 5 4

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 1 0

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

4 9 / 4

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

Chapter 6 Assessment Answer KeyStandardized Test PracticePage 373 Page 374

Documents

Chapter 6 Resource Masters - Glencoe/McGraw-Hill v Glencoe Algebra 2 Assessment Options The assessment masters in the Chapter 6 Resource Mastersoffer a wide range of assessment tools