Chapter 5 Resource Masters - No-IP

Chapter 5Resource Masters

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Copyright © by the McGraw-Hill Companies, Inc. All rights reserved. Permission is granted toreproduce the material contained herein on the condition that such material be reproduced only forclassroom use; be provided to students, teachers, and families without charge; and be used solelyin conjunction with Glencoe Algebra 2. Any other reproduction, for use or sale, is prohibited withoutprior written permission of the publisher.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240

ISBN13: 978-0-07-873975-0ISBN10: 0-07-873975-6 Algebra 2 CRM5

Printed in the United States of America

1 2 3 4 5 6 7 8 9 10 005 13 12 11 10 09 08 07 06

Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters areavailable as consumable workbooks in both English and Spanish.

ISBN10 ISBN13Study Guide and Intervention Workbook 0-07-877355-5 978-0-07-877355-6Skills Practice Workbook 0-07-877357-1 978-0-07-877357-0Practice Workbook 0-07-877358-X 978-0-07-877358-7Word Problem Practice Workbook 0-07-877360-1 978-0-07-877360-0

Spanish VersionsStudy Guide and Intervention Workbook 0-07-877356-3 978-0-07-877356-3Practice Workbook 0-07-877359-8 978-0-07-877359-4

Answers for Workbooks The answers for Chapter 5 of these workbooks can be found in the back ofthis Chapter Resource Masters booklet.

StudentWorks PlusTM This CD-ROM includes the entire Student Edition test along with the Englishworkbooks listed above.

TeacherWorks PlusTM All of the materials found in this booklet are included for viewing, printing, andediting in this CD-ROM.

Spanish Assessment Masters (ISBN10: 0-07-0-07-877361-X, ISBN13: 978-0-07-877361-7)These masters contain a Spanish version of Chapter 5 Test Form 2A and Form 2C.

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Chapter 5 iii Glencoe Algebra 2

Contents

Teacher's Guide to Using the Chapter 5Resource Masters . . . . . . . . . . . . . . . . . . . . .iv

Chapter Resources Student-Built Glossary . . . . . . . . . . . . . . . . . . .1Anticipation Guide (English) . . . . . . . . . . . . . . .3Anticipation Guide (Spanish) . . . . . . . . . . . . . . .4

Lesson 5-1Graphing Quadratic FunctionsLesson Reading Guide . . . . . . . . . . . . . . . . . . .5Study Guide and Intervention . . . . . . . . . . . . . .6Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . . .8Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9Word Problem Practice . . . . . . . . . . . . . . . . . .10Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . .11

Lesson 5-2Solving Quadratic Equations by GraphingLesson Reading Guide . . . . . . . . . . . . . . . . . .12Study Guide and Intervention . . . . . . . . . . . . .13Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . .15Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16Word Problem Practice . . . . . . . . . . . . . . . . . .17Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . .18

Lesson 5-3Solving Quadratic Equations by FactoringLesson Reading Guide . . . . . . . . . . . . . . . . . .19Study Guide and Intervention . . . . . . . . . . . . .20Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . .22Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23Word Problem Practice . . . . . . . . . . . . . . . . . .24Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . .25Graphing Calculator . . . . . . . . . . . . . . . . . . . .26

Lesson 5-4Complex NumbersLesson Reading Guide . . . . . . . . . . . . . . . . . .27Study Guide and Intervention . . . . . . . . . . . . .28Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . .30Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31Word Problem Practice . . . . . . . . . . . . . . . . . .32Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . .33

Lesson 5-5Completing the SquareLesson Reading Guide . . . . . . . . . . . . . . . . . .34Study Guide and Intervention . . . . . . . . . . . . .35Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . .37Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38

Word Problem Practice . . . . . . . . . . . . . . . . . .39Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . .40

Lesson 5-6The Quadratic Formula and the DiscriminantLesson Reading Guide . . . . . . . . . . . . . . . . . .41Study Guide and Intervention . . . . . . . . . . . . .42Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . .44Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45Word Problem Practice . . . . . . . . . . . . . . . . . .46Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . .47Spreadsheet . . . . . . . . . . . . . . . . . . . . . . . . . .48

Lesson 5-7Analyzing Graphs of Quadratic FunctionsLesson Reading Guide . . . . . . . . . . . . . . . . . .49Study Guide and Intervention . . . . . . . . . . . . .50Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . .52Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53Word Problem Practice . . . . . . . . . . . . . . . . . .54Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . .55

Lesson 5-8Graphing and Solving Quadratic InequalitiesLesson Reading Guide . . . . . . . . . . . . . . . . . .56Study Guide and Intervention . . . . . . . . . . . . .57Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . .59Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60Word Problem Practice . . . . . . . . . . . . . . . . . .61Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . .62Graphing Calculator . . . . . . . . . . . . . . . . . . . .63

AssessmentStudent Recording Sheet . . . . . . . . . . . . . . . .65rubric for Pre-AP . . . . . . . . . . . . . . . . . . . . . . .66Chapter 5 Quizzes 1 and 2 . . . . . . . . . . . . . . .67Chapter 5 Quizzes 3 and 4 . . . . . . . . . . . . . . .68Chapter 5 Mid-Chapter Test . . . . . . . . . . . . . .69Chapter 5 Vocabulary Test . . . . . . . . . . . . . . .70Chapter 5 Test, Form 1 . . . . . . . . . . . . . . . . . .71Chapter 5 Test, Form 2A . . . . . . . . . . . . . . . . .73Chapter 5 Test, Form 2B . . . . . . . . . . . . . . . . .75Chapter 5 Test, Form 2C . . . . . . . . . . . . . . . .77Chapter 5 Test, Form 2D . . . . . . . . . . . . . . . .79Chapter 5 Test, Form 3 . . . . . . . . . . . . . . . . . .81Chapter 5 Extended Response Test . . . . . . . .83Standardized Test Practice . . . . . . . . . . . . . . .84

Answers . . . . . . . . . . . . . . . . . . . . . .A1–A40

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Chapter 5 iv Glencoe Algebra 2

Teacher’s Guide to Using the Chapter 5 Resource Masters

The Chapter 5 Resource Masters includes the core materials needed for Chapter 5.These materials include worksheets, extensions, and assessment options. Theanswers for these pages appear at the back of this booklet.

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

Chapter ResourcesStudent-Built Glossary (pages 1–2)These masters are a student study tool thatpresents up to twenty of the key vocabularyterms from the chapter. Students are torecord definitions and/or examples for eachterm. You may suggest that studentshighlight or star the terms with which theyare not familiar. Give this to studentsbefore beginning Lesson 5-1. Encouragethem to add these pages to theirmathematics study notebooks. Remindthem to complete the appropriate words asthey study each lesson.

Anticipation Guide (pages 3–4) Thismaster, presented in both English andSpanish, is a survey used before beginningthe chapter to pinpoint what students mayor may not know about the concepts in thechapter. Students will revisit this surveyafter they complete the chapter to see iftheir perceptions have changed.

Lesson ResourcesLesson Reading Guide Get Ready for theLesson extends the discussion from thebeginning of the Student Edition lesson.Read the Lesson asks students to interpretthe context of and relationships amongterms in the lesson. Finally, RememberWhat You Learned asks students tosummarize what they have learned usingvarious representation techniques. Use as astudy tool for note taking or as an informalreading assignment. It is also a helpfultool for ELL (English Language Learners).

Study Guide and Intervention Thesemasters provide vocabulary, key concepts,additional worked-out examples and Check Your Progress exercises to use as areteaching activity. It can also be used inconjunction with the Student Edition as aninstructional tool for students who havebeen absent.

Skills Practice This master focuses moreon the computational nature of the lesson.Use as an additional practice option or ashomework for second-day teaching of thelesson.

Practice This master closely follows thetypes of problems found in the Exercisessection of the Student Edition and includesword problems. Use as an additionalpractice option or as homework for second-day teaching of the lesson.

Word Problem Practice This masterincludes additional practice in solving wordproblems that apply the concepts of thelesson. Use as an additional practice or ashomework for second-day teaching of thelesson.

Enrichment These activities may extendthe concepts of the lesson, offer an historicalor multicultural look at the concepts, orwiden students’ perspectives on themathematics they are learning. They arewritten for use with all levels of students.

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Chapter 5 v Glencoe Algebra 2

Graphing Calculator, ScientificCalculator, or Spreadsheet ActivitiesThese activities present ways in whichtechnology can be used with the concepts in some lessons of this chapter. Use as analternative approach to some concepts or as an integral part of your lessonpresentation.

Assessment OptionsThe assessment masters in the Chapter 5Resource Masters offer a wide range ofassessment tools for formative (monitoring)assessment and summative (final)assessment.

Student Recording Sheet This mastercorresponds with the standardized testpractice at the end of the chapter.

Pre-AP Rubric This master providesinformation for teachers and students onhow to assess performance on open-endedquestions.

Quizzes Four free-response quizzes offerassessment at appropriate intervals in thechapter.

Mid-Chapter Test This 1-page testprovides an option to assess the first half ofthe chapter. It parallels the timing of theMid-Chapter Quiz in the Student Editionand includes both multiple-choice and free-response questions.

Vocabulary Test This test is suitable forall students. It includes a list of vocabularywords and 10 questions to assess students’knowledge of those words. This can also beused in conjunction with one of the leveledchapter tests.

Leveled Chapter Tests

• Form 1 contains multiple-choicequestions and is intended for use withbelow grade level students.

• Forms 2A and 2B contain multiple-choicequestions aimed at on grade levelstudents. These tests are similar informat to offer comparable testingsituations.

• Forms 2C and 2D contain free-responsequestions aimed at on grade levelstudents. These tests are similar informat to offer comparable testingsituations.

• Form 3 is a free-response test for usewith above grade level students.

All of the above mentioned tests include afree-response Bonus question.

Extended-Response Test Performanceassessment tasks are suitable for allstudents. Sample answers and a scoringrubric are included for evaluation.

Standardized Test Practice These threepages are cumulative in nature. It includesthree parts: multiple-choice questions with bubble-in answer format, griddablequestions with answer grids, and short-answer free-response questions.

Answers• The answers for the Anticipation Guide

and Lesson Resources are provided asreduced pages with answers appearing in red.

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

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Chap

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sThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 5.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

axis of symmetry

completing the square

complex conjugates

complex number

constant term

discriminant

dihs·KRIH·muh·nuhnt

imaginary unit

linear term

maximum value

(continued on the next page)

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Chapter 5 1 Glencoe Algebra 2

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Vocabulary Term Found on Page Definition/Description/Example

minimum value

parabola

puh·RA·buh·luh

pure imaginary number

quadratic equation

kwah·DRA·tihk

Quadratic

quadratic function

quadratic inequality

quadratic term

root

square root

vertex

vertex form

zero

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STEP 1


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Anticipation GuideQuadratic Functions and Inequalities

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Before you begin Chapter 5

• Read each statement.

• Decide whether you Agree (A) or Disagree (D) with the statement.

• Write A or D in the first column OR if you are not sure whether you agree or disagree,write NS (Not Sure).

After you complete Chapter 5

• Reread each statement and complete the last column by entering an A or a D.

• Did any of your opinions about the statements change from the first column?

• For those statements that you mark with a D, use a piece of paper to write an example ofwhy you disagree.

STEP 2

STEP 1 STEP 2A, D, or NS

StatementA or D

1. All quadratic functions have a term with the variable to the second power.

2. If the graph of the quadratic function y � ax2 � c opens up then c � 0.

3. A quadratic equation whose graph does not intersect the x-axis has no real solution.

4. Since graphing shows the exact solutions to a quadratic equation, no other method is necessary for solving.

5. If (x � 3)(x � 4) � 0, then either x � 3 � 0 or x � 4 � 0.

6. An imaginary number contains i, which equals the square root of �1.

7. A method called completing the square can be used to rewrite a quadratic expression as a perfect square.

8. The quadratic formula can only be used for quadratic equations that cannot be solved by graphing or completing the square.

9. The discriminant of a quadratic equation can be used to determine the direction the graph will open.

10. The graph of y � 2x2 is a dilation of the graph of y � x 2.

11. The graph of y � (x � 2)2 will be two units to the right of the graph of y � x2.

12. The graph of a quadratic inequality containing the symbol � will be a parabola opening downward.

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NOMBRE ______________________________________ FECHA ____________ PERÍODO ___

Ejercicios preparatoriosFunciones cuadráticas y desigualdades

Capítulo 5 4 Álgebra 2 de Glencoe

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PASO 1 Antes de comenzar el Capítulo 5

• Lee cada enunciado.

• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.

• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta,escribe NS (No estoy seguro(a).

Después de completar el Capítulo 5

• Vuelve a leer cada enunciado y completa la última columna con una A o una D.

• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?

• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con losenunciados que marcaste con una D.

PASO 2

PASO 1Enunciado

PASO 2A, D o NS A o D

1. Todas las funciones cuadráticas tienen un término con la variable elevada a la segunda potencia.

2. Si la gráfica de la función cuadrática y � ax2 � c se abre hacia arriba, entonces c � 0.

3. Una ecuación cuadrática cuya gráfica no interseca el eje x no tiene solución real.

4. Dado que graficar muestra las soluciones exactas de una ecuación cuadrática, no se necesita ningún otro método para resolverla.

5. Si (x � 3)(x � 4) � 0, entonces x � 3 � 0 ó x � 4 � 0.

6. Un número imaginario contiene i, la cual es igual a la raíz cuadrada de �1.

7. El método de completar el cuadrado se puede usar para volver a plantear una expresión cuadrática como un cuadrado perfecto.

8. La fórmula cuadrática se puede usar sólo para ecuaciones cuadráticas que no pueden resolverse mediante la completación del cuadrado o una gráfica.

9. Se puede usar el discriminante de una ecuación cuadrática para determinar la dirección en que se abrirá la gráfica.

10. La gráfica de y � 2x2 es una dilatación de la gráfica de y � x 2.

11. La gráfica de y � (x � 2)2 estará dos unidades a la derecha de la gráfica de y � x2.

12. La gráfica de una desigualdad cuadrática con el símbolo �será una parábola que se abre hacia abajo.

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Get Ready for the LessonRead the introduction to Lesson 5-1 in your textbook.• Based on the graph in your textbook, for what ticket price is the income the greatest?

• Use the graph to estimate the maximum income.

Read 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 .

Remember What You Learned4. 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?

x

f(x)

O(0, –1)

(–2, 4)

Lesson Reading GuideGraphing Quadratic Functions

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Exercises

Example

Study Guide and InterventionGraphing Quadratic Functions

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

x

f(x)

O

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8

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4 8–4

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O

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–8

4 8–8 –4

x

f(x)

O 4–4

4

8

–8

12

–4

11�4

3�2

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3�2

3�2

3�2

x

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�b�2a

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Exercises

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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, and find the maximum or minimum value of each function. Then state thedomain and range of the function.

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. Thedomain is all real numbers. The range isall reals greater than or equal to theminimum value, that is {f(x) | f(x) � 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.The domain is all real numbers. Therange is all reals less than or equal to themaximum value, that is {f(x) | f(x) 101}.

�2�2(�1)

�b�2a

Determine whether each function has a maximum or minimum value, and find themaximum or minimum value. Then state the domain and range of the function.

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

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

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

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

Study Guide and Intervention (continued)

Graphing Quadratic Functions

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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 � 15

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

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 � 8

Determine whether each function has a maximum or a minimum value, and findthe maximum or minimum value. Then state the domain and range of the function.

10. f(x) � 6x2 11. f(x) � �8x2 12. f(x) � x2 � 2x

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

16. f(x) � �2x2 � 4x � 3 17. f(x) � 3x2 � 12x � 3 18. f(x) � 2x2 � 4x � 1

x

f(x)

Ox

f(x)

O

16

12

8

4

2–2 4 6

x

f(x)

O

5-1 Skills PracticeGraphing Quadratic Functions

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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 � 1

Determine whether each function has a maximum or a minimum value, and findthe maximum or minimum value of each function. Then state the domain andrange of the function.

4. f(x) � x2 � 2x � 8 5. f(x) � x2 � 6x � 14 6. v(x) � �x2 � 14x � 57

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

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.

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?

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

f(x)

xO

16

12

8

4

x

f(x)

O 2–2–4–6x

f(x)

O

16

12

8

4

2 4 6 8

PracticeGraphing Quadratic Functions

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Word Problem PracticeGraphing Quadratic Functions

1. TRAJECTORIES A cannonball islaunched from a cannon at the top of acliff. If the path of the cannonball istraced on apiece of graphpaper aligned sothat the cannonis situated on the y-axis, theequation thatdescribes the path is

y � ��16

100�x2 � �

12

�x � 47,

where x is the horizontal distance fromthe cliff and y is the vertical distanceabove the ground in feet. How highabove the ground is the cannon?

2. TICKETING The manager of asymphony computes that the symphonywill earn �40P2 � 1100P dollars perconcert if they charge P dollars fortickets. What ticket price should thesymphony charge in order to maximizeits profits?

3. ARCHES An architect decides to use aparabolic arch for the main entrance of ascience museum. In one of his plans, thetop edge of the arch is described by the

graph of y � � x2 � x � . What

are the coordinates of the vertex of thisparabola?

4. FRAMING A frame company offers aline of square frames. If the side lengthof the frame is s, then the area of theopening in the frame is given by thefunction a(s) � s2 � 10s � 24.Graph a(s).

WALKING For Exercises 5–7, use thefollowing information.Canal Street and Walker Street areperpendicular to each other. Evita is drivingsouth on Canal Street and is currently 5miles north of the intersection with WalkerStreet. Jack is at the intersection of Canaland Walker Streets and heading east onWalker. Jack and Evita are both driving 30miles per hour.

5. When Jack is x miles east of theintersection, where is Evita?

6. The distance between Jack and Evita isgiven by the formula �x2 � (�5 � x)�2�. Forwhat value of x are Jack and Evita attheir closest? (Hint: Minimize the squareof the distance.)

7. What is the distance of closestapproach?

a

sO

5

5

75�4

5�2

1�4

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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

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NAME ______________________________________________ DATE______________ PERIOD _____

Get Ready for the LessonRead the introduction to Lesson 5-2 in your textbook.

Write a quadratic function that describes the height of a ball t seconds after it is dropped from a height of 125 feet.

Read 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. 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.

Remember What You Learned

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

x

y

Ox

y

Ox

y

O

x

y

O

Lesson Reading GuideSolving Quadratic Equations by Graphing

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Study Guide and InterventionSolving Quadratic Equations by Graphing

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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. x2 � 4x � 5 � 0 3. x2 � 5x � 4 � 0

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

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

Example

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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

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

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

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Skills PracticeSolving Quadratic Equations By Graphing

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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

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 � 0

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.

x

f(x)

O 6–6 12–12

36

24

12x

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

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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

Solve 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 � 0

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?

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?

xf(x)

O

x

f(x)

O

x

f(x)

O

x

f(x)

Ox

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

PracticeSolving Quadratic Equations By Graphing

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Word Problem PracticeSolving Quadratic Equations by Graphing

1. TRAJECTORIES David threw a baseballinto the air. The function of the height ofthe baseball in feet is h � 80t � 16t2,where t represents the time in secondsafter the ball was thrown. Use thisgraph of the function to determine howlong it took for the ball to fall back tothe ground.

2. BRIDGES The main support for abridge is a large parabolic arch. Theheight of the arch above the ground is

given by the function h � 32 � x2,

where h is the height in meters and x isthe distance in meters from the center ofthe bridge. Graph this equation anddescribe where the arch touches theground.

3. LOGIC Wilma is thinking of twonumbers. The sum is 2 and the productis �24. Use a quadratic equation to findthe two numbers.

4. RADIO TELESCOPES The cross-sectionof a large radio telescope is a parabola.The dish is set into the ground. Theequation that describes the cross-section

is d � x2 � x � , where d gives

the depth of the dish below ground andx is the distance from the control center,both in meters. If the dish does notextend above the ground level, what isthe diameter of the dish? Solve bygraphing.

BOATS For Exercises 5 and 6, use thefollowing information.The distance between two boats is

d � �t2 � 1�0t � 3�5�,

where d is distance in meters and t is timein seconds.

5. Make a graph of d2 versus t.

6. Do the boats ever collide?

d

tO 5

y

xO 4010

�6

32�3

4�3

2�75

h

xO

20

�20 20

1�50

a

tO 1 2 3 4 5�1

�40

40

80

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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

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

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

10. Explain how solving absolute value equations algebraically and finding zeros of absolute value functions graphically are related.

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Lesson Reading GuideSolving Quadratic Equations by Factoring


Write two different quadratic equations in intercept form that have corresponding graphswith the same x-intercepts.

Read 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

(x � 3)(x � 7) � 0

x � 3 � 0 or x � 7 � 0

x � 3 x � 7

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?

Explain the errors in the other two students’ work.


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.

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NAME ______________________________________________ DATE______________ PERIOD _____

Study Guide and InterventionSolving Quadratic Equations by Factoring

Solve Equations by Factoring When you use factoring to solve a quadraticequation, 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.a. 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

Solve each equation by factoring.

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

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

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

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

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

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

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

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

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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 theequation in standard form.


Solving Quadratic Equations by Factoring

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

Write a quadratic equation with the given roots. Write the equation in standard form.

1. 3, �4 2. �8, �2 3. 1, 9

4. �5 5. 10, 7 6. �2, 15

7. � , 5 8. 2, 9. �7,

10. 3, 11. � , �1 12. 9,

13. , � 14. , � 15. ,

16. � , 17. , 18. , 1�6

1�8

3�4

1�2

7�2

7�8

1�5

3�7

1�2

5�4

2�3

2�3

1�6

4�9

2�5

3�4

2�3

1�3

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Skills PracticeSolving Quadratic Equations by Factoring


1. 1, 4 2. 6, �9

3. �2, �5 4. 0, 7

5. � , �3 6. � ,

Factor each polynomial.

7. m2 � 7m � 18 8. 2x2 � 3x � 5

9. 4z2 � 4z � 15 10. 4p2 � 4p � 24

11. 3y2 � 21y � 36 12. c2 � 100


13. x2 � 64 14. x2 � 100 � 0

15. x2 � 3x � 2 � 0 16. x2 � 4x � 3 � 0

17. x2 � 2x � 3 � 0 18. x2 � 3x � 10 � 0

19. x2 � 6x � 5 � 0 20. x2 � 9x � 0

21. x2 � 4x � 21 22. 2x2 � 5x � 3 � 0

23 4x2 � 5x � 6 � 0 24. 3x2 � 13x � 10 � 0

25. NUMBER THEORY Find two consecutive integers whose product is 272.

3�4

1�2

1�3

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1. 7, 2 2. 0, 3 3. �5, 8

4. �7, �8 5. �6, �3 6. 3, �4

7. 1, 8. , 2 9. 0, �


10. r3 � 3r2 � 54r 11. 8a2 � 2a � 6 12. c2 � 49

13. x3 � 8 14. 16r2 � 169 15. b4 � 81


16. x2 � 4x � 12 � 0 17. x2 � 16x � 64 � 0

18. x2 � 6x � 8 � 0 19. x2 � 3x � 2 � 0

20. x2 � 4x � 0 21. 7x2 � 4x

22. 10x2 � 9x 23. x2 � 2x � 99

24. x2 � 12x � �36 25. 5x2 � 35x � 60 � 0

26. 36x2 � 25 27. 2x2 � 8x � 90 � 0

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

29. NUMBER THEORY Find two consecutive odd positive integers whose product is 323.

30. GEOMETRY The length of a rectangle is 2 feet more than its width. Find thedimensions of the rectangle if its area is 63 square feet.

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

7�2

1�3

1�2

PracticeSolving Quadratic Equations by Factoring

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Word Problem PracticeSolving Quadratic Equations by Factoring

1. FLASHLIGHTS When Dora shines herflashlight on the wall at a certain angle,the edge of the lit area is in the shape ofa parabola. The equation of the parabolais y � 2x2 � 2x � 60. Factor thisquadratic equation.

2. SIGNS David was looking through anold algebra book and came across thisequation.

x2 6x � 8 � 0

The sign in front of the 6 was blottedout. How does the missing sign dependon the signs of the roots?

3. ROOTS In the same algebra book thathe was looking through in Exercise 2,David found another partially blottedout equation.

x2 � 21x � 100 � 0

The book claims that one of the roots ofthe equation is 4. What must the otherroot be and what number is covered bythe blot?

4. PROGRAMMING Ray is a computerprogrammer. He needs to find thequadratic function of this graph for analgorithm related to a game involvingdice. Provide such a function.

ANIMATION For Exercises 5–7, use thefollowing information.A computer graphics animator would like tomake a realistic simulation of tossed ball.The animator wants the ball to follow theparabolic trajectory represented by thequadratic equation f(x) � �0.2(x � 5)(x � 5).

5. What are the solutions of f(x) � 0?

6. Write f(x) in standard form.

7. If the animator changes the equation tof(x) � �0.2x2 � 20, what are thesolutions of f(x) � 0?

y

x

O

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Using Patterns to FactorStudy the patterns below for factoring the sum and the difference of cubes.

a3 � b3 � (a � b)(a2 � ab � b2)a3 � b3 � (a � b)(a2 � ab � b2)

This pattern can be extended to other odd powers. Study these examples.

Factor a5 � b5.Extend the first pattern to obtain a5 � b5 � (a � b)(a4 � a3b � a2b2 � ab3 � b4).Check: (a � b)(a4 � a3b � a2b2 � ab3 � b4) � a5 � a4b � a3b2 � a2b3 � ab4

� a4b � a3b2 � a2b3 � ab4 � b5

� a5 � b5

Factor a5 � b5.Extend the second pattern to obtain a5 � b5 � (a � b)(a4 � a3b � a2b2 � ab3 � b4).Check: (a � b)(a4 � a3b � a2b2 � ab3 � b4) � a5 � a4b � a3b2 � a2b3 � ab4

� a4b � a3b2 � a2b3 � ab4 � b5

� a5 � b5

In general, if n is an odd integer, when you factor an � bn or an � bn, one factor will beeither (a � b) or (a � b), depending on the sign of the original expression. The other factorwill have the following properties:

• The first term will be an � 1 and the last term will be bn � 1.• The exponents of a will decrease by 1 as you go from left to right.• The exponents of b will increase by 1 as you go from left to right.• The degree of each term will be n � 1.• If the original expression was an � bn, the terms will alternately have � and � signs.• If the original expression was an � bn, the terms will all have � signs.

Use the patterns above to factor each expression.

1. a7 � b7

2. c9 � d9

3. e11 � f 11

To factor x10 � y10, change it to (x5 � y5)(x5 � y5) and factor each binomial. Usethis approach to factor each expression.

4. x10 � y10

5. a14 � b14

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Graphing Calculator ActivityUsing Tables to Factor by Grouping

The TABLE feature of a graphing calculator can be used to help factor apolynomial of the form ax2 � bx � c.

Factor 10x2 � 43x � 28 by grouping.

Make a table of the negative factors of 10 28 or 280. Look for a pairof factors whose sum is �43.

Enter the equation y � �28

x0

� in Y1 to find the factors of 280. Then,

find the sum of the factors using y � �28

x0

� � x in Y2. Set up the table

to display the negative factors of 280 by setting �Tbl = to �1.Examine the results.

Keystrokes: 280 [TBLSET] 1 1

[TABLE].

The last line of the table shows that �43x may be replaced with �8x +(�35x).

10x2 � 43x � 28 � 10x2 � 8x � (�35x) � 28� 2x(5x � 4) � (�7)(5x � 4)� (5x � 4)(2x � 7)

Thus, 10x2 � 43x � 28 � (5x � 4)(2x � 7).

2ndENTER(–)ENTER(–)2ndENTER

+ENTERENTERVARSENTERY=


1. y2 � 20y � 96 2. 4z2 � 33z � 35 3. 4y2 � y �18 4. 6a2 � 2a � 15

5. 6m2 � 17m � 12 6. 24z2 � 46z � 15 7. 36y2 � 84y � 49 8. 4b2 � 36b � 403

Factor 12x2 � 7x � 12.

Look at the factors of 12 �12 or �144 for a pair whose sum is �7.Enter an equation to determine the factors in Y1 and an equation tofind the sum of factors in Y2. Examine the table to find a sum of �7.Keystrokes: 144

[TBLSET] 1 1 [TABLE].

12x2 � 7x � 12 � 12x2 � 9x � (�16x) � 12� 3x(4x � 3) � 4(4x � 3)� (4x � 3)(3x � 4)

Thus, 12x2 � 7x � 12 � (4x � 3)(3x � 4).

2ndENTERENTER2ndENTER+

ENTERENTERVARSENTER(–)Y=

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Lesson Reading GuideComplex Numbers


Suppose the number i is defined such that i2 � �1. Complete each equation.

2i2 � (2i)2 � i4 �

Read the Lesson

1. Complete each statement.

a. The form a � bi is called the of a complex number.

b. In the complex number 4 � 5i, the real part is and the imaginary part is .

This is an example of a complex number that is also a(n) number.

c. In the complex number 3, the real part is and the imaginary part is .

This is example of complex number that is also a(n) number.

d. In the complex number 7i, the real part is and the imaginary part is .

This is an example of a complex number that is also a(n) number.

2. Give the complex conjugate of each number.

a. 3 � 7i

b. 2 � i

3. Why are complex conjugates used in dividing complex numbers?

4. Explain how you would use complex conjugates to find (3 � 7i) � (2 � i).


5. How can you use what you know about simplifying an expression such as to

help you remember how to simplify fractions with imaginary numbers in thedenominator?

1 � �3��2 � �5�

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Study Guide and InterventionComplex Numbers

SQUARE ROOTS A square root of a number n is a number whose square is n. For

nonnegative real numbers a and b, �ab� � �a� �b� and � , b � 0. The imaginary

unit i is defined to have the property that i2 � �1. Simplified square root expressions donot have radicals in the denominator, and any number remaining under the square root hasno perfect square factor other than 1.

�a��b�

a�b

a. Simplify �48�.�48� � �16 3�

� �16� �3�� 4�3�

b. Simplify ��63�.��63� � ��1 7� 9�

� ��1� �7� �9�� 3i�7�

a. Simplify �125x2�y5�.�125x2�y5� � �5 25�x2y4y�

� �25� �5� �x2� �y4� �y�� 5xy2 �5y�

b. Simplify ��44x�6�.��44x�6� � ��1 4� 11 � x6�

� ��1� �4� �11� �x6�� 2i�11�x3

Solve x2 � 5 � 0.x2 � 5 � 0 Original equation.

x2 � �5 Subtract 5 from each side.

x � ��5�i Square Root Property.

Simplify.

1. �72� 2. ��24�

3. 4. �75x4y�7�

5. ��84� 6. ��32x�y4�

Solve each equation.

7. 5x2 � 45 � 0 8. 4x2 � 24 � 0

9. �9x2 � 9 10. 7x2 � 84 � 0

128�147

Example 2

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Complex Numbers

Simplify.

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

4. (�11 � 4i) � (1 � 5i) 5. (8 � 4i) � (8 � 4i) 6. (5 � 2i) � (�6 � 3i)

7. (2 � i)(3 � i) 8. (5 � 2i)(4 � i) 9. (4 � 2i)(1 � 2i)

10. 11. 12.6 � 5i�3i

7 � 13i�2i

5�3 � i

Operations with Complex Numbers

A complex number is any number that can be written in the form a � bi, Complex Number where a and b are real numbers and i is the imaginary unit (i 2 � �1).

a is called the real part, and b is called the imaginary part.

Addition and Combine like terms.Subtraction of (a � bi) � (c � di) � (a � c) � (b � d )iComplex Numbers (a � bi) � (c � di) � (a � c) � (b � d )i

Multiplication of Use the definition of i2 and the FOIL method:Complex Numbers (a � bi)(c � di) � (ac � bd ) � (ad � bc)i

Complex Conjugatea � bi and a � bi are complex conjugates. The product of complex conjugates is always a real number.

To divide by a complex number, first multiply the dividend and divisor by the complexconjugate of the divisor.

Simplify (6 � i) � (4 � 5i).

(6 � i) � (4 � 5i)� (6 � 4) � (1 � 5)i� 10 � 4i

Simplify (2 � 5i) (�4 � 2i).

(2 � 5i) (�4 � 2i)� 2(�4) � 2(2i) � (�5i)(�4) � (�5i)(2i)� �8 � 4i � 20i � 10i2

� �8 � 24i � 10(�1)� 2 � 24i

Simplify (8 � 3i) � (6 � 2i).

(8 � 3i) � (6 � 2i)� (8 � 6) � [3 � (�2)]i� 2 � 5i

Simplify .

�

�

�

� � i11�13

3�13

3 � 11i�13

6 � 9i � 2i � 3i2��

4 � 9i2

2 � 3i�2 � 3i

3 � i�2 � 3i

3 � i�2 � 3i

3 � i�2 � 3i

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Skills PracticeComplex Numbers

Simplify.

1. �99� 2. 3. �52x3y�5� 4. ��108�x7�

5. ��81x6� 6. ��23� ��46�

7. (3i)(�2i)(5i) 8. i11

9. i65 10. (7 � 8i) � (�12 � 4i)

11. (�3 � 5i) � (18 � 7i) 12. (10 � 4i) � (7 � 3i)

13. (7 � 6i)(2 � 3i) 14. (3 � 4i)(3 � 4i)

15. 16.


17. 3x2 � 3 � 0 18. 5x2 � 125 � 0

19. 4x2 � 20 � 0 20. �x2 � 16 � 0

21. x2 � 18 � 0 22. 8x2 � 96 � 0

Find the values of m and n that make each equation true.

23. 20 � 12i � 5m � 4ni 24. m � 16i � 3 � 2ni

25. (4 � m) � 2ni � 9 � 14i 26. (3 � n) � (7m � 14)i � 1 � 7i

3i�4 � 2i

8 � 6i�3i

27�49

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Simplify.

1. ��36a�3b4� 2. ��8� ��32� 3. ��15� ��25�

4. 5. 6. 7. (�3i)(4i)(�5i) 8. (7i)2(6i) 9. i42

10. i55 11. i89 12. (5 � 2i) � (�13 � 8i)

13. (7 � 6i) � (9 � 11i) 14. (�12 � 48i) � (15 � 21i) 15. (10 � 15i) � (48 � 30i)

16. (28 � 4i) � (10 � 30i) 17. (6 � 4i)(6 � 4i) 18. (8 � 11i)(8 � 11i)

19. (4 � 3i)(2 � 5i) 20. (7 � 2i)(9 � 6i) 21.

22. 23. 24.


25. 5n2 � 35 � 0 26. 2m2 � 10 � 0

27. 4m2 � 76 � 0 28. �2m2 � 6 � 0

29. �5m2 � 65 � 0 30. x2 � 12 � 0

Find the values of m and n that make each equation true.

31. 15 � 28i � 3m � 4ni 32. (6 � m) � 3ni � �12 � 27i

33. (3m � 4) � (3 � n)i � 16 � 3i 34. (7 � n) � (4m � 10)i � 3 � 6i

35. ELECTRICITY The impedance in one part of a series circuit is 1 � 3j ohms and theimpedance in another part of the circuit is 7 � 5j ohms. Add these complex numbers tofind the total impedance in the circuit.

36. ELECTRICITY Using the formula E � IZ, find the voltage E in a circuit when thecurrent I is 3 � j amps and the impedance Z is 3 � 2j ohms.

3�4

2 � 4i�1 � 3i

3 � i�2 � i

2�7 � 8i

6 � 5i�

�2i

17�81

a6b3�98

550x�

49

PracticeComplex Numbers

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1. SIGN ERRORS Jennifer and Jessicacome up with different answers to thesame problem. They had to multiply (4 � i)(4 � i) and give their answer as a complex number. Jennifer claims thatthe answer is 15 and Jessica claims that the answer is 17. Who is correct?Explain.

2. COMPLEX CONJUGATES You haveseen that the product of complexconjugates is always a real number.Show that the sum of complexconjugates is also always a real number.

3. PYTHAGOREAN TRIPLES If threeintegers a, b, and c, satisfy a2 � a2 � c2,then they are called a PythagoreanTriple. Suppose that a, b, and c are aPythagorean triple. Show that the realand imaginary parts of (a � bi)2,together with the number c2, formanother Pythagorean triple.

4. ROTATIONS Complex numbers can beused to perform rotations in the plane.For example, if (x, y) are the coordinatesof a point in the plane, then the real and imaginary parts of i(x � yi) are the horizontal and vertical coordinatesof the 90° counterclockwise rotation of(x, y) about the origin. What are the realand imaginary parts of i(x � yi)?

ELECTRICAL ENGINEERING ForExercises 5–7, use the followinginformation.Complex numbers can be used to describethe alternating current (AC) in an electriccircuit like the one used in your home. Z,the impedance in an AC circuit, is related tothe voltage V and the current I by the

formula Z � .

5. Find Z if V � 5 � 2i and I � 3i.

6. Find Z if V � 2 � 3i and I � �3i.

7. Find V if Z � and I � 3i.2 � 3i�

3

V�I

Word Problem PracticeComplex Numbers

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Conjugates and Absolute ValueWhen studying complex numbers, it is often convenient to represent a complex number by a single variable. For example, we might let z � x � yi. We denote the conjugate of z by z�. Thus, z� � x � yi.

We can define the absolute value of a complex number as follows.

� z� � � x � yi� � �x2 � y�2�

There are many important relationships involving conjugates and absolute values of complex numbers.

Show �z �2 � zz� for any complex number z.

Let z � x � yi. Then,zz� � (x � yi)(x � yi)

� x2 � y2

� �(x2 � y2�)2�� z�2

Show is the multiplicative inverse for any nonzero

complex number z.

We know � z�2 � zz�. If z � 0, then we have z� � � 1.

Thus, is the multiplicative inverse of z.

For each of the following complex numbers, find the absolute value andmultiplicative inverse.

1. 2i 2. �4 � 3i 3. 12 � 5i

4. 5 � 12i 5. 1 � i 6. �3� � i

7. � i 8. � i 9. �12� � i

�3��2

�2��2

�2��2

�3��3

�3��3

z�� z�2

z�� z�2

z��z �2

Enrichment

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Explain what it means to say that the driver accelerates at a constant rate of 8 feet per second squared.

Read 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

x � 6 � ��81�

x � 6 � �9

x � 6 � 9 or x � 6 � �9

x � 15 x � �3

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

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

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


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?

Lesson Reading GuideCompleting the Square

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Square Root Property Use the Square Root Property to solve a quadratic equationthat is in the form “perfect square trinomial � constant.”

Solve each equation by using the Square Root Property.

a. x2 � 8x � 16 � 25x2 � 8x � 16 � 25

(x � 4)2 � 25x � 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


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

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

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

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

Study Guide and InterventionCompleting the Square

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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

Find the valueof c 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

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

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

Solve each equation by completing the square.

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

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

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

1�2


Completing the Square

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Skills PracticeCompleting the Square


1. x2 � 8x � 16 � 1 2. x2 � 4x � 4 � 1

3. x2 � 12x � 36 � 25 4. 4x2 � 4x � 1 � 9

5. x2 � 4x � 4 � 2 6. x2 � 2x � 1 � 5

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


9. x2 � 10x � c 10. x2 � 14x � c

11. x2 � 24x � c 12. x2 � 5x � c

13. x2 � 9x � c 14. x2 � x � c


15. x2 � 13x � 36 � 0 16. x2 � 3x � 0

17. x2 � x � 6 � 0 18. x2 � 4x � 13 � 0

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

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

23. x2 � �11 24. x2 � 2x � 4 � 0

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

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

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


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

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

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


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

22. x2 � 18 � 9x 23. x2 � 14x � 19 � 0 24. x2 � 16x � 7 � 0

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?

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

5�3

1�4

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PracticeCompleting the Square

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1. COMPLETING THE SQUARESamantha needs to solve the equation

x2 � 12x � 40.

What must she do to each side of theequation to complete the square?

2. SQUARE ROOTS Evan is asked tosolve the equation x2 � 8x � 16 � 25.He recognizes that the left-hand side ofthe equation is a perfect squaretrinomial. Factor the left-hand side.

3. COMPOUND INTEREST Nikkiinvested $1000 in a savings accountwith interest compounded annually.After two years the balance in theaccount is $1210. Use the compoundinterest formula A � P(1 � r)t to findthe annual interest rate.

4. REACTION TIME Lauren was eatinglunch when she saw her friend Jasonapproach. The room was crowded andJason had to lift his tray to avoidobstacles. Suddenly, a glass on Jason’slunch tray tipped and fell off the tray.Lauren lunged forward and managed tocatch the glass just before it hit theground. The height h, in feet, of theglass t seconds after it was dropped isgiven by h � �16t2 � 4.5. Laurencaught the glass when it was six inchesoff the ground. How long was the glassin the air before Lauren caught it?

5. PARABOLAS A parabola is modeled byy � x2 � 10x � 28. Jane’s homeworkproblem requires that she find thevertex of the parabola. She uses thecompleting square method to expressthe function in the form y � (x � h)2 � k, where (h,k) is thevertex of the parabola. Write thefunction in the form used by Jane.

AUDITORIUM SEATING For Exercises6–8, use the following information.The seats in an auditorium are arranged ina square grid pattern. There are 45 rowsand 45 columns of chairs. For a specialconcert, organizers decide to increaseseating by adding n rows and n columns to make a square pattern of seating 45 � n seats on a side.

6. How many seats are there after theexpansion?

7. What is n if organizers wish to add 1000seats?

8. If organizers do add 1000 seats, what isthe seating capacity of the auditorium?

Word Problem PracticeCompleting the Square

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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

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Lesson Reading GuideThe Quadratic Formula and the Discriminant


Describe how you would calculate the position of the diver after 1 second using the equationin your textbook.

Read the Lesson

1. a. Write the Quadratic Formula.

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

�8

21

0


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?

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Study Guide and InterventionThe Quadratic Formula and the Discriminant

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

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

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

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

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

2�25

3r�5

5 � 9�2

5 � �81��2

�(�5) � �(�5)2�� 4(1�)(�14�)��2(1)

�b � �b2 � 4�ac��2a

�b � �b2 ��4ac��

2a

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Example

Roots and the Discriminant

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

Roots of a Quadratic Equation

Discriminant Type and Number of Roots

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

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

b2 � 4ac � 0 1 rational root

b2 � 4ac � 0 2 complex roots

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


The Quadratic Formula and the Discriminant

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.

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 2. 9x2 � 6x � 1 � 0 3. 2x2 � 7x � 4 � 0

4. x2 � 4x � 4 � 0 5. 5x2 � 36x � 7 � 0 6. 4x2 � 4x � 11 � 0

7. x2 � 7x � 6 � 0 8. m2 � 8m � �14 9. 25x2 � 40x � �16

10. 4x2 � 20x � 29 � 0 11. 6x2 � 26x � 8 � 0 12. 4x2 � 4x � 11 � 0

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Skills PracticeThe Quadratic Formula and the Discriminant

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

3. 3x2 � 2x � 0 4. 20x2 � 7x � 3 � 0

5. 5x2 � 6 � 0 6. x2 � 6 � 0

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

9. x2 � 2x � 17 � 0 10. x2 � 49 � 0

11. x2 � x � 1 � 0 12. 2x2 � 3x � �2

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

13. x2 � 64 14. x2 � 30 � 0

15. x2 � x � 30 16. 16x2 � 24x � 27 � 0

17. x2 � 4x � 11 � 0 18. x2 � 8x � 17 � 0

19. x2 � 25 � 0 20. 3x2 � 36 � 0

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?

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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

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

7. 5x2 � 2x � 4 � 0 8. 12x2 � x � 6 � 0 9. 7x2 � 6x � 2 � 0

10. 12x2 � 2x � 4 � 0 11. 6x2 � 2x � 1 � 0 12. x2 � 3x � 6 � 0

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

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

16. 7x2 � 5x � 0 17. 4x2 � 9 � 0

18. 3x2 � 8x � 3 19. x2 � 21 � 4x

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

22. x2 � 6x � 3 � 0 23. x2 � 14x � 53 � 0

24. 3x2 � �54 25. 25x2 � 20x � 6 � 0

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

28. x2 � 4x � 15 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?

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?

PracticeThe Quadratic Formula and the Discriminant

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1. PARABOLAS The graph of a quadraticequation of the form y � ax2 � bx � c isshown below.

Is the discriminant b2 � 4ac positive,negative, or zero?

2. TANGENT Kathleen is trying to find bso that the x-axis is tangent to theparabola y � x2 � bx � 4. She finds onevalue that works, b � 4. Is this the onlyvalue that works? Explain.

3. AREA Conrad has a triangle whosebase has length x � 3 and whose heightis 2x � 4. What is the area of thistriangle? For what values of x is thisarea equal to 210? Do all the solutionsmake sense?

4. EXAMPLES Give an example of aquadratic function f(x) that has thefollowing properties.

I. The discriminant of f is zero.

II. There is no real solution of theequation f(x) � 10.

Sketch the graph of x � f(x).

TANGENTS For Exercises 5 and 6, usethe following information.The graph of y � x2 is a parabola thatpasses through the point at (1, 1). The line y � mx � m � 1, where m is a constant,also passes through the point at (1, 1).

5. To find the points of intersectionbetween the line y � mx � m � 1 and the parabola y � x2, set x2 �mx � m � 1 and then solve for x.Rearranging terms, this equationbecomes x2 � mx � m � 1 � 0. What isthe discriminant of this equation?

6. For what value of m is there only onepoint of intersection? Explain themeaning of this in terms of thecorresponding line and the parabola.

yx

O

y

xO

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

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

then s1 � s2 � ��ba

� and s1 s2 � �ac

�.

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

4. 4 � �3� 6. ��25�, �

27� 6.

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

�2 � 3�5��7

x

y

O

(–5–2, –301–

4)

10

–10

–20

–30

2 4–2–4–6–8

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You have learned the Location Principle, which can be used to approximatethe real zeros of a polynomial.

In the spreadsheet above, the positive real zero of ƒ(x) � x2 � 2 can beapproximated in the following way. Set the spreadsheet preference to manualcalculation. The values in A2 and B2 are the endpoints of a range of values.The values in D2 through J2 are values equally in the interval from A2 toB2. The formulas for these values are A2, A2 � (B2 � A2) 6, A2 � 2*(B2 �A2)/6, A2 � 3*(B2 � A2)/6, A2 � 4*(B2 � A2)/6, A2 � 5*(B2 � A2)/6, and B2,respectively.

Row 3 gives the function values at these points. The function ƒ(x) � x2 � 2 isentered into the spreadsheet in Cell D3 as D2^2 � 2. This function is thencopied to the remaining cells in the row.

You can use this spreadsheet to study the function values at the points incells D2 through J2. The value in cell F3 is positive and the value in cell G3is negative, so there must be a zero between �1.6667 and 0. Enter these values in cells A2 and B2, respectively, and recalculate the spreadsheet. (Youwill have to recalculate a number of times.) The result is a new table fromwhich you can see that there is a zero between 1.41414 and 1.414306.Because these values agree to three decimal places, the zero is about 1.414.This can be verified by using algebra.

By solving x2 � 2 � 0, we obtain x � ��2�. The positive root is x � ��2� � 1.414213. . . , which verifies the result.

Spreadsheet InvestigationApproximating the Real Zeros of Polynomials

A12345

C

xf(x)

B

5

G

0–2

D

–523

J

523

E

–3.33333339.1111111

H

1.66666679.1111111

I

3.33333330.7777778

F

–1.66666670.7777778

–5

Sheet 1 Sheet 2 Sheet 3

Exercises

1. Use a spreadsheet like the one above to approximate the zero of ƒ(x) � 3x � 2 to threedecimal places. Then verify your answer by using algebra to find the exact value of theroot.

2. Use a spreadsheet like the one above to approximate the real zeros of f(x) � x2 � 2x � 0.5.Round your answer to four decimal places. Then, verify your answer by using the quadratic formula.

3. Use a spreadsheet like the one above to approximate the real zero of ƒ(x) � x3 � �32�x2 � 6x � 2

between � 0.4 and � 0.3.

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Lesson Reading GuideAnalyzing Graphs of Quadratic Equations


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

• What does subtracting a positive number to x before squaring do to the graph of y � x2?

Read 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?

b. What is the equation of the axis of symmetry?

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

d. What do you know about the graph if ⏐a⏐ � 1?

If ⏐a⏐ � 1?

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

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

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

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?

x

y

Ox

y

Ox

y

Ox

y

O

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Study Guide and InterventionAnalyzing Graphs of Quadratic Functions

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 opens up.

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 opens down.

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

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

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

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

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

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Exercises

Example

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 � 6

4. y � �4x2 � 16x � 11 5. y � 3x2 � 12x � 5 6. y � 5x2 � 10x � 9

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Skills PracticeAnalyzing Graphs of Quadratic Functions

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 � 6

4. y � �3(x � 5)2 5. y � �5x2 � 9 6. y � (x � 2)2 � 18

7. y � x2 � 2x � 5 8. y � x2 � 6x � 2 9. y � �3x2 � 24x

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)

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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 � 8x

4. y � x2 � 10x � 20 5. y � 2x2 � 12x � 18 6. y � 3x2 � 6x � 5

7. y � �2x2 � 16x � 32 8. y � �3x2 � 18x � 21 9. y � 2x2 � 16x � 29

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)

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

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

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?

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.

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PracticeAnalyzing Graphs of Quadratic Functions

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1. ARCHES A parabolic arch is used as abridge support. The graph of the arch isshown below.

If the equation that corresponds to this graph is written in the form y � a(x � h)2 � k, what are h and k?

2. TRANSLATIONS For a computeranimation, Barbara uses the quadraticfunction f(x) � �42(x � 20)2 � 16800 tohelp her simulate an object tossed onanother planet. For one skit, she had touse the function f(x � 5) � 8000 insteadof f(x). Where is the vertex of the graphof y � f(x � 5) � 8000?

3. MIRRORS The cross-section of areflecting telescope mirror is described

by the parabola y � �110� (x � 5)2 � .

Graph this parabola.

4. WATER JETS The graph shows thepath of a jet of water.

The equation corresponding to thisgraph is y � a(x � h) 2 � k. What are a,h, and k?

PROFIT For Exercises 5–7, use thefollowing information.A theater operator predicts that the theatercan make �4x2 � 160x dollars per show iftickets are priced at x dollars.

5. Rewrite the equation y � �4x2 � 16x inthe form y � a(x � h) 2 � k.

6. What is the vertex of the parabola andwhat is its axis of symmetry?

7. Graph the parabola.

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Word Problem PracticeAnalyzing Graphs of Quadratic Functions

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Enrichment

A Shortcut to Complex RootsWhen graphing a quadratic function, the real roots are shown in the graph.You have learned that quadratic functions can also have imaginary roots that cannot be seen on the graph of the function. However, there is a way to graphically represent the complex roots of a quadratic function.

Find the complex roots of the quadratic function y � x2 � 4x � 5.

Step 1: Graph the function.

Step 2: Reflect the graph over the horizontal line containing the vertex. In this example,the vertex is (2, 1).

Step 3: The real part of the complex root is the point halfway between the x-intercepts ofthe reflected graph and the imaginary part of the complex roots are � and � halfthe distance between the x-intercepts of the reflected graph. So, in this example,the complex roots are 2 � 1i and 2 � 1i.

Using this method, find the complex roots of the following quadratic functions.

1. y � x2 � 2x � 5 2. y � x2 � 4x � 8

3. y � x2 � 6x � 13 4. y � x2 � 2x � 17

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Lesson Reading GuideGraphing and Solving Quadratic Inequalities


• How far above the ground is the trampoline surface?

• Using the quadratic function given in the introduction, write a quadratic inequality that describes the times at which the trampolinist is more than 20 feet above the ground.

Read 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?

b. Should the parabola be solid or dashed? How do you know?

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?

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

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 i. {x⏐x � 0 or x � 4}

b. �x2 � 4x 0 ii. {x⏐0 � x � 4}

c. �x2 � 4x � 0 iii. {x⏐x 0 or x � 4}

d. �x2 � 4x � 0 iv. {x⏐0 x 4}


3. A quadratic inequality in two variables may have the form y � ax2 � bx � c,y � ax2 � bx � c, y � ax2 � bx � c, or y ax2 � bx � c. Describe a way to rememberwhich region to shade by looking at the inequality symbol and without using a test point.

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Study Guide and InterventionGraphing and Solving Quadratic Inequalities

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

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

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

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

Find the roots of the related quadratic equation by factoring,

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

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

Solve the inequality x2 � x � 6 � 0.

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

Solve each inequality.

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

4. c2 4 5. 2m2 � m � 1 6. y2 � �8

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

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

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

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Skills PracticeGraphing and Solving Quadratic Inequalities


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

Solve each inequality algebraically.

7. x2 � 3x � 10 � 0 8. x2 � 2x � 35 � 0

9. x2 � 18x � 81 0 10. x2 36

11. x2 � 7x � 0 12. x2 � 7x � 6 � 0

13. x2 � x � 12 � 0 14. x2 � 9x � 18 0

15. x2 � 10x � 25 � 0 16. �x2 � 2x � 15 � 0

17. x2 � 3x � 0 18. 2x2 � 2x � 4

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

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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

Solve each inequality algebraically.

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

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

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

16. 4x2 � 4x � 1 � 0 17. 5x2 � 10 � 27x 18. 9x2 � 31x � 12 0

19. FENCING Vanessa has 180 feet of fencing that she intends to use to build a rectangularplay area for her dog. She wants the play area to enclose at least 1800 square feet. Whatare the possible widths of the play area?

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?

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1. HUTS The space inside a hut is shadedin the graph. The parabola is described

by the equation y � � (x � 1)2 � 4.

Write an inequality that describes theshaded region.

2. DISCRIMINANTS Consider theequation ax2 � bx � c � 0. Assume thatthe discriminant is zero and that a ispositive. What are the solutions of theinequality ax2 � bx � c 0?

3. TOSSING Gail and Veronica are fixinga leak in a roof. Gail is working on theroof and Veronica is tossing up suppliesto Gail. When Gail tosses up a tapemeasure, the height h, in feet, of theobject above the ground t seconds afterGail tosses it is h � �16t2 � 32t � 5.Gail can catch the object any time it isabove 17 feet. How much time does Gailhave to try to catch the tape measure?

4. KIOSKS Caleb is designing a kiosk bywrapping a piece of sheet metal withdimensions x � 5 inches by 4x � 8inches into a cylindrical shape. Ignoringcost, Caleb would like a kiosk that has asurface area of at least 4480 squareinches. What values of x satisfy thiscondition?

TUNNELS For Exercises 5 and 6, usethe following information.An architect wants to use a parabolic archas the entrance of a tunnel. She sketchesthe plan on a piece of graph paper. Shewould like the maximum height of thetunnel to be located at (4, 4), and she wouldlike the origin to be on the parabola as well.

5. Write an equation for the desiredparabola.

6. Write an inequality that describes theregion above the parabola, part of whichwill be filled in with concrete. Graphthis inequality.

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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

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

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

Enrichment

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Graphing Calculator ActivityQuadratic Inequalities and the Test Menu

The inequality symbols, called relational operators, in the TEST menu can beused to display the solution of a quadratic inequality. Another method that canbe used to find the solution set of a quadratic inequality is to graph each sideof an inequality separately. Examine the graphs and use the intersect functionto determine the range of values for which the inequality is true.

Solve each inequality.

1. �x2 � 10x � 21 � 0 2. x2 � 9 � 0 3. x2 � 10x � 25 0

4. x2 � 3x 28 5. 2x2 � x � 3 6. 4x2 � 12x � 9 � 0

7. 23 � �x2 � 10x 8. x2 � 4x � 13 0 9. (x � 1)(x �3) � 0

Solve x2 � x � 6.

Place the calculator in Dot mode. Enter the inequality into Y1.Then trace the graph and describe the solution as an inequality.Keystrokes: [TEST] 4 6 4.

Use TRACE to determine the endpoints of the segments.Theses values are used to express the solution of the inequality,{ x | x � � 3 or x � 2 }.

ZOOM2nd+x 2Y=

Solve 2x2 � 4x � 5 � 3.

Place the left side of the inequality in Y1 and the right side in Y2.Determine the points of intersection. Use the intersection points to express the solution set of the inequality. Be sure to set the calculator to Connected mode.Keystrokes: 2 4 5 3

6.

Press [CALC] 5 and use the key to move the cursor to the left of the first intersection point. Press . Then move the cursor to the right of the intersection point and press

. One of the values used in the solution set is displayed.Repeat the procedure on the other intersection point.

The solution is { x | �3.24 x 1.24}.

ENTER

ENTER

ENTER

2nd

ZOOM

ENTERENTER—+x 2Y=

[�4.7, 4.7] scl:1 by [�3.1, 3.1] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1

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Student Recording SheetUse this recording sheet with pages 308–309 of the Student Edition.

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10. Record your answer and fill in thebubbles in the grid below. Be sure to use the correct place value.

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Record your answers for Question 12on the back of this paper.

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A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

Read each question. Then fill in the correct answer.

Ass

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5NAME ______________________________________________ DATE______________ PERIOD _____

Rubric for Scoring Pre-AP(Use to score the Pre-AP question on page 309 of the Student Edition.)

General Scoring Guidelines

• If a student gives only a correct numerical answer to a problem but does not show howhe or she arrived at the answer, the student will be awarded only 1 credit. All extendedresponse questions require the student to show work.

• A fully correct answer for a multiple-part question requires correct responses for allparts of the question. For example, if a question has three parts, the correct response toone or two parts of the question that required work to be shown is not considered afully correct response.

• Students who use trial and error to solve a problem must show their method. Merelyshowing that the answer checks or is correct is not considered a complete response forfull credit.

Exercise 10 Rubric

Score Specific Criteria

4 Part a shows that the maximum height of a rocket is 160 meters because the

vertex form of the equation is h � �4.9�t � �470��

2� 160, so the vertex is at

��470�, 160�. Part b shows that the time is the x coordinate of the vertex, �

470� or

5.7. It will take the rocket 5.7 seconds to reach the maximum height.

3 A generally correct solution, but may contain minor flaws in reasoning orcomputation.

2 A partially correct interpretation and/or solution to the problem.

1 A correct solution with no evidence or explanation.

0 An incorrect solution indicating no mathematical understanding of theconcept or task, or no solution given.

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Chapter 5 Quiz 1(Lessons 5–1 and 5–2)


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

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

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

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

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

4. x2 � 2x � 3 5. x2 � 4x � 7 � 0 4.

5.

3, �1

minimum, 1

�3; x � �1; �1


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

3. Solve 5x2 � 100 � 0.

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

4. �6 and 2 5. �23� and �4

Simplify.6. ��80� 7. ��6� � ��12�

8. (6 � 9i) � (17 � 12i) 9. (7 � 3i)(8 � 4i)

10. �32 �

�ii�

NAME DATE PERIOD

SCORE

xO

f(x )

(0, �3)(�1, �4)

f(x) � x2 � 2x � 3

x � �1

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

68 � 4i

�11 � 3i

3x2 � 10x � 8 � 0

x2 � 4x � 12 � 0

{�9, 5}

between 1 and 2;between �6 and �5

��5, �23��

4i �5�

�12

� � �12

�i

�6�2�

�2i �5�

066-086 A2-05-873975 5/11/06 7:26 AM Page 67

5

5

NAME DATE PERIOD

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NAME DATE PERIOD

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

2.

3. What is the solution of x2 � 10x � 11? 3.

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

5. Find the value of the discriminant for 3x2 � 6x � 11. Then 5.describe the number and type of roots for the equation.

2 ��5�

{�1, 11}

1 � 3�5�

{�10, 2}


NAME DATE PERIOD

SCORE


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

2. Write an equation for the parabola whose vertex is at 2.(�5, 0) and passes through (0, 50).

3. Graph y � ��13�(x � 2)2 � 3. 3.

4. Use the graph of its related function to write the 4.solutions of �x2 � 6x � 5 � 0.

5. MULTIPLE CHOICE What is the solution of 4x2 � 1 � 4x? 5.

A. all reals B. empty set C. �x � x � �12�� D. �x � x � �

12��

A

{x �1 � x � 5}

y

xO

y � 2(x � 5)2

xO

y

(2, �1)

x � 2

�96; 2 complex roots

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Chapter 5 Mid-Chapter Test (Lessons 5–1 through 5–4)


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

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

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

F. x � 1 or x � 5 H. x � �12� or x � 5

G. x � ��12� or x � �5 J. x � �1 or x � �5 2.

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

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

4. The current in one part of a series circuit is 3 � 2j amps.The current in another part of the circuit is 2 � 4j amps.Find the total amps in the circuit.F. 5 � 2j H. 1 � 2jG. 6 � 8j J. 7j 4.

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

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

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

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

For Questions 8 and 9, solve each equation by factoring.

8. x2 � 7x � 18 9. 4x2 � x 9.

10. Simplify �3 �5i

5i�. 10.

y

xO

1, 3Part II

D

F

A

H

B

Part I

xO

f(x )

8. {�2, 9}

�0, �14

��

minimum, �9 �12

�

��2354� � �

1354i

�

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5

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

1. Two complex numbers of the form a � bi and a � bi 1.are called imaginary units.

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

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

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

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

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

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

8. Pure imaginary numbers are square roots of negative real numbers. 8.

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

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

Define each term in your own words.

11. parabola Sample answer: A parabola is a smoothcurve that is the graph of a quadratic equation.

12. axis of symmetry Sample answer: An axis of symmetryis a line along which you can fold a graph and getmatching parts on both sides of the line.

true

true

true

false; roots

axis of symmetrycompleting the squarecomplex conjugatescomplex numberconstant termdiscriminantimaginary unit

linear termmaximum valueminimum valueparabolapure imaginary numberquadratic equationquadratic function

quadratic inequalityquadratic termrootssquare root vertexvertex formzeros

NAME DATE PERIOD

SCORE Chapter 5 Vocabulary Test NAME DATE PERIOD

SCORE


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false; constant term

false; quadratic inequality

false;quadratic term

false; minimum value

false; discriminant

false; complex conjugates

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Chapter 5 Test, Form 1NAME DATE PERIOD

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

2. What is the equation of the axis of symmetry of y � �3(x � 6)2 � 12?F. x � 2 G. x � �6 H. x � 6 J. x � �18 2.

3. The graph of f(x) � �2x2 � x opens _____ and has a _____ value.A. down; maximum C. up; maximumB. down; minimum D. up; minimum 3.

4. The related graph of a quadratic equation is shown at the right.Use the graph to determine the solutions of the equation.F. �2, 3 H. �3, 2G. 0, �6 J. 0, 2 4.

5. The quadratic function f(x) � x2 has _____.A. no zeros C. exactly two zerosB. exactly one zero D. more than two zeros 5.

6. Solve x2 � 3x � 10 � 0 by factoring.F. {�5, 2} G. (�2, �5) H. {�2, 5} J. {�10, 1} 6.

7. Which quadratic equation has roots �2 and 3?A. x2 � x � 6 � 0 C. x2 � 6x � 1 � 0B. x2 � x � 6 � 0 D. x2 � x � 6 � 0 7.

8. Simplify (5 � 2i)(1 � 3i).F. 5 � 6i G. �1 H. �1 � 17i J. 11 � 17i 8.

9. ELECTRICITY The total impedance of a series circuit is the sum of the impedances of all parts of the circuit. A technician determined that the impedance of the first part of a particular circuit was 2 � 5j ohms.The impedance of the remaining part of the circuit was 3 � 2j ohms.What was the total impedance of the circuit?A. 5 � 3j ohms C. �1 � 7j ohmsB. 5 � 7j ohms D. 16 � 11j ohms 9.

10. To solve x2 � 8x � 16 � 25 by using the Square Root Property, you would first rewrite the equation as _____.F. (x � 4)2 � 25 H. x2 � 8x � 9 � 0G. (x � 4)2 � 5 J. x2 � 8x � 9 10. F

A

H

B

H

B

H

A

G

B

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f(x )

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5 Chapter 5 Test, Form 1 (continued)

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

12. The quadratic equation x2 � 6x � 1 is to be solved by completing the square.Which equation would be the first step in that solution?F. x2 � 6x � 1 � 0 H. x2 � 6x � 36 � 1 � 36G. x(x � 6) � 1 J. x2 � 6x � 9 � 1 � 9 12.

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

A. ��3

2�5�

� B. �3

2�13�� C. ��3

2�13�� D. �

3 2�5�� 13.

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

14. x2 � 3x � 7 � 0F. 2 complex roots H. 2 real, irrational rootsG. 2 real, rational roots J. 1 real, rational root 14.

15. x2 � 4x � 4A. 2 real, rational roots C. 1 real, rational rootB. 2 real, irrational roots D. no real roots 15.

16. What is the vertex of y � 2(x � 3)2 � 6?F. (�3, �6) G. (3, �6) H. (�3, 6) J. (3, 6) 16.

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

18. Which quadratic function has its vertex at (2, 3) and passes through (1, 0)?F. y � 2(x � 2)2 � 3 H. y � �3(x � 2)2 � 3G. y � �3(x � 2)2 � 3 J. y � 2(x � 2)2 � 3 18.

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

20. Solve (x � 4)(x � 2) � 0.F. {x � x � �2 or x � 4} H. {x � �4 � x � 2}G. {x � �2 � x � 4} J. {x � x � �2 or x � 4} 20.

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

1 and 7; 14

G

B

G

B

J

C

F

D

J

B

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

2. Identify the quadratic function graphed at the right.F. f(x) � �x2 � 2xG. f(x) � �x2 � 2xH. f(x) � x2 � 2xJ. f(x) � �(x � 2)2 2.

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

4. Solve �x2 � 4x by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.F. 4, 0 H. �4, 0G. between �4 and 4 J. �2, 4 4.

5. Solve x2 � 3x � 18 by factoring.A. {6} B. {�6, 3} C. {�9, 2} D. {�3, 6} 5.

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

F. x2 � 4x � 4 � 0 H. 5x2 � 9x � 2 � 0G. 5x2 � 9x � 2 � 0 J. 5x2 � 11x � 2 � 0 6.

7. Simplify (4 � 12i) � (�8 � 4i).A. 12 � 8 B. 28 C. 12 � 16i D. 12 � 16i 7.

8. Simplify �47��

23ii�.

F. �129

1� � �

1239�

i G. �2191� � �

1249�

i H. �1239�

� �1279�

i J. �1279�

� �1239�

i 8.

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

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

F. �841� G. �

92� H. ��

841� J. 81 10. F

D

F

C

G

D

H

A

G

C

Chapter 5 Test, Form 2ANAME DATE PERIOD

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f(x )

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

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

F. ��5

6�13�� G. �

5 3�13�� H. �

5 6�37�� J. �

5 6�13�� 12.


13. 2x2 � 7x � 9 � 0A. 2 real, rational C. 2 complexB. 2 real, irrational D. 1 real, rational 13.

14. x2 � 20 � 12x � 16F. 1 real, irrational H. 2 real, rationalG. no real J. 1 real, rational 14.

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

y � �12�(x � 8)2 � 2.

A. (�8, 2); x � �8; up C. (8, �2); x � 8; upB. (�8, �2); x � �8; down D. (8, 2); x � 8; up 15.

16. Which quadratic function has its vertex at (�2, 7) and opens down?F. y � �3(x � 2)2 � 7 H. y � (x � 2)2 � 7G. y � �12(x � 2)2 � 7 J. y � �2(x � 2)2 � 7 16.

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

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

F. y � ��32�(x � 8)2 � 4 H. y � ��

14�(x � 8)2 � 4

G. y � �32�(x � 6)2 � 2 J. y � ��

32�(x � 8)2 � 4 18.

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

20. Solve x2 � 2x � 24.F. {x � �4 � x � 6} H. {x � �6 � x � 4}G. {x � x � �6 or x � 4} J. {x � x � �4 or x � 6} 20.

Bonus Write a quadratic equation with roots �i�

43�

�. B:

J

C

F

B

F

D

J

C

J

B

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Chapter 5 Test, Form 2A (continued)

NAME DATE PERIOD

yxO

Sample answer:16x2 � 3 � 0

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

2. Identify the quadratic function graphed at the right.F. f(x) � x2 � 4xG. f(x) � �x2 � 4xH. f(x) � �x2 � 4xJ. f(x) � �(x � 4)2 2.

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

4. Solve x2 � 4x by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.F. �4, 0 H. between �4 and 4G. 2, �4 J. 0, 4 4.

5. Solve x2 � 3x � 28 by factoring.A. {�4, 7} B. {�14, 2} C. {�7, 4} D. {�2, 14} 5.

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

F. 2x2 � 11x � 21 � 0 H. 3x2 � 19x � 14 � 0G. 3x2 � 23x � 14 � 0 J. 2x2 � 11x � 21 � 0 6.

7. Simplify (15 � 13i) � (�1 � 17i).A. 16 � 30i B. 16 � 4i C. 16 � 30i D. 46 7.

8. Simplify �12 ��

32ii

�.

F. �87� � �

17�i G. �

87� � i H. �4 � 7i J. ��1

43�

� �173�

i 8.

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

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

F. �2156�

G. �54� H. �

245� J. �

52� 10. H

A

J

A

H

A

J

B

H

B

Chapter 5 Test, Form 2B

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5

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

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

F. ��5

4�17�� G. �

5 4�17�� H. �

5 4�33�� J. �

5 2�17�� 12.


13. 3x2 � x � 12 � 0A. 2 complex roots C. 2 real, rational rootsB. 1 real, rational root D. 2 real, irrational roots 13.

14. x2 � 10 � 3x � 3F. 2 complex roots H. 2 real, irrational rootsG. 1 real, rational root J. 2 real, rational roots 14.

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

16. Which quadratic function has its vertex at (�3, 5) and opens down?F. y � (x � 3)2 � 5 H. y � (x � 3)2 � 5G. y � �(x � 3)2 � 5 J. y � �(x � 3)2 � 5 16.

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

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

F. y � ��111�

(x � 5)2 � 7 H. y � �2(x � 5)2 � 7

G. y � ��12�(x � 5)2 � 7 J. y � ��

12�(x � 5)2 � 7 18.

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

20. Solve 2x � 3 � x2.F. {x � �1 � x � 3} H. {x � �3 � x � 1}G. {x � x � �1 or x � 3} J. {x � x � �3 or x � 1} 20.

Bonus Write a quadratic equation with roots �i�

32�

�. B:

F

A

H

D

G

B

F

D

G

C

NAME DATE PERIOD


Copyright ©

Glencoe/M

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raw-H

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panies, Inc.

Chapter 5 Test, Form 2B (continued)

NAME DATE PERIOD

y

xO

Sample answer:9x2 � 2 � 0

066-086 A2-05-873975 5/11/06 7:26 AM Page 76

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

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

3. Solve x2 � 6x � 8 by graphing. If exact roots cannot be 3.found, state the consecutive integers between which the roots are located.

4. Solve 5x2 � 13x � 6 by factoring. 4.

5. GEOMETRY The length of a rectangle is 7 inches longer 5.than its width. If the area of the rectangle is 144 square inches, what are its dimensions?

6. ELECTRICITY The total impedance of a series circuit is the 6.sum of the impedances of all parts of the circuit. Suppose that the first part of a circuit has an impedance of 6 � 5j ohms and that the total impedance of the circuit was 12 � 7j ohms. What is the impedance of the remainder of the circuit?

7. ELECTRICITY In an AC circuit, the voltage E (in volts), 7.current I (in amps), and impedance Z (in ohms) are related by the formula E � I � Z. Find the current in a circuit with voltage 10 � 3j volts and impedance 4 � j ohms.

8. Write a quadratic equation with �6 and �34� as its roots. 8.

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

9. Solve x2 � 6x � 9 � 25 by using the Square Root Property. 9. {�8, 2}

4x2 � 21x � 18

6 � 12j ohms

9 in. by 16 in.

y

xO

2, 4

maximum; 4

xO

f(x )

(3, 0)

(1, 5)

f(x) � �5x2 � 10x

x � 1

NAME DATE PERIOD

SCORE

Ass

essm

ent

Chapter 5 Test, Form 2C


��3, �25

��

�3177� � �

2127�j amps

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

10. x2 � 4x � 9 � 0 10.

11. 2x2 � 3x � 2 � 0 11.

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

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

13. 9x2 � 12x � 4 � 0 13.

14. 4x2 � 1 � 9x � 2 14.

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

opening for y � ��23�(x � 5)2 � 7.

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

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

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

19. Graph y x2 � 6x � 9. 19.

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

Bonus Write a quadratic equation with roots ��37�

�. Write the B:

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

9x2 � 7 � 0

y

xO

y � (x � 3)2 � 1

{�2 � �13�}

NAME DATE PERIOD


Chapter 5 Test, Form 2C (continued)

NAME DATE PERIOD

0; 1 real,rational root

33; 2 real,irrational roots

(�5, �7);x � �5; down

y � �32

�(x � 2)2 � 1

h(t ) � �16(t � 1.5)2 � 51; 51ft

�x �x � ��12

� or x � 3�

�3 �

1i0�31��

��2, �12

��

066-086 A2-05-873975 5/11/06 7:26 AM Page 78

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

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

3. Solve x2 � 2x � 3 � 0 by graphing. If exact roots cannot be 3.found, state the consecutive integers between which the roots are located.

4. Solve 3x2 � x � 4 by factoring. 4.

5. GEOMETRY The length of a rectangle is 10 inches longer 5.than its width. If the area of the rectangle is 144 square inches, what are its dimensions?

6. ELECTRICITY The total impedance of a series circuit is the 6.sum of the impedances of all parts of the circuit. Suppose that the first part of a circuit has an impedance of 7 � 4j ohms and that the total impedance of the circuit was 16 � 2j ohms. What is the impedance of the remainder of the circuit?

7. ELECTRICITY In an AC circuit, the voltage E (in volts), 7.current I (in amps), and impedance Z (in ohms) are related by the formula E � I � Z. Find the impedance in a circuit with voltage 12 � 2j volts and current 3 � 5j amps.

8. Write a quadratic equation with �4 and �32� as its roots. 8.


2x2 � 5x � 12

9 � 6j ohms

8 in. by 18 in.

y

xO

1, �3

minimum; �17

xO

f(x )

(0, 3)

(2, �1)

x � 2f(x) � x2 �4x � 3

NAME DATE PERIOD

SCORE

Ass

essm

ent

Chapter 5 Test, Form 2D


�2137� � �

2177�j ohms

��1, �43��

066-086 A2-05-873975 5/12/06 2:08 PM Page 79

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9. Solve 9x2 � 12x � 4 � 6 by using the Square Root Property. 9.


10. x2 � 8x � 14 � 0 10.

11. 3x2 � x � 2 � 0 11.

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

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

13. 25x2 � 20x � 4 � 0 13.

14. 2x2 � 10x � 9 � 2x 14.

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

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

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

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

19. Graph y � x2 � 4x � 4.19.

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

Bonus Write a quadratic equation with roots ��45�

�. B:

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

16x2 � 5 � 0

y

xO

NAME DATE PERIOD


Chapter 5 Test, Form 2D (continued)

NAME DATE PERIOD


(6,�5); x � 6; down

y � ��14

�(x � 4)2 � 2

y � (x � 2)2 � 4

��2 �3

�6��

��9 �4�41��

�4 � �2��

��1, �23

��

0; 1 real, rational root

h(t ) � �16(t � 2)2 � 76; 76 ft

�x ��32

� � x � 5�

066-086 A2-05-873975 5/12/06 1:48 PM Page 80

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

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

34�x2

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

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

For Questions 4 and 5, solve each equation by graphing. 4.If exact roots cannot be found, state the consecutive integers between which the roots are located.

4. 0.5x2 � 9 � 4.5x

5. �23�x � 3 � �

13�x2 5.

6. Solve 18x2 � 15 � 39x by factoring. 6.

7. Write a quadratic equation with ��23� and 1.75 as its roots. 7.


8. Simplify (5 � i) � (2 � 4i) � (3 � i). 8.

9. Simplify �22��

ii

��

5�5�� . 9.

4 � 6i

y

xO

2

2

y

xO

2

2

3, 6

$8.00; $6400

NAME DATE PERIOD

SCORE

Ass

essm

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Chapter 5 Test, Form 3


xO

f(x )

(0, 3)(� ),1

383

f(x) � 3x2 � 2x � 3

� 13x �

� �19

� � �49�5�

� i

minimum; �2225�

12x2 � 13x � 14 � 0

between �3 and �2;between 4 and 5

��12

�, �53

��

066-086 A2-05-873975 5/15/06 1:10 PM Page 81

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

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

11. 2x2 � �52�x � 2 � 0 11.

12. x2 � 2.5x � 3 � 0.5 12.

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

Quadratic Formula.

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

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

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

y � ��35��x � �

12��

2� �

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

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

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

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

vertex as y � �13�x2 � 6x � �

823� and x-intercept 1.

19. Graph y � �(x2 � 2x) � 5.25. 19.

20. Solve �x � �72��(x � 1)2 � 0. 20.

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

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

y

xO

�2 � k � 2

6 � 4�2�

{�3.5, 1}

{�0.35, 0.85}

NAME DATE PERIOD


Chapter 5 Test, Form 3 (continued)

NAME DATE PERIOD

��5 � i8��39�

y � ��35� �x � �

72��2 � �

12�;

��72�, ��

12��; x � ��

72�;

down

1.2; two real,irrational roots

y � ��22090

�(x � 9)2 � �229�

16x2 � 24x � 29 � 0

�x �x � ��72

� or x � 1�

h(t ) � �9.1(t � 32.5)2 �30,000; 30,000 ft

066-086 A2-05-873975 5/12/06 1:50 PM Page 82

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

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

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

problem.

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

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

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

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

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

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

Ass

essm

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Chapter 5 Extended-Response Test


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NAME DATE PERIOD

SCORE

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Standardized Test Practice (Chapters 1–5)

1. If �ab� � �

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

A 16b B 12b C �32b� D �

83�b 1.

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

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

4. Which of the following is the greatest?

F �23� G �

79� H �

1105�

J �181�

4.

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

A 7b B b C �72b� D 14b 5.

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

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

A T � (xd � c) B T � �xcd� C �xd

T� c� D �

T �c

xd� 7.

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

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

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

A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

NAME DATE PERIOD

6

3

15

Part 1: Multiple Choice

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


NAME DATE PERIOD

066-086 A2-05-873975 5/12/06 2:14 PM Page 84

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NAME DATE PERIOD

Ass

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Standardized Test Practice (continued)

15. The bar graph shows the distribution of votes 15.among the candidates for senior class president.If 220 seniors voted in all, how many students voted for either Theo or Pam?

16. Find the median of x, 2x � 1, �2x

� � 13, 45, and 16.

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

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

08

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

451

Part 2: Griddable

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

JoeyAnaPamTheo

20

30

10

0

40

Perc

ent

of

vote

s re

ceiv

ed 50

Candidates

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

2?

A �3 B 12 �14� C 6 �

14� D �12 �

14� 11.

12. All four walls of a rectangular room that is 14 feet wide,20 feet long, and 8 feet high, are to be painted. What is the minimum cost of paint if one gallon covers at most 130 square feet and the paint costs $22 per gallon?F $92 G $102 H $110 J $190 12.

13. If i2 � �1, then what is the value of i32?

A �1 B 1 C �i D i 13.

14. Which of the following is the sum of both solutions of the equation x2 � x �42 � 0?

F 13 G �1 H �13 J 1 14. F G H J

A B C D

F G H J

A B C D

066-086 A2-05-873975 5/12/06 2:14 PM Page 85

NAME DATE PERIOD


NAME DATE PERIOD NAME DATE PERIOD

17. Find the value of 12 � 36 4 � (5 � 7)2. 17.

18. Find the slope of the line that is parallel to the line with 18.equation 3x � 4y � 10.

19. Describe the system 2x � 3y � 21 and y � 5 � �23�x as 19.

consistent and independent, consistent and dependent, or inconsistent.

20. Find the coordinates of the vertices of the figure formed 20.by the system of inequalities.x � �2 x � y � 6y � �2 x � y � �2

21. Find the value of � �. 21.

22. Solve � � � � � by using inverse matrices. 22.

23. Solve 2x2 � 40 � 0. 23.

24. PHYSICS An object is thrown straight up from the top of 24.a 100-foot platform at a velocity of 48 feet per second. The

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

25. Solve x2 � 2x � 3 by graphing.

25.

26. Solve 4x2 � 4x � 24 by factoring. 26.

27. Find the value of the discriminant for 7x2 � 5x � 1 � 0. 27.Then describe the number and type of roots for the equation.

28. Use y � x2 � 7x � 5 for parts a�c.

a. Write the equation in vertex form. 28a.

b. Identify the vertex. 28b.

c. Identify the axis of symmetry. 28c.

(�2, 3)

y

xO

�1, 3

136 ft; 1.5 s

�2i �5�

(2, �3)11�13

ab

�13

4�2

92124

5�6

inconsistent

17

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Standardized Test Practice (continued)

(�2, 0), (�2, 8),(0, �2), (8, �2)

��34

�

�3, 2 complex roots

y � �x � �72

��2

� �249�

��72

� , ��249��

x � �72

�

Part 3: Short Answer

Instructions: Write your answers in the space provided.

066-086 A2-05-873975 5/11/06 7:26 AM Page 86

Chapter 5 A1 Glencoe Algebra 2

An

swer

s

Answers (Anticipation Guide and Lesson 5-1)

STEP

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Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

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,w

rite

NS

(N

ot S

ure

).

Aft

er y

ou c

omp

lete

Ch

ap

ter

5

•R

erea

d ea

ch s

tate

men

t an

d co

mpl

ete

the

last

col

um

n b

y en

teri

ng

an A

or a

D.

•D

id a

ny

of y

our

opin

ion

s ab

out

the

stat

emen

ts c

han

ge f

rom

th

e fi

rst

colu

mn

?

•F

or t

hos

e st

atem

ents

th

at y

ou m

ark

wit

h a

D, u

se a

pie

ce o

f pa

per

to w

rite

an

exa

mpl

e of

wh

y yo

u d

isag

ree.

STEP

2

ST

EP

1S

TE

P 2

A,D

,or

NS

Sta

tem

ent

A o

r D

1.A

ll q

uad

rati

c fu

nct

ion

s h

ave

a te

rm w

ith

th

e va

riab

le t

o th

e se

con

d po

wer

.A

2.If

th

e gr

aph

of

the

quad

rati

c fu

nct

ion

y�

ax2

�c

open

s u

p th

en c

�0.

D

3.A

quad

rati

c eq

uat

ion

wh

ose

grap

h d

oes

not

in

ters

ect

the

x-ax

is h

as n

o re

al s

olu

tion

. A

4.S

ince

gra

phin

g sh

ows

the

exac

t so

luti

ons

to a

qu

adra

tic

equ

atio

n, n

o ot

her

met

hod

is

nec

essa

ry f

or s

olvi

ng.

D

5.If

(x

�3)

(x�

4) �

0, t

hen

eit

her

x�

3 �

0 or

x�

4 �

0.

A6.

An

im

agin

ary

nu

mbe

r co

nta

ins

i, w

hic

h e

qual

s th

e sq

uar

e ro

ot o

f �

1.A

7.A

met

hod

cal

led

com

plet

ing

the

squ

are

can

be

use

d to

re

wri

te a

qu

adra

tic

expr

essi

on a

s a

perf

ect

squ

are.

A

8.T

he

quad

rati

c fo

rmu

la c

an o

nly

be

use

d fo

r qu

adra

tic

equ

atio

ns

that

can

not

be

solv

ed b

y gr

aph

ing

or

Dco

mpl

etin

g th

e sq

uar

e.

9.T

he

disc

rim

inan

t of

a q

uad

rati

c eq

uat

ion

can

be

use

d to

de

term

ine

the

dire

ctio

n t

he

grap

h w

ill

open

.D

10.

Th

e gr

aph

of

y�

2x2

is a

dil

atio

n o

f th

e gr

aph

of

y�

x2 .

A11

.T

he

grap

h o

f y

�(x

�2)

2w

ill

be t

wo

un

its

to t

he

righ

t of

th

e gr

aph

of

y�

x2.

D

12.

Th

e gr

aph

of

a qu

adra

tic

ineq

ual

ity

con

tain

ing

the

sym

bol

�w

ill

be a

par

abol

a op

enin

g do

wn

war

d.D

Lesson 5-1

Cha

pter

55

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-1


Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 5

-1 i

n y

our

text

boo

k.

•B

ased

on

the

grap

h in

you

r te

xtbo

ok, f

or w

hat

tick

et p

rice

is

the

inco

me

the

grea

test

?$4

0•

Use

th

e gr

aph

to

esti

mat

e th

e m

axim

um

in

com

e.ab

ou

t $7

2,00

0

Rea

d t

he

Less

on

1.a.

For

th

e qu

adra

tic

fun

ctio

n f

(x)

�2x

2�

5x�

3, 2

x2is

th

e te

rm,

5xis

th

e te

rm, a

nd

3 is

th

e te

rm.

b.

For

th

e qu

adra

tic

fun

ctio

n f

(x)

��

4 �

x�

3x2 ,

a�

, b�

, an

d

c�

.

2.C

onsi

der

the

quad

rati

c fu

nct

ion

f(x

) �

ax2

�bx

�c,

wh

ere

a�

0.

a.T

he

grap

h o

f th

is f

un

ctio

n i

s a

.

b.

Th

e y-

inte

rcep

t is

.

c.T

he

axis

of

sym

met

ry i

s th

e li

ne

.

d.

If a

�0,

th

en t

he

grap

h o

pen

s an

d th

e fu

nct

ion

has

a

valu

e.

e.If

a�

0, t

hen

th

e gr

aph

ope

ns

and

the

fun

ctio

n h

as a

valu

e.

3.R

efer

to

the

grap

h a

t th

e ri

ght

as y

ou c

ompl

ete

the

foll

owin

g se

nte

nce

s.

a.T

he

curv

e is

cal

led

a .

b.

Th

e li

ne

x�

�2

is c

alle

d th

e .

c.T

he

poin

t (�

2, 4

) is

cal

led

the

.

d.

Bec

ause

th

e gr

aph

con

tain

s th

e po

int

(0, �

1), �

1 is

the

.

Rem

emb

er W

hat

Yo

u L

earn

ed4.

How

can

you

rem

embe

r th

e w

ay t

o u

se t

he

x2te

rm o

f a

quad

rati

c fu

nct

ion

to

tell

wh

eth

er t

he

fun

ctio

n h

as a

max

imu

m o

r a

min

imu

m v

alu

e?S

amp

le a

nsw

er:

Rem

emb

er t

hat

th

e g

rap

h o

f f(

x) �

x2

(wit

h a

�0)

is a

U-s

hap

ed c

urv

eth

at o

pen

s u

p a

nd

has

a m

inim

um

.Th

e g

rap

h o

f g

(x)

��

x2

(wit

h a

�0)

is ju

st t

he

op

po

site

.It

op

ens

do

wn

an

d h

as a

max

imu

m.

y-in

terc

ept

vert

ex

axis

of

sym

met

ry

par

abo

la

x

f(x)

O ( 0, –

1)

( –2,

4)

max

imu

md

ow

nw

ard

min

imu

mu

pw

ard

x�

�� 2b a�

c

par

abo

la

�4

1�

3

con

stan

tlin

ear

qu

adra

tic

Less

on R

eadi

ng G

uide

Gra

ph

ing

Qu

adra

tic

Fu

nct

ion

s

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A1


Answers (Lesson 5-1)

Cha

pter

56

Gle

ncoe

Alg

ebra

2

5-1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Exer

cise

s

Exam

ple

Stud

y G

uide

and

Inte

rven

tion

Gra

ph

ing

Qu

adra

tic

Fu

nct

ion

sG

rap

h Q

uad

rati

c Fu

nct

ion

s

Qu

adra

tic

Fu

nct

ion

Afu

nctio

n de

fined

by

an e

quat

ion

of t

he f

orm

f(x

) �

ax2

�bx

�c,

whe

re a

�0

Gra

ph

of

a Q

uad

rati

cA

par

abo

law

ith t

hese

cha

ract

eris

tics:

yin

terc

ept:

c;

axis

of

sym

met

ry:

x�

;F

un

ctio

nx-

coor

dina

te o

f ve

rtex

:

Fin

d t

he

y-in

terc

ept,

th

e eq

uat

ion

of

the

axis

of

sym

met

ry, a

nd

th

ex-

coor

din

ate

of t

he

vert

ex f

or t

he

grap

h o

f f(

x) �

x2�

3x�

5. U

se t

his

in

form

atio

nto

gra

ph

th

e fu

nct

ion

.

a�

1, b

��

3, a

nd

c�

5, s

o th

e y-

inte

rcep

t is

5. T

he

equ

atio

n o

f th

e ax

is o

f sy

mm

etry

is

x�

or

. Th

e x-

coor

din

ate

of t

he

vert

ex i

s .

Nex

t m

ake

a ta

ble

of v

alu

es f

or x

nea

r .

xx

2�

3x�

5f(

x)

(x,f

(x))

002

�3(

0) �

55

(0,

5)

112

� 3

(1)

�5

3(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, t

he

equ

atio

n o

f th

e ax

is o

f sy

mm

etry

, an

d t

he

x-co

ord

inat

eof

th

e ve

rtex

.b

.M

ake

a ta

ble

of

valu

es t

hat

in

clu

des

th

e ve

rtex

.c.

Use

th

is i

nfo

rmat

ion

to

grap

h t

he

fun

ctio

n.

1.f(

x) �

x2�

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

3

f(x

)1

33

9

x�

10

�2

1

f(x

)3

22

�1

x�

3�

2�

1�

4

f(x

)�

10

30

11 � 43 � 2

11 � 43 � 2

3 � 23 � 2

x

f(x)

O

3 � 2

3 � 23 � 2

�(�

3)�

2(1)

�b

� 2a

�b

� 2a

Exer

cise

s

Exam

ple

Lesson 5-1

Cha

pter

57

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-1


Max

imu

m a

nd

Min

imu

m V

alu

esT

he

y-co

ordi

nat

e of

th

e ve

rtex

of

a qu

adra

tic

fun

ctio

n i

s th

e m

axim

um

or

min

imu

m v

alu

e of

th

e fu

nct

ion

.

Max

imu

m o

r M

inim

um

Val

ue

The

gra

ph o

f f(

x)

�ax

2�

bx�

c, w

here

a�

0, o

pens

up

and

has

a m

inim

umo

f a

Qu

adra

tic

Fu

nct

ion

whe

n a

�0.

The

gra

ph o

pens

dow

n an

d ha

s a

max

imum

whe

n a

�0.

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

has

a m

axim

um

or

min

imu

mva

lue,

an

d f

ind

th

e m

axim

um

or

min

imu

m v

alu

e of

eac

h f

un

ctio

n. T

hen

sta

te t

he

dom

ain

an

d r

ange

of

the

fun

ctio

n.

a.f(

x) �

3x2

�6x

�7

For

th

is f

un

ctio

n, a

�3

and

b�

�6.

Sin

ce a

�0,

th

e gr

aph

ope

ns

up,

an

d th

efu

nct

ion

has

a m

inim

um

val

ue.

Th

e m

inim

um

val

ue

is t

he

y-co

ordi

nat

eof

th

e ve

rtex

. Th

e x-

coor

din

ate

of t

he

vert

ex i

s �

��

1.

Eva

luat

e th

e fu

nct

ion

at

x�

1 to

fin

d th

em

inim

um

val

ue.

f(1)

�3(

1)2

�6(

1) �

7 �

4, s

o th

em

inim

um

val

ue

of t

he

fun

ctio

n i

s 4.

Th

edo

mai

n i

s al

l re

al n

um

bers

. Th

e ra

nge

is

all

real

s gr

eate

r th

an o

r eq

ual

to

the

min

imu

m v

alu

e, t

hat

is

{f(x

) | f

(x)

�4}

.

�6

� 2(3)

�b

� 2a

b. f

(x)

�10

0 �

2x�

x2

For

th

is f

un

ctio

n, a

��

1 an

d b

��

2.S

ince

a�

0, t

he

grap

h o

pen

s do

wn

, an

dth

e fu

nct

ion

has

a m

axim

um

val

ue.

Th

e m

axim

um

val

ue

is t

he

y-co

ordi

nat

e of

the

vert

ex. T

he

x-co

ordi

nat

e of

th

e ve

rtex

is

�

��

�1.

Eva

luat

e th

e fu

nct

ion

at

x�

�1

to f

ind

the

max

imu

m v

alu

e.f(

�1)

�10

0 �

2(�

1) �

(�1)

2�

101,

so

the

min

imu

m v

alu

e of

th

e fu

nct

ion

is

101.

Th

e do

mai

n i

s al

l re

al n

um

bers

. Th

era

nge

is

all

real

s le

ss t

han

or

equ

al t

o th

em

axim

um

val

ue,

th

at i

s {f

(x)

| f(x

)

101}

.

�2

� 2(�

1)�

b� 2a

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

has

a m

axim

um

or

min

imu

m v

alu

e, a

nd

fin

d t

he

max

imu

m o

r m

inim

um

val

ue.

Th

en s

tate

th

e d

omai

n a

nd

ran

ge o

f th

e fu

nct

ion

.

1.f(

x) �

2x2

�x

�10

2.f(

x) �

x2�

4x�

73.

f(x)

�3x

2�

3x�

1

min

.,9

;al

l rea

ls;

min

.,�

11;

all r

eals

;m

in.,

;al

l rea

ls;

{f(x

)|f(x

) �

9}

{f(x

) |f(x

) �

�11

}{f

(x)|f

(x)

�}

4.f(

x) �

16 �

4x�

x2

5.f(

x) �

x2�

7x�

116.

f(x)

��

x2�

6x�

4

max

.,20

;al

l rea

ls;

min

.,�

;al

l rea

ls;

max

.,5;

all r

eals

;{f

(x)|f

(x)

�20

}{f

(x)|f

(x)

��

}{f

(x) |f

(x)

�5}

7.f(

x) �

x2�

5x�

28.

f(x)

�20

�6x

�x2

9.f(

x) �

4x2

�x

�3

min

.,�

;al

l rea

ls;

max

.,29

;al

l rea

ls;

min

.,2

;al

l rea

ls;

{f(x

)|f(x

) �

�}

{f(x

) |f(x

) �

29}

{f(x

)|f(x

) �

2}

10.f

(x)

��

x2�

4x�

1011

.f(x

) �

x2�

10x

�5

12.f

(x)

��

6x2

�12

x�

21m

ax.,

14;

all r

eals

;m

in.,

�20

;al

l rea

ls;

max

.,27

;al

l rea

ls;

{f(x

)|f(x

) �

14}

{f(x

)|f(x

) �

�20

}{f

(x)|f

(x)

�27

}25 � 2617 � 4

15 � 1617 � 4

5 � 4

5 � 4

1 � 47 � 8

1 � 47 � 8St

udy

Gui

de a

nd In

terv

enti

on(c

ontin

ued)

Gra

ph

ing

Qu

adra

tic

Fu

nct

ion

s

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A2


An

swer

s


Cha

pter

58

Gle

ncoe

Alg

ebra

2


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

For

eac

h q

uad

rati

c fu

nct

ion

, fin

d t

he

y-in

terc

ept,

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.

1.f(

x) �

3x2

2.f(

x) �

x2�

13.

f(x)

��

x2�

6x�

150;

x�

0;0

1;x

�0;

0�

15;

x�

3;3

4.f(

x) �

2x2

�11

5.f(

x) �

x2�

10x

�5

6.f(

x) �

�2x

2�

8x�

7�

11;

x�

0;0

5;x

�5;

57;

x�

2;2

Com

ple

te p

arts

a–c

for

eac

h q

uad

rati

c fu

nct

ion

.a.

Fin

d t

he

y-in

terc

ept,

th

e eq

uat

ion

of

the

axis

of

sym

met

ry, a

nd

th

e x-

coor

din

ate

of t

he

vert

ex.

b.

Mak

e a

tab

le o

f va

lues

th

at i

ncl

ud

es t

he

vert

ex.

c.U

se t

his

in

form

atio

n t

o gr

aph

th

e fu

nct

ion

.

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,

an

d f

ind

the

max

imu

m o

r m

inim

um

val

ue.

Th

en s

tate

th

e d

omai

n a

nd

ran

ge o

f th

e fu

nct

ion

.

10.f

(x)

�6x

211

.f(x

) �

�8x

212

.f(x

) �

x2�

2xm

in.;

0;al

l rea

ls;

max

.;0;

all r

eals

;m

in.;

�1;

all r

eals

;{f

(x)|f

(x)

�0}

{f(x

)|f(x

) �

0}{f

(x)|f

(x)

��

1}13

.f(x

) �

x2�

2x�

1514

.f(x

) �

�x2

�4x

�1

15.f

(x)

�x2

�2x

�3

min

.;14

;al

l rea

ls;

max

.;3;

all r

eals

;m

in.;

�4;

all r

eals

;{f

(x)|f

(x)

�14

}{f

(x)|f

(x)

�3}

{f(x

)|f(x

) �

�4}

16.f

(x)

��

2x2

�4x

�3

17.f

(x)

�3x

2�

12x

�3

18.f

(x)

�2x

2�

4x�

1m

ax.;

�1;

all r

eals

;m

in.;

�9;

all r

eals

;m

in.;

�1;

all r

eals

;{f

(x) |f

(x)

��

1}{f

(x)|f

(x)

��

9}{f

(x)|f

(x)

��

1}

( 3, –

1)x

f(x)

O( 2

, 0)

x

f(x)

O16 12 8 4

2–2

46

( 0, 0

)x

f(x)

O

x0

23

46

f(x

)8

0�

10

8

x�

20

24

6

f(x

)16

40

416

x�

2�

10

12

f(x

)�

8�

20

�2

�8

5-1

Skill

s Pr

acti

ceG

rap

hin

g Q

uad

rati

c F

un

ctio

ns

Lesson 5-1

Cha

pter

59

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-1


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,

th

e eq

uat

ion

of

the

axis

of

sym

met

ry, a

nd

th

e x-

coor

din

ate

of t

he

vert

ex.

b.

Mak

e a

tab

le o

f va

lues

th

at i

ncl

ud

es t

he

vert

ex.

c.U

se t

his

in

form

atio

n t

o gr

aph

th

e fu

nct

ion

.

1.f(

x) �

x2�

8x�

152.

f(x)

��

x2�

4x�

123.

f(x)

�2x

2�

2x�

115

;x

�4;

412

;x

��

2;�

21;

x�

0.5;

0.5

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

has

a m

axim

um

or

a m

inim

um

val

ue,

an

d f

ind

the

max

imu

m o

r m

inim

um

val

ue

of e

ach

fu

nct

ion

. Th

en s

tate

th

e d

omai

n a

nd

ran

ge o

f th

e fu

nct

ion

.

4.f(

x) �

x2�

2x�

85.

f(x)

�x2

�6x

�14

6.v(

x) �

�x2

�14

x�

57m

in.;

�9;

all r

eals

;m

in.;

5;al

l rea

ls;

max

.;�

8;al

l rea

ls;

{f(x

)|f(x

) �

�9}

{f(x

)|f(x

) �

5}{f

(x)|f

(x)

��

8}

7.f(

x) �

2x2

�4x

�6

8.f(

x) �

�x2

�4x

�1

9.f(

x) �

��2 3� x

2�

8x�

24m

in.;

�8;

all r

eals

;m

ax.;

3;al

l rea

ls;

max

.;0;

all r

eals

;{f

(x)|f

(x)

��

8}{f

(x)|f

(x)

�3}

{f(x

)|f(x

) �

0}

10.G

RA

VIT

ATI

ON

Fro

m 4

fee

t ab

ove

a sw

imm

ing

pool

, Su

san

th

row

s a

ball

upw

ard

wit

h a

velo

city

of

32 f

eet

per

seco

nd.

Th

e h

eigh

t h

(t)

of t

he

ball

tse

con

ds a

fter

Su

san

th

row

s it

is g

iven

by

h(t

) �

�16

t2�

32t

�4.

Fin

d th

e m

axim

um

hei

ght

reac

hed

by

the

ball

an

dth

e ti

me

that

th

is h

eigh

t is

rea

ched

.20

ft;

1 s

11.H

EALT

H C

LUB

SL

ast

year

, th

e S

port

sTim

e A

thle

tic

Clu

b ch

arge

d $2

0 to

par

tici

pate

in

an a

erob

ics

clas

s. S

even

ty p

eopl

e at

ten

ded

the

clas

ses.

Th

e cl

ub

wan

ts t

o in

crea

se t

he

clas

s pr

ice

this

yea

r. T

hey

exp

ect

to l

ose

one

cust

omer

for

eac

h $

1 in

crea

se i

n t

he

pric

e.

a.W

hat

pri

ce s

hou

ld t

he

clu

b ch

arge

to

max

imiz

e th

e in

com

e fr

om t

he

aero

bics

cla

sses

?$4

5b

.W

hat

is

the

max

imu

m i

nco

me

the

Spo

rtsT

ime

Ath

leti

c C

lub

can

exp

ect

to m

ake?

$202

5

f(x)

xO

( 0.5

, 0.5

)

16 12 8 4

( –2,

16)

x

f(x)

O2

–2–4

–6( 4

, –1)

x

f(x)

O16 12 8 4

24

68

x�

10

0.5

12

f(x

)5

10.

51

5

x�

6�

4�

20

2

f(x

)0

1216

120

x0

24

68

f(x

)15

3�

13

15

Prac

tice

Gra

ph

ing

Qu

adra

tic

Fu

nct

ion

s

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A3



Cha

pter

510

Gle

ncoe

Alg

ebra

2

5-1


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Wor

d Pr

oble

m P

ract

ice

Gra

ph

ing

Qu

adra

tic

Fu

nct

ion

s1.

TRA

JEC

TOR

IES

Aca

nn

onba

ll i

sla

un

ched

fro

m a

can

non

at

the

top

of a

clif

f. I

f th

e pa

th o

f th

e ca

nn

onba

ll i

str

aced

on

api

ece

of g

raph

pape

r al

ign

ed s

oth

at t

he

can

non

is s

itu

ated

on

th

e y-

axis

, th

eeq

uat

ion

th

atde

scri

bes

the

path

is y

��

� 161 00�

x2�

�1 2� x �

47,

wh

ere

xis

th

e h

oriz

onta

l di

stan

ce f

rom

the

clif

f an

d y

is t

he

vert

ical

dis

tan

ceab

ove

the

grou

nd

in f

eet.

How

hig

hab

ove

the

grou

nd

is t

he

can

non

?47

ft

2.TI

CK

ETIN

GT

he

man

ager

of

asy

mph

ony

com

pute

s th

at t

he

sym

phon

yw

ill

earn

�40

P2

�11

00P

doll

ars

per

con

cert

if

they

ch

arge

Pdo

llar

s fo

rti

cket

s. W

hat

tic

ket

pric

e sh

ould

th

esy

mph

ony

char

ge i

n o

rder

to

max

imiz

eit

s pr

ofit

s?$1

3.75

3.A

RC

HES

An

arc

hit

ect

deci

des

to u

se a

para

boli

c ar

ch f

or t

he

mai

n e

ntr

ance

of

asc

ien

ce m

use

um

. In

on

e of

his

pla

ns,

th

eto

p ed

ge o

f th

e ar

ch i

s de

scri

bed

by t

he

grap

h o

f y

��

x2�

x �

. Wh

at

are

the

coor

din

ates

of

the

vert

ex o

f th

ispa

rabo

la?

(5,2

5)

4.FR

AM

ING

Afr

ame

com

pan

y of

fers

ali

ne

of s

quar

e fr

ames

. If

the

side

len

gth

of t

he

fram

e is

s, t

hen

th

e ar

ea o

f th

eop

enin

g in

th

e fr

ame

is g

iven

by

the

fun

ctio

n a

(s)

�s2

�10

s�

24.

Gra

ph a

(s).

WA

LKIN

GF

or E

xerc

ises

5–7

, use

th

efo

llow

ing

info

rmat

ion

.C

anal

Str

eet

and

Wal

ker

Str

eet

are

perp

endi

cula

r to

eac

h o

ther

. Evi

ta i

s dr

ivin

gso

uth

on

Can

al S

tree

t an

d is

cu

rren

tly

5m

iles

nor

th o

f th

e in

ters

ecti

on w

ith

Wal

ker

Str

eet.

Jac

k is

at

the

inte

rsec

tion

of

Can

alan

d W

alke

r S

tree

ts a

nd

hea

din

g ea

st o

nW

alke

r. J

ack

and

Evi

ta a

re b

oth

dri

vin

g 30

mil

es p

er h

our.

5.W

hen

Jac

k is

xm

iles

eas

t of

th

ein

ters

ecti

on, w

her

e is

Evi

ta?

5 �

xm

i no

rth

of

the

inte

rsec

tio

n

6.T

he

dist

ance

bet

wee

n J

ack

and

Evi

ta i

sgi

ven

by

the

form

ula

�x2

�(

�5

�x)

�2 �.

For

wh

at v

alu

e of

xar

e Ja

ck a

nd

Evi

taat

th

eir

clos

est?

(H

int:

Min

imiz

e th

esq

uar

e of

th

e di

stan

ce.)

x�

2.5

7.W

hat

is

the

dist

ance

of

clos

est

appr

oach

?

mi

5�2�

�2

a

sO5

5

75 � 45 � 2

1 � 4

Exer

cise

s

Exam

ple

Lesson 5-1

Cha

pter

511

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-1


Fin

din

g t

he

Axi

s o

f S

ymm

etry

of

a P

arab

ola

As

you

kn

ow, i

f f(

x) �

ax2

�bx

�c

is a

qu

adra

tic

fun

ctio

n, t

he

valu

es o

f x

that

mak

e f(

x) e

qual

to

zero

are

an

d .

Th

e av

erag

e of

th

ese

two

nu

mbe

r va

lues

is

�� 2b a�

.

Th

e fu

nct

ion

f(x

) h

as i

ts m

axim

um

or

min

imu

m

valu

e w

hen

x�

�� 2b a�

. Sin

ce t

he

axis

of

sym

met

ry

of t

he

grap

h o

f f(

x) p

asse

s th

rou

gh t

he

poin

t w

her

e th

e m

axim

um

or

min

imu

m o

ccu

rs, t

he

axis

of

sym

met

ry h

as t

he

equ

atio

n x

��

� 2b a�.

Fin

d t

he

vert

ex a

nd

axi

s of

sym

met

ry f

or f

(x)

�5x

2�

10x

�7.

Use

x�

�� 2b a�

.

x�

�� 21 (0 5)�

��

1T

he

x-co

ordi

nat

e of

th

e ve

rtex

is

�1.

Su

bsti

tute

x�

�1

in f

(x)

�5x

2�

10x

�7.

f(�

1) �

5(�

1)2

�10

(�1)

�7

��

12T

he

vert

ex i

s (�

1,�

12).

Th

e ax

is o

f sy

mm

etry

is

x�

�� 2b a�

, or

x�

�1.

Fin

d t

he

vert

ex a

nd

axi

s of

sym

met

ry f

or t

he

grap

h o

f ea

ch f

un

ctio

n

usi

ng

x�

�� 2b a�

.

1.f(

x) �

x2�

4x�

8(2

,�12

);x

�2

2.g(

x) �

�4x

2�

8x�

3(�

1,7)

;x

��

1

3.y

��

x2�

8x�

3(4

,19)

;x

�4

4.f(

x) �

2x2

�6x

�5

��3 2� ,

�1 2� �;x

��

�3 2�

5.A

(x)

�x2

�12

x�

36(�

6,0)

;x

��

66.

k(x)

��

2x2

�2x

�6

��1 2� ,�

5�1 2� �;

x�

�1 2�

O

f(x)

x

––

,f

((

((

b –– 2a b –– 2a

b –– 2ax

= –

f(x

) =

ax

2 +

bx

+ c

�b

��

b2�

4�

ac��

��

2a�

b�

�b2

�4

�ac�

��

�2a

Enri

chm

ent

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A4


An

swer

s


Cha

pter

512

Gle

ncoe

Alg

ebra

2

5-2


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 5

-2 i

n y

our

text

boo

k.

Wri

te a

qu

adra

tic

fun

ctio

n t

hat

des

crib

es t

he

hei

ght

of a

bal

l t

seco

nds

aft

er i

t is

dr

oppe

d fr

om a

hei

ght

of 1

25 f

eet.

h(t

) �

�16

t2�

125

Rea

d t

he

Less

on

1.T

he

grap

h o

f th

e qu

adra

tic

fun

ctio

n f

(x)

��

x2�

x�

6 is

sh

own

at

the

righ

t. U

se t

he

grap

h t

o fi

nd

the

solu

tion

s of

th

equ

adra

tic

equ

atio

n �

x2�

x�

6 �

0.�

2 an

d 3

2.S

ketc

h a

gra

ph t

o il

lust

rate

eac

h s

itu

atio

n.

a.A

para

bola

th

at o

pen

s b

.A

para

bola

th

at o

pen

s c.

Apa

rabo

la t

hat

ope

ns

dow

nw

ard

and

repr

esen

ts a

u

pwar

d an

d re

pres

ents

a

dow

nw

ard

and

qu

adra

tic

fun

ctio

n w

ith

tw

o qu

adra

tic

fun

ctio

n w

ith

re

pres

ents

a

re

al z

eros

, bot

h o

f w

hic

h a

reex

actl

y on

e re

al z

ero.

Th

e

quad

rati

c fu

nct

ion

n

egat

ive

nu

mbe

rs.

zero

is

a po

siti

ve n

um

ber.

w

ith

no

real

zer

os.

Rem

emb

er W

hat

Yo

u L

earn

ed

3.T

hin

k of

a m

emor

y ai

d th

at c

an h

elp

you

rec

all

wh

at i

s m

ean

t by

th

e ze

ros

of a

qu

adra

tic

fun

ctio

n.

Sam

ple

an

swer

:Th

e b

asic

fac

ts a

bo

ut

a su

bje

ct a

re s

om

etim

es c

alle

d t

he

AB

Cs.

In t

he

case

of

zero

s,th

e A

BC

s ar

e th

e X

YZ

s,b

ecau

se t

he

zero

sar

e th

e x-

valu

es t

hat

mak

e th

e y-

valu

es e

qu

al t

o z

ero

.

x

y

Ox

y

Ox

y

O

x

y

O

Less

on R

eadi

ng G

uide

So

lvin

g Q

uad

rati

c E

qu

atio

ns

by G

rap

hin

g

Exer

cise

s

Stud

y G

uide

and

Inte

rven

tion

So

lvin

g Q

uad

rati

c E

qu

atio

ns

by G

rap

hin

g

Lesson 5-2

Cha

pter

513

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-2


Solv

e Q

uad

rati

c Eq

uat

ion

s

Qu

adra

tic

Eq

uat

ion

Aqu

adra

tic e

quat

ion

has

the

form

ax

2�

bx�

c�

0, w

here

a�

0.

Ro

ots

of

a Q

uad

rati

c E

qu

atio

nso

lutio

n(s)

of

the

equa

tion,

or

the

zero

(s)

of t

he r

elat

ed q

uadr

atic

fun

ctio

n

Th

e ze

ros

of a

qu

adra

tic

fun

ctio

n a

re t

he

x-in

terc

epts

of

its

grap

h. T

her

efor

e, f

indi

ng

the

x-in

terc

epts

is

one

way

of

solv

ing

the

rela

ted

quad

rati

c eq

uat

ion

.

Sol

ve x

2�

x �

6 �

0 b

y gr

aph

ing.

Gra

ph t

he

rela

ted

fun

ctio

n f

(x)

�x2

�x

�6.

Th

e x-

coor

din

ate

of t

he

vert

ex i

s �

�, a

nd

the

equ

atio

n o

f th

e

axis

of

sym

met

ry i

s x

��

.

Mak

e a

tabl

e of

val

ues

usi

ng

x-va

lues

aro

un

d �

.

x�

1�

01

2

f(x

)�

6�

6�

6�

40

Fro

m t

he

tabl

e an

d th

e gr

aph

, we

can

see

th

at t

he

zero

s of

th

e fu

nct

ion

are

2 a

nd

�3.

Sol

ve e

ach

eq

uat

ion

by

grap

hin

g.

1.x2

�2x

�8

�0

2,�

42.

x2�

4x�

5 �

05,

�1

3.x2

�5x

�4

�0

1,4

4.x2

�10

x�

21 �

05.

x2�

4x�

6 �

06.

4x2

�4x

�1

�0

3,7

no

rea

l so

luti

on

s�

1 � 2

x

f(x)

Ox

f(x)

O

x

f(x)

O

x

f(x)

O

x

f(x)

Ox

f(x)

O

1 � 41 � 2

1 � 2

1 � 2

1 � 2�

b� 2a

x

f(x)

O

Exam

ple

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A5



Exam

ple

Exer

cise

s

Cha

pter

514

Gle

ncoe

Alg

ebra

2

5-2


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Esti

mat

e So

luti

on

sO

ften

, you

may

not

be

able

to

fin

d ex

act

solu

tion

s to

qu

adra

tic

equ

atio

ns

by g

raph

ing.

Bu

t yo

u c

an u

se t

he

grap

h t

o es

tim

ate

solu

tion

s.

Sol

ve x

2�

2x�

2 �

0 b

y gr

aph

ing.

If

exac

t ro

ots

can

not

be

fou

nd

,st

ate

the

con

secu

tive

in

tege

rs b

etw

een

wh

ich

th

e ro

ots

are

loca

ted

.

Th

e eq

uat

ion

of

the

axis

of

sym

met

ry o

f th

e re

late

d fu

nct

ion

is

x�

��

1, s

o th

e ve

rtex

has

x-c

oord

inat

e 1.

Mak

e a

tabl

e of

val

ues.

x�

10

12

3

f(x

)1

�2

�3

�2

1

Th

e x-

inte

rcep

ts o

f th

e gr

aph

are

bet

wee

n 2

an

d 3

and

betw

een

0 a

nd

�1.

So

one

solu

tion

is

betw

een

2 a

nd

3, a

nd

the

oth

er s

olu

tion

is

betw

een

0 a

nd

�1.

Sol

ve t

he

equ

atio

ns

by

grap

hin

g. I

f ex

act

root

s ca

nn

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.

1.x2

�4x

�2

�0

2.x2

�6x

�6

�0

3.x2

�4x

�2�

0

bet

wee

n 0

an

d 1

;b

etw

een

�2

and

�1;

bet

wee

n �

1 an

d 0

;b

etw

een

3 a

nd

4b

etw

een

�5

and

�4

bet

wee

n �

4 an

d �

3

4.�

x2�

2x�

4 �

05.

2x2

�12

x�

17 �

06.

�x2

�x

��

0

bet

wee

n 3

an

d 4

;b

etw

een

2 a

nd

3;

bet

wee

n �

2 an

d �

1;b

etw

een

�2

and

�1

bet

wee

n 3

an

d 4

bet

wee

n 3

an

d 4x

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

Stud

y G

uide

and

Inte

rven

tion

(con

tinue

d)

So

lvin

g Q

uad

rati

c E

qu

atio

ns

by G

rap

hin

g

Skill

s Pr

acti

ceS

olv

ing

Qu

adra

tic

Eq

uat

ion

s B

y G

rap

hin

g

Lesson 5-2

Cha

pter

515

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-2


Use

th

e re

late

d g

rap

h o

f ea

ch e

qu

atio

n t

o d

eter

min

e it

s so

luti

ons.

1.x2

�2x

�3

�0

2.�

x2�

6x�

9 �

03.

3x2

�4x

�3

�0

�3,

1�

3n

o r

eal s

olu

tio

ns

Sol

ve e

ach

eq

uat

ion

by

grap

hin

g. I

f ex

act

root

s ca

nn

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 �

0

1,5

no

rea

l so

luti

on

sb

etw

een

0 a

nd

1;

bet

wee

n 5

an

d 6

Use

a q

uad

rati

c eq

uat

ion

to

fin

d t

wo

real

nu

mb

ers

that

sat

isfy

eac

h s

itu

atio

n, o

rsh

ow t

hat

no

such

nu

mb

ers

exis

t.

7.T

hei

r su

m i

s �

4, a

nd

thei

r pr

odu

ct i

s 0.

8.T

hei

r su

m i

s 0,

an

d th

eir

prod

uct

is

�36

.

�x

2�

4x�

0;0,

�4

�x

2�

36 �

0;�

6,6

f(x) �

�x2

� 3

6

x

f (x)

O6

–612

–12

36 24 12

f(x) �

�x2

� 4

x

x

f (x)

O

f(x) �

x2

� 6

x �

4

x

f (x)

Of(x

) � �

x2 �

2x

� 4x

f (x)

O

f(x) �

x2

� 6

x �

5

x

f (x)

O

x

f(x) O

f(x) �

3x2

� 4

x �

3

x

f(x)

O

f(x) �

�x2

� 6

x �

9

x

f(x)

O

f(x) �

x2

� 2

x �

3

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A6


An

swer

s


Cha

pter

516

Gle

ncoe

Alg

ebra

2

5-2


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Use

th

e re

late

d g

rap

h o

f ea

ch e

qu

atio

n t

o d

eter

min

e it

s so

luti

ons.

1.�

3x2

�3

�0

2.3x

2�

x�

3 �

03.

x2�

3x�

2 �

0

�1,

1n

o r

eal s

olu

tio

ns

1,2

Sol

ve e

ach

eq

uat

ion

by

grap

hin

g. I

f ex

act

root

s ca

nn

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.�

2x2

�6x

�5

�0

5.x2

�10

x�

24 �

06.

2x2

�x

�6

�0

bet

wee

n 0

an

d 1

;�

6,�

4b

etw

een

�2

and

�1,

bet

wee

n �

4 an

d �

32

Use

a q

uad

rati

c eq

uat

ion

to

fin

d t

wo

real

nu

mb

ers

that

sat

isfy

eac

h s

itu

atio

n, o

rsh

ow t

hat

no

such

nu

mb

ers

exis

t.

7.T

hei

r su

m i

s 1,

an

d th

eir

prod

uct

is

�6.

8.T

hei

r su

m i

s 5,

an

d th

eir

prod

uct

is

8.

For

Exe

rcis

es 9

an

d 1

0, u

se t

he

form

ula

h(t

) �

v 0t

�16

t2, w

her

e h

(t)

is t

he

hei

ght

of a

n o

bje

ct i

n f

eet,

v0

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

60 f

eet

per

seco

nd.

Igno

ring

Mar

ta’s

heig

ht, h

ow lo

ng a

fter

she

rel

ease

s th

e ba

ll w

ill it

hit

the

gro

und?

3.75

s10

.VO

LCA

NO

ESA

volc

anic

eru

ptio

n b

last

s a

bou

lder

upw

ard

wit

h a

n i

nit

ial

velo

city

of

240

feet

per

sec

ond.

How

lon

g w

ill

it t

ake

the

bou

lder

to

hit

th

e gr

oun

d if

it

lan

ds a

t th

esa

me

elev

atio

n f

rom

wh

ich

it

was

eje

cted

?15

s

�x2

�5x

�8

�0;

no

su

ch r

eal

nu

mb

ers

exis

tx

f (x)

Of(x

) � �

x2 �

5x

� 8

�x

2�

x�

6 �

0;3,

�2

f(x) �

�x2

� x

� 6 x

f (x)

O

x

f (x)

O

f(x) �

2x2

� x

� 6

f(x) �

x2

� 1

0x �

24

x

f (x)

O

f(x) �

�2x

2 �

6x

� 5

x

f (x)

O–4

–2–6

12 8 4

x

f(x)

O

f(x) �

x2

� 3

x �

2

x

f(x) O

f(x) �

3x2

� x

� 3

x

f(x)

O

f(x) �

�3x

2 �

3Prac

tice

So

lvin

g Q

uad

rati

c E

qu

atio

ns

By

Gra

ph

ing

Lesson 5-2

Cha

pter

517

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-2


Wor

d Pr

oble

m P

ract

ice

So

lvin

g Q

uad

rati

c E

qu

atio

ns

by G

rap

hin

g1.

TRA

JEC

TOR

IES

Dav

id t

hre

w a

bas

ebal

lin

to t

he

air.

Th

e fu

nct

ion

of

the

hei

ght

ofth

e ba

seba

ll i

n f

eet

is h

�80

t�

16t2

,w

her

e t

repr

esen

ts t

he

tim

e in

sec

onds

afte

r th

e ba

ll w

as t

hro

wn

. Use

th

isgr

aph

of

the

fun

ctio

n t

o de

term

ine

how

lon

g it

too

k fo

r th

e ba

ll t

o fa

ll b

ack

toth

e gr

oun

d.

5 se

con

ds

2.B

RID

GES

Th

e m

ain

su

ppor

t fo

r a

brid

ge i

s a

larg

e pa

rabo

lic

arch

. Th

eh

eigh

t of

th

e ar

ch a

bove

th

e gr

oun

d is

give

n b

y th

e fu

nct

ion

h�

32 �

x2,

wh

ere

his

th

e h

eigh

t in

met

ers

and

xis

the

dist

ance

in

met

ers

from

th

e ce

nte

r of

the

brid

ge. G

raph

th

is e

quat

ion

an

dde

scri

be w

her

e th

e ar

ch t

ouch

es t

he

grou

nd.

40 m

fro

m t

he

cen

ter

of

the

bri

dg

e o

n e

ach

sid

e.

3.LO

GIC

Wil

ma

is t

hin

kin

g of

tw

on

um

bers

. Th

e su

m i

s 2

and

the

prod

uct

is �

24. U

se a

qu

adra

tic

equ

atio

n t

o fi

nd

the

two

nu

mbe

rs.

6 an

d �

4

4.R

AD

IO T

ELES

CO

PES

Th

e cr

oss-

sect

ion

of a

lar

ge r

adio

tel

esco

pe i

s a

para

bola

.T

he

dish

is

set

into

th

e gr

oun

d. T

he

equ

atio

n t

hat

des

crib

es t

he

cros

s-se

ctio

n

is d

�x2

�x

�

, wh

ere

dgi

ves

the

dept

h o

f th

e di

sh b

elow

gro

un

d an

dx

is t

he

dist

ance

fro

m t

he

con

trol

cen

ter,

both

in

met

ers.

If

the

dish

doe

s n

otex

ten

d ab

ove

the

grou

nd

leve

l, w

hat

is

the

diam

eter

of

the

dish

? S

olve

by

grap

hin

g.

30 m

BO

ATS

For

Exe

rcis

es 5

an

d 6

, use

th

efo

llow

ing

info

rmat

ion

.T

he

dist

ance

bet

wee

n t

wo

boat

s is

d�

�t2

�1

�0t

�3

�5�,

wh

ere

dis

dis

tan

ce i

n m

eter

s an

d t

is t

ime

in s

econ

ds.

5.M

ake

a gr

aph

of

d2

vers

us

t.

6.D

o th

e bo

ats

ever

col

lide

?N

o

d

tO

5

y

xO

4010

�6

32 � 34 � 3

2 � 75

h

xO20

�20

20

1 � 50

a

tO

12

34

5�

1

�40

4080

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A7


Answers (Lessons 5-2 and 5-3)

Cha

pter

518

Gle

ncoe

Alg

ebra

2

5-2


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Gra

ph

ing

Ab

solu

te V

alu

e E

qu

atio

ns

You

can

sol

ve a

bsol

ute

val

ue

equ

atio

ns

in m

uch

th

e sa

me

way

you

sol

ved

quad

rati

c eq

uat

ion

s. G

raph

th

e re

late

d ab

solu

te v

alu

e fu

nct

ion

for

eac

h

equ

atio

n u

sin

g a

grap

hin

g ca

lcu

lato

r. T

hen

use

th

e ZE

ROfe

atu

re i

n t

he

CALC

men

u t

o fi

nd

its

real

sol

uti

ons,

if

any.

Rec

all

that

sol

uti

ons

are

poin

ts

wh

ere

the

grap

h i

nte

rsec

ts t

he

x-ax

is.

For

eac

h e

qu

atio

n, m

ake

a sk

etch

of

the

rela

ted

gra

ph

an

d f

ind

th

e so

luti

ons

rou

nd

ed t

o th

e n

eare

st h

un

dre

dth

.

1.|x

�5|

�0

2.|4

x�

3| �

5 �

03.

|x�

7| �

0

�5

No

so

luti

on

s7

4.|x

�3|

�8

�0

5.�

|x�

3| �

6 �

06.

|x�

2| �

3 �

0

�11

,5�

9,3

�1,

5

7.|3

x �

4| �

28.

|x �

12| �

109.

|x|�

3 �

0

�2,

��2 3�

�22

,�2

�3,

3

10.E

xpla

in h

ow s

olvi

ng

abso

lute

val

ue

equ

atio

ns

alge

brai

call

y an

d fi

ndi

ng

zero

s of

abs

olu

te v

alu

e fu

nct

ion

s gr

aph

ical

ly a

re r

elat

ed.

Sam

ple

an

swer

:va

lues

of

xw

hen

so

lvin

g a

lgeb

raic

ally

are

th

e x-

inte

rcep

ts (

or

zero

s) o

f th

e fu

nct

ion

wh

en g

rap

hed

.

Enri

chm

ent

Lesson 5-3

Cha

pter

519

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3


Less

on R

eadi

ng G

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qu

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acto

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.

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ple

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swer

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arla

use

d t

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fac

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.Lar

ry u

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rs b

ut

mu

ltip

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corr

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emb

er W

hat

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u L

earn

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3.A

good

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rem

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r a

con

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epre

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mor

e th

an o

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crib

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reco

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f th

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exa

ctly

on

e x-

inte

rcep

t,th

en t

he

equ

atio

n h

as a

do

ub

le r

oo

t.

{3,7

}

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A8


An

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A1-A30 A2-05-873975 5/11/06 7:28 AM Page A9



Cha

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A1-A30 A2-05-873975 5/11/06 7:28 AM Page A10


An

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Cha

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Gle

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GN

SD

avid

was

loo

kin

g th

rou

gh a

nol

d al

gebr

a bo

ok a

nd

cam

e ac

ross

th

iseq

uat

ion

.

x26x

�8

�0

Th

e si

gn i

n f

ron

t of

th

e 6

was

blo

tted

out.

How

doe

s th

e m

issi

ng

sign

dep

end

on t

he

sign

s of

th

e ro

ots?

T

he

mis

sin

g s

ign

is t

he

sam

e as

the

sig

n o

f th

e tw

o r

oo

ts,

bec

ause

th

eir

pro

du

ct is

ap

osi

tive

nu

mb

er,8

.

3.R

OO

TSIn

th

e sa

me

alge

bra

book

th

ath

e w

as l

ooki

ng

thro

ugh

in

Exe

rcis

e 2,

Dav

id f

oun

d an

oth

er p

arti

ally

blo

tted

out

equ

atio

n.

x2�

21x

�10

0 �

0

Th

e bo

ok c

laim

s th

at o

ne

of t

he

root

s of

the

equ

atio

n i

s 4.

Wh

at m

ust

th

e ot

her

root

be

and

wh

at n

um

ber

is c

over

ed b

yth

e bl

ot?

Th

e o

ther

ro

ot

is �

25 a

nd

th

eb

lott

ed o

ut

nu

mb

er is

�10

0.

4.PR

OG

RA

MM

ING

Ray

is

a co

mpu

ter

prog

ram

mer

. He

nee

ds t

o fi

nd

the

quad

rati

c fu

nct

ion

of

this

gra

ph f

or a

nal

gori

thm

rel

ated

to

a ga

me

invo

lvin

gdi

ce. P

rovi

de s

uch

a f

un

ctio

n.

f(x)

�x

2�

18x

�77

AN

IMA

TIO

NF

or E

xerc

ises

5–7

, use

th

efo

llow

ing

info

rmat

ion

.A

com

pute

r gr

aphi

cs a

nim

ator

wou

ld l

ike

tom

ake

a re

alis

tic

sim

ulat

ion

of t

osse

d ba

ll.

The

ani

mat

or w

ants

the

bal

l to

fol

low

the

para

boli

c tr

ajec

tory

rep

rese

nted

by

the

quad

rati

c eq

uati

on f

(x)

��

0.2(

x�

5)(x

�5)

.

5.W

hat

are

th

e so

luti

ons

of f

(x)

�0?

x�

�5

or

x�

5

6.W

rite

f(x

) in

sta

nda

rd f

orm

.f(

x) �

�0.

2x2

�5

7.If

th

e an

imat

or c

han

ges

the

equ

atio

n t

of(

x) �

�0.

2x2

�20

, wh

at a

re t

he

solu

tion

s of

f(x

) �

0?x

��

10 o

r x

�10

y

x

O

Exam

ple

1

Lesson 5-3

Cha

pter

525

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-3


Usi

ng

Pat

tern

s to

Fac

tor

Stu

dy t

he

patt

ern

s be

low

for

fac

tori

ng

the

sum

an

d th

e di

ffer

ence

of

cube

s.

a3�

b3�

(a�

b)(a

2�

ab�

b2)

a3�

b3�

(a�

b)(a

2�

ab�

b2)

Th

is p

atte

rn c

an b

e ex

ten

ded

to o

ther

odd

pow

ers.

Stu

dy t

hes

e ex

ampl

es.

Fac

tor

a5

�b5

.E

xten

d th

e fi

rst

patt

ern

to

obta

in a

5�

b5�

(a�

b)(a

4�

a3b

�a2

b2�

ab3

�b4

).C

hec

k:(

a�

b)(a

4�

a3b

�a2

b2�

ab3

�b4

) �

a5�

a4b

�a3

b2�

a2b3

�ab

4

�a4

b�

a3b2

�a2

b3�

ab4

�b5

�a5

�b5

Fac

tor

a5

�b5

.E

xten

d th

e se

con

d pa

tter

n t

o ob

tain

a5

�b5

�(a

�b)

(a4

�a3

b�

a2b2

�ab

3�

b4).

Ch

eck

:(a

�b)

(a4

�a3

b�

a2b2

�ab

3�

b4)

�a5

�a4

b �

a3b2

�a2

b3�

ab4

�a4

b�

a3b2

�a2

b3�

ab4

�b5

�a5

�b5

In g

ener

al, i

f n

is a

n o

dd i

nte

ger,

wh

en y

ou f

acto

r an

�bn

or a

n�

bn, o

ne

fact

or w

ill

beei

ther

(a

�b)

or

(a�

b), d

epen

din

g on

th

e si

gn o

f th

e or

igin

al e

xpre

ssio

n. T

he

oth

er f

acto

rw

ill

hav

e th

e fo

llow

ing

prop

erti

es:

•T

he

firs

t te

rm w

ill

be a

n�

1an

d th

e la

st t

erm

wil

l be

bn

�1 .

•T

he

expo

nen

ts o

f a

wil

l de

crea

se b

y 1

as y

ou g

o fr

om l

eft

to r

igh

t.•

Th

e ex

pon

ents

of

bw

ill

incr

ease

by

1 as

you

go

from

lef

t to

rig

ht.

•T

he

degr

ee o

f ea

ch t

erm

wil

l be

n�

1.•

If t

he

orig

inal

exp

ress

ion

was

an

�bn

, th

e te

rms

wil

l al

tern

atel

y h

ave

�an

d �

sign

s.•

If t

he

orig

inal

exp

ress

ion

was

an

�bn

, th

e te

rms

wil

l al

l h

ave

�si

gns.

Use

th

e p

atte

rns

abov

e to

fac

tor

each

exp

ress

ion

.

1.a7

�b7

(a�

b)(

a6

�a

5 b�

a4 b

2�

a3 b

3�

a2 b

4�

ab5

�b

6 )

2.c9

�d

9(c

�d

)(c

8�

c7 d

�c

6 d2

�c

5 d3

�c4

d4

�c

3 d5

�c

2 d6

�cd

7�

d8 )

3.e1

1�

f11

(e�

f)(e

10�

e9 f

�e

8 f2

�e

7 f3

�e

6 f4

�e

5 f5

�e

4 f6

�e

3 f7

�e

2 f8

�ef

9�

f10 )

To

fact

or x

10�

y10 ,

ch

ange

it

to (

x5

�y5

)(x

5�

y5)

and

fac

tor

each

bin

omia

l. U

se t

his

app

roac

h t

o fa

ctor

eac

h e

xpre

ssio

n.

4.x1

0�

y10

(x�

y)(

x4

�x

3 y�

x2y

2�

xy3

�y

4 )(x

�y

)(x

4�

x3 y

�x

2 y2

�xy

3�

y4 )

5.a1

4�

b14

(a�

b)(

a6�

a5 b

�a

4 b2

�a

3 b3

�a

2 b4

�ab

5�

b6 )

(a�

b)

(a6

�a5

b�

a4b

2�

a3b

3�

a2b

4�

ab5

�b

6 )

Enri

chm

ent

Exam

ple

2

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A11



Exer

cise

s

Exam

ple

2

Exam

ple

1

Cha

pter

526

Gle

ncoe

Alg

ebra

2

5-3


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Gra

phin

g Ca

lcul

ator

Act

ivit

yU

sin

g T

able

s to

Fac

tor

by G

rou

pin

gT

he

TA

BL

Efe

atu

re o

f a

grap

hin

g ca

lcu

lato

r ca

n b

e u

sed

to h

elp

fact

or a

poly

nom

ial

of t

he

form

ax2

�bx

�c.

Fac

tor

10x2

�43

x�

28 b

y gr

oup

ing.

Mak

e a

tabl

e of

th

e n

egat

ive

fact

ors

of 1

0

28 o

r 28

0. L

ook

for

a pa

irof

fac

tors

wh

ose

sum

is

�43

.

En

ter

the

equ

atio

n y

��28 x0 �

in Y

1to

fin

d th

e fa

ctor

s of

280

. Th

en,

fin

d th

e su

m o

f th

e fa

ctor

s u

sin

g y

��28 x0 �

�x

in Y

2. S

et u

p th

e ta

ble

to d

ispl

ay t

he

neg

ativ

e fa

ctor

s of

280

by

sett

ing

�T

bl

= to

�1.

Exa

min

e th

e re

sult

s.

Key

stro

kes:

28

0 [T

BL

SE

T]

1 1

[TA

BL

E].

Th

e la

st l

ine

of t

he

tabl

e sh

ows

that

�43

xm

ay b

e re

plac

ed w

ith

�

8x+(

�35

x).

10x2

�43

x�

28�

10x2

�8x

�(�

35x)

�28

�2x

(5x

�4)

�(�

7)(5

x�

4)�

(5x

�4)

(2x

�7)

Th

us,

10x

2�

43x

�28

�(5

x�

4)(2

x�

7).

2nd

EN

TER

(–)

EN

TER

(–)

2nd

EN

TER

+E

NTE

RE

NTE

RV

AR

SE

NTE

R

Y=

Fac

tor

each

pol

ynom

ial.

1.y2

�20

y�

962.

4z2

�33

z�

353.

4y2

�y

�18

4.6a

2�

2a�

15(y

�4)

(y�

24)

(4z

�5)

(z�

7)(4

y �

9)(y

�2)

prim

e

5.6m

2�

17m

�12

6.24

z2�

46z

�15

7.36

y2�

84y

�49

8.4b

2�

36b

�40

3(2

m�

3)(3

m�

4)(1

2z�

5)(2

z�

3)(6

y�

7)2

(2b

�31

)(2b

�13

)

Fac

tor

12x2

�7x

�12

.

Loo

k at

th

e fa

ctor

s of

12

�

12 o

r�

144

for

a pa

ir w

hos

e su

m i

s�

7.E

nte

r an

equ

atio

n t

o de

term

ine

the

fact

ors

in Y

1an

d an

equ

atio

n t

ofi

nd

the

sum

of

fact

ors

in Y

2. E

xam

ine

the

tabl

e to

fin

d a

sum

of�

7.K

eyst

roke

s:

144

[TB

LS

ET

] 1

1 [T

AB

LE

].

12x2

�7x

�12

� 1

2x2

�9x

�(�

16x)

�12

�3x

(4x

�3)

�4(

4x�

3)�

(4x

�3)

(3x

�4)

T

hu

s, 1

2x2

�7x

�12

�(4

x�

3)(3

x�

4).

2nd

EN

TER

EN

TER

2nd

EN

TER

+

EN

TER

EN

TER

VA

RS

EN

TER

(–

)Y

=

Lesson 5-4

Cha

pter

527

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-4


Less

on R

eadi

ng G

uide

Co

mp

lex

Nu

mb

ers

Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 5

-4 i

n y

our

text

boo

k.

Sup

pose

the

num

ber

iis

def

ined

suc

h th

at i

2�

�1.

Com

plet

e ea

ch e

quat

ion.

2i2

�(2

i)2

�i4

�

Rea

d t

he

Less

on

1.C

ompl

ete

each

sta

tem

ent.

a.T

he

form

a�

biis

cal

led

the

of a

com

plex

nu

mbe

r.

b.

In t

he c

ompl

ex n

umbe

r 4

�5i

, the

rea

l par

t is

an

d th

e im

agin

ary

part

is

.

Th

is i

s an

exa

mpl

e of

a c

ompl

ex n

um

ber

that

is

also

a(n

) n

um

ber.

c.In

th

e co

mpl

ex n

um

ber

3, t

he

real

par

t is

an

d th

e im

agin

ary

part

is

.

Th

is i

s ex

ampl

e of

com

plex

nu

mbe

r th

at i

s al

so a

(n)

nu

mbe

r.

d.

In t

he

com

plex

nu

mbe

r 7i

, th

e re

al p

art

is

and

the

imag

inar

y pa

rt i

s .

Th

is i

s an

exa

mpl

e of

a c

ompl

ex n

um

ber

that

is

also

a(n

) n

um

ber.

2.G

ive

the

com

plex

con

juga

te o

f ea

ch n

um

ber.

a.3

�7i

b.

2 �

i

3.W

hy

are

com

plex

con

juga

tes

use

d in

div

idin

g co

mpl

ex n

um

bers

?T

he

pro

du

ct o

fco

mp

lex

con

jug

ates

is a

lway

s a

real

nu

mb

er.

4.E

xpla

in h

ow y

ou w

ould

use

com

plex

con

juga

tes

to f

ind

(3 �

7i)

�(2

�i)

.W

rite

th

ed

ivis

ion

in f

ract

ion

fo

rm.T

hen

mu

ltip

ly n

um

erat

or

and

den

om

inat

or

by

2 �

i.

Rem

emb

er W

hat

Yo

u L

earn

ed

5.H

ow c

an y

ou u

se w

hat

you

kn

ow a

bou

t si

mpl

ifyi

ng

an e

xpre

ssio

n s

uch

as

to

hel

p yo

u r

emem

ber

how

to

sim

plif

y fr

acti

ons

wit

h i

mag

inar

y n

um

bers

in

th

ede

nom

inat

or?

Sam

ple

an

swer

:In

bo

th c

ases

,yo

u c

an m

ult

iply

th

en

um

erat

or

and

den

om

inat

or

by t

he

con

jug

ate

of

the

den

om

inat

or.

1 �

�3�

� 2 �

�5�

2 �

i

3 �

7i

pu

re im

agin

ary

70

real

03

imag

inar

y

54

stan

dar

d f

orm

1�

4�

2

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A12


An

swer

s


Exer

cise

s

Exam

ple

3

Exam

ple

1

Cha

pter

528

Gle

ncoe

Alg

ebra

2

5-4


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Stud

y G

uide

and

Inte

rven

tion

Co

mp

lex

Nu

mb

ers

SQU

AR

E R

OO

TSA

squ

are

root

of a

nu

mbe

r n

is a

nu

mbe

r w

hos

e sq

uar

e is

n. F

or

nonn

egat

ive

real

num

bers

aan

d b,

�ab�

� �

a�

�b�

and

�

, b �

0. T

he i

mag

inar

y

un

iti

is d

efin

ed t

o h

ave

the

prop

erty

th

at i

2�

�1.

Sim

plif

ied

squ

are

root

exp

ress

ion

s do

not

hav

e ra

dica

ls i

n t

he

den

omin

ator

, an

d an

y n

um

ber

rem

ain

ing

un

der

the

squ

are

root

has

no

perf

ect

squ

are

fact

or o

ther

th

an 1

.

�a�

� �b�

a � b

a.S

imp

lify

�48�

.�

48 � �

�16

3

��

�16�

�

3��

4�3�

b.

Sim

pli

fy �

�63

�.

��

63�

��

�1

7

�

9�

��

�1

�

�7�

�

9��

3i�

7�

a.S

imp

lify

�12

5x2

�y5 �

.�

125x

2�

y5 ��

�5

25

�x2

y4y

��

�25�

�

5�

�x2 �

�

y4 �

�y�

�5x

y2 �

5y�

b.

Sim

pli

fy �

�44

x�

6 �.�

�44

x�

6 ��

��

1

4�

11

�

x6 ��

��

1�

�

4�

�11�

�

x6 ��

2i�

11�x3

Sol

ve x

2�

5 �

0.x2

�5

�0

Orig

inal

equ

atio

n.

x2�

�5

Sub

trac

t 5

from

eac

h si

de.

x�

��

5�iS

quar

e R

oot

Pro

pert

y.

Sim

pli

fy.

1.�

72�6�

2�2.

��

24�

2i�

6�

3.

4.�

75x4

y�

7 �5x

2 y3 �

3y�

5.�

�84

�2i

�21�

6.�

�32

x�

y4 �4y

2 i�

2x�

Sol

ve e

ach

eq

uat

ion

.

7.5x

2�

45 �

0�

3i8.

4x2

�24

�0

�i�

6�

9.�

9x2

�9

�i

10.

7x2

�84

�0

�2i

�3�

8�6�

�21

128

� 147

Exam

ple

2

Exer

cise

s

Exam

ple

4

Exam

ple

2

Exam

ple

3

Exam

ple

1

Lesson 5-4

Cha

pter

529

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

E__

____

____

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ER

IOD

____

_

5-4


Stud

y G

uide

and

Inte

rven

tion

(con

tinue

d)

Co

mp

lex

Nu

mb

ers

Sim

pli

fy.

1.(�

4 �

2i)

�(6

�3i

)2.

(5 �

i) �

(3 �

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) �

(4 �

2i)

2 �

i2

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10 �

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11 �

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Op

erat

ion

s w

ith

Co

mp

lex

Nu

mb

ers

Aco

mpl

ex n

umbe

r is

any

num

ber

that

can

be

writ

ten

in t

he f

orm

a�

bi,

Co

mp

lex

Nu

mb

erw

here

aan

d b

are

real

num

bers

and

iis

the

imag

inar

y un

it (i

2�

�1)

.a

is c

alle

d th

e re

al p

art,

and

bis

cal

led

the

imag

inar

y pa

rt.

Ad

dit

ion

an

d

Com

bine

like

ter

ms.

Su

btr

acti

on

of

(a�

bi)

�(c

�di

) �

(a�

c) �

(b�

d)i

Co

mp

lex

Nu

mb

ers

(a�

bi)

�(c

�di

) �

(a�

c) �

(b�

d)i

Mu

ltip

licat

ion

of

Use

the

def

initi

on o

f i2

and

the

FO

ILm

etho

d:C

om

ple

x N

um

ber

s(a

�bi

)(c

�di

) �

(ac

�bd

) �

(ad

�bc

)i

Co

mp

lex

Co

nju

gat

ea

�bi

and

a�

biar

e co

mpl

ex c

onju

gate

s. T

he p

rodu

ct o

f co

mpl

ex c

onju

gate

s is

al

way

s a

real

num

ber.

To d

ivid

e by

a c

ompl

ex n

um

ber,

fir

st m

ult

iply

th

e di

vide

nd

and

divi

sor

by t

he

com

ple

xco

nju

gate

of t

he

divi

sor.

Sim

pli

fy (

6 �

i)

�(4

�5i

).

(6 �

i) �

(4 �

5i)

�(6

�4)

�(1

�5)

i�

10 �

4i

Sim

pli

fy (

2 �

5i)

(�

4 �

2i).

(2 �

5i)

(�

4 �

2i)

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�4)

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2i)

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5i)(

�4)

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5i)(

2i)

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8 �

4i�

20i

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i2

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8 �

24i

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(�1)

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i

Sim

pli

fy (

8 �

3i)

�(6

�2i

).

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3i)

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�2i

)�

(8 �

6) �

[3 �

(�2)

]i�

2 �

5i

Sim

pli

fy

.

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i11 � 13

3 � 133 �

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6 �

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A1-A30 A2-05-873975 5/11/06 7:28 AM Page A13



Cha

pter

530

Gle

ncoe

Alg

ebra

2

5-4


NA

ME

____

____

____

____

____

____

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____

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AT

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ER

IOD

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Skill

s Pr

acti

ceC

om

ple

x N

um

ber

sS

imp

lify

.

1.�

99 �3�

11�2.

3.�

52x3

y�

5 �2||

x|| y

2�

13x

y�

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|| x3 || �

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i)(�

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i

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i

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i14

.(3

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15.

16.

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ve e

ach

eq

uat

ion

.

17.3

x2�

3 �

0�

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19.4

x2�

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0�

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x2�

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0�

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�0

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Fin

d t

he

valu

es o

f m

and

nth

at m

ake

each

eq

uat

ion

tru

e.

23.2

0 �

12i

�5m

�4n

i4,

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24.m

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i�

3 �

2ni

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4 �

m)

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n)

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27 � 49

Lesson 5-4

Cha

pter

531

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ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

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AT

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ER

IOD

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5-4


Sim

pli

fy.

1.�

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a�

3 b4

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��

8�

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15�

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i42

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i�

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5511

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5 �

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8i)

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7 �

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i)14

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57 �

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21.

23 �

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22.

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ve e

ach

eq

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.

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m2

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Fin

d t

he

valu

es o

f m

and

nth

at m

ake

each

eq

uat

ion

tru

e.

31.1

5 �

28i

�3m

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i5,

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32.(

6 �

m)

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i�

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,9

33.(

3m�

4) �

(3 �

n)i

�16

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4,6

34.(

7 �

n)

�(4

m�

10)i

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1,�

4

35.E

LEC

TRIC

ITY

Th

e im

peda

nce

in

on

e pa

rt o

f a

seri

es c

ircu

it i

s 1

�3j

ohm

s an

d th

eim

peda

nce

in

an

oth

er p

art

of t

he

circ

uit

is

7 �

5joh

ms.

Add

th

ese

com

plex

nu

mbe

rs t

ofi

nd

the

tota

l im

peda

nce

in

th

e ci

rcu

it.

8 �

2j

oh

ms

36.E

LEC

TRIC

ITY

Usi

ng

the

form

ula

E�

IZ, f

ind

the

volt

age

Ein

a c

ircu

it w

hen

th

ecu

rren

t I

is 3

�j

amps

an

d th

e im

peda

nce

Zis

3 �

2joh

ms.

11 �

3j

volt

s

3 � 4

2 �

4i� 1

�3i

7 �

i�

53

�i

� 2 �

i14

�16

i�

�11

32

� 7 �

8i

�5

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6 �

5i�

�2i

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17 � 81|a

3 |b

�2b�

�� 14

a6 b3

�98

15�

2x��

755

0x�

49

Prac

tice

Co

mp

lex

Nu

mb

ers

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A14


An

swer

s


Cha

pter

532

Gle

ncoe

Alg

ebra

2

5-4


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1.SI

GN

ER

RO

RS

Jen

nif

er a

nd

Jess

ica

com

e u

p w

ith

dif

fere

nt

answ

ers

to t

he

sam

e pr

oble

m. T

hey

had

to

mu

ltip

ly

(4 �

i)(4

�i)

an

d gi

ve t

hei

r an

swer

as

a co

mpl

ex n

um

ber.

Jen

nif

er c

laim

s th

atth

e an

swer

is

15 a

nd

Jess

ica

clai

ms

that

th

e an

swer

is

17. W

ho

is c

orre

ct?

Exp

lain

.Je

ssic

a is

co

rrec

t;(4

�i)

(4 �

i) �

z�

�i

16 �

4i�

4i�

i2

� 1

6 �

(�1)

�16

�1

�17

.

2.C

OM

PLEX

CO

NJU

GA

TES

You

hav

ese

en t

hat

th

e pr

odu

ct o

f co

mpl

exco

nju

gate

s is

alw

ays

a re

al n

um

ber.

Sh

ow t

hat

th

e su

m o

f co

mpl

exco

nju

gate

s is

als

o al

way

s a

real

nu

mbe

r.a

�b

ian

d a

�b

iare

co

mp

lex

con

jug

ates

an

d t

hei

r su

m is

2a,

wh

ich

is r

eal.

3.PY

THA

GO

REA

N T

RIP

LES

If t

hre

ein

tege

rs a

, b, a

nd

c, s

atis

fy a

2�

a2�

c2,

then

th

ey a

re c

alle

d a

Pyt

hag

orea

nTr

iple

. Su

ppos

e th

at a

, b, a

nd

car

e a

Pyt

hag

orea

n t

ripl

e. S

how

th

at t

he

real

and

imag

inar

y pa

rts

of (

a�

bi)2

,to

geth

er w

ith

th

e n

um

ber

c2, f

orm

anot

her

Pyt

hag

orea

n t

ripl

e.(a

�b

i)2

�a

2�

b2

�2a

bi;

a2

�b

2an

d 2

abar

e in

teg

ers

and

(a2

�b

2 )2

�(2

ab)2

�a

4�

2a2 b

2�

b4

�4a

2 b2

�a

4�

2a2 b

2�

b4

�(a

2�

b2 )

2�

(c2 )

2 ,so

a2

�b

2�

c2

as d

esir

ed.

4.R

OTA

TIO

NS

Com

plex

nu

mbe

rs c

an b

eu

sed

to p

erfo

rm r

otat

ion

s in

th

e pl

ane.

For

exa

mpl

e, i

f (x

, y)

are

the

coor

din

ates

of a

poi

nt

in t

he

plan

e, t

hen

th

e re

al

and

imag

inar

y pa

rts

of i

(x�

yi)

are

the

hor

izon

tal

and

vert

ical

coo

rdin

ates

of t

he

90°

cou

nte

rclo

ckw

ise

rota

tion

of

(x, y

) ab

out

the

orig

in. W

hat

are

th

e re

alan

d im

agin

ary

part

s of

i(x

�yi

)?T

he

real

par

t is

�y

and

imag

inar

yp

art

is x

.

ELEC

TRIC

AL

ENG

INEE

RIN

GF

orE

xerc

ises

5–7

, use

th

e fo

llow

ing

info

rmat

ion

.C

ompl

ex n

um

bers

can

be

use

d to

des

crib

eth

e al

tern

atin

g cu

rren

t (A

C)

in a

n e

lect

ric

circ

uit

lik

e th

e on

e u

sed

in y

our

hom

e. Z

,th

e im

peda

nce

in

an

AC

cir

cuit

, is

rela

ted

toth

e vo

ltag

e V

and

the

curr

ent

Iby

th

e

form

ula

Z�

.

5.F

ind

Zif

V�

5 �

2ian

d I

�3i

.

Z�

6.F

ind

Zif

V�

2 �

3ian

d I

��

3i.

Z�

7.F

ind

Vif

Z�

and

I�

3i.

V�

3 �

2i

2 �

3i�

3

2�

3i�

3

5i �

2�

�3V � I

�2�

�2

�2 �

�2

Wor

d Pr

oble

m P

ract

ice

Co

mp

lex

Nu

mb

ers

Exer

cise

s

Exam

ple

2

Exam

ple

1

Lesson 5-4

Cha

pter

533

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-4


Co

nju

gat

es a

nd

Ab

solu

te V

alu

eW

hen

stu

dyin

g co

mpl

ex n

um

bers

, it

is o

ften

con

ven

ien

t to

rep

rese

nt

a co

mpl

ex

nu

mbe

r by

a s

ingl

e va

riab

le. F

or e

xam

ple,

we

mig

ht

let

z�

x�

yi. W

e de

not

e th

e co

nju

gate

of

zby

z �. T

hu

s, z �

�x

�yi

.

We

can

def

ine

the

abso

lute

val

ue

of a

com

plex

nu

mbe

r as

fol

low

s.

�z��

�x�

yi��

�x2

�y

�2 �

Th

ere

are

man

y im

port

ant

rela

tion

ship

s in

volv

ing

con

juga

tes

and

abso

lute

va

lues

of

com

plex

nu

mbe

rs.

Sh

ow �z

�2�

zz �fo

r an

y co

mp

lex

nu

mb

er z

.

Let

z�

x�

yi. T

hen

,zz �

�(x

�yi

)(x

�yi

)�

x2�

y2

��

( x2

�y2

�)2 �

��z

�2

Sh

ow

is t

he

mu

ltip

lica

tive

in

vers

e fo

r an

y n

onze

ro

com

ple

x n

um

ber

z.

We

know

�z�2

�zz �.

If

z �

0, t

hen

we

hav

e z �

��1.

Th

us,

is

th

e m

ult

ipli

cati

ve i

nve

rse

of z

.

For

eac

h o

f th

e fo

llow

ing

com

ple

x n

um

ber

s, f

ind

th

e ab

solu

te v

alu

e an

dm

ult

ipli

cati

ve i

nve

rse.

1.2i

2;�� 2i �

2.�

4 �

3i5;

��4 2� 5

3i�

3.12

�5i

13;�12

1� 695i

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4.5

�12

i13

;�5

� 16912

i�

5.1

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i�

6.�

3��

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�i

9.�1 2�

�i

;1;

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1;�1 2�

�i

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�2

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6��

3

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z �� z

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z �� z

�2

z �� z

�2

Enri

chm

ent

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A15



Cha

pter

534

Gle

ncoe

Alg

ebra

2

5-5


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 5

-5 i

n y

our

text

boo

k.

Exp

lain

wh

at i

t m

ean

s to

say

th

at t

he

driv

er a

ccel

erat

es a

t a

con

stan

t ra

te o

f 8

feet

per

sec

ond

squ

ared

.

If t

he

dri

ver

is t

rave

ling

at

a ce

rtai

n s

pee

d a

t a

par

ticu

lar

mo

men

t,th

en o

ne

seco

nd

late

r,th

e d

rive

r is

tra

velin

g 8

fee

t p

er s

eco

nd

fas

ter.

Rea

d t

he

Less

on

1.G

ive

the

reas

on f

or e

ach

ste

p in

th

e fo

llow

ing

solu

tion

of

an e

quat

ion

by

usi

ng

the

Squ

are

Roo

t P

rope

rty.

x2�

12x

�36

�81

Orig

inal

equ

atio

n

(x�

6)2

�81

Fact

or

the

per

fect

sq

uar

e tr

ino

mia

l.

x�

6 �

��

81�S

qu

are

Ro

ot

Pro

per

ty

x�

6 �

�9

81 �

9

x�

6 �

9 or

x�

6 �

�9

Rew

rite

as

two

eq

uat

ion

s.

x�

15

x�

�3

So

lve

each

eq

uat

ion

.

2.E

xpla

in h

ow t

o fi

nd

the

con

stan

t th

at m

ust

be

adde

d to

mak

e a

bin

omia

l in

to a

per

fect

squ

are

trin

omia

l.

Sam

ple

an

swer

:Fin

d h

alf

of

the

coef

ficie

nt

of

the

linea

r te

rm a

nd

sq

uar

e it.

3.a.

Wh

at i

s th

e fi

rst

step

in

sol

vin

g th

e eq

uat

ion

3x2

�6x

�5

by c

ompl

etin

g th

e sq

uar

e?D

ivid

e th

e eq

uat

ion

by

3.

b.

Wh

at i

s th

e fi

rst

step

in

sol

vin

g th

e eq

uat

ion

x2

�5x

�12

�0

by c

ompl

etin

g th

esq

uar

e?A

dd

12

to e

ach

sid

e.

Rem

emb

er W

hat

Yo

u L

earn

ed

4.H

ow c

an y

ou u

se t

he

rule

s fo

r sq

uar

ing

a bi

nom

ial

to h

elp

you

rem

embe

r th

e pr

oced

ure

for

chan

gin

g a

bin

omia

l in

to a

per

fect

squ

are

trin

omia

l?

On

e o

f th

e ru

les

for

squ

arin

g a

bin

om

ial i

s (x

�y

)2�

x2

�2x

y�

y2 .

Inco

mp

leti

ng

th

e sq

uar

e,yo

u a

re s

tart

ing

wit

h x

2�

bx

and

nee

d t

o f

ind

y2 .

Th

is s

ho

ws

you

th

at b

�2y

,so

y�

.Th

at is

why

yo

u m

ust

tak

e h

alf

of

the

coef

fici

ent

and

sq

uar

e it

to

get

th

e co

nst

ant

that

mu

st b

e ad

ded

to

com

ple

te t

he

squ

are.

b � 2

Less

on R

eadi

ng G

uide

Co

mp

leti

ng

th

e S

qu

are

Exam

ple

Exer

cise

s

Lesson 5-5

Cha

pter

535

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-5


Squ

are

Ro

ot

Pro

per

tyU

se t

he

Squ

are

Roo

t P

rope

rty

to s

olve

a q

uad

rati

c eq

uat

ion

that

is

in t

he

form

“pe

rfec

t sq

uar

e tr

inom

ial

�co

nst

ant.

”

Sol

ve e

ach

eq

uat

ion

by

usi

ng

the

Sq

uar

e R

oot

Pro

per

ty.

a. x

2�

8x�

16 �

25x2

�8x

�16

�25

(x�

4)2

�25

x�

4 �

�25�

or x

�4

��

�25�

x�

5 �

4 �

9or

x

��

5 �

4 �

�1

Th

e so

luti

on s

et i

s {9

, �1}

.

b. 4

x2�

20x

�25

�32

4x2

�20

x�

25�

32(2

x�

5)2

�32

2x�

5 �

�32�

or 2

x�

5 �

��

32�2x

�5

�4�

2�or

2x

�5

��

4�2�

x�

Th

e so

luti

on s

et i

s �

�.5

�4�

2��

� 2

5 �

4�2�

�� 2

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ve e

ach

eq

uat

ion

by

usi

ng

the

Sq

uar

e R

oot

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per

ty.

1.x2

�18

x�

81 �

492.

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20x

�10

0 �

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4x2

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{2,1

6}{�

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�

4.36

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5.9x

2�

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x2�

40x

�16

�28

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��

7.4x

2�

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24x

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x�

9 �

121

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��

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{�0.

8,1.

4}

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x�

4 �

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.36x

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��

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15 � 2

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5 � 23 � 2

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y G

uide

and

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rven

tion

Co

mp

leti

ng

th

e S

qu

are

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A16


An

swer

s


Exam

ple

2Ex

amp

le 1

Exer

cise

s

Cha

pter

536

Gle

ncoe

Alg

ebra

2

5-5


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Co

mp

lete

th

e Sq

uar

eTo

com

plet

e th

e sq

uar

e fo

r a

quad

rati

c ex

pres

sion

of

the

form

x2

�bx

, fo

llow

th

ese

step

s.

1.F

ind

.➞

2.S

quar

e .

➞3.

Add

��2

to x

2�

bx.

b � 2b � 2

b � 2

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d t

he

valu

eof

cth

at m

akes

x2

�22

x�

ca

per

fect

sq

uar

e tr

inom

ial.

Th

enw

rite

th

e tr

inom

ial

as t

he

squ

are

of a

bin

omia

l.

Ste

p 1

b�

22;

�11

Ste

p 2

112

�12

1S

tep

3c

�12

1

The

tri

nom

ial

is x

2�

22x

�12

1,w

hic

h c

an b

e w

ritt

en a

s (x

�11

)2.

b � 2

Sol

ve 2

x2�

8x�

24 �

0 b

yco

mp

leti

ng

the

squ

are.

2x2

�8x

�24

�0

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inal

equ

atio

n

�D

ivid

e ea

ch s

ide

by 2

.

x2�

4x�

12 �

0x2

�4x

�12

is n

ot a

per

fect

squ

are.

x2�

4x�

12A

dd 1

2 to

eac

h si

de.

x2�

4x�

4 �

12 �

4S

ince

��2

�4,

add

4 t

o ea

ch s

ide.

(x�

2)2

�16

Fac

tor

the

squa

re.

x�

2 �

�4

Squ

are

Roo

t P

rope

rty

x�

6 or

x�

�2

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ve e

ach

equa

tion.

Th

e so

luti

on s

et i

s {6

, �2}

.

4 � 2

0 � 22x

2�

8x�

24�

� 2

Fin

d t

he

valu

e of

cth

at m

akes

eac

h t

rin

omia

l a

per

fect

sq

uar

e. T

hen

wri

te t

he

trin

omia

l as

a p

erfe

ct s

qu

are.

1.x2

�10

x�

c2.

x2�

60x

�c

3.x2

�3x

�c

25;

(x�

5)2

900;

(x�

30)2

;�x

��2

4.x2

�3.

2x�

c5.

x2�

x�

c6.

x2�

2.5x

�c

2.56

;(x

� 1

.6)2

;�x

��2

1.56

25;

(x�

1.25

)2

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ve e

ach

eq

uat

ion

by

com

ple

tin

g th

e sq

uar

e.

7.y2

�4y

�5

�0

8.x2

�8x

�65

�0

9.s2

�10

s�

21 �

0�

1,5

�5,

133,

7

10.2

x2�

3x�

1 �

011

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�13

x�

7 �

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.25x

2�

40x

�9

�0

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2�

4x�

1 �

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.y2

�12

y�

4 �

015

.t2

�3t

�8

�0

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��

3��

6 �

4�2�

�3

��

41��

� 29 � 51 � 5

1 � 21 � 2

1 � 41 � 16

1 � 2

3 � 29 � 4

Stud

y G

uide

and

Inte

rven

tion

(con

tinue

d)

Co

mp

leti

ng

th

e S

qu

are

Lesson 5-5

Cha

pter

537

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

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ER

IOD

____

_

5-5


Skill

s Pr

acti

ceC

om

ple

tin

g t

he

Sq

uar

eS

olve

eac

h e

qu

atio

n b

y u

sin

g th

e S

qu

are

Roo

t P

rop

erty

.

1.x2

�8x

�16

�1

3,5

2.x2

�4x

�4

�1

�1,

�3

3.x2

�12

x�

36 �

25�

1,�

114.

4x2

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�1

�9

�1,

2

5.x2

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�4

�2

�2

��

2�6.

x2�

2x�

1 �

51

��

5�

7.x2

�6x

�9

�7

3 �

�7�

8.x2

�16

x�

64 �

15�

8 �

�15�

Fin

d t

he

valu

e of

cth

at m

akes

eac

h t

rin

omia

l a

per

fect

sq

uar

e. T

hen

wri

te t

he

trin

omia

l as

a p

erfe

ct s

qu

are.

9.x2

�10

x�

c25

;(x

�5)

210

.x2

�14

x�

c49

;(x

�7)

2

11.x

2�

24x

�c

144;

(x�

12)2

12.x

2�

5x�

c;�x

��2

13.x

2�

9x�

c;�x

��2

14.x

2�

x�

c;�x

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ve e

ach

eq

uat

ion

by

com

ple

tin

g th

e sq

uar

e.

15.x

2�

13x

�36

�0

4,9

16.x

2�

3x�

00,

�3

17.x

2�

x�

6 �

02,

�3

18.x

2�

4x�

13 �

02

��

17�

19.2

x2�

7x�

4 �

0�

4,20

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�0

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21.x

2�

3x�

6 �

022

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�x

�3

�0

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1 �

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��

33��

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1 � 31 � 2

1 � 21 � 4

9 � 281 � 4

5 � 225 � 4

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A17



Cha

pter

538

Gle

ncoe

Alg

ebra

2

5-5


NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

____

_

Sol

ve e

ach

eq

uat

ion

by

usi

ng

the

Sq

uar

e R

oot

Pro

per

ty.

1.x2

�8x

�16

�1

2.x2

�6x

�9

�1

3.x2

�10

x�

25 �

16

�5,

�3

�4,

�2

�9,

�1

4.x2

�14

x�

49 �

95.

4x2

�12

x�

9 �

46.

x2�

8x�

16 �

8

4,10

�,�

4 �

2�2�

7.x2

�6x

�9

�5

8.x2

�2x

�1

�2

9.9x

2�

6x�

1 �

2

3 �

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1 �

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Fin

d t

he

valu

e of

cth

at m

akes

eac

h t

rin

omia

l a

per

fect

sq

uar

e. T

hen

wri

te t

he

trin

omia

l as

a p

erfe

ct s

qu

are.

10.x

2�

12x

�c

11.x

2�

20x

�c

12.x

2�

11x

�c

36;

(x�

6)2

100;

(x�

10)2

;�x

��2

13.x

2�

0.8x

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14.x

2�

2.2x

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15.x

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x�

c

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;(x

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;(x

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1)2

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24;

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0.18

)2

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x�

c

;�x

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;�x

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;�x

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ve e

ach

eq

uat

ion

by

com

ple

tin

g th

e sq

uar

e.

19.x

2�

6x�

8 �

0�

4,�

220

.3x2

�x

�2

�0

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21.3

x2�

5x�

2 �

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19 �

024

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026

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6 �

029

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30.7

x2�

6x�

2 �

0

31.G

EOM

ETRY

Wh

en t

he

dim

ensi

ons

of a

cu

be a

re r

edu

ced

by 4

in

ches

on

eac

h s

ide,

th

esu

rfac

e ar

ea o

f th

e n

ew c

ube

is

864

squ

are

inch

es. W

hat

wer

e th

e di

men

sion

s of

th

eor

igin

al c

ube

?16

in.b

y 16

in.b

y 16

in.

32.I

NV

ESTM

ENTS

Th

e am

oun

t of

mon

ey A

in a

n a

ccou

nt

in w

hic

h P

doll

ars

is i

nve

sted

for

2 ye

ars

is g

iven

by

the

form

ula

A�

P(1

�r)

2 , w

her

e r

is t

he

inte

rest

rat

e co

mpo

un

ded

ann

ual

ly. I

f an

in

vest

men

t of

$80

0 in

th

e ac

cou

nt

grow

s to

$88

2 in

tw

o ye

ars,

at

wh

atin

tere

st r

ate

was

it

inve

sted

?5%

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�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

Prac

tice

Co

mp

leti

ng

th

e S

qu

are

Lesson 5-5

Cha

pter

539

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

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ER

IOD

____

_

5-5


1.C

OM

PLET

ING

TH

E SQ

UA

RE

Sam

anth

a n

eeds

to

solv

e th

e eq

uat

ion

x2�

12x

�40

.

Wh

at m

ust

sh

e do

to

each

sid

e of

th

eeq

uat

ion

to

com

plet

e th

e sq

uar

e?A

dd

36.

2.SQ

UA

RE

RO

OTS

Eva

n i

s as

ked

toso

lve

the

equ

atio

n x

2�

8x�

16 �

25.

He

reco

gniz

es t

hat

th

e le

ft-h

and

side

of

the

equ

atio

n i

s a

perf

ect

squ

are

trin

omia

l. F

acto

r th

e le

ft-h

and

side

.(x

�4)

2

3.C

OM

POU

ND

IN

TER

EST

Nik

kiin

vest

ed $

1000

in

a s

avin

gs a

ccou

nt

wit

h i

nte

rest

com

pou

nde

d an

nu

ally

.A

fter

tw

o ye

ars

the

bala

nce

in

th

eac

cou

nt

is $

1210

. Use

th

e co

mpo

un

din

tere

st f

orm

ula

A�

P(1

�r)

tto

fin

dth

e an

nu

al i

nte

rest

rat

e.10

%

4.R

EAC

TIO

N T

IME

Lau

ren

was

eat

ing

lun

ch w

hen

sh

e sa

w h

er f

rien

d Ja

son

appr

oach

. Th

e ro

om w

as c

row

ded

and

Jaso

n h

ad t

o li

ft h

is t

ray

to a

void

obst

acle

s. S

udd

enly

, a g

lass

on

Jas

on’s

lun

ch t

ray

tipp

ed a

nd

fell

off

th

e tr

ay.

Lau

ren

lu

nge

d fo

rwar

d an

d m

anag

ed t

oca

tch

th

e gl

ass

just

bef

ore

it h

it t

he

grou

nd.

Th

e h

eigh

t h

, in

fee

t, o

f th

egl

ass

tse

con

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fter

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dro

pped

is

give

n b

y h

��

16t2

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5. L

aure

nca

ugh

t th

e gl

ass

wh

en i

t w

as s

ix i

nch

esof

f th

e gr

oun

d. H

ow l

ong

was

th

e gl

ass

in t

he

air

befo

re L

aure

n c

augh

t it

?0.

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con

d

5.PA

RA

BO

LAS

Apa

rabo

la i

s m

odel

ed b

yy

�x2

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ane’

s h

omew

ork

prob

lem

req

uir

es t

hat

sh

e fi

nd

the

vert

ex o

f th

e pa

rabo

la. S

he

use

s th

eco

mpl

etin

g sq

uar

e m

eth

od t

o ex

pres

sth

e fu

nct

ion

in

th

e fo

rm

y�

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wh

ere

(h,k

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th

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rtex

of

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para

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te t

he

fun

ctio

n i

n t

he

form

use

d by

Jan

e.y

�(x

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DIT

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IUM

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or E

xerc

ises

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th

e fo

llow

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rmat

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.T

he

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s in

an

au

dito

riu

m a

re a

rran

ged

ina

squ

are

grid

pat

tern

. Th

ere

are

45 r

ows

and

45 c

olu

mn

s of

ch

airs

. For

a s

peci

alco

nce

rt, o

rgan

izer

s de

cide

to

incr

ease

seat

ing

by a

ddin

g n

row

s an

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s to

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squ

are

patt

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of

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ats

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ats?

10

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aniz

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dd 1

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s, w

hat

is

the

seat

ing

capa

city

of

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audi

tori

um

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d Pr

oble

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ract

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mp

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A1-A30 A2-05-873975 5/11/06 7:28 AM Page A18


An

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ncoe

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ebra

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5-5


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Th

e G

old

en Q

uad

rati

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qu

atio

ns

Ago

lden

rec

tan

gle

has

th

e pr

oper

ty t

hat

its

len

gth

ca

n b

e w

ritt

en a

s a

�b,

wh

ere

ais

th

e w

idth

of

the

rect

angl

e an

d �a

� ab

��

�a b� . A

ny

gold

en r

ecta

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n b

e

divi

ded

into

a s

quar

e an

d a

smal

ler

gold

en r

ecta

ngl

e,

as s

how

n.

Th

e pr

opor

tion

use

d to

def

ine

gold

en r

ecta

ngl

es c

an b

e u

sed

to d

eriv

e tw

o qu

adra

tic

equ

atio

ns.

The

se a

reso

met

imes

call

ed g

old

en q

uad

rati

c eq

uat

ion

s.

Sol

ve e

ach

pro

ble

m.

1.In

th

e pr

opor

tion

for

th

e go

lden

rec

tan

gle,

let

aeq

ual

1. W

rite

th

e re

sult

ing

quad

rati

c eq

uat

ion

an

d so

lve

for

b.

b2

�b

�1

�0

b�

2.In

th

e pr

opor

tion

, let

beq

ual

1. W

rite

th

e re

sult

ing

quad

rati

c eq

uat

ion

an

d so

lve

for

a.

a2

�a

�1

�0

a�

3.D

escr

ibe

the

diff

eren

ce b

etw

een

the

two

gold

en q

uad

rati

c eq

uat

ion

s yo

u

fou

nd

in e

xerc

ises

1 a

nd

2.

Th

e si

gn

s o

f th

e fi

rst-

deg

ree

term

s ar

e o

pp

osi

te.

4.S

how

th

at t

he

posi

tive

sol

uti

ons

of t

he

two

equ

atio

ns

in e

xerc

ises

1 a

nd

2 ar

e re

cipr

ocal

s.

��

��

��1 4�

5�

�1

5.U

se t

he

Pyt

hag

orea

n T

heo

rem

to

fin

d a

radi

cal

expr

essi

on f

or t

he

diag

onal

of

a g

olde

n r

ecta

ngl

e w

hen

a�

1.

d�

6.F

ind

a ra

dica

l ex

pres

sion

for

th

e di

agon

al o

f a

gold

en r

ecta

ngl

e w

hen

b�

1.

d�

�10

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��

5��

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a

a a

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a

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chm

ent

NA

ME

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pter

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5-6


Less

on R

eadi

ng G

uide

Th

e Q

uad

rati

c F

orm

ula

an

d t

he

Dis

crim

inan

tG

et R

ead

y fo

r th

e Le

sso

nR

ead

th

e in

trod

uct

ion

to

Les

son

5-6

in

you

r te

xtb

ook

.

Des

crib

e h

ow y

ou w

ould

cal

cula

te t

he

posi

tion

of

the

dive

r af

ter

1 se

con

d u

sin

g th

e eq

uat

ion

in y

our

text

book

.S

amp

le a

nsw

er:

Su

bst

itu

te 1

fo

r t

in t

he

equ

atio

n a

nd

eval

uat

e th

e ex

pre

ssio

n.

Rea

d t

he

Less

on

1.a.

Wri

te t

he

Qu

adra

tic

For

mu

la.

x�

b.

Iden

tify

th

e va

lues

of

a, b

, an

d c

that

you

wou

ld u

se t

o so

lve

2x2

�5x

��

7, b

ut

don

ot a

ctu

ally

sol

ve t

he

equ

atio

n.

a�

b�

c�

2.S

upp

ose

that

you

are

sol

vin

g fo

ur

quad

rati

c eq

uat

ion

s w

ith

rat

ion

al c

oeff

icie

nts

an

dh

ave

fou

nd

the

valu

e of

th

e di

scri

min

ant

for

each

equ

atio

n. I

n e

ach

cas

e, g

ive

the

nu

mbe

r of

roo

ts a

nd

desc

ribe

th

e ty

pe o

f ro

ots

that

th

e eq

uat

ion

wil

l h

ave.

Val

ue

of

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crim

inan

tN

um

ber

of

Ro

ots

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e o

f R

oo

ts

642

real

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ion

al

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lex

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atio

nal

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real

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al

Rem

emb

er W

hat

Yo

u L

earn

ed

3.H

ow c

an l

ooki

ng

at t

he

Qu

adra

tic

For

mu

la h

elp

you

rem

embe

r th

e re

lati

onsh

ips

betw

een

th

e va

lue

of t

he

disc

rim

inan

t an

d th

e n

um

ber

of r

oots

of

a qu

adra

tic

equ

atio

nan

d w

het

her

th

e ro

ots

are

real

or

com

plex

?

Sam

ple

an

swer

:Th

e d

iscr

imin

ant

is t

he

exp

ress

ion

un

der

th

e ra

dic

al in

the

Qu

adra

tic

Fo

rmu

la.L

oo

k at

th

e Q

uad

rati

c F

orm

ula

an

d c

on

sid

er w

hat

hap

pen

s w

hen

yo

u t

ake

the

pri

nci

pal

sq

uar

e ro

ot

of

b2

�4a

can

d a

pp

ly�

in f

ron

t o

f th

e re

sult

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b2

�4a

cis

po

siti

ve,i

ts p

rin

cip

al s

qu

are

roo

tw

ill b

e a

po

siti

ve n

um

ber

an

d a

pp

lyin

g �

will

giv

e tw

o d

iffe

ren

t re

also

luti

on

s,w

hic

h m

ay b

e ra

tio

nal

or

irra

tio

nal

.If

b2

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c�

0,it

sp

rin

cip

al s

qu

are

roo

t is

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o a

pp

lyin

g �

in t

he

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adra

tic

Fo

rmu

la w

illo

nly

lead

to

on

e so

luti

on

,wh

ich

will

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rati

on

al (

assu

min

g a

,b,a

nd

car

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.If

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cis

neg

ativ

e,si

nce

th

e sq

uar

e ro

ots

of

neg

ativ

en

um

ber

s ar

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ot

real

nu

mb

ers,

you

will

get

tw

o c

om

ple

x ro

ots

,co

rres

po

nd

ing

to

th

e �

and

�in

th

e �

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A1-A30 A2-05-873975 5/11/06 7:28 AM Page A19



Exer

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Exam

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Stud

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n b

e u

sed

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olve

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nce

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ritt

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5-6


Exer

cise

s

Exam

ple

Ro

ots

an

d t

he

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crim

inan

t

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crim

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he e

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pe

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end

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mb

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equ

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uide

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e Q

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A1-A30 A2-05-873975 5/11/06 7:28 AM Page A20


An

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�11

x�

26 �

0

0;1

rati

on

al r

oo

t;4

225;

2 ra

tio

nal

ro

ots

;�

2,13

3.3x

2�

2x�

04.

20x2

�7x

�3

�0

4;2

rati

on

al r

oo

ts;

0,28

9;2

rati

on

al r

oo

ts;

�,

5.5x

2�

6 �

06.

x2�

6 �

0

120;

2 ir

rati

on

al r

oo

ts;

�24

;2

irra

tio

nal

ro

ots

;�

�6�

7.x2

�8x

�13

�0

8.5x

2�

x�

1 �

0

12;

2 ir

rati

on

al r

oo

ts;

�4

��

3�21

;2

irra

tio

nal

ro

ots

;

9.x2

�2x

�17

�0

10.x

2�

49 �

0

72;

2 ir

rati

on

al r

oo

ts;

1 �

3�2�

�19

6;2

com

ple

x ro

ots

;�

7i

11.x

2�

x�

1 �

012

.2x2

�3x

��

2

�3;

2 co

mp

lex

roo

ts;

�7;

2 co

mp

lex

roo

ts;

Sol

ve e

ach

eq

uat

ion

by

usi

ng

the

met

hod

of

you

r ch

oice

. Fin

d e

xact

sol

uti

ons.

13.x

2�

64�

814

.x2

�30

�0

��

30�

15.x

2�

x�

30�

5,6

16.1

6x2

�24

x�

27 �

0,�

17.x

2�

4x�

11 �

02

� �

15�18

.x2

�8x

�17

�0

4 �

�33�

19.x

2�

25 �

0�

5i20

.3x2

�36

�0

�2i

�3�

21.2

x2�

10x

�11

�0

22.2

x2�

7x�

4 �

0

23.8

x2�

1 �

4x24

.2x2

�2x

�3

�0

25.P

AR

AC

HU

TIN

GIg

nor

ing

win

d re

sist

ance

, th

e di

stan

ce d

(t)

in f

eet

that

a p

arac

hu

tist

fall

s in

tse

con

ds c

an b

e es

tim

ated

usi

ng

the

form

ula

d(t

) �

16t2

. If

a pa

rach

uti

st ju

mps

from

an

air

plan

e an

d fa

lls

for

1100

fee

t be

fore

ope

nin

g h

er p

arac

hu

te, h

ow m

any

seco

nds

pass

bef

ore

she

open

s th

e pa

rach

ute

?ab

ou

t 8.

3 s

�1

�i�

5��

� 21

�i

�4

7 �

�17�

�� 4

�5

��

3��

� 2

3 � 49 � 4

3 �

i �7�

�� 4

1 �

i �3�

�� 2

1 �

�21�

�� 10

�30�

�5

1 � 43 � 5

2 � 3

Lesson 5-6

Cha

pter

545

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-6


Com

ple

te p

arts

a�

c fo

r ea

ch q

uad

rati

c eq

uat

ion

.a.

Fin

d t

he

valu

e of

th

e d

iscr

imin

ant.

b.

Des

crib

e th

e n

um

ber

an

d t

ype

of r

oots

.c.

Fin

d t

he

exac

t so

luti

ons

by

usi

ng

the

Qu

adra

tic

For

mu

la.

1.x2

�16

x�

64 �

02.

x2�

3x3.

9x2

�24

x�

16 �

0

0;1

rati

on

al;

89;

2 ra

tio

nal

;0,

30;

1 ra

tio

nal

;

4.x2

�3x

�40

5.3x

2�

9x�

2 �

010

5;6.

2x2

�7x

�0

169;

2 ra

tio

nal

;�

5,8

2 ir

rati

on

al;

49;

2 ra

tio

nal

;0,

�

7.5x

2�

2x�

4 �

0�

76;

8.12

x2�

x�

6 �

028

9;9.

7x2

�6x

�2

�0

�20

;

2 co

mp

lex;

2 ra

tio

nal

;,�

2 co

mp

lex;

10.1

2x2

�2x

�4

�0

196;

11.6

x2�

2x�

1 �

028

;12

.x2

�3x

�6

�0

�15

;

2 ra

tio

nal

;,�

2 ir

rati

on

al;

2 co

mp

lex;

13.4

x2�

3x2

�6

�0

105;

14.1

6x2

�8x

�1

�0

15.2

x2�

5x�

6 �

073

;

2 ir

rati

on

al;

0;1

rati

on

al;

2 ir

rati

on

al;

Sol

ve e

ach

eq

uat

ion

by

usi

ng

the

met

hod

of

you

r ch

oice

. Fin

d e

xact

sol

uti

ons.

16.7

x2�

5x�

00,

17.4

x2�

9 �

0�

18.3

x2�

8x�

3,�

319

.x2

�21

�4x

�3,

7

20.3

x2�

13x

�4

�0

,421

.15x

2�

22x

��

8�

,�

22.x

2�

6x�

3 �

03

��

6�23

.x2

�14

x�

53 �

07

�2i

24.3

x2�

�54

�3i

�2�

25.2

5x2

�20

x�

6 �

0

26.4

x2�

4x�

17 �

027

.8x

�1

�4x

2

28.x

2�

4x�

152

�i�

11�29

.4x2

�12

x�

7 �

0

30. G

RA

VIT

ATI

ON

The

hei

ght

h(t)

in f

eet

of a

n ob

ject

tse

cond

s af

ter

it is

pro

pelle

d st

raig

ht u

pfr

om t

he

grou

nd

wit

h a

n i

nit

ial

velo

city

of

60 f

eet

per

seco

nd

is m

odel

ed b

y th

e eq

uat

ion

h(t

) �

�16

t2�

60t.

At

wh

at t

imes

wil

l th

e ob

ject

be

at a

hei

ght

of 5

6 fe

et?

1.75

s,2

s

31.S

TOPP

ING

DIS

TAN

CE

Th

e fo

rmu

la d

�0.

05s2

�1.

1ses

tim

ates

th

e m

inim

um

sto

ppin

gdi

stan

ce d

in f

eet

for

a ca

r tr

avel

ing

sm

iles

per

hou

r. I

f a

car

stop

s in

200

fee

t, w

hat

is t

hefa

stes

t it

cou

ld h

ave

been

tra

velin

g w

hen

the

driv

er a

pplie

d th

e br

akes

?ab

ou

t 53

.2 m

i/h

3 �

�2�

�2

2 �

�3�

�2

1 �

4i�

2

2 �

�10�

�� 54 � 5

2 � 31 � 3

1 � 3

3 � 25 � 7

5 �

�73�

�� 4

1 � 43

��

105

��

� 8

�3

�i �

15��

� 21

��

7��

62 � 3

1 � 2

�3

�i �

5��

� 72 � 3

3 � 41

�i�

19��

� 5

7 � 2�

9 �

�10

5�

�� 6

4 � 3

Prac

tice

Th

e Q

uad

rati

c F

orm

ula

an

d t

he

Dis

crim

inan

t

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A21



Cha

pter

546

Gle

ncoe

Alg

ebra

2

5-6


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

1.PA

RA

BO

LAS

Th

e gr

aph

of

a qu

adra

tic

equ

atio

n o

f th

e fo

rm y

�ax

2�

bx�

cis

show

n b

elow

.

Is t

he

disc

rim

inan

t b2

�4a

cpo

siti

ve,

neg

ativ

e,or

zer

o?n

egat

ive

2.TA

NG

ENT

Kat

hle

en i

s tr

yin

g to

fin

d b

so t

hat

th

e x-

axis

is

tan

gen

t to

th

epa

rabo

la y

�x2

�bx

�4.

Sh

e fi

nds

on

eva

lue

that

wor

ks,b

�4.

Is t

his

th

e on

lyva

lue

that

wor

ks?

Exp

lain

.N

o,b

��

4 al

so w

ork

s;th

e x-

axis

will

be

tan

gen

t w

hen

th

ed

iscr

imin

ant

b2

�16

is z

ero

.T

his

hap

pen

s w

hen

b�

4 o

r �

4.

3.A

REA

Con

rad

has

a t

rian

gle

wh

ose

base

has

len

gth

x�

3 an

d w

hos

e h

eigh

tis

2x

�4.

Wh

at i

s th

e ar

ea o

f th

istr

ian

gle?

For

wh

at v

alu

es o

f x

is t

his

area

equ

al t

o 21

0? D

o al

l th

e so

luti

ons

mak

e se

nse

?x2

�5x

�6;

x�

12 o

r x

��

17;

on

ly x

�12

mak

es s

ense

bec

ause

th

e le

ng

ths

are

no

nn

egat

ive.

4.EX

AM

PLES

Giv

e an

exa

mpl

e of

aqu

adra

tic

fun

ctio

n f

(x)

that

has

th

efo

llow

ing

prop

erti

es.

I.T

he

disc

rim

inan

t of

fis

zer

o.

II.T

her

e is

no

real

sol

uti

on o

f th

eeq

uat

ion

f(x

) �

10.

Ske

tch

th

e gr

aph

of

x�

f(x)

.S

amp

le a

nsw

er:

f(x)

��

x2

TAN

GEN

TSF

or E

xerc

ises

5 a

nd

6,u

seth

e fo

llow

ing

info

rmat

ion

.T

he

grap

h o

f y

�x2

is a

par

abol

a th

atpa

sses

th

rou

gh t

he

poin

t at

(1,

1).T

he

lin

e y

�m

x�

m�

1,w

her

e m

is a

con

stan

t,al

so p

asse

s th

rou

gh t

he

poin

t at

(1,

1).

5.T

o fi

nd

the

poin

ts o

f in

ters

ecti

onbe

twee

n t

he

lin

e y

�m

x�

m�

1 an

d th

e pa

rabo

la y

�x2

,set

x2

�m

x�

m�

1 an

d th

en s

olve

for

x.

Rea

rran

gin

g te

rms,

this

equ

atio

nbe

com

es x

2�

mx

�m

�1

�0.

Wh

at i

sth

e di

scri

min

ant

of t

his

equ

atio

n?

x2�

4m �

4

6.F

or w

hat

val

ue

of m

is t

her

e on

ly o

ne

poin

t of

in

ters

ecti

on?

Exp

lain

th

em

ean

ing

of t

his

in

ter

ms

of t

he

corr

espo

ndi

ng

lin

e an

d th

e pa

rabo

la.

m�

2;th

e p

arab

ola

y�

x2an

dth

e lin

e y

�2x

�1

hav

e ex

actl

yo

ne

po

int

of

inte

rsec

tio

n a

t (1

,1).

In o

ther

wo

rds,

this

lin

e is

tan

gen

t to

th

e p

arab

ola

at

(1,1

).

yx

O

y

xO5

-5

Wor

d Pr

oble

m P

ract

ice

Th

e Q

uad

rati

c F

orm

ula

an

d t

he

Dis

crim

inan

t

Exer

cise

s

Exam

ple

Lesson 5-6

Cha

pter

547

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-6


Su

m a

nd

Pro

du

ct o

f R

oo

ts

Som

etim

es y

ou m

ay k

now

th

e ro

ots

of a

qu

adra

tic

equ

atio

n w

ith

out

know

ing

the

equ

atio

nit

self

. Usi

ng

you

r kn

owle

dge

of f

acto

rin

g to

sol

ve a

n e

quat

ion

, you

can

wor

k ba

ckw

ard

tofi

nd

the

quad

rati

c eq

uat

ion

. Th

e ru

le f

or f

indi

ng

the

sum

an

d pr

odu

ct o

f ro

ots

is a

s fo

llow

s:

Su

m a

nd

Pro

du

ct o

f R

oo

tsIf

the

root

s of

ax2

�bx

�c

�0,

with

a≠

0, a

re s

1an

d s 2

,

then

s1

�s 2

��

�b a�an

d s 1

�s 2

�� ac � .

Aro

ad w

ith

an

in

itia

l gr

adie

nt,

or

slop

e, o

f 3%

can

be

rep

rese

nte

db

y th

e fo

rmu

la y

�a

x2�

0. 0

3x�

c, w

her

e y

is t

he

elev

atio

n a

nd

xis

th

e d

ista

nce

alon

g th

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

0 fe

et a

lon

g th

e cu

rve.

You

can

fin

d t

he

equ

atio

n o

f th

e tr

ansi

tion

cu

rve.

Eq

uat

ion

s of

tra

nsi

tion

cu

rves

are

use

d b

y ci

vil

engi

nee

rs t

o d

esig

n s

moo

th a

nd

safe

roa

ds.

Th

e ro

ots

are

x�

3 an

d x

��

8.

3 �

(�8)

��

5A

dd t

he r

oots

.

3(�

8) �

�24

Mul

tiply

the

roo

ts.

Equ

atio

n: x

2�

5x�

24 �

0

Wri

te a

qu

adra

tic

equ

atio

n t

hat

has

th

e gi

ven

roo

ts.

1.6,

�9

2.5,

�1

3.6,

6

x2

�3x

�54

�0

x2

�4x

�5

�0

x2

�12

x�

36 �

0

4.4

��

3�6.

��2 5� ,

�2 7�6.

x2

�8x

�13

�0

35x

2�

4x�

4 �

049

x2

�42

x�

205

�0

Fin

d k

such

th

at t

he

nu

mb

er g

iven

is

a ro

ot o

f th

e eq

uat

ion

.

7.7;

2x2

�kx

�21

�0

8.�

2; x

2�

13x

�k

�0

�11

�30

�2

�3�

5��

� 7

x

y

O

(–5 – 2,

–30

1 – 4)

10 –10

–20

–30

24

–2–4

–6–8

Enri

chm

ent

A1-A30 A2-05-873975 5/12/06 1:43 PM Page A22


An

swer

s


Cha

pter

548

Gle

ncoe

Alg

ebra

2

5-6


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

You

hav

e le

arn

ed t

he

Loc

atio

n P

rin

cipl

e, w

hic

h c

an b

e u

sed

to a

ppro

xim

ate

the

real

zer

os o

f a

poly

nom

ial.

In t

he

spre

adsh

eet

abov

e, t

he

posi

tive

rea

l ze

ro o

f ƒ(

x) �

x2�

2 ca

n b

eap

prox

imat

ed i

n t

he

foll

owin

g w

ay. S

et t

he

spre

adsh

eet

pref

eren

ce t

o m

anu

-al

cal

cula

tion

. Th

e va

lues

in

A2

and

B2

are

the

endp

oin

ts o

f a

ran

ge o

f va

l-u

es. T

he

valu

es i

n D

2 th

rou

gh J

2 ar

e va

lues

equ

ally

in

th

e in

terv

al f

rom

A2

to B

2. T

he fo

rmul

as fo

r th

ese

valu

es a

re A

2,A

2�

(B2

�A

2) 6

, A2

� 2

*(B

2�

A2)

/6,A

2 �

3*(

B2

�A

2)/6

, A2

� 4

*(B

2�

A2)

/6, A

2 �

5*(

B2

�A

2)/6

, an

d B

2,re

spec

tive

ly.

Row

3 g

ives

th

e fu

nct

ion

val

ues

at

thes

e po

ints

. Th

e fu

nct

ion

ƒ(x

) �

x2�

2 is

ente

red

into

th

e sp

read

shee

t in

Cel

l D

3 as

D2^

2 �

2. T

his

fu

nct

ion

is

then

copi

ed t

o th

e re

mai

nin

g ce

lls

in t

he

row

.

You

can

use

th

is s

prea

dsh

eet

to s

tudy

th

e fu

nct

ion

val

ues

at

the

poin

ts i

nce

lls

D2

thro

ugh

J2.

Th

e va

lue

in c

ell

F3

is p

osit

ive

and

the

valu

e in

cel

l G

3is

neg

ativ

e, s

o th

ere

mu

st b

e a

zero

bet

wee

n �

1.66

67 a

nd

0. E

nte

r th

ese

valu

es i

n c

ells

A2

and

B2,

res

pect

ivel

y, a

nd

reca

lcu

late

th

e sp

read

shee

t. (

You

wil

l h

ave

to r

ecal

cula

te a

nu

mbe

r of

tim

es.)

Th

e re

sult

is

a n

ew t

able

fro

mw

hic

h y

ou c

an s

ee t

hat

th

ere

is a

zer

o be

twee

n 1

.414

14 a

nd

1.41

4306

.B

ecau

se t

hes

e va

lues

agr

ee t

o th

ree

deci

mal

pla

ces,

th

e ze

ro i

s ab

out

1.41

4.T

his

can

be

veri

fied

by

usi

ng

alge

bra.

By

solv

ing

x2�

2 �

0, w

e ob

tain

x�

��

2�. T

he

posi

tive

roo

t is

x

��

�2�

�1.

4142

13. .

. , w

hic

h v

erif

ies

the

resu

lt.

Spre

adsh

eet

Inve

stig

atio

nA

pp

roxi

mat

ing

th

e R

eal Z

ero

s o

f P

oly

no

mia

ls

A1 2 3 4 5

C

x f(x)

B 5

G 0 –2

D –5 23

J 5 23

E

–3.3333333

9.1111111

H

1.6666667

9.1111111

I

3.3333333

0.7777778

F

–1.6666667

0.7777778

–5

Sh

eet

1S

hee

t 2

Sh

eet

3

Exer

cise

s

1.U

se a

spr

eads

hee

t li

ke t

he

one

abov

e to

app

roxi

mat

e th

e ze

ro o

f ƒ(

x) �

3x�

2 to

th

ree

deci

mal

pla

ces.

Th

en v

erif

y yo

ur

answ

er b

y u

sin

g al

gebr

a to

fin

d th

e ex

act

valu

e of

th

ero

ot.

Th

e sp

read

shee

t g

ives

x�

0.66

7.B

y so

lvin

g f

or

xal

geb

raic

ally

,x

��2 3�.

So

,th

e ap

pro

xim

atio

n is

co

rrec

t.

2.U

se a

spr

eads

heet

like

the

one

abo

ve t

o ap

prox

imat

e th

e re

al z

eros

of

f(x)

�x2

�2x

�0.

5.R

oun

d yo

ur

answ

er t

o fo

ur

deci

mal

pla

ces.

Th

en, v

erif

y yo

ur

answ

er b

y u

sin

g th

e qu

adra

tic

form

ula

.T

he

pro

cess

giv

es�

1.70

71 a

nd

�0.

2929

to

th

e n

eare

st

ten

-th

ou

san

dth

.Th

e q

uad

rati

c fo

rmu

la g

ives

x�

�1

�� 22 � �

.�

1�

�� 22 � ��

�1.

7071

an

d �

1�

�� 22 � ��

�0.

2929

.3.

Use

a s

prea

dshe

et li

ke t

he o

ne a

bove

to

appr

oxim

ate

the

real

zer

o of

ƒ(x

) �x3

��3 2�x

2�

6x�

2be

twee

n �

0.4

and

�0.

3.�

0.37

81 t

o t

he

nea

rest

ten

-th

ou

san

dth

Lesson 5-7

Cha

pter

549

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-7


Less

on R

eadi

ng G

uide

An

alyz

ing

Gra

ph

s o

f Q

uad

rati

c E

qu

atio

ns

Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 5

-7 i

n y

our

text

boo

k.

•W

hat

doe

s ad

din

g a

posi

tive

nu

mbe

r to

x2

do t

o th

e gr

aph

of

y�

x2?

It m

oves

th

e g

rap

h u

p.

•W

hat

doe

s su

btra

ctin

g a

posi

tive

nu

mbe

r to

xbe

fore

squ

arin

g do

to

the

grap

h o

f y

�x2

?It

mov

es t

he

gra

ph

to

th

e ri

gh

t.

Rea

d t

he

Less

on

1.C

ompl

ete

the

foll

owin

g in

form

atio

n a

bou

t th

e gr

aph

of

y�

a(x

�h

)2�

k.

a.W

hat

are

th

e co

ordi

nat

es o

f th

e ve

rtex

?(h

,k)

b.

Wh

at i

s th

e eq

uat

ion

of

the

axis

of

sym

met

ry?

x�

h

c.In

wh

ich

dir

ecti

on d

oes

the

grap

h o

pen

if

a�

0? I

f a

�0?

up

;d

ow

n

d.

Wh

at d

o yo

u k

now

abo

ut

the

grap

h i

f ⏐a

⏐�

1?It

is w

ider

th

an t

he

gra

ph

of

y�

x2 .

If ⏐

a⏐�

1?It

is n

arro

wer

th

an t

he

gra

ph

of

y�

x2 .

2.M

atch

eac

h g

raph

wit

h t

he

desc

ript

ion

of

the

con

stan

ts i

n t

he

equ

atio

n i

n v

erte

x fo

rm.

a.a

�0,

h�

0, k

�0

iiib

.a�

0, h

�0,

k�

0iv

c.a

�0,

h�

0, k

�0

iid

.a�

0, h

�0,

k�

0i

i.ii

.ii

i.iv

.

Rem

emb

er W

hat

Yo

u L

earn

ed

3.W

hen

grap

hing

qua

drat

ic f

unct

ions

suc

h as

y�

(x�

4)2

and

y�

(x�

5)2 ,

man

y st

uden

tsha

ve t

roub

le r

emem

beri

ng w

hich

rep

rese

nts

a tr

ansl

atio

n of

the

gra

ph o

f y

�x2

to t

he le

ftan

d w

hich

rep

rese

nts

a tr

ansl

atio

n to

the

rig

ht. W

hat

is a

n ea

sy w

ay t

o re

mem

ber

this

?

Sam

ple

an

swer

:In

fu

nct

ion

s lik

e y

�(x

�4)

2 ,th

e p

lus

sig

n p

uts

th

eg

rap

h “

ahea

d”

so t

hat

th

e ve

rtex

co

mes

“so

on

er”

than

th

e o

rig

in a

nd

th

etr

ansl

atio

n is

to

th

e le

ft.I

n f

un

ctio

ns

like

y�

(x�

5)2 ,

the

min

us

pu

ts t

he

gra

ph

“b

ehin

d”

so t

hat

th

e ve

rtex

co

mes

“la

ter”

than

th

e o

rig

in a

nd

th

etr

ansl

atio

n is

to

th

e ri

gh

t.

x

y

Ox

y

Ox

y

Ox

y

O

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A23



Exer

cise

s

Exam

ple

Cha

pter

550

Gle

ncoe

Alg

ebra

2

5-7


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Stud

y G

uide

and

Inte

rven

tion

An

alyz

ing

Gra

ph

s o

f Q

uad

rati

c F

un

ctio

ns

An

alyz

e Q

uad

rati

c Fu

nct

ion

s

The

gra

ph o

f y

�a

(x�

h)2

�k

has

the

follo

win

g ch

arac

teris

tics:

•V

erte

x: (

h, k

)V

erte

x F

orm

•A

xis

of s

ymm

etry

: x

�h

of

a Q

uad

rati

c•

Ope

ns u

p if

a�

0F

un

ctio

n•

Ope

ns d

own

if a

�0

•N

arro

wer

tha

n th

e gr

aph

of y

�x

2if

⏐a⏐

�1

•W

ider

tha

n th

e gr

aph

of y

�x

2if

⏐a⏐

�1

Iden

tify

th

e ve

rtex

, axi

s of

sym

met

ry, a

nd

dir

ecti

on o

f op

enin

g of

each

gra

ph

.

a.y

�2(

x�

4)2

�11

Th

e ve

rtex

is

at (

h, k

) or

(�

4, �

11),

an

d th

e ax

is o

f sy

mm

etry

is

x�

�4.

Th

e gr

aph

ope

ns

up.

a.y

��

(x�

2)2

�10

Th

e ve

rtex

is

at (

h, k

) or

(2,

10)

, an

d th

e ax

is o

f sy

mm

etry

is

x�

2. T

he

grap

h

open

s do

wn

.

Eac

h q

uad

rati

c fu

nct

ion

is

give

n i

n v

erte

x fo

rm. I

den

tify

th

e ve

rtex

, axi

s of

sym

met

ry, a

nd

dir

ecti

on o

f op

enin

g of

th

e gr

aph

.

1.y

�(x

�2)

2�

162.

y�

4(x

�3)

2�

73.

y�

(x�

5)2

�3

(2,1

6);

x�

2;u

p(�

3,�

7);

x�

�3;

up

(5,3

);x

�5;

up

4.y

��

7(x

�1)

2�

95.

y�

(x�

4)2

�12

6.y

�6(

x�

6)2

�6

(�1,

�9)

;x

��

1;d

ow

n(4

,�12

);x

�4;

up

(�6,

6);

x�

�6;

up

7.y

�(x

�9)

2�

128.

y�

8(x

�3)

2�

29.

y�

�3(

x�

1)2

�2

(9,1

2);

x�

9;u

p(3

,�2)

;x

�3;

up

(1,�

2);

x�

1;d

ow

n

10.y

��

(x�

5)2

�12

11.y

�(x

�7)

2�

2212

.y�

16(x

�4)

2�

1

(�5,

12);

x�

�5;

do

wn

(7,2

2);

x�

7;u

p(4

,1);

x�

4;u

p

13.y

�3(

x�

1.2)

2�

2.7

14.y

��

0.4(

x�

0.6)

2�

0.2

15.y

�1.

2(x

�0.

8)2

�6.

5

(1.2

,2.7

);x

�1.

2;u

p(0

.6,�

0.2)

;x

�0.

6;(�

0.8,

6.5)

;x

��

0.8;

do

wn

up

4 � 35 � 2

2 � 5

1 � 5

1 � 2

1 � 4

Lesson 5-7

Cha

pter

551

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-7


Exer

cise

s

Exam

ple

Wri

te Q

uad

rati

c Fu

nct

ion

s in

Ver

tex

Form

Aqu

adra

tic

fun

ctio

n i

s ea

sier

to

grap

h w

hen

it

is i

n v

erte

x fo

rm. Y

ou c

an w

rite

a q

uad

rati

c fu

nct

ion

of

the

form

y

�ax

2�

bx�

cin

ver

tex

from

by

com

plet

ing

the

squ

are.

Wri

te y

�2x

2�

12x

�25

in

ver

tex

form

. Th

en g

rap

h t

he

fun

ctio

n.

y�

2x2

�12

x�

25y

�2(

x2�

6x)

�25

y�

2(x2

�6x

�9)

�25

�18

y�

2(x

�3)

2�

7

Th

e ve

rtex

for

m o

f th

e eq

uat

ion

is

y�

2(x

�3)

2�

7.

Wri

te e

ach

qu

adra

tic

fun

ctio

n i

n v

erte

x fo

rm. T

hen

gra

ph

th

e fu

nct

ion

.

1.y

�x2

�10

x �

322.

y �

x2�

6x3.

y�

x2�

8x�

6y

�(x

�5)

2�

7y

�(x

�3)

2�

9y

�(x

�4)

2�

10

4.y

��

4x2

�16

x�

115.

y�

3x2

�12

x�

56.

y�

5x2

�10

x�

9y

��

4(x

�2)

2�

5y

�3(

x�

2)2

�7

y�

5(x�

1)2

�4 x

y

O

x

y

O

x

y

O

x

y

O4

–48

8 4 –4 –8 –12

x

y

O

x

y

O

x

y

O

Stud

y G

uide

and

Inte

rven

tion

(con

tinue

d)

An

alyz

ing

Gra

ph

s o

f Q

uad

rati

c F

un

ctio

ns

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A24


An

swer

s


Cha

pter

552

Gle

ncoe

Alg

ebra

2

5-7


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Skill

s Pr

acti

ceA

nal

yzin

g G

rap

hs

of

Qu

adra

tic

Fu

nct

ion

sW

rite

eac

h q

uad

rati

c fu

nct

ion

in

ver

tex

form

, if

not

alr

ead

y in

th

at f

orm

. Th

enid

enti

fy t

he

vert

ex, a

xis

of s

ymm

etry

, an

d d

irec

tion

of

open

ing.

1.y

�(x

�2)

22.

y�

�x2

�4

3.y

�x2

�6

y�

(x�

2)2

�0;

y�

�(x

�0)

2�

4;y

�(x

�0)

2�

6;(2

,0);

x�

2;u

p(0

,4);

x�

0;d

ow

n(0

,�6)

;x

�0;

up

4.y

��

3(x

�5)

25.

y�

�5x

2�

96.

y�

(x�

2)2

�18

y�

�3(

x�

5)2

�0;

y�

�5(

x�

0)2

�9;

y�

(x�

2)2

�18

;(�

5,0)

;x

��

5;d

ow

n(0

,9);

x�

0;d

ow

n(2

,�18

);x

�2;

up

7.y

�x2

�2x

�5

8.y

�x2

�6x

�2

9.y

��

3x2

�24

xy

�(x

�1)

2�

6;y

�(x

�3)

2�

7;y

��

3(x

�4)

2�

48;

(1,�

6);

x�

1;u

p(�

3,�

7);

x�

�3;

up

(4,4

8);

x�

4;d

ow

n

Gra

ph

eac

h f

un

ctio

n.

10.y

�(x

�3)

2�

111

.y�

(x�

1)2

�2

12.y

��

(x�

4)2

�4

13.y

��

(x�

2)2

14.y

��

3x2

�4

15.y

�x2

�6x

�4

Wri

te a

n e

qu

atio

n f

or t

he

par

abol

a w

ith

th

e gi

ven

ver

tex

that

pas

ses

thro

ugh

th

egi

ven

poi

nt.

16.v

erte

x: (

4, �

36)

17.v

erte

x: (

3, �

1)18

.ver

tex:

(�

2, 2

)po

int:

(0,

�20

)po

int:

(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

Lesson 5-7

Cha

pter

553

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-7


Wri

te e

ach

qu

adra

tic

fun

ctio

n i

n v

erte

x fo

rm, i

f n

ot a

lrea

dy

in t

hat

for

m. T

hen

iden

tify

th

e ve

rtex

, axi

s of

sym

met

ry, a

nd

dir

ecti

on o

f op

enin

g.

1.y

��

6(x

�2)

2�

12.

y�

2x2

�2

3.y

��

4x2

�8x

y�

�6(

x�

2)2

�1;

y�

2(x

�0)

2�

2;y

��

4(x

�1)

2�

4;(�

2,�

1);

x�

�2;

do

wn

(0,2

);x

�0;

up

(1,4

);x

�1;

do

wn

4.y

�x2

�10

x�

205.

y�

2x2

�12

x�

186.

y�

3x2

�6x

�5

y�

(x�

5)2

�5;

y�

2(x

�3)

2 ;(�

3,0)

;y

�3(

x�

1)2

�2;

(�5,

�5)

;x

��

5;u

px

��

3;u

p(1

,2);

x�

1;u

p

7.y

��

2x2

�16

x�

328.

y�

�3x

2�

18x

�21

9.y

�2x

2�

16x

�29

y�

�2(

x�

4)2 ;

y�

�3(

x�

3)2

�6;

y�

2(x

�4)

2�

3;(�

4,0)

;x

��

4;d

ow

n(3

,6);

x�

3;d

ow

n(�

4,�

3);

x�

�4;

up

Gra

ph

eac

h f

un

ctio

n.

10.y

�(x

�3)

2�

111

.y�

�x2

�6x

�5

12.y

�2x

2�

2x�

1

Wri

te a

n e

qu

atio

n f

or t

he

par

abol

a w

ith

th

e gi

ven

ver

tex

that

pas

ses

thro

ugh

th

egi

ven

poi

nt.

13.v

erte

x: (

1, 3

)14

.ver

tex:

(�

3, 0

) 15

.ver

tex:

(10

, �4)

poin

t: (

�2,

�15

)po

int:

(3,

18)

poin

t: (

5, 6

)y

��

2(x

�1)

2�

3y

�(x

�3)

2y

�(x

�10

)2�

4

16.W

rite

an

equ

atio

n f

or a

par

abol

a w

ith

ver

tex

at (

4, 4

) an

d x-

inte

rcep

t 6.

y�

�(x

�4)

2�

4

17.W

rite

an

equ

atio

n f

or a

par

abol

a w

ith

ver

tex

at (

�3,

�1)

an

d y-

inte

rcep

t 2.

y�

(x�

3)2

�1

18.B

ASE

BA

LLT

he

hei

ght

hof

a b

aseb

all

tse

con

ds a

fter

bei

ng

hit

is

give

n b

y h

(t)

��

16t2

�80

t�

3. W

hat

is

the

max

imu

m h

eigh

t th

at t

he

base

ball

rea

ches

, an

dw

hen

doe

s th

is o

ccu

r?10

3 ft

;2.

5 s

19.S

CU

LPTU

RE

Am

oder

n sc

ulpt

ure

in a

par

k co

ntai

ns a

par

abol

ic a

rc t

hat

star

ts a

t th

e gr

oun

d an

d re

ach

es a

max

imu

m h

eigh

t of

10

feet

aft

er a

hor

izon

tal

dist

ance

of

4 fe

et. W

rite

a q

uad

rati

c fu

nct

ion

in

ver

tex

form

that

des

crib

es t

he

shap

e of

th

e ou

tsid

e of

th

e ar

c, w

her

e y

is t

he

hei

ght

of a

poi

nt

on t

he

arc

and

xis

its

hor

izon

tal

dist

ance

fro

m t

he

left

-han

dst

arti

ng

poin

t of

th

e ar

c.y

��

(x�

4)2

�10

5 � 8

10 ft

4 ft

1 � 3

2 � 51 � 2

x

y O

x

y

O

x

y

O

Prac

tice

An

alyz

ing

Gra

ph

s o

f Q

uad

rati

c F

un

ctio

ns

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A25



Cha

pter

554

Gle

ncoe

Alg

ebra

2

5-7


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

1.A

RC

HES

A p

arab

olic

arc

h i

s u

sed

as a

brid

ge s

upp

ort.

Th

e gr

aph

of

the

arch

is

show

n b

elow

.

If t

he

equ

atio

n t

hat

cor

resp

onds

to

this

gra

ph i

s w

ritt

en i

n t

he

form

y

�a(

x�

h)2

�k,

wh

at a

re h

and

k?h

��

1 an

d k

�5

2.TR

AN

SLA

TIO

NS

For

a c

ompu

ter

anim

atio

n,B

arba

ra u

ses

the

quad

rati

cfu

nct

ion

f(x

) �

�42

(x�

20)2

�16

800

toh

elp

her

sim

ula

te a

n o

bjec

t to

ssed

on

anot

her

pla

net

.For

on

e sk

it,s

he

had

to

use

th

e fu

nct

ion

f(x

�5)

�80

00 i

nst

ead

of f

(x).

Wh

ere

is t

he

vert

ex o

f th

e gr

aph

of y

�f(

x�

5) �

8000

?(1

5,88

00)

3.M

IRR

OR

ST

he

cros

s-se

ctio

n o

f a

refl

ecti

ng

tele

scop

e m

irro

r is

des

crib

ed

by t

he

para

bola

y �

� 11 0�(x

� 5

)2�

.

Gra

ph t

his

par

abol

a.

4.W

ATE

R J

ETS

Th

e gr

aph

sh

ows

the

path

of

a je

t of

wat

er.

Th

e eq

uat

ion

cor

resp

ondi

ng

to t

his

grap

h i

s y

�a(

x�

h)

2�

k.W

hat

are

a,

h,a

nd

k?a

��

2,h

�2,

k�

6

PRO

FIT

For

Exe

rcis

es 5

–7,u

se t

he

foll

owin

g in

form

atio

n.

A t

hea

ter

oper

ator

pre

dict

s th

at t

he

thea

ter

can

mak

e �

4x2

�16

0xdo

llar

s pe

r sh

ow i

fti

cket

s ar

e pr

iced

at

xdo

llar

s.

5.R

ewri

te t

he

equ

atio

n y

��

4x2

�16

xin

the

form

y�

a(x

�h

) 2

�k.

y�

�4(

x�

2)2

�16

6.W

hat

is

the

vert

ex o

f th

e pa

rabo

la a

nd

wh

at i

s it

s ax

is o

f sy

mm

etry

?ve

rtex

at

(20,

1600

);ax

is is

x�

20

7.G

raph

th

e pa

rabo

la.

y

xO

800

1600

2040

y

xO

5

y

O10

x

5 � 2

y

xO

5

5

-5Wor

d Pr

oble

m P

ract

ice

An

alyz

ing

Gra

ph

s o

f Q

uad

rati

c F

un

ctio

ns

Exam

ple

Exer

cise

s

Lesson 5-7

Cha

pter

555

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-7


Enri

chm

ent

A S

ho

rtcu

t to

Co

mp

lex

Ro

ots

Wh

en g

raph

ing

a qu

adra

tic

fun

ctio

n, t

he

real

roo

ts a

re s

how

n i

n t

he

grap

h.

You

hav

e le

arn

ed t

hat

qu

adra

tic

fun

ctio

ns

can

als

o h

ave

imag

inar

y ro

ots

that

can

not

be

seen

on

th

e gr

aph

of

the

fun

ctio

n. H

owev

er, t

her

e is

a w

ay t

o gr

aph

ical

ly r

epre

sen

t th

e co

mpl

ex r

oots

of

a qu

adra

tic

fun

ctio

n.

Fin

d t

he

com

ple

x ro

ots

of t

he

qu

adra

tic

fun

ctio

ny

�x2

�4x

�5.

Ste

p 1

:G

raph

th

e fu

nct

ion

.

Ste

p 2

:R

efle

ct t

he

grap

h o

ver

the

hor

izon

tal

lin

e co

nta

inin

g th

e ve

rtex

. In

th

is e

xam

ple,

the

vert

ex i

s (2

, 1).

Ste

p 3

:T

he

real

par

t of

th

e co

mpl

ex r

oot

is t

he

poin

t h

alfw

ay b

etw

een

th

e x-

inte

rcep

ts o

fth

e re

flec

ted

grap

h a

nd

the

imag

inar

y pa

rt o

f th

e co

mpl

ex r

oots

are

�an

d�

hal

fth

e di

stan

ce b

etw

een

th

e x-

inte

rcep

ts o

f th

e re

flec

ted

grap

h. S

o, i

n t

his

exa

mpl

e,th

e co

mpl

ex r

oots

are

2�

1ian

d 2

�1i

.

Usi

ng

this

met

hod

, fin

d t

he

com

ple

x ro

ots

of t

he

foll

owin

g q

uad

rati

c fu

nct

ion

s.

1.y

�x2

�2x

�5

2.y

�x2

�4x

�8

�1

�2i

,�1

�2i

�2

�2i

,�2

�2i

3.y

�x2

�6x

�13

4.y

�x2

�2x

�17

�3

�2i

,�3

�2i

�1

�4i

,�1

�4i

y

xO

y

xO6

5

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A26


An

swer

s


Cha

pter

556

Gle

ncoe

Alg

ebra

2

5-8


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Less

on R

eadi

ng G

uide

Gra

ph

ing

an

d S

olv

ing

Qu

adra

tic

Ineq

ual

itie

sG

et R

ead

y fo

r th

e Le

sso

nR

ead

th

e in

trod

uct

ion

to

Les

son

5-8

in

you

r te

xtb

ook

.

•H

ow f

ar a

bove

th

e gr

oun

d is

th

e tr

ampo

lin

e su

rfac

e?3.

75 f

eet

•U

sin

g th

e qu

adra

tic

fun

ctio

n g

iven

in

th

e in

trod

uct

ion

, wri

te a

qu

adra

tic

ineq

ual

ity

that

des

crib

es t

he

tim

es a

t w

hic

h t

he

tram

poli

nis

t is

mor

e th

an

20 f

eet

abov

e th

e gr

oun

d.�

16t2

�42

t�

3.75

�20

Rea

d t

he

Less

on

1.A

nsw

er t

he

foll

owin

g qu

esti

ons

abou

t h

ow y

ou w

ould

gra

ph t

he

ineq

ual

ity

y�

x2�

x�

6.

a.W

hat

is

the

rela

ted

quad

rati

c eq

uat

ion

?y

�x

2�

x�

6

b.

Sh

ould

th

e pa

rabo

la b

e so

lid

or d

ash

ed?

How

do

you

kn

ow?

solid

;Th

e in

equ

alit

y sy

mb

ol i

s �

.

c.T

he

poin

t (0

, 2)

is i

nsi

de t

he

para

bola

. To

use

th

is a

s a

test

poi

nt,

su

bsti

tute

for

xan

d fo

r y

in t

he

quad

rati

c in

equ

alit

y.

d.

Is t

he

stat

emen

t 2

�02

�0

�6

tru

e or

fal

se?

tru

e

e.S

hou

ld t

he

regi

on i

nsi

de o

r ou

tsid

e th

e pa

rabo

la b

e sh

aded

?in

sid

e

2.T

he

grap

h o

f y

��

x2�

4xis

sh

own

at

the

righ

t. M

atch

eac

h

of t

he

foll

owin

g re

late

d in

equ

alit

ies

wit

h i

ts s

olu

tion

set

.

a.�

x2�

4x�

0ii

i.{x

⏐x�

0 or

x�

4}

b.

�x2

�4x

0

iiiii

.{x⏐

0 �

x�

4}

c.�

x2�

4x�

0iv

iii.

{x⏐x

0

or x

�4}

d.

�x2

�4x

�0

iiv

.{x

⏐0

x

4}

Rem

emb

er W

hat

Yo

u L

earn

ed

3.A

quad

rati

c in

equ

alit

y in

tw

o va

riab

les

may

hav

e th

e fo

rm y

�ax

2�

bx�

c,

y�

ax2

�bx

�c,

y�

ax2

�bx

�c,

or

y

ax2

�bx

�c.

Des

crib

e a

way

to

rem

embe

rw

hich

reg

ion

to s

hade

by

look

ing

at t

he i

nequ

alit

y sy

mbo

l an

d w

itho

ut u

sing

a t

est

poin

t.S

amp

le a

nsw

er:

If t

he

sym

bo

l is

�o

r �

,sh

ade

the

reg

ion

ab

ove

the

par

abo

la.I

f th

e sy

mb

ol i

s �

or

�,s

had

e th

e re

gio

n b

elo

w t

he

par

abo

la.x

y

O( 0

, 0)

( 4, 0

)

( 2, 4

)

20

Exer

cise

s

Exam

ple

Lesson 5-8

Cha

pter

557

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-8


Stud

y G

uide

and

Inte

rven

tion

Gra

ph

ing

an

d S

olv

ing

Qu

adra

tic

Ineq

ual

itie

sG

rap

h Q

uad

rati

c In

equ

alit

ies

To g

raph

a q

uad

rati

c in

equ

alit

y in

tw

o va

riab

les,

use

the

foll

owin

g st

eps:

1.G

raph

th

e re

late

d qu

adra

tic

equ

atio

n, y

�ax

2�

bx�

c.U

se a

das

hed

lin

e fo

r �

or �

; use

a s

olid

lin

e fo

r

or �

.

2.Te

st a

poi

nt

insi

de t

he

para

bola

.If

it

sati

sfie

s th

e in

equ

alit

y, s

had

e th

e re

gion

in

side

th

e pa

rabo

la;

oth

erw

ise,

sh

ade

the

regi

on o

uts

ide

the

para

bola

.

Gra

ph

th

e in

equ

alit

y y

�x2

�6x

�7.

Fir

st g

raph

th

e eq

uat

ion

y�

x2�

6x�

7. B

y co

mpl

etin

g th

e sq

uar

e, y

ou g

et t

he

vert

ex f

orm

of

the

equ

atio

n y

�(x

�3)

2�

2,

so t

he

vert

ex i

s (�

3, �

2). M

ake

a ta

ble

of v

alu

es a

rou

nd

x�

�3,

an

d gr

aph

. Sin

ce t

he

ineq

ual

ity

incl

ude

s �

, use

a d

ash

ed l

ine.

Test

th

e po

int

(�3,

0),

wh

ich

is

insi

de t

he

para

bola

. Sin

ce

(�3)

2�

6(�

3) �

7 �

�2,

an

d 0

��

2, (

�3,

0)

sati

sfie

s th

e in

equ

alit

y. T

her

efor

e, s

had

e th

e re

gion

in

side

th

e pa

rabo

la.

Gra

ph

eac

h i

neq

ual

ity.

1.y

�x2

�8x

�17

2.y

x2

�6x

�4

3.y

�x2

�2x

�2

4.y

��

x2�

4x�

65.

y�

2x2

�4x

6.y

��

2x2

�4x

�2

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A27



Exer

cise

s

Cha

pter

558

Gle

ncoe

Alg

ebra

2

5-8


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Solv

e Q

uad

rati

c In

equ

alit

ies

Qu

adra

tic

ineq

ual

itie

s in

on

e va

riab

le c

an b

e so

lved

grap

hic

ally

or

alge

brai

call

y.

To s

olve

ax

2�

bx�

c�

0:F

irst

grap

h y

�ax

2�

bx�

c. T

he s

olut

ion

cons

ists

of

the

x-va

lues

Gra

ph

ical

Met

ho

dfo

r w

hich

the

gra

ph is

bel

ow

the

x-ax

is.

To s

olve

ax

2�

bx�

c�

0:F

irst

grap

h y

�ax

2�

bx�

c. T

he s

olut

ion

cons

ists

the

x-v

alue

s fo

r w

hich

the

gra

ph is

ab

ove

the

x-ax

is.

Fin

d th

e ro

ots

of t

he r

elat

ed q

uadr

atic

equ

atio

n by

fac

torin

g,

Alg

ebra

ic M

eth

od

com

plet

ing

the

squa

re,

or u

sing

the

Qua

drat

ic F

orm

ula.

2 ro

ots

divi

de t

he n

umbe

r lin

e in

to 3

inte

rval

s.Te

st a

val

ue in

eac

h in

terv

al t

o se

e w

hich

inte

rval

s ar

e so

lutio

ns.

If t

he

ineq

ual

ity

invo

lves

or

�, t

he

root

s of

th

e re

late

d eq

uat

ion

are

in

clu

ded

in t

he

solu

tion

set

.

Sol

ve t

he

ineq

ual

ity

x2�

x�

6 �

0.

Fir

st f

ind

the

root

s of

th

e re

late

d eq

uat

ion

x2

�x

�6

�0.

Th

eeq

uat

ion

fac

tors

as

(x�

3)(x

�2)

�0,

so

the

root

s ar

e 3

and

�2.

T

he

grap

h o

pen

s u

p w

ith

x-i

nte

rcep

ts 3

an

d �

2, s

o it

mu

st b

e on

or

bel

ow t

he

x-ax

is f

or �

2

x

3. T

her

efor

e th

e so

luti

on s

et i

s {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

�o

r x

�4 �

��x⏐⏐

�x

�5 �

1 � 33 � 2

1 � 63 � 4

1 � 2

1 � 2

x

y

O

Stud

y G

uide

and

Inte

rven

tion

(con

tinue

d)

Gra

ph

ing

an

d S

olv

ing

Qu

adra

tic

Ineq

ual

itie

s

Exam

ple

Lesson 5-8

Cha

pter

559

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-8


Skill

s Pr

acti

ceG

rap

hin

g a

nd

So

lvin

g Q

uad

rati

c In

equ

alit

ies

Gra

ph

eac

h i

neq

ual

ity.

1.y

�x2

�4x

�4

2.y

x2

�4

3.y

�x2

�2x

�5

Use

th

e gr

aph

of

its

rela

ted

fu

nct

ion

to

wri

te t

he

solu

tion

s of

eac

h i

neq

ual

ity.

4.x2

�6x

�9

0

5.�

x2�

4x�

32 �

06.

x2�

x�

20 �

0

3�

8 �

x�

4x

��

5 o

r x

�4

Sol

ve e

ach

in

equ

alit

y al

geb

raic

ally

.

7.x2

�3x

�10

�0

8.x2

�2x

�35

�0

{x⏐⏐ �

2 �

x�

5}{x

⏐⏐ x�

�7

or

x�

5}

9.x2

�18

x�

81

010

.x2

36

{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 o

r x

�3}

{x⏐⏐ �

6 �

x�

�3}

15.x

2�

10x

�25

�0

16.�

x2�

2x�

15 �

0al

l rea

ls{x

⏐⏐ �5

�x

�3}

17.x

2�

3x�

018

.2x2

�2x

�4

{x⏐⏐ x

��

3 o

r x

�0}

{x⏐⏐ x

��

2 o

r x

�1}

19.�

x2�

64

�16

x20

.9x2

�12

x�

9 �

0al

l rea

ls�

x

y O2

5

x

y O2

6

x

y O

x

y

O

x

y

O

x

y

O

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A28


An

swer

s


Cha

pter

560

Gle

ncoe

Alg

ebra

2

5-8


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Gra

ph

eac

h i

neq

ual

ity.

1.y

x2

�4

2.y

�x2

�6x

�6

3.y

�2x

2�

4x�

2

Use

th

e gr

aph

of

its

rela

ted

fu

nct

ion

to

wri

te t

he

solu

tion

s of

eac

h i

neq

ual

ity.

4.x2

�8x

�0

5.�

x2�

2x�

3 �

06.

x2�

9x�

14

0

x�

0 o

r x

�8

�3

�x

�1

2 �

x�

7

Sol

ve e

ach

in

equ

alit

y al

geb

raic

ally

.

7.x2

�x

�20

�0

8.x2

�10

x�

16 �

09.

x2�

4x�

5

0

{x⏐⏐ x

��

4 o

r 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 o

r 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 o

r x

��

16.4

x2�

4x�

1 �

017

.5x2

�10

�27

x18

.9x2

�31

x�

12

0

�x⏐⏐x

�

��x⏐⏐

x�

or

x�

5 ��x⏐⏐

�3

�x

��

�19

.FEN

CIN

GV

anes

sa h

as 1

80 f

eet

of f

enci

ng

that

sh

e in

ten

ds t

o u

se t

o bu

ild

a re

ctan

gula

rpl

ay a

rea

for

her

dog

. Sh

e w

ants

th

e pl

ay a

rea

to e

ncl

ose

at l

east

180

0 sq

uar

e fe

et. W

hat

are

the

poss

ible

wid

ths

of t

he

play

are

a?30

ft

to 6

0 ft

20.B

USI

NES

SA

bicy

cle

mak

er s

old

300

bicy

cles

last

yea

r at

a p

rofi

t of

$30

0 ea

ch. T

he m

aker

wan

ts t

o in

crea

se t

he

prof

it m

argi

n t

his

yea

r, b

ut

pred

icts

th

at e

ach

$20

in

crea

se i

npr

ofit

wil

l re

duce

the

num

ber

of b

icyc

les

sold

by

10. H

ow m

any

$20

incr

ease

s in

pro

fit

can

the

mak

er a

dd i

n a

nd

expe

ct t

o m

ake

a to

tal

prof

it o

f at

lea

st $

100,

000?

fro

m 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

OPrac

tice

Gra

ph

ing

an

d S

olv

ing

Qu

adra

tic

Ineq

ual

itie

s

Lesson 5-8

Cha

pter

561

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-8


1.H

UTS

Th

e sp

ace

insi

de a

hu

t is

sh

aded

in t

he

grap

h. T

he

para

bola

is

desc

ribe

d

by t

he

equ

atio

n y

��

(x�

1)2

�4.

Wri

te a

n i

neq

ual

ity

that

des

crib

es t

he

shad

ed r

egio

n.

0 �

y�

�(x

�1)

2�

4

2.D

ISC

RIM

INA

NTS

Con

side

r th

eeq

uat

ion

ax2

�bx

�c

�0.

Ass

um

e th

atth

e di

scri

min

ant

is z

ero

and

that

ais

posi

tive

. Wh

at a

re t

he

solu

tion

s of

th

ein

equ

alit

y ax

2�

bx�

c

0?

x�

�

3.TO

SSIN

GG

ail

and

Ver

onic

a ar

e fi

xin

ga

leak

in

a r

oof.

Gai

l is

wor

kin

g on

th

ero

of a

nd

Ver

onic

a is

tos

sin

g u

p su

ppli

esto

Gai

l. W

hen

Gai

l to

sses

up

a ta

pem

easu

re, t

he

hei

ght

h, i

n f

eet,

of

the

obje

ct a

bove

th

e gr

oun

d t

seco

nds

aft

erG

ail

toss

es i

t is

h�

�16

t2�

32t

�5.

Gai

l ca

n c

atch

th

e ob

ject

an

y ti

me

it i

sab

ove

17 f

eet.

How

mu

ch t

ime

does

Gai

lh

ave

to t

ry t

o ca

tch

th

e ta

pe m

easu

re?

1 se

con

d

4.K

IOSK

SC

aleb

is

desi

gnin

g a

kios

k by

wra

ppin

g a

piec

e of

sh

eet

met

al w

ith

dim

ensi

ons

x�

5 in

ches

by

4x�

8in

ches

in

to a

cyl

indr

ical

sh

ape.

Ign

orin

gco

st, C

aleb

wou

ld l

ike

a ki

osk

that

has

asu

rfac

e ar

ea o

f at

lea

st 4

480

squ

are

inch

es. W

hat

val

ues

of

xsa

tisf

y th

isco

ndi

tion

?x

�30

(N

ote

th

at t

he

valu

es o

f x

��

37 r

esu

lt in

a h

igh

erp

rod

uct

,bu

t n

egat

ive

len

gth

s d

o n

ot

mak

e se

nse

.)

TUN

NEL

SF

or E

xerc

ises

5 a

nd

6, u

seth

e fo

llow

ing

info

rmat

ion

.A

n a

rch

itec

t w

ants

to

use

a p

arab

olic

arc

has

th

e en

tran

ce o

f a

tun

nel

. Sh

e sk

etch

esth

e pl

an o

n a

pie

ce o

f gr

aph

pap

er. S

he

wou

ld l

ike

the

max

imu

m h

eigh

t of

th

etu

nn

el t

o be

loc

ated

at

(4, 4

), a

nd

she

wou

ldli

ke t

he

orig

in t

o be

on

th

e pa

rabo

la a

s w

ell.

5.W

rite

an

equ

atio

n f

or t

he

desi

red

para

bola

.y

��

0.25

(x�

4)2

�4

6.W

rite

an

in

equ

alit

y th

at d

escr

ibes

th

ere

gion

abo

ve t

he

para

bola

, par

t of

wh

ich

wil

l be

fil

led

in w

ith

con

cret

e. G

raph

this

in

equ

alit

y.y

��

0.25

(x�

4)2

�4

y

xO

b � 2a

4 � 5

y

xO

4 � 5

Wor

d Pr

oble

m P

ract

ice

Gra

ph

ing

an

d S

olv

ing

of

Qu

adra

tic

Ineq

ual

itie

s

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A29



Cha

pter

562

Gle

ncoe

Alg

ebra

2

5-8


NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

Gra

ph

ing

Ab

solu

te V

alu

e In

equ

alit

ies

You

can

sol

ve a

bsol

ute

val

ue

ineq

ual

itie

s by

gra

phin

g in

mu

ch t

he

sam

e m

ann

er y

ou g

raph

ed q

uad

rati

c in

equ

alit

ies.

Gra

ph t

he

rela

ted

abso

lute

fu

nct

ion

fo

r ea

ch i

neq

ual

ity

by u

sin

g a

grap

hin

g ca

lcu

lato

r. F

or �

and

�, i

den

tify

th

e x-

valu

es, i

f an

y, f

or w

hic

h t

he

grap

h l

ies

belo

wth

e x-

axis

. For

�an

d

, ide

nti

fy

the

xva

lues

, if

any,

for

wh

ich

th

e gr

aph

lie

s ab

ove

the

x-ax

is.

For

eac

h i

neq

ual

ity,

mak

e a

sket

ch o

f th

e re

late

d g

rap

h a

nd

fin

d t

he

solu

tion

s ro

un

ded

to

the

nea

rest

hu

nd

red

th.

1.|x

�3|

�0

2.|x|

�6

�0

3.�

|x �

4| �

8 �

0

x�

3 o

r x

�3

�6

�x

�6

�12

�x

�4

4.2|x

�6|

�2

�0

5.|3x

�3|

�0

6.|x

�7|

�5

x�

�7

or

x�

�5

all r

eal n

um

ber

s2

�x

�12

7.|7x

�1|

�13

8.|x

�3.

6|

4.2

9.|2x

�5|

7

x�

�1.

71 o

r x

�2

�0.

6 �

x�

7.8

�6

�x

�1

Enri

chm

ent

Exer

cise

s

Exam

ple

2

Exam

ple

1

Lesson 5-8

Cha

pter

563

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

5-8


Gra

phin

g Ca

lcul

ator

Act

ivit

yQ

uad

rati

c In

equ

alit

ies

and

th

e Te

st M

enu

The

ine

qual

ity

sym

bols

, cal

led

rela

tion

al o

pera

tors

, in

the

TE

ST

men

u ca

n be

used

to

disp

lay

the

solu

tion

of

a qu

adra

tic

ineq

uali

ty. A

noth

er m

etho

d th

at c

anbe

use

d to

fin

d th

e so

luti

on s

et o

f a

quad

rati

c in

equ

alit

y is

to

grap

h e

ach

sid

eof

an

ineq

ualit

y se

para

tely

. Exa

min

e th

e gr

aphs

and

use

the

in

ters

ect

func

tion

to d

eter

min

e th

e ra

nge

of

valu

es f

or w

hic

h t

he

ineq

ual

ity

is t

rue.

Sol

ve e

ach

in

equ

alit

y.

1.�

x2�

10x

�21

�0

2.x2

�9

�0

3.x2

�10

x�

25

0{x

| x�

�7

or

x�

�3}

{x| �

3 �

x�

3}{x

| x�

�5}

4.x2

�3x

28

5.2x

2�

x�

36.

4x2

�12

x�

9 �

0{x

| �7

�x

��

4 }

{x| x

��

1.5

or

x�

1}{x

| x�

�1.

5 o

r x

��

1.5}

7.23

��

x2�

10x

8.x2

�4x

�13

0

9.(x

�1)

(x�

3)�

0{x

| x�

3.58

or

x�

6.41

}{x

| �2.

12 �

x�

6.12

}{x

| x�

�1

or

x�

3}

Sol

ve x

2�

x�

6.

Pla

ce t

he

calc

ula

tor

in D

ot m

ode.

En

ter

the

ineq

ual

ity

into

Y1.

Th

en t

race

th

e gr

aph

an

d de

scri

be t

he

solu

tion

as

an i

neq

ual

ity.

Key

stro

kes:

[TE

ST] 4

6

4. Use

TR

AC

Eto

det

erm

ine

the

endp

oin

ts o

f th

e se

gmen

ts.

Th

eses

val

ues

are

use

d to

exp

ress

th

e so

luti

on o

f th

e in

equ

alit

y,

ZO

OM

2nd

+x

2Y

=

Sol

ve 2

x2�

4x�

5�

3.

Pla

ce t

he

left

sid

e of

th

e in

equ

alit

y in

Y1

and

the

righ

t si

de i

n Y

2.D

eter

min

e th

e po

ints

of

inte

rsec

tion

. Use

th

e in

ters

ecti

on p

oin

ts

to e

xpre

ss t

he

solu

tion

set

of

the

ineq

ual

ity.

Be

sure

to

set

the

calc

ula

tor

to C

onn

ecte

dm

ode.

Key

stro

kes:

2

4 5

3 6.

Pre

ss

[CA

LC

] 5

and

use

th

e ke

y to

mov

e th

e cu

rsor

to

th

e le

ft o

f th

e fi

rst

inte

rsec

tion

poi

nt.

Pre

ss

. Th

en m

ove

the

curs

or t

o th

e ri

ght

of t

he

inte

rsec

tion

poi

nt

and

pres

s . O

ne

of t

he

valu

es u

sed

in t

he

solu

tion

set

is

disp

laye

d.

Rep

eat

the

proc

edu

re o

n t

he

oth

er i

nte

rsec

tion

poi

nt.

Th

e so

luti

on i

s {

x|

�3.

24

x

1.24

}.

EN

TER

EN

TER

EN

TER

2nd

ZO

OM

EN

TER

EN

TER

—+

x2

Y=

[�4.

7, 4

.7] s

cl:1

by

[�3.

1, 3

.1] s

cl:1

[�10

, 10]

scl

:1 b

y [�

10, 1

0] s

cl:1

[�10

, 10]

scl

:1 b

y [�

10, 1

0] s

cl:1

A1-A30 A2-05-873975 5/11/06 7:28 AM Page A30


Chapter 5 Assessment Answer KeyQuiz 1 (Lessons 5–1 and 5–2) Quiz 3 (Lessons 5–5 and 5–6) Mid-Chapter TestPage 67 Page 68 Page 69

An

swer

s

Quiz 2 (Lessons 5–3 and 5–4)

Page 67Quiz 4 (Lessons 5–7 and 5–8)

Page 68

1.

2.

3.

e

4.

5.

3, �1

minimum, 1

�3; x � �1; �1

xO

f(x )

(0, �3)(�1, �4)

f(x) � x2 � 2x � 3

x � �1

between 1 and 2;between �6 and �5

1.

2.

3.

4.

5.

2 ��5�

{�1, 11}

1 � 3�5�

{�10, 2}


1.

2.

3.

4.

5.

2 ��5�

{�1, 11}

1 � 3�5�

{�10, 2}


1.

_.

2.

3.

4.

�1�1 5.

6.

7.

9.

10.

y

xO

1, 3

D

F

A

H

B

8. {�2, 9}

�0, �14

��

minimum, �9 �12

�

��2354� � �

1354i

�

1.

2.

3.

4.

5. A

{x �1 � x � 5}

y

xO

y � 2(x � 5)2

xO

y

(2, �1)

x � 2

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

68 � 4i

�11 � 3i

3x2 � 10x � 8 � 0

x2 � 4x � 12 � 0

{�9, 5}

��5, �23��

4i �5�

�12

� � �12

�i

�6�2�

�2i �5�

A31-A40 A2-05-873975 5/11/06 7:46 AM Page 31


Chapter 5 Assessment Answer KeyVocabulary Test Form 1Page 70 Page 71 Page 72

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

true

true

true

false; roots

false; constant term

false; quadratic inequality

false;quadratic term

false; minimum value

false; discriminant

false; complex conjugates 1.

2.

3.

4.

5.

6.

7.

8.

9.

10. F

A

H

B

H

B

H

A

G

B11.

12.

13.

14.

15.

16.

17.

18.

19.

20. G

B

G

B

J

C

F

D

J

B

B: 1 and 7; 14

11. Sample answer:A parabola is asmooth curve that isthe graph of aquadratic equation.

12. Sample answer: Anaxis of symmetry is aline along which youcan fold a graph andget matching parts onboth sides of the line.

A31-A40 A2-05-873975 5/11/06 7:46 AM Page 32


Chapter 5 Assessment Answer KeyForm 2A Form 2BPage 73 Page 74 Page 75 Page 76

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10. F

D

F

C

G

D

H

A

G

C 11.

12.

13.

14.

15.

16.

17.

18.

19.

20. J

C

F

B

F

D

J

C

J

B 1.

2.

3.

4.

5.

6.

7.

8.

9.

10. H

A

J

A

H

A

J

B

H

B 11.

12.

13.

14.

15.

16.

17.

18.

19.

20. F

A

H

D

G

B

F

D

G

C

B: Sample answer:9x2 � 2 � 0

B: Sample answer:16x2 � 3 � 0

A31-A40 A2-05-873975 5/11/06 7:46 AM Page 33


Chapter 5 Assessment Answer KeyForm 2CPage 77 Page 78

1.

2.

3.

4.

5.

6.

t

7.

8.

9. {�8, 2}

4x2 � 21x � 18

6 � 12j ohms

9 in. by 16 in.

y

xO

2, 4

maximum; 4

xO

f(x )

(3, 0)

(1, 5)

f(x) � �5x2 � 10x

x � 1

��3, �25

��

�3177� � �

2127�j amps

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: 9x2 � 7 � 0

y

xO

y � (x � 3)2 � 1

0; 1 real,rational root

33; 2 real,irrational roots

(�5, �7);x � �5; down

y � �32

�(x � 2)2 � 1

h(t ) � �16(t � 1.5)2 � 51; 51ft

�x �x � ��12

� or x � 3�

�3 �

1i0�31��

��2, �12

��{�2 � �13�}

A31-A40 A2-05-873975 5/12/06 1:53 PM Page 34


An

swer

s

Chapter 5 Assessment Answer KeyForm 2DPage 79 Page 80

1.

2.

3.

4.

5.

6.

7.

8. 2x2 � 5x � 12

9 � 6j ohms

8 in. by 18 in.

y

xO

1, �3

minimum; �17

xO

f(x )

(0, 3)

(2, �1)

x � 2f(x) � x2 �4x � 3

�2137� � �

2177�j ohms

��1, �43��

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: 16x2 � 5 � 0

y

xO


(6,�5); x � 6; down

y � ��14

�(x � 4)2 � 2

y � (x � 2)2 � 4

��2 �3

�6��

��9 �4�41��

�4 � �2��

��1, �23

��

0; 1 real, rational root

h(t ) � �16(t � 2)2 � 76; 76 ft

�x ��32

� � x � 5�

A31-A40 A2-05-873975 5/11/06 7:46 AM Page 35


Chapter 5 Assessment Answer KeyForm 3Page 81 Page 82

1.

2.

3.

4.

5.

6.

7.

8.

9.

4 � 6i

y

xO

2

2

y

xO

2

2

3, 6

$8.00; $6400

At

xO

f(x )

(0, 3)(� ),1

383

f(x) � 3x2 � 2x � 3

� 13x �

� �19

� � �49�5�

� i

minimum; �2225�

12x2 � 13x � 14 � 0

between �3 and �2;between 4 and 5

��12

�, �53

��

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

y

xO

�2 � k � 2

6 � 4�2�

{�3.5, 1}

{�0.35, 0.85}

��5 � i8��39�

y � ��35� �x � �

72��2 � �

12�;

��72�, ��

12��; x � ��

72�;

down

1.2; two real,irrational roots

y � ��22090

�(x � 9)2 � �229�

16x2 � 24x � 29 � 0

�x �x � ��72

� or x � 1�

h(t ) � �9.1(t � 32.5)2 �30,000; 30,000 ft

A31-A40 A2-05-873975 5/11/06 7:46 AM Page 36


An

swer

s

Chapter 5 Assessment Answer KeyPage 83, Extended-Response Test

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts ofgraphing, analyzing, and finding the maximum andminimum values of quadratic functions; solving quadraticequations; computing with complex numbers; and solvinginequalities.

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

• Shows an understanding of the concepts of graphing,analyzing, and finding the maximum and minimum valuesof quadratic functions; solving quadratic equations;computing with complex numbers; and solving inequalities.

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

• Shows an understanding of most of the concepts ofgraphing, analyzing, and finding the maximum andminimum values of quadratic functions; solving quadraticequations; computing with complex numbers; and solvinginequalities.

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

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

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

• Shows little or no understanding of most of the conceptsof graphing, analyzing, and finding the maximum andminimum values of quadratic functions; solving quadraticequations; computing with complex numbers; and solvinginequalities.

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

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

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

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

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

A31-A40 A2-05-873975 5/11/06 7:46 AM Page 37


Chapter 5 Assessment Answer KeyPage 83, Extended-Response Test

Sample Answers

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

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

2b. Student responses will vary. One of thethree conditions must be omitted ormodified. Sample answer: “...thatpasses through (�1, �12).”

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

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

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

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

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

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

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

In addition to the scoring rubric found on page A37, the following sample answers may be used as guidance in evaluating open-ended assessment items.

A31-A40 A2-05-873975 5/11/06 7:46 AM Page 38


Chapter 5 Assessment Answer KeyStandardized Test PracticePage 84 Page 85

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10. F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

A B C D 11.

12.

13.

14. F G H J

A B C D

F G H J

A B C D

15.

16.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

08

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

451

A31-A40 A2-05-873975 5/11/06 7:46 AM Page 39


Chapter 5 Assessment Answer KeyStandardized Test PracticePage 86

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28a.

28b.

28c.

(�2, 3)

y

xO

�1, 3

136 ft; 1.5 s

�2i �5�

(2, �3)

92

inconsistent

17

(�2, 0), (�2, 8),(0, �2), (8, �2)

��34

�

�3, 2 complex roots

y � �x � �72

��2

� �249�

��72

� , ��249��

x � �72

�

A31-A40 A2-05-873975 5/11/06 7:46 AM Page 40

Chapter 5 Resource Masters - No-IP

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