Chapter 6 Resource Masters - Math Class › uploads › 6 › ...©Glencoe/McGraw-Hill 314 Glencoe Algebra 2 Maximum and Minimum Values The y-coordinate of the vertex of a quadratic

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 3f (x) 1 3 3 9

x "1 0 "2 1f (x) 3 2 2 "1

x "3 "2 "1 "4f (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 6f (x) 8 0 "1 0 8

x "2 0 2 4 6f (x) 16 4 0 4 16

x "2 "1 0 1 2f (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 2f (x) 5 1 0.5 1 5

x "6 "4 "2 0 2f (x) 0 12 16 12 0

x 0 2 4 6 8f (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 income

the 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

vertexaxis 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?

Read the introduction to Lesson 6-2 at the top of page 294 in your textbook.

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$

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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

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1%2

7%2

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2%3

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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

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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 ! 034. NUMBER THEORY Find two consecutive even positive integers whose product is 624.

24, 2635. NUMBER THEORY Find two consecutive odd positive integers whose product is 323.

17, 1936. GEOMETRY The length of a rectangle is 2 feet more than its width. Find the

dimensions of the rectangle if its area is 63 square feet. 7 ft by 9 ft37. PHOTOGRAPHY The length and width of a 6-inch by 8-inch photograph are reduced by

the 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.

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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

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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

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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%

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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 Propertyx $ 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$$$

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$b ) #b2 $$4ac$%%%2a

ExampleExample

ExercisesExercises


Roots and the Discriminant

Discriminant The 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;

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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

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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

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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 complex21 2 real, irrational0 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 ! %a

c%.

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

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Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5

ExampleExample

Study Guide and InterventionAnalyzing Graphs of Quadratic Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6


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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

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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

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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


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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

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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

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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

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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 ! hc. In which direction does the graph open if a & 0? If a ' 0? up; downd. 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 ivc. 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.

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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

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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

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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 Method completing 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} {x⏐1 & 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} {x⏐x & "7 or x % "2} {x⏐2 ( x ( 5}

10. 2x2 " 5x$ 3 * 0 11. 4x2 $ 23x " 15 & 0 12. $6x2 $ 11x " 2 ' 0

!x⏐"3 ( x ( " !x⏐x & or x % 5" !x⏐x & "2 or x % "13. 2x2 $ 11x " 12 + 0 14. x2 $ 4x " 5 ' 0 15. 3x2 $ 16x " 5 ' 0

!x⏐x & or x % 4" ) !x⏐ & x & 5"1$

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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} {x⏐x ( "7 or x * 5}

9. x2 $ 18x " 81 * 0 10. x2 * 36{x⏐x ! 9} {x⏐"6 & x & 6}

11. x2 $ 7x & 0 12. x2 " 7x " 6 ' 0{x⏐x & 0 or x % 7} {x⏐"6 & x & "1}

13. x2 " x $ 12 & 0 14. x2 " 9x " 18 * 0{x⏐x & "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{x⏐x & "3 or x % 0} {x⏐x & "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

{x⏐x & "4 or x % 5} {x⏐2 & x & 8} )

10. x2 " 14x " 49 + 0 11. x2 $ 5x & 14 12. $x2 $ 15 + 8x

all reals {x⏐x & "2 or x % 7} {x⏐"5 ( x ( "3}

13. $x2 " 5x $ 7 * 0 14. 9x2 " 36x " 36 * 0 15. 9x * 12x2

all reals {x⏐x ! "2} !x⏐x ( 0 or x * "16. 4x2 " 4x " 1 & 0 17. 5x2 " 10 + 27x 18. 9x2 " 31x " 12 * 0

!x⏐x + " " !x⏐x ( 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 " 6b. 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? truee. 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. {xx ' 0 or x & 4}

b. $x2 " 4x * 0 iii ii. {x0 ' x ' 4}

c. $x2 " 4x + 0 iv iii. {xx * 0 or x + 4}

d. $x2 " 4x ' 0 i iv. {x0 * 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

© Glencoe/McGraw-Hill A2 Glencoe Algebra 2

Answers (Lesson 6-1)

Stu

dy

Gu

ide

and I

nte

rven

tion

Gra

phin

g Q

uadr

atic

Fun

ctio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-1

6-1

©G

lenc

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w-H

ill31

3G

lenc

oe A

lgeb

ra 2

Lesson 6-1

Gra

ph

Qu

adra

tic

Fun

ctio

ns

Qua

drat

ic F

unct

ion

Afu

nctio

n de

fined

by

an e

quat

ion

of th

e fo

rm f

(x) !

ax2

"bx

"c,

whe

re a

#0

Gra

ph o

f a Q

uadr

atic

Apa

rabo

law

ith th

ese

char

acte

ristic

s: y

inte

rcep

t: c;

axis

of s

ymm

etry

: x!

;Fu

nctio

nx-

coor

dina

te o

f ver

tex:

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

-int

erce

pt is

5.T

he e

quat

ion

of t

he a

xis

of s

ymm

etry

is

x!

or

.The

x-c

oord

inat

e of

the

ver

tex

is

.

Nex

t m

ake

a ta

ble

of v

alue

s fo

r x

near

.

xx2

"3x

#5

f(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.M

ake

a ta

ble

of v

alu

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"

6x"

82.

f(x)

!$

x2$

2x"

23.

f(x)

!2x

2$

4x"

38,

x!

"3,

"3

2,x

!"

1,"

13,

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

3f(

x)1

33

9x

"1

0"

21

f(x)

32

2"

1x

"3

"2

"1

"4

f(x)

"1

03

0

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

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ill31

4G

lenc

oe A

lgeb

ra 2

Max

imu

m a

nd

Min

imu

m V

alu

esT

he y

-coo

rdin

ate

of t

he v

erte

x of

a q

uadr

atic

func

tion

is t

he m

axim

um o

r m

inim

um v

alue

of

the

func

tion

.

Max

imum

or M

inim

um V

alue

Th

e gr

aph

of f(

x) !

ax2

"bx

"c,

whe

re a

#0,

ope

ns u

p an

d ha

s a

min

imum

of a

Qua

drat

ic F

unct

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

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Gra

phin

g Q

uadr

atic

Fun

ctio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-1

6-1

Exam

ple

Exam

ple

a.f(

x) !

3x2

"6x

#7

For

this

fun

ctio

n,a

!3

and

b!

$6.

Sinc

e a

&0,

the

grap

h op

ens

up,a

nd t

hefu

ncti

on h

as a

min

imum

val

ue.

The

min

imum

val

ue is

the

y-c

oord

inat

eof

the

ver

tex.

The

x-c

oord

inat

e of

the

ve

rtex

is

!$

!1.

Eva

luat

e th

e fu

ncti

on a

t x

!1

to f

ind

the

min

imum

val

ue.

f(1)

!3(

1)2

$6(

1) "

7 !

4,so

the

min

imum

val

ue o

f th

e fu

ncti

on is

4.

$6

% 2(3)

$b

% 2a

b.f(

x) !

100

"2x

"x2

For

this

fun

ctio

n,a

!$

1 an

d b

!$

2.Si

nce

a'

0,th

e gr

aph

open

s do

wn,

and

the

func

tion

has

a m

axim

um v

alue

.T

he m

axim

um v

alue

is t

he y

-coo

rdin

ate

ofth

e ve

rtex

.The

x-c

oord

inat

e of

the

ver

tex

is

!$

!$

1.

Eva

luat

e th

e fu

ncti

on a

t x

!$

1 to

fin

dth

e m

axim

um v

alue

.f(

$1)

!10

0 $

2($

1) $

($1)

2!

101,

soth

e m

inim

um v

alue

of

the

func

tion

is 1

01.

$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

"10

11.f

(x) !

x2$

10x

"5

12.f

(x) !

$6x

2"

12x

"21

max

.,14

min

.,"

20m

ax.,

27

13.f

(x) !

25x2

"10

0x"

350

14.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

phin

g Q

uadr

atic

Fun

ctio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-1

6-1

©G

lenc

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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;

05;

x!

5;5

7;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.

Mak

e a

tabl

e of

val

ues

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

.

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) !

6x2

11.f

(x) !

$8x

212

.f(x

) !x2

"2x

min

.;0

max

.;0

min

.;"

1

13.f

(x) !

x2"

2x"

1514

.f(x

) !$

x2"

4x$

115

.f(x

) !x2

"2x

$3

min

.;14

max

.;3

min

.;"

4

16.f

(x) !

$2x

2"

4x$

317

.f(x

) !3x

2"

12x

"3

18.f

(x) !

2x2

"4x

"1

max

.;"

1m

in.;

"9

min

.;"

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)

80

"1

08

x"

20

24

6f(

x)16

40

416

x"

2"

10

12

f(x)

"8

"2

0"

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.

Mak

e a

tabl

e of

val

ues

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$

8x"

152.

f(x)

!$

x2$

4x"

123.

f(x)

!2x

2$

2x"

115

;x!

4;4

12;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

,Sus

an t

hrow

s a

ball

upw

ard

wit

h a

velo

city

of

32 f

eet

per

seco

nd.T

he h

eigh

t h(

t) o

f th

e ba

ll t

seco

nds

afte

r Su

san

thro

ws

itis

giv

en b

y h(

t) !

$16

t2"

32t

"4.

Fin

d th

e m

axim

um h

eigh

t re

ache

d by

the

bal

l and

the

tim

e th

at t

his

heig

ht is

rea

ched

.20

ft;1

s

11.H

EALT

H C

LUB

SL

ast

year

,the

Spo

rtsT

ime

Ath

leti

c C

lub

char

ged

$20

to p

arti

cipa

te in

an a

erob

ics

clas

s.Se

vent

y pe

ople

att

ende

d th

e cl

asse

s.T

he c

lub

wan

ts t

o in

crea

se t

hecl

ass

pric

e th

is y

ear.

The

y ex

pect

to

lose

one

cus

tom

er f

or e

ach

$1 in

crea

se in

the

pri

ce.

a.W

hat

pric

e sh

ould

the

clu

b ch

arge

to

max

imiz

e th

e in

com

e fr

om t

he a

erob

ics

clas

ses?

$45

b.W

hat

is t

he m

axim

um in

com

e th

e Sp

orts

Tim

e A

thle

tic

Clu

b ca

n ex

pect

to

mak

e?$2

025

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)

51

0.5

15

x"

6"

4"

20

2f(

x)0

1216

120

x0

24

68

f(x)

153

"1

315

Pra

ctic

e (A

vera

ge)

Gra

phin

g Q

uadr

atic

Fun

ctio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-1

6-1



Rea

din

g t

o L

earn

Math

emati

csG

raph

ing

Qua

drat

ic F

unct

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

ucti

on t

o L

esso

n 6-

1 at

the

top

of

page

286

in y

our

text

book

.•

Bas

ed o

n th

e gr

aph

in y

our

text

book

,for

wha

t ti

cket

pri

ce is

the

inco

me

the

grea

test

?$4

0•

Use

the

gra

ph t

o es

tim

ate

the

max

imum

inco

me.

abou

t $72

,000

Rea

din

g t

he

Less

on

1.a.

For

the

quad

rati

c fu

ncti

on f

(x) !

2x2

"5x

"3,

2x2

is t

he

term

,

5xis

the

te

rm,a

nd 3

is t

he

term

.

b.Fo

r th

e qu

adra

tic

func

tion

f(x

) !$

4 "

x$

3x2 ,

a!

,b!

,and

c!

.

2.C

onsi

der

the

quad

rati

c fu

ncti

on f

(x) !

ax2

"bx

"c,

whe

re a

#0.

a.T

he g

raph

of

this

fun

ctio

n is

a

.

b.T

he y

-int

erce

pt is

.

c.T

he a

xis

of s

ymm

etry

is t

he li

ne

.

d.If

a&

0,th

en t

he g

raph

ope

ns

and

the

func

tion

has

a

valu

e.

e.If

a'

0,th

en t

he g

raph

ope

ns

and

the

func

tion

has

a

valu

e.

3.R

efer

to

the

grap

h at

the

rig

ht a

s yo

u co

mpl

ete

the

follo

win

g se

nten

ces.

a.T

he c

urve

is c

alle

d a

.

b.T

he li

ne x

!$

2 is

cal

led

the

.

c.T

he p

oint

($2,

4) is

cal

led

the

.

d.B

ecau

se t

he g

raph

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 us

e th

e x2

term

of

a qu

adra

tic

func

tion

to

tell

whe

ther

the

fun

ctio

n ha

s a

max

imum

or

a m

inim

um v

alue

?Sa

mpl

e an

swer

:R

emem

ber t

hat t

he g

raph

of f

(x) !

x2(w

ith a

%0)

is a

U-s

hape

d cu

rve

that

ope

ns u

p an

d ha

s a

min

imum

.The

gra

ph o

f g(x

) !"

x2(w

ith a

&0)

is ju

st th

e op

posi

te.I

t ope

ns d

own

and

has

a m

axim

um.

y-in

terc

ept

vert

exax

is o

f sym

met

rypa

rabo

la

x

f(x)

O ( 0, –

1)

( –2,

4)

max

imum

dow

nwar

d

min

imum

upw

ard

x!

"$ 2b a$

c

para

bola

"4

1"

3co

nsta

ntlin

ear

quad

ratic

©G

lenc

oe/M

cGra

w-H

ill31

8G

lenc

oe A

lgeb

ra 2

Find

ing

the

Axi

s of

Sym

met

ry o

f a P

arab

ola

As

you

know

,if f

(x) !

ax2

"bx

"c

is a

qua

drat

ic f

unct

ion,

the

valu

es o

f x

that

mak

e f(

x) e

qual

to

zero

are

an

d .

The

ave

rage

of

thes

e tw

o nu

mbe

r va

lues

is $

% 2b a%.

The

fun

ctio

n f(

x) h

as it

s m

axim

um o

r m

inim

um

valu

e w

hen

x!

$% 2b a%

.Sin

ce t

he a

xis

of s

ymm

etry

of t

he g

raph

of f

(x)

pass

es t

hrou

gh t

he p

oint

whe

re

the

max

imum

or

min

imum

occ

urs,

the

axis

of

sym

met

ry h

as t

he e

quat

ion

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

-coo

rdin

ate

of t

he v

erte

x is

$1.

Subs

titu

te x

!$

1 in

f(x

) !5x

2"

10x

$7.

f($

1) !

5($

1)2

"10

($1)

$7

!$

12T

he v

erte

x is

($1,

$12

).T

he a

xis

of s

ymm

etry

is x

!$

% 2b a%,o

r 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!

44.

f(x)

!2x

2"

6x"

5%"

$3 2$ ,$1 2$ &;

x!

"$3 2$

5.A

(x) !

x2"

12x

"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) =

ax2 +

bx

+ c

$b

$#

b2$

4$

ac$%

%%

2a$

b"

#b2

$4

$ac$

%%

%2a

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-1

6-1

Exam

ple

Exam

ple


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Solv

ing

Qua

drat

ic E

quat

ions

by

Gra

phin

g

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

Qua

drat

ic E

quat

ion

Aqu

adra

tic e

quat

ion

has

the

form

ax2

"bx

"c

!0,

whe

re a

#0.

Roo

ts o

f a Q

uadr

atic

Equ

atio

nso

lutio

n(s)

of t

he e

quat

ion,

or t

he z

ero(

s) o

f the

rela

ted

quad

ratic

func

tion

The

zer

os o

f a

quad

rati

c fu

ncti

on a

re t

he x

-int

erce

pts

of it

s gr

aph.

The

refo

re,f

indi

ng t

he

x-in

terc

epts

is o

ne w

ay o

f so

lvin

g th

e re

late

d qu

adra

tic

equa

tion

.

Sol

ve x

2#

x "

6 !

0 by

gra

ph

ing.

Gra

ph t

he r

elat

ed f

unct

ion

f(x)

!x2

"x

$6.

The

x-c

oord

inat

e of

the

ver

tex

is

!$

,and

the

equ

atio

n of

the

axis

of

sym

met

ry is

x!

$.

Mak

e a

tabl

e of

val

ues

usin

g x-

valu

es a

roun

d $

.

x$

1$

01

2

f(x)

$6

$6

$6

$4

0

Fro

m t

he t

able

and

the

gra

ph,w

e ca

n se

e th

at t

he z

eros

of

the

func

tion

are

2 a

nd $

3.

Sol

ve e

ach

equ

atio

n b

y gr

aph

ing.

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 re

al s

olut

ions

"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

find

exa

ct s

olut

ions

to

quad

rati

ceq

uati

ons

by g

raph

ing.

But

you

can

use

the

gra

ph t

o es

tim

ate

solu

tion

s.

Sol

ve x

2"

2x"

2 !

0 by

gra

ph

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.

The

equ

atio

n of

the

axi

s of

sym

met

ry o

f th

e re

late

d fu

ncti

on 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$

21

The

x-i

nter

cept

s of

the

gra

ph a

re b

etw

een

2 an

d 3

and

betw

een

0 an

d$

1.So

one

sol

utio

n is

bet

wee

n 2

and

3,an

d th

e ot

her

solu

tion

isbe

twee

n 0

and

$1.

Sol

ve t

he

equ

atio

ns

by g

rap

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

betw

een

0 an

d 1;

betw

een

"2

and

"1;

betw

een

"1

and

0;be

twee

n 3

and

4be

twee

n "

5 an

d "

4be

twee

n "

4 an

d "

3

4.$

x2"

2x"

4 !

05.

2x2

$12

x"

17 !

06.

$x2

"x

"!

0

betw

een

3 an

d 4;

betw

een

2 an

d 3;

betw

een

"2

and

"1;

betw

een

"2

and

"1

betw

een

3 an

d 4

betw

een

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

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Solv

ing

Qua

drat

ic E

quat

ions

by

Gra

phin

g

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-2

6-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Solv

ing

Qua

drat

ic E

quat

ions

By

Gra

phin

g

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

quat

ion

to

det

erm

ine

its

solu

tion

s.

1.x2

"2x

$3

!0

2.$

x2$

6x$

9 !

03.

3x2

"4x

"3

!0

"3,

1"

3no

real

sol

utio

ns

Sol

ve e

ach

equ

atio

n b

y gr

aph

ing.

If e

xact

roo

ts c

ann

ot b

e fo

un

d,s

tate

th

eco

nse

cuti

ve i

nte

gers

bet

wee

n w

hic

h t

he

root

s ar

e lo

cate

d.

4.x2

$6x

"5

!0

5.$

x2"

2x$

4 !

06.

x2$

6x"

4 !

01,

5no

real

sol

utio

nsbe

twee

n 0

and

1;be

twee

n 5

and

6

Use

a q

uad

rati

c eq

uat

ion

to

fin

d t

wo

real

nu

mbe

rs t

hat

sat

isfy

eac

h s

itu

atio

n,o

rsh

ow t

hat

no

such

nu

mbe

rs e

xist

.

7.T

heir

sum

is $

4,an

d th

eir

prod

uct

is 0

.8.

The

ir s

um is

0,a

nd t

heir

pro

duct

is $

36.

"x2

"4x

!0;

0,"

4"

x2#

36 !

0;"

6,6

f(x) !

"x2 #

36

x

f (x)

O6

–612

–12

36 24 12

f(x) !

"x2 "

4x

x

f (x)

O

f(x) !

x2 " 6

x # 4

x

f (x)

Of(x

) ! "

x2 # 2

x " 4x

f (x)

O

f(x) !

x2 " 6

x # 5

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 # 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

quat

ion

to

det

erm

ine

its

solu

tion

s.

1.$

3x2

"3

!0

2.3x

2"

x"

3 !

03.

x2$

3x"

2 !

0

"1,

1no

real

sol

utio

ns1,

2S

olve

eac

h e

quat

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

betw

een

0 an

d 1;

"6,

"4

betw

een

"2

and

"1,

betw

een

"4

and

"3

2

Use

a q

uad

rati

c eq

uat

ion

to

fin

d t

wo

real

nu

mbe

rs t

hat

sat

isfy

eac

h s

itu

atio

n,o

rsh

ow t

hat

no

such

nu

mbe

rs e

xist

.

7.T

heir

sum

is 1

,and

the

ir p

rodu

ct is

$6.

8.T

heir

sum

is 5

,and

the

ir p

rodu

ct 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

bjec

t in

fee

t,v 0

is t

he

obje

ct’s

in

itia

l ve

loci

ty i

n f

eet

per

sec

ond

,an

d t

is t

he

tim

e in

sec

ond

s.

9.B

ASE

BA

LLM

arta

thr

ows

a ba

seba

ll w

ith

an in

itia

l upw

ard

velo

city

of 6

0 fe

et p

er s

econ

d.Ig

nori

ng M

arta

’s he

ight

,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

erup

tion

bla

sts

a bo

ulde

r up

war

d w

ith

an in

itia

l vel

ocit

y of

240

feet

per

sec

ond.

How

long

will

it t

ake

the

boul

der

to h

it t

he g

roun

d if

it la

nds

at t

hesa

me

elev

atio

n fr

om w

hich

it w

as e

ject

ed?

15 s

"x2

#5x

"8

!0;

no s

uch

real

num

bers

exi

stx

f (x)

Of(x

) ! "

x2 # 5

x " 8

"x2

#x

#6

!0;

3,"

2f(x

) ! "

x2 # x

# 6 x

f (x)

O

x

f (x)

O

f(x) !

2x2 "

x "

6f(x

) ! x2 #

10x

# 2

4x

f (x)

O

f(x) !

"2x

2 " 6

x # 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 (A

vera

ge)

Solv

ing

Qua

drat

ic E

quat

ions

By

Gra

phin

g

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-2

6-2


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csSo

lvin

g Q

uadr

atic

Equ

atio

ns b

y G

raph

ing

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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 qu

adra

tic

fun

ctio

n m

odel

a f

ree-

fall

rid

e?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 6-

2 at

the

top

of

page

294

in y

our

text

book

.

Wri

te a

qua

drat

ic f

unct

ion

that

des

crib

es t

he h

eigh

t of

a b

all t

seco

nds

afte

rit

is d

ropp

ed f

rom

a h

eigh

t of

125

fee

t.h(

t) !

"16

t2#

125

Rea

din

g t

he

Less

on

1.T

he g

raph

of

the

quad

rati

c fu

ncti

on f

(x) !

$x2

"x

"6

is s

how

n at

the

rig

ht.U

se t

he g

raph

to

find

the

sol

utio

ns o

f th

equ

adra

tic

equa

tion

$x2

"x

"6

!0.

"2

and

3

2.Sk

etch

a g

raph

to

illus

trat

e ea

ch s

itua

tion

.

a.A

par

abol

a th

at o

pens

b.

A p

arab

ola

that

ope

ns

c.A

par

abol

a th

at o

pens

dow

nwar

d an

d re

pres

ents

a

upw

ard

and

repr

esen

ts a

do

wnw

ard

and

qu

adra

tic

func

tion

wit

h tw

o qu

adra

tic

func

tion

wit

h

repr

esen

ts a

re

al z

eros

,bot

h of

whi

ch a

reex

actl

y on

e re

al z

ero.

The

qu

adra

tic

func

tion

ne

gati

ve n

umbe

rs.

zero

is a

pos

itiv

e nu

mbe

r.w

ith

no r

eal z

eros

.

Hel

pin

g Y

ou

Rem

emb

er

3.T

hink

of

a m

emor

y ai

d th

at c

an h

elp

you

reca

ll w

hat

is m

eant

by

the

zero

sof

a q

uadr

atic

func

tion

.

Sam

ple

answ

er:T

he b

asic

fact

s ab

out a

sub

ject

are

som

etim

es c

alle

d th

eA

BC

s.In

the

case

of z

eros

,the

AB

Cs

are

the

XYZs

,bec

ause

the

zero

sar

e th

e x-

valu

es th

at m

ake

the

y-va

lues

equ

al to

zer

o.

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

phin

g A

bsol

ute

Valu

e Eq

uatio

ns

You

can

solv

e ab

solu

te v

alue

equ

atio

ns in

muc

h th

e sa

me

way

you

sol

ved

quad

rati

c eq

uati

ons.

Gra

ph t

he r

elat

ed a

bsol

ute

valu

e fu

ncti

on f

or e

ach

equa

tion

usi

ng a

gra

phin

g ca

lcul

ator

.The

n us

e th

e ZE

ROfe

atur

e in

the

CA

LCm

enu

to f

ind

its

real

sol

utio

ns,i

f an

y.R

ecal

l tha

t so

luti

ons

are

poin

ts

whe

re t

he g

raph

inte

rsec

ts t

he x

-axi

s.

For

eac

h e

quat

ion

,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

"5|

!0

2.|4

x$

3| "

5 !

03.

|x$

7| !

0

"5

No

solu

tions

7

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 a

bsol

ute

valu

e eq

uati

ons

alge

brai

cally

and

fin

ding

ze

ros

of a

bsol

ute

valu

e fu

ncti

ons

grap

hica

lly a

re r

elat

ed.

Sam

ple

answ

er:v

alue

s of

xw

hen

solv

ing

alge

brai

cally

are

the

x-in

terc

epts

(or z

eros

) of t

he fu

nctio

n w

hen

grap

hed.

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-2

6-2



Stu

dy

Gu

ide

and I

nte

rven

tion

Solv

ing

Qua

drat

ic E

quat

ions

by

Fact

orin

g

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

fact

orin

g to

sol

ve a

qua

drat

ic e

quat

ion,

you

use

the

follo

win

g pr

oper

ty.

Zero

Pro

duct

Pro

pert

yFo

r any

real

num

bers

aan

d b,

if a

b!

0, th

en e

ither

a!

0 or

b!

0, o

r bot

h a

and

b!

0.

Sol

ve e

ach

equ

atio

n b

y fa

ctor

ing.

Exam

ple

Exam

ple

a.3x

2!

15x

3x2

!15

xO

rigin

al e

quat

ion

3x2

$15

x!

0Su

btra

ct 1

5xfro

m b

oth

side

s.

3x(x

$5)

!0

Fact

or th

e bi

nom

ial.

3x !

0or

x$

5 !

0Ze

ro P

rodu

ct P

rope

rty

x!

0or

x!

5So

lve

each

equ

atio

n.

The

sol

utio

n se

t is

{0,

5}.

b.4x

2"

5x!

214x

2$

5x!

21O

rigin

al e

quat

ion

4x2

$5x

$21

!0

Subt

ract

21

from

bot

h si

des.

(4x

"7)

(x$

3)!

0Fa

ctor

the

trino

mia

l.

4x"

7 !

0or

x$

3 !

0Ze

ro P

rodu

ct P

rope

rty

x!

$or

x

!3

Solv

e ea

ch e

quat

ion.

The

sol

utio

n se

t is

%$,3

&.7 % 4

7 % 4

Exer

cises

Exer

cises

Sol

ve e

ach

equ

atio

n b

y fa

ctor

ing.

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

sTo

wri

te a

qua

drat

ic e

quat

ion

wit

h ro

ots

pan

d q,

let

(x$

p)(x

$q)

!0.

The

n m

ulti

ply

usin

g F

OIL

.

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

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Solv

ing

Qua

drat

ic E

quat

ions

by

Fact

orin

g

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

, qw

ith $

5.

(x$

3)(x

"5)

!0

Sim

plify

.

x2"

2x$

15 !

0U

se F

OIL

.

The

equ

atio

n x2

"2x

$15

!0

has

root

s 3

and

$5.

b."

,

(x$

p)(x

$q)

!0

'x$

!$"(!x

$"!

0

!x"

"!x$

"!0

(!

0

!24

(0

24x2

"13

x$

7 !

0

The

equ

atio

n 24

x2"

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 !

0x2

#10

x#

16 !

0x2

"10

x#

9 !

04.

$5

5.10

,76.

$2,

15x2

#10

x#

25 !

0x2

"17

x#

70 !

0x2

"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

35x2

"22

x#

3 !

0

16.$

,17

.,

18.

,

16x2

"42

x"

498x

2"

10x

#3

!0

48x2

"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

Solv

ing

Qua

drat

ic E

quat

ions

by

Fact

orin

g

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

equ

atio

n b

y fa

ctor

ing.

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

,4x2

"5x

#4

!0

20.6

,$9

x2#

3x"

54 !

0

21.$

2,$

5x2

#7x

#10

!0

22.0

,7x2

"7x

!0

23.$

,$3

3x2

#10

x#

3 !

024

.$,

8x2

"2x

"3

!0

25.F

ind

two

cons

ecut

ive

inte

gers

who

se p

rodu

ct is

272

.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

equ

atio

n b

y fa

ctor

ing.

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,

8x2

"9x

#14

!0

x2"

3x!

0x2

"3x

"40

!0

25.$

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TH

EORY

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ive

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gers

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se p

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624

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,26

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BER

TH

EORY

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d tw

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siti

ve in

tege

rs w

hose

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23.

17,1

936

.GEO

MET

RYT

he le

ngth

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fee

t m

ore

than

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wid

th.F

ind

the

dim

ensi

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ecta

ngle

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s ar

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63

squa

re f

eet.

7 ft

by 9

ft37

.PH

OTO

GR

APH

YT

he le

ngth

and

wid

th o

f a

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ch b

y 8-

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f th

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igin

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es w

ill t

he d

imen

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s of

the

pho

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to

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educ

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2 in

.

4 % 52 % 3

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2 $ 33 $ 2

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6-3

6-3



Rea

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atio

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6-3

6-3

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Lesson 6-3

Pre-

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ow i

s th

e Z

ero

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use

d i

n g

eom

etry

?

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d th

e in

trod

ucti

on t

o L

esso

n 6-

3 at

the

top

of

page

301

in y

our

text

book

.

Wha

t do

es t

he e

xpre

ssio

n x(

x"

5) m

ean

in t

his

situ

atio

n?

It re

pres

ents

the

area

of t

he re

ctan

gle,

sinc

e th

e ar

ea is

the

prod

uct o

f the

wid

th a

nd le

ngth

.

Rea

din

g t

he

Less

on

1.T

he s

olut

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of a

qua

drat

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quat

ion

by f

acto

ring

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n be

low

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e th

e re

ason

for

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ste

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rigin

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ach

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ctor

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omia

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$3

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Pro

duct

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pert

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lve

each

equ

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n.T

he s

olut

ion

set

is

.

2.O

n an

alg

ebra

qui

z,st

uden

ts w

ere

aske

d to

wri

te a

qua

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quat

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wit

h $

7 an

d 5

asit

s ro

ots.

The

wor

k th

at t

hree

stu

dent

s in

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ss w

rote

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r pa

pers

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how

n be

low

.

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laR

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rror

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the

oth

er t

wo

stud

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’ wor

k.

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ple

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arla

use

d th

e w

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fact

ors.

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ed th

e co

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s bu

t mul

tiplie

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em in

corr

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.

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

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ay t

o re

mem

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ncep

t is

to

repr

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t it

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ore

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one

way

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raph

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para

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d t

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val

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um

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13.D

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rim

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mbe

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reat

er t

han

40?

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e ex

ampl

es

in y

our

answ

er.

No.

Am

ong

the

prim

es o

mitt

ed a

re 5

9,67

,73,

79,8

9,10

1,10

3,10

7,10

9,an

d 12

7.

14.E

uler

’s f

orm

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prod

uces

pri

mes

for

man

y va

lues

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,but

it d

oes

not

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r al

l of

them

.Fin

d th

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rst

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xfo

r w

hich

the

for

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a fa

ils.

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t:T

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x!

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168

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412 .

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6-3

6-3


An

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Lesson 6-4

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o so

lve

a qu

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equa

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tha

t is

in t

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“pe

rfec

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uare

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cons

tant

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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

equ

atio

n b

y co

mp

leti

ng

the

squ

are.

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

Whe

n th

e di

men

sion

s of

a c

ube

are

redu

ced

by 4

inch

es o

n ea

ch s

ide,

the

surf

ace

area

of

the

new

cub

e is

864

squ

are

inch

es.W

hat

wer

e th

e di

men

sion

s of

the

orig

inal

cub

e?16

in.b

y 16

in.b

y 16

in.

32.I

NV

ESTM

ENTS

The

am

ount

of

mon

ey A

in a

n ac

coun

t in

whi

ch P

dolla

rs is

inve

sted

for

2 ye

ars

is g

iven

by

the

form

ula

A!

P(1

"r)

2 ,w

here

ris

the

inte

rest

rat

e co

mpo

unde

dan

nual

ly.I

f an

inve

stm

ent

of $

800

in t

he a

ccou

nt g

row

s to

$88

2 in

tw

o ye

ars,

at w

hat

inte

rest

rat

e w

as it

inve

sted

?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 (A

vera

ge)

Com

plet

ing

the

Squa

re

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-4

6-4


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csC

ompl

etin

g th

e Sq

uare

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

ucti

on t

o L

esso

n 6-

4 at

the

top

of

page

306

in y

our

text

book

.

Exp

lain

wha

t it

mea

ns t

o sa

y th

at t

he d

rive

r ac

cele

rate

s at

a c

onst

ant

rate

of 8

fee

t pe

r se

cond

squ

are.

If th

e dr

iver

is tr

avel

ing

at a

cer

tain

spe

ed a

t a p

artic

ular

mom

ent,

then

one

sec

ond

late

r,th

e dr

iver

is tr

avel

ing

8 fe

etpe

r sec

ond

fast

er.

Rea

din

g t

he

Less

on

1.G

ive

the

reas

on f

or e

ach

step

in t

he f

ollo

win

g so

luti

on o

f an

equ

atio

n by

usi

ng t

heSq

uare

Roo

t P

rope

rty.

x2$

12x

"36

!81

Orig

inal

equ

atio

n

(x$

6)2

!81

Fact

or th

e pe

rfect

squ

are

trin

omia

l.x

$6

!)

#81$

Squa

re R

oot P

rope

rty

x$

6 !

)9

81 !

9x

$6

!9

or x

$6

!$

9R

ewrit

e as

two

equa

tions

.x

!15

x

!$

3So

lve

each

equ

atio

n.

2.E

xpla

in h

ow t

o fi

nd t

he c

onst

ant

that

mus

t be

add

ed t

o m

ake

a bi

nom

ial i

nto

a pe

rfec

tsq

uare

tri

nom

ial.

Sam

ple

answ

er:F

ind

half

of th

e co

effic

ient

of t

he li

near

term

and

squ

are

it.

3.a.

Wha

t is

the

fir

st s

tep

in s

olvi

ng t

he e

quat

ion

3x2

"6x

!5

by c

ompl

etin

g th

e sq

uare

?D

ivid

e th

e eq

uatio

n by

3.

b.W

hat

is t

he f

irst

ste

p in

sol

ving

the

equ

atio

n x2

"5x

$12

!0

by c

ompl

etin

g th

esq

uare

?A

dd 1

2 to

eac

h si

de.

Hel

pin

g Y

ou

Rem

emb

er

4.H

ow c

an y

ou u

se t

he r

ules

for

squ

arin

g a

bino

mia

l to

help

you

rem

embe

r th

e pr

oced

ure

for

chan

ging

a b

inom

ial i

nto

a pe

rfec

t sq

uare

tri

nom

ial?

One

of t

he ru

les

for s

quar

ing

a bi

nom

ial i

s (x

#y)

2!

x2#

2xy

#y2

.In

com

plet

ing

the

squa

re,y

ou a

re s

tart

ing

with

x2

#bx

and

need

to fi

nd y

2 .Th

is s

how

s yo

u th

at b

!2y

,so

y!

.Tha

t is

why

you

mus

t tak

e ha

lf of

th

e co

effic

ient

and

squ

are

it to

get

the

cons

tant

that

mus

t be

adde

d to

com

plet

e th

e sq

uare

.

b $ 2

©G

lenc

oe/M

cGra

w-H

ill33

6G

lenc

oe A

lgeb

ra 2

The

Gol

den

Qua

drat

ic E

quat

ions

A g

old

en r

ecta

ngl

eha

s th

e pr

oper

ty t

hat

its

leng

th

can

be w

ritt

en a

s a

"b,

whe

re a

is t

he w

idth

of

the

rect

angl

e an

d %a

" ab

%!

%a b% .A

ny g

olde

n re

ctan

gle

can

be

divi

ded

into

a s

quar

e an

d a

smal

ler

gold

en r

ecta

ngle

,as

sho

wn.

The

pro

port

ion

used

to

defi

ne g

olde

n re

ctan

gles

can

be

used

to

deri

ve t

wo

quad

rati

c eq

uati

ons.

The

se a

reso

met

imes

calle

d go

lden

qua

drat

ic e

quat

ions

.

Sol

ve e

ach

pro

blem

.

1.In

the

pro

port

ion

for

the

gold

en r

ecta

ngle

,let

aeq

ual 1

.Wri

te t

he r

esul

ting

qu

adra

tic

equa

tion

and

sol

ve f

or b

.

b2#

b"

1!

0 b

!

2.In

the

pro

port

ion,

let

beq

ual 1

.Wri

te t

he r

esul

ting

qua

drat

ic e

quat

ion

and

solv

e fo

r a.

a2"

a"

1!

0a

!

3.D

escr

ibe

the

diff

eren

ce b

etw

een

the

two

gold

en q

uadr

atic

equ

atio

ns y

ou

foun

d in

exe

rcis

es 1

and

2.

The

sign

s of

the

first

-deg

ree

term

s ar

e op

posi

te.

4.Sh

ow t

hat

the

posi

tive

sol

utio

ns o

f th

e tw

o eq

uati

ons

in e

xerc

ises

1 a

nd 2

ar

e re

cipr

ocal

s.

'('

(!!

$"1 4#

5$

!1

5.U

se t

he P

ytha

gore

an T

heor

em t

o fi

nd a

rad

ical

exp

ress

ion

for

the

diag

onal

of

a g

olde

n re

ctan

gle

whe

n a

!1.

d!

6.F

ind

a ra

dica

l exp

ress

ion

for

the

diag

onal

of

a go

lden

rec

tang

le 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

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-4

6-4



Stu

dy

Gu

ide

and I

nte

rven

tion

The

Qua

drat

ic F

orm

ula

and

the

Dis

crim

inan

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

The

Qu

adra

tic

For

mu

laca

n be

use

d to

sol

ve a

nyqu

adra

tic

equa

tion

onc

e it

is w

ritt

en in

the

for

m a

x2"

bx"

c!

0.

Qua

drat

ic F

orm

ula

The

solu

tions

of a

x2"

bx"

c!

0, w

ith a

#0,

are

giv

en b

y x

!.

Sol

ve x

2"

5x!

14 b

y u

sin

g th

e Q

uad

rati

c F

orm

ula

.

Rew

rite

the

equ

atio

n as

x2

$5x

$14

!0.

x!

Qua

drat

ic F

orm

ula

!R

epla

ce a

with

1, b

with

$5,

and

cw

ith $

14.

!Si

mpl

ify.

! !7

or $

2

The

sol

utio

ns a

re $

2 an

d 7.

Sol

ve e

ach

equ

atio

n b

y u

sin

g th

e Q

uad

rati

c F

orm

ula

.

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$

$4a

c$

%%

%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

tTh

e ex

pres

sion

und

er th

e ra

dica

l sig

n, b

2$

4ac,

in th

e Q

uadr

atic

For

mul

a is

cal

led

the

disc

rimin

ant.

Ro

ots

of

a Q

uad

rati

c Eq

uat

ion

Dis

crim

inan

tTy

pe a

nd N

umbe

r of R

oots

b2$

4ac

&0

and

a pe

rfect

squ

are

2 ra

tiona

l roo

ts

b2$

4ac

&0,

but

not

a pe

rfect

squ

are

2 irr

atio

nal r

oots

b2$

4ac

!0

1 ra

tiona

l roo

t

b2$

4ac

'0

2 co

mpl

ex ro

ots

Fin

d t

he

valu

e of

th

e d

iscr

imin

ant

for

each

equ

atio

n.T

hen

des

crib

eth

e n

um

ber

and

typ

es o

f ro

ots

for

the

equ

atio

n.

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

The

Qua

drat

ic F

orm

ula

and

the

Dis

crim

inan

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-5

6-5

Exam

ple

Exam

ple

a.2x

2#

5x#

3T

he d

iscr

imin

ant

is

b2$

4ac

!52

$4(

2)(3

) or

1.

The

dis

crim

inan

t is

a p

erfe

ct s

quar

e,so

the

equa

tion

has

2 r

atio

nal r

oots

.

b.3x

2"

2x#

5T

he d

iscr

imin

ant

is

b2$

4ac

!($

2)2

$4(

3)(5

) or

$56

.T

he d

iscr

imin

ant

is n

egat

ive,

so t

heeq

uati

on 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.D

escr

ibe

the

nu

mbe

r an

d t

ype

of r

oots

.c.

Fin

d t

he

exac

t so

luti

ons

by u

sin

g th

e Q

uad

rati

c F

orm

ula

.

1.p2

"12

p!

$4

128;

2.9x

2$

6x"

1 !

00;

3.2x

2$

7x$

4 !

081

;tw

o irr

atio

nal

root

s;on

e ra

tiona

l roo

t;2

ratio

nal r

oots

;",4

"6

'4 #

2$

4.x2

"4x

$4

!0

32;

5.5x

2$

36x

"7

!0

1156

;6.

4x2

$4x

"11

!0

2 irr

atio

nal r

oots

;2

ratio

nal r

oots

;"

160;

2 co

mpl

ex ro

ots;

"2

'2 #

2$,7

7.x2

$7x

"6

!0

25;

8.m

2$

8m!

$14

8;9.

25x2

$40

x!

$16

0;2

ratio

nal r

oots

;2

irrat

iona

l roo

ts;

1 ra

tiona

l roo

t;1,

64

'#

2$

10.4

x2"

20x

"29

!0

"64

;11

.6x2

"26

x"

8 !

048

4;12

.4x2

$4x

$11

!0

192;

2 co

mpl

ex ro

ots;

2 ra

tiona

l roo

ts;

2 irr

atio

nal r

oots

;4 $ 5

1 '

i#10$

$$ 2

1 $ 5

1 $ 21 $ 3


An

swer

s


Skil

ls P

ract

ice

The

Qua

drat

ic F

orm

ula

and

the

Dis

crim

inan

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

__PE

RIO

D__

___

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.D

escr

ibe

the

nu

mbe

r an

d t

ype

of r

oots

.c.

Fin

d t

he

exac

t so

luti

ons

by u

sin

g th

e Q

uad

rati

c F

orm

ula

.

1.x2

$8x

"16

!0

2.x2

$11

x$

26 !

0

0;1

ratio

nal r

oot;

422

5;2

ratio

nal r

oots

;"2,

13

3.3x

2$

2x!

04.

20x2

"7x

$3

!0

4;2

ratio

nal r

oots

;0,

289;

2 ra

tiona

l roo

ts;"

,

5.5x

2$

6 !

06.

x2$

6 !

0

120;

2 irr

atio

nal r

oots

;'24

;2 ir

ratio

nal r

oots

;'#

6$

7.x2

"8x

"13

!0

8.5x

2$

x$

1 !

0

12;2

irra

tiona

l roo

ts;"

4 '

#3$

21;2

irra

tiona

l roo

ts;

9.x2

$2x

$17

!0

10.x

2"

49 !

0

72;2

irra

tiona

l roo

ts;1

'3#

2$"

196;

2 co

mpl

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sin

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tse

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n be

est

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sing

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for

mul

a d(

t) !

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.If

a pa

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mps

from

an

airp

lane

and

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ls f

or 1

100

feet

bef

ore

open

ing

her

para

chut

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w m

any

seco

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e pa

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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.D

escr

ibe

the

nu

mbe

r an

d t

ype

of r

oots

.c.

Fin

d t

he

exac

t so

luti

ons

by u

sin

g th

e Q

uad

rati

c F

orm

ula

.

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6 !

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plex

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plex

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f yo

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ind

exa

ct s

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tion

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30. G

RA

VIT

ATI

ON

The

hei

ght

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in fe

et o

f an

obje

ct t

seco

nds

afte

r it

is p

rope

lled

stra

ight

up

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gro

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wit

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ial v

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0 fe

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eled

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at

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ight

of

56 f

eet?

1.75

s,2

s31

.STO

PPIN

G D

ISTA

NC

ET

he f

orm

ula

d!

0.05

s2"

1.1s

esti

mat

es t

he m

inim

um s

topp

ing

dist

ance

din

feet

for

a ca

r tr

avel

ing

sm

iles

per

hour

.If a

car

sto

ps in

200

feet

,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?

abou

t 53.

2 m

i/h

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Pra

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vera

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The

Qua

drat

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orm

ula

and

the

Dis

crim

inan

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

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___

6-5

6-5



Rea

din

g t

o L

earn

Math

emati

csTh

e Q

uadr

atic

For

mul

a an

d th

e D

iscr

imin

ant

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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 bl

ood

pre

ssu

re r

elat

ed t

o ag

e?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 6-

5 at

the

top

of

page

313

in y

our

text

book

.

Des

crib

e ho

w y

ou w

ould

cal

cula

te y

our

norm

al b

lood

pre

ssur

e us

ing

one

ofth

e fo

rmul

as in

you

r te

xtbo

ok.

Sam

ple

answ

er:S

ubst

itute

you

r age

for A

in th

e ap

prop

riate

form

ula

(for f

emal

es o

r mal

es) a

nd e

valu

ate

the

expr

essi

on.

Rea

din

g t

he

Less

on

1.a.

Wri

te t

he Q

uadr

atic

For

mul

a.x

!

b.Id

enti

fy t

he v

alue

s of

a,b

,and

cth

at y

ou w

ould

use

to

solv

e 2x

2$

5x!

$7,

but

dono

t ac

tual

ly s

olve

the

equ

atio

n.

a!

b!

c!

2.Su

ppos

e th

at y

ou a

re s

olvi

ng f

our

quad

rati

c eq

uati

ons

wit

h ra

tion

al c

oeff

icie

nts

and

have

fou

nd t

he v

alue

of

the

disc

rim

inan

t fo

r ea

ch e

quat

ion.

In e

ach

case

,giv

e th

enu

mbe

r of

roo

ts a

nd d

escr

ibe

the

type

of

root

s th

at t

he e

quat

ion

will

hav

e.

Valu

e of

Dis

crim

inan

tN

umbe

r of R

oots

Type

of R

oots

642

real

,rat

iona

l$

82

com

plex

212

real

,irr

atio

nal

01

real

,rat

iona

l

Hel

pin

g Y

ou

Rem

emb

er

3.H

ow c

an lo

okin

g at

the

Qua

drat

ic F

orm

ula

help

you

rem

embe

r th

e re

lati

onsh

ips

betw

een

the

valu

e of

the

dis

crim

inan

t an

d th

e nu

mbe

r of

roo

ts o

f a

quad

rati

c eq

uati

onan

d w

heth

er t

he r

oots

are

rea

l or

com

plex

?Sa

mpl

e an

swer

:The

dis

crim

inan

t is

the

expr

essi

on u

nder

the

radi

cal i

nth

e Q

uadr

atic

For

mul

a.Lo

ok a

t the

Qua

drat

ic F

orm

ula

and

cons

ider

wha

tha

ppen

s w

hen

you

take

the

prin

cipa

l squ

are

root

of b

2"

4ac

and

appl

y'

in fr

ont o

f the

resu

lt.If

b2"

4ac

is p

ositi

ve,i

ts p

rinci

pal s

quar

e ro

otw

ill b

e a

posi

tive

num

ber a

nd a

pply

ing

'w

ill g

ive

two

diffe

rent

real

solu

tions

,whi

ch m

ay b

e ra

tiona

l or i

rrat

iona

l.If

b2"

4ac

!0,

itspr

inci

pal s

quar

e ro

ot is

0,s

o ap

plyi

ng '

in th

e Q

uadr

atic

For

mul

a w

illon

ly le

ad to

one

sol

utio

n,w

hich

will

be

ratio

nal (

assu

min

g a,

b,an

d c

are

inte

gers

).If

b2"

4ac

is n

egat

ive,

sinc

e th

e sq

uare

root

s of

neg

ativ

enu

mbe

rs a

re n

ot re

al n

umbe

rs,y

ou w

ill g

et tw

o co

mpl

ex ro

ots,

corr

espo

ndin

g to

the

#an

d "

in th

e '

sym

bol.7

"5

2

"b

'#

b2"

4$

ac $$

$ 2a

©G

lenc

oe/M

cGra

w-H

ill34

2G

lenc

oe A

lgeb

ra 2

Sum

and

Pro

duct

of R

oots

So

met

imes

you

may

kno

w t

he r

oots

of

a qu

adra

tic

equa

tion

wit

hout

kno

win

g th

e eq

uati

onit

self.

Usi

ng y

our

know

ledg

e of

fac

tori

ng t

o so

lve

an e

quat

ion,

you

can

wor

k ba

ckw

ard

tofi

nd t

he q

uadr

atic

equ

atio

n.T

he r

ule

for

find

ing

the

sum

and

pro

duct

of

root

s is

as

follo

ws:

Sum

and

Pro

duct

of R

oots

If th

e ro

ots

of a

x2"

bx"

c!

0, w

ith a

≠0,

are

s1

and

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 g

rad

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.

Equ

atio

ns

of t

ran

siti

on c

urv

es a

re u

sed

by

civi

l en

gin

eers

to

des

ign

sm

ooth

an

d s

afe

road

s.

The

roo

ts a

re x

!3

and

x!

$8.

3 "

($8)

!$

5Ad

d th

e ro

ots.

3($

8) !

$24

Mul

tiply

the

root

s.

Equ

atio

n:x2

"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 !

0x2

"4x

"5

!0

x2"

12x

#36

!0

4.4

)#

3$6.

$%2 5% ,

%2 7%6.

x2"

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13 !

035

x2#

4x"

4 !

049

x2"

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#20

5 !

0

Fin

d k

such

th

at t

he

nu

mbe

r gi

ven

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, –3

01 – 4)

10 –10

–20

–30

24

–2–4

–6–8

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-5

6-5

Exam

ple

Exam

ple


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Ana

lyzi

ng G

raph

s of

Qua

drat

ic F

unct

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

grap

h of

y!

a(x

$h)

2"

kha

s th

e fo

llow

ing

char

acte

ristic

s:•

Verte

x: (h

, k)

Vert

ex F

orm

•Ax

is o

f sym

met

ry: x

!h

of a

Qua

drat

ic•

Ope

ns u

p if

a&

0Fu

nctio

n•

Ope

ns d

own

if a

'0

•N

arro

wer

than

the

grap

h of

y!

x2if a

&

1•

Wid

er th

an th

e gr

aph

of y

!x2

if 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

The

ver

tex

is a

t (h

,k)

or ($

4,$

11),

and

the

axis

of

sym

met

ry is

x!

$4.

The

gra

ph o

pens

up,a

nd is

nar

row

er t

han

the

grap

h of

y !

x2.

a.y

!"

(x"

2)2

#10

The

ver

tex

is a

t (h

,k)

or (

2,10

),an

d th

e ax

is o

f sy

mm

etry

is x

!2.

The

gra

ph o

pens

dow

n,an

d is

wid

er t

han

the

grap

h of

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;

up("

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;

dow

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;

up(3

,"2)

;x!

3;up

(1,"

2);x

!1;

dow

n

10.y

!$

(x"

5)2

"12

11.y

!(x

$7)

2"

2212

.y!

16(x

$4)

2"

1

("5,

12);

x!

"5;

dow

n(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;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

uadr

atic

fun

ctio

n is

eas

ier

togr

aph

whe

n it

is in

ver

tex

form

.You

can

wri

te a

qua

drat

ic f

unct

ion

of t

he f

orm

y

!ax

2"

bx"

cin

ver

tex

from

by

com

plet

ing

the

squa

re.

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) "

25y

!2(

x2$

6x"

9) "

25 $

18y

!2(

x$

3)2

"7

The

ver

tex

form

of

the

equa

tion

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

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Ana

lyzi

ng G

raph

s of

Qua

drat

ic F

unct

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-6

6-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Ana

lyzi

ng G

raph

s of

Qua

drat

ic F

unct

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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;up

(0,4

);x

!0;

dow

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;

dow

n(0

,9);

x!

0;do

wn

(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;

up("

3,"

7);x

!"

3;up

(4,4

8);x

!4;

dow

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

quat

ion

for

th

e p

arab

ola

wit

h t

he

give

n v

erte

x th

at p

asse

s th

rou

gh t

he

give

n p

oin

t.

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;

upx

!"

3;up

(1,2

);x

!1;

up7.

y!

$2x

2$

16x

$32

8.y

!$

3x2

"18

x$

219.

y!

2x2

"16

x"

29y

!"

2(x

#4)

2 ;y

!"

3(x

"3)

2#

6;y

!2(

x#

4)2

"3;

("4,

0);x

!"

4;do

wn

(3,6

);x

!3;

dow

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

quat

ion

for

th

e p

arab

ola

wit

h t

he

give

n v

erte

x th

at p

asse

s th

rou

gh t

he

give

n p

oin

t.

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

equa

tion

for

a p

arab

ola

wit

h ve

rtex

at

(4,4

) an

d x-

inte

rcep

t 6.

y!

"(x

"4)

2#

417

.Wri

te a

n eq

uati

on f

or a

par

abol

a w

ith

vert

ex a

t ($

3,$

1) a

nd y

-int

erce

pt 2

.y

!(x

#3)

2"

118

.BA

SEB

ALL

The

hei

ght

hof

a b

aseb

all t

seco

nds

afte

r be

ing

hit

is g

iven

by

h(t)

!$

16t2

"80

t"

3.W

hat

is t

he m

axim

um h

eigh

t th

at t

he b

aseb

all r

each

es,a

ndw

hen

does

thi

s oc

cur?

103

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

ound

and

rea

ches

a m

axim

um h

eigh

t of

10

feet

aft

er a

hori

zont

al d

ista

nce

of 4

fee

t.W

rite

a q

uadr

atic

fun

ctio

n in

ver

tex

form

that

des

crib

es t

he s

hape

of

the

outs

ide

of t

he a

rc,w

here

yis

the

hei

ght

of a

poi

nt o

n th

e ar

c an

dx

is it

s ho

rizo

ntal

dis

tanc

e fr

om t

he le

ft-h

and

star

ting

poi

nt o

f 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 (A

vera

ge)

Ana

lyzi

ng G

raph

s of

Qua

drat

ic F

unct

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-6

6-6


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csA

naly

zing

Gra

phs

of Q

uadr

atic

Equ

atio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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 u

sed

to

grap

h a

ny

quad

rati

cfu

nct

ion

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 6-

6 at

the

top

of

page

322

in y

our

text

book

.

•W

hat

does

add

ing

a po

siti

ve n

umbe

r to

x2

do t

o th

e gr

aph

of y

!x2

?It

mov

es th

e gr

aph

up.

•W

hat

does

sub

trac

ting

a p

osit

ive

num

ber

to x

befo

re s

quar

ing

do t

o th

egr

aph

of y

!x2

?It

mov

es th

e gr

aph

to th

e rig

ht.

Rea

din

g t

he

Less

on

1.C

ompl

ete

the

follo

win

g in

form

atio

n ab

out

the

grap

h of

y!

a(x

$h)

2"

k.

a.W

hat

are

the

coor

dina

tes

of t

he v

erte

x?(h

,k)

b.W

hat

is t

he e

quat

ion

of t

he a

xis

of s

ymm

etry

?x

!h

c.In

whi

ch d

irec

tion

doe

s th

e gr

aph

open

if a

&0?

If

a'

0?up

;dow

nd.

Wha

t do

you

kno

w a

bout

the

gra

ph if

a

'1?

It is

wid

er th

an th

e gr

aph

of y

!x2

.If

a

&1?

It is

nar

row

er th

an th

e gr

aph

of y

!x2

.

2.M

atch

eac

h gr

aph

wit

h th

e de

scri

ptio

n of

the

con

stan

ts in

the

equ

atio

n in

ver

tex

form

.

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 fu

ncti

ons

such

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!

x2to

the

left

and

whi

ch r

epre

sent

s a

tran

slat

ion

to t

he r

ight

.Wha

t is

an

easy

way

to

rem

embe

r th

is?

Sam

ple

answ

er:I

n fu

nctio

ns li

ke y

!(x

#4)

2 ,th

e pl

us s

ign

puts

the

grap

h “a

head

”so

that

the

vert

ex c

omes

“so

oner

”th

an th

e or

igin

and

the

tran

slat

ion

is to

the

left.

In fu

nctio

ns li

ke y

!(x

"5)

2 ,th

e m

inus

put

s th

egr

aph

“beh

ind”

so th

at th

e ve

rtex

com

es “

late

r”th

an th

e or

igin

and

the

tran

slat

ion

is to

the

right

.

x

y

Ox

y

Ox

y

Ox

y

O

©G

lenc

oe/M

cGra

w-H

ill34

8G

lenc

oe A

lgeb

ra 2

Patte

rns

with

Diff

eren

ces

and

Sum

s of

Squ

ares

Som

e w

hole

num

bers

can

be

wri

tten

as

the

diff

eren

ce o

f tw

o sq

uare

s an

dso

me

cann

ot.F

orm

ulas

can

be

deve

lope

d to

des

crib

e th

e se

ts o

f nu

mbe

rsal

gebr

aica

lly.

If p

ossi

ble,

wri

te e

ach

nu

mbe

r as

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

cann

ot4.

322

"12

5.4

22"

026.

532

"22

7.6

cann

ot8.

742

"32

9.8

32"

1210

.932

"02

11.1

0ca

nnot

12.1

162

"52

13.1

242

"22

14.1

372

"62

15.1

4ca

nnot

16.1

542

"12

Eve

n n

um

bers

can

be

wri

tten

as

2n,w

her

e n

is o

ne

of t

he

nu

mbe

rs

0,1,

2,3,

and

so

on.O

dd

nu

mbe

rs c

an b

e w

ritt

en 2

n#

1.U

se t

hes

e ex

pre

ssio

ns

for

thes

e p

robl

ems.

17.S

how

tha

t an

y od

d nu

mbe

r ca

n be

wri

tten

as

the

diff

eren

ce o

f tw

o sq

uare

s.2n

#1

!(n

#1)

2"

n2

18.S

how

tha

t th

e ev

en n

umbe

rs c

an b

e di

vide

d in

to t

wo

sets

:tho

se t

hat

can

be w

ritt

en in

the

for

m 4

nan

d th

ose

that

can

be

wri

tten

in t

he f

orm

2 "

4n.

Find

4n

for n

!0,

1,2,

and

so o

n.Yo

u ge

t {0,

4,8,

12,…

}.Fo

r 2 #

4n,y

ouge

t {2,

6,10

,12,

…}.

Toge

ther

thes

e se

ts in

clud

e al

l eve

n nu

mbe

rs.

19.D

escr

ibe

the

even

num

bers

tha

t ca

nnot

be

wri

tten

as

the

diff

eren

ce o

f tw

o sq

uare

s.2

#4n

,for

n!

0,1,

2,3,

…20

.Sho

w t

hat

the

othe

r ev

en n

umbe

rs c

an b

e w

ritt

en a

s th

e di

ffer

ence

of

two

squa

res.

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

mbe

rs f

rom

0 t

hro

ugh

15

by w

riti

ng

each

on

e as

th

e su

m o

f th

e le

ast

nu

mbe

r 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

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-6

6-6



Stu

dy

Gu

ide

and I

nte

rven

tion

Gra

phin

g an

d So

lvin

g Q

uadr

atic

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

sTo

gra

ph a

qua

drat

ic in

equa

lity

in t

wo

vari

able

s,us

eth

e fo

llow

ing

step

s:

1.G

raph

the

rel

ated

qua

drat

ic e

quat

ion,

y!

ax2

"bx

"c.

Use

a d

ashe

d lin

e fo

r '

or &

;use

a s

olid

line

for

*or

+.

2.Te

st a

poi

nt in

side

the

par

abol

a.If

it s

atis

fies

the

ineq

ualit

y,sh

ade

the

regi

on in

side

the

par

abol

a;ot

herw

ise,

shad

e th

e re

gion

out

side

the

par

abol

a.

Gra

ph

th

e in

equ

alit

y y

%x2

#6x

#7.

Fir

st g

raph

the

equ

atio

n y

!x2

"6x

"7.

By

com

plet

ing

the

squa

re,y

ou g

et t

he v

erte

x fo

rm o

f th

e eq

uati

on y

!(x

"3)

2$

2,so

the

ver

tex

is ($

3,$

2).M

ake

a ta

ble

of v

alue

s ar

ound

x!

$3,

and

grap

h.Si

nce

the

ineq

ualit

y in

clud

es &

,use

a d

ashe

d lin

e.Te

st t

he p

oint

($3,

0),w

hich

is in

side

the

par

abol

a.Si

nce

($3)

2"

6($

3) "

7 !

$2,

and

0 &

$2,

($3,

0) s

atis

fies

the

in

equa

lity.

The

refo

re,s

hade

the

reg

ion

insi

de t

he p

arab

ola.

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

Qua

drat

ic in

equa

litie

s in

one

var

iabl

e ca

n be

sol

ved

grap

hica

lly o

r al

gebr

aica

lly.

To s

olve

ax2

"bx

"c

'0:

Firs

t gra

ph y

!ax

2"

bx"

c. T

he s

olut

ion

cons

ists

of t

he x

-val

ues

Gra

phic

al M

etho

dfo

r whi

ch th

e gr

aph

is b

elow

the

x-ax

is.

To s

olve

ax2

"bx

"c

&0:

Firs

t gra

ph y

!ax

2"

bx"

c. T

he s

olut

ion

cons

ists

the

x-va

lues

fo

r whi

ch th

e gr

aph

is a

bove

the

x-ax

is.

Find

the

root

s of

the

rela

ted

quad

ratic

equ

atio

n by

fact

orin

g,

Alg

ebra

ic M

etho

dco

mpl

etin

g th

e sq

uare

, or u

sing

the

Qua

drat

ic F

orm

ula.

2 ro

ots

divi

de th

e nu

mbe

r lin

e in

to 3

inte

rval

s.Te

st a

val

ue in

eac

h in

terv

al to

see

whi

ch in

terv

als

are

solu

tions

.

If t

he in

equa

lity

invo

lves

*or

+,t

he r

oots

of

the

rela

ted

equa

tion

are

incl

uded

in t

heso

luti

on s

et.

Sol

ve t

he

ineq

ual

ity

x2"

x"

6 (

0.

Fir

st f

ind

the

root

s of

the

rel

ated

equ

atio

n x2

$x

$6

!0.

The

equa

tion

fac

tors

as

(x$

3)(x

"2)

!0,

so t

he r

oots

are

3 a

nd $

2.T

he g

raph

ope

ns u

p w

ith

x-in

terc

epts

3 a

nd $

2,so

it m

ust

be o

n or

bel

ow t

he x

-axi

s fo

r $

2 *

x*

3.T

here

fore

the

sol

utio

n se

t 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}

{x⏐1

&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}{x

⏐2 (

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

!x⏐x

&or

x%

4 ")

!x⏐&

x&

5 "1 $ 3

3 $ 2

1 $ 63 $ 4

1 $ 2

1 $ 2

x

y

O

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Gra

phin

g an

d So

lvin

g Q

uadr

atic

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-7

6-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Gra

phin

g an

d So

lvin

g Q

uadr

atic

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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 or

x%

4

Sol

ve e

ach

in

equ

alit

y al

gebr

aica

lly.

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

{x⏐x

!9}

{x⏐"

6 &

x&

6}

11.x

2$

7x&

012

.x2

"7x

"6

'0

{x⏐x

&0

or x

%7}

{x⏐"

6 &

x&

"1}

13.x

2"

x$

12 &

014

.x2

"9x

"18

*0

{x⏐x

&"

4 or

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

{x⏐x

&"

3 or

x%

0}{x

⏐x&

"2

or 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 or

x%

8"

3 (

x(

12

(x

(7

Sol

ve e

ach

in

equ

alit

y al

gebr

aica

lly.

7.x2

$x

$20

&0

8.x2

$10

x"

16 '

09.

x2"

4x"

5 *

0

{x⏐x

&"

4 or

x%

5}{x

⏐2 &

x&

8})

10.x

2"

14x

"49

+0

11.x

2$

5x&

1412

.$x2

$15

+8x

all r

eals

{x⏐x

&"

2 or

x%

7}{x

⏐"5

(x

("

3}

13.$

x2"

5x$

7 *

014

.9x2

"36

x"

36 *

015

.9x

*12

x2

all r

eals

{x⏐x

!"

2}!x⏐

x(

0 or

x*

"16

.4x2

"4x

"1

&0

17.5

x2"

10 +

27x

18.9

x2"

31x

"12

*0

!x⏐x

+"

"!x⏐

x(

or x

*5 "

!x⏐"

3 (

x(

""

19.F

ENC

ING

Vane

ssa

has

180

feet

of

fenc

ing

that

she

inte

nds

to u

se t

o bu

ild a

rec

tang

ular

play

are

a fo

r he

r do

g.Sh

e w

ants

the

pla

y ar

ea t

o en

clos

e at

leas

t 18

00 s

quar

e fe

et.W

hat

are

the

poss

ible

wid

ths

of t

he p

lay

area

?30

ft to

60

ft20

.BU

SIN

ESS

A b

icyc

le m

aker

sol

d 30

0 bi

cycl

es la

st y

ear

at a

pro

fit o

f $30

0 ea

ch.T

he m

aker

wan

ts t

o in

crea

se t

he p

rofi

t m

argi

n th

is y

ear,

but

pred

icts

tha

t ea

ch $

20 in

crea

se in

prof

it w

ill r

educ

e th

e nu

mbe

r of

bic

ycle

s so

ld b

y 10

.How

man

y $2

0 in

crea

ses

in p

rofit

can

the

mak

er a

dd in

and

exp

ect

to m

ake

a to

tal p

rofi

t of

at

leas

t $1

00,0

00?

from

5 to

10

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 (A

vera

ge)

Gra

phin

g an

d So

lvin

g Q

uadr

atic

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-7

6-7



Rea

din

g t

o L

earn

Math

emati

csG

raph

ing

and

Solv

ing

Qua

drat

ic In

equa

litie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

ucti

on t

o L

esso

n 6-

7 at

the

top

of

page

329

in y

our

text

book

.

•H

ow f

ar a

bove

the

gro

und

is t

he t

ram

polin

e su

rfac

e?3.

75 fe

et•

Usi

ng t

he q

uadr

atic

fun

ctio

n gi

ven

in t

he in

trod

ucti

on,w

rite

a q

uadr

atic

ineq

ualit

y th

at d

escr

ibes

the

tim

es a

t w

hich

the

tra

mpo

linis

t is

mor

eth

an 2

0 fe

et a

bove

the

gro

und.

"16

t2#

42t#

3.75

%20

Rea

din

g t

he

Less

on

1.A

nsw

er t

he f

ollo

win

g qu

esti

ons

abou

t ho

w y

ou w

ould

gra

ph t

he in

equa

lity

y+

x2"

x$

6.

a.W

hat

is t

he r

elat

ed q

uadr

atic

equ

atio

n?y

!x2

#x

"6

b.Sh

ould

the

par

abol

a be

sol

id o

r da

shed

? H

ow d

o yo

u kn

ow?

solid

;The

ineq

ualit

y sy

mbo

l is

*.

c.T

he p

oint

(0,

2) is

insi

de t

he p

arab

ola.

To u

se t

his

as a

tes

t po

int,

subs

titu

te

for

xan

d fo

r y

in t

he q

uadr

atic

ineq

ualit

y.

d.Is

the

sta

tem

ent

2 +

02"

0 $

6 tr

ue o

r fa

lse?

true

e.Sh

ould

the

reg

ion

insi

de o

r ou

tsid

e th

e pa

rabo

la b

e sh

aded

?in

side

2.T

he g

raph

of y

!$

x2"

4xis

sho

wn

at t

he r

ight

.Mat

ch e

ach

of t

he f

ollo

win

g re

late

d in

equa

litie

s w

ith

its

solu

tion

set

.

a.$

x2"

4x&

0ii

i.{xx

'0

or x

&4}

b.$

x2"

4x*

0iii

ii.{

x0

'x

'4}

c.$

x2"

4x+

0iv

iii.

{xx

*0

or x

+4}

d.$

x2"

4x'

0i

iv.

{x0

*x

*4}

Hel

pin

g Y

ou

Rem

emb

er

3.A

qua

drat

ic in

equa

lity

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

ineq

ualit

y sy

mbo

l and

wit

hout

usi

ng a

tes

t po

int.

Sam

ple

answ

er:I

f the

sym

bol i

s %

or *

,sha

de th

e re

gion

abo

ve th

epa

rabo

la.I

f the

sym

bol i

s &

or (

,sha

de th

e re

gion

bel

ow th

e pa

rabo

la.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

phin

g A

bsol

ute

Valu

e In

equa

litie

s Yo

u ca

n so

lve

abso

lute

val

ue in

equa

litie

s by

gra

phin

g in

muc

h th

e sa

me

man

ner

you

grap

hed

quad

rati

c in

equa

litie

s.G

raph

the

rel

ated

abs

olut

e fu

ncti

on

for

each

ineq

ualit

y by

usi

ng a

gra

phin

g ca

lcul

ator

.For

&an

d +

,ide

ntif

y th

e x-

valu

es,i

f an

y,fo

r w

hich

the

gra

ph li

es b

elow

the

x-ax

is.F

or '

and

*,i

dent

ify

the

xva

lues

,if

any,

for

whi

ch t

he g

raph

lies

abo

veth

e x-

axis

.

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 or

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

*"

5al

l rea

l num

bers

2 &

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

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

6-7

6-7

Documents

Chapter 6 Resource Masters - Math Class › uploads › 6 › ...©Glencoe/McGraw-Hill 314 Glencoe Algebra 2 Maximum and Minimum Values The y-coordinate of the vertex of a quadratic