Holt McDougal Florida Larson Algebra 1

Holt McDougalFlorida

Larson Algebra 1

Chapter ResourcesVolume 2: Chapters 7-12

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ISBN 13: 978-0-547-25018-2

ISBN 10: 0-547-25018-5

1 2 3 4 5 6 7 8 9 XXX 15 14 13 12 11 10 09

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Algebra 1Chapter Resource Book iii

ContentsChapter 7 Systems of Equations and Inequalities

Chapter Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–4

7.1 Solve Linear Systems by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–15

7.2 Solve Linear Systems by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16–21

7.3 Solve Linear Systems by Adding or Subtracting . . . . . . . . . . . . . . . . . . . . . . 22–28

7.4 Solve Linear Systems by Multiplying First . . . . . . . . . . . . . . . . . . . . . . . . . . 29–35

7.5 Solve Special Types of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36–44

7.6 Solve Systems of Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45–55

Chapter Review Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Chapter 8 Exponents and Exponential Functions Chapter Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57–60

8.1 Apply Exponent Properties Involving Products . . . . . . . . . . . . . . . . . . . . . . . 61–67

8.2 Apply Exponent Properties Involving Quotients . . . . . . . . . . . . . . . . . . . . . . 68–73

8.3 Defi ne and Use Zero and Negative Exponents . . . . . . . . . . . . . . . . . . . . . . . . 74–83

8.4 Use Scientifi c Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84–90

8.5 Write and Graph Exponential Growth Functions . . . . . . . . . . . . . . . . . . . . . 91–101

8.6 Write and Graph Exponential Decay Functions . . . . . . . . . . . . . . . . . . . . . 102–114

Chapter Review Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Chapter 9 Polynomials and Factoring Chapter Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117–120

9.1 Add and Subtract Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121–128

9.2 Multiply Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129–134

9.3 Find Special Products of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135–140

9.4 Solve Polynomial Equations in Factored Form . . . . . . . . . . . . . . . . . . . . . 141–147

9.5 Factor x2 1 bx 1 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148–153

9.6 Factor ax2 1 bx 1 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154–160

9.7 Factor Special Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161–166

9.8 Factor Polynomials Completely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167–173

Chapter Review Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

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Algebra 1Chapter Resource Bookiv

Chapter 10 Quadratic Equations and Functions Chapter Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175–178

10.1 Graph y 5 ax2 1 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179–187

10.2 Graph y 5 ax2 1 bx 1 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188–201

10.3 Solve Quadratic Equations by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . 202–211

10.4 Use Square Roots to Solve Quadratic Equations . . . . . . . . . . . . . . . . . . . 212–218

10.5 Solve Quadratic Equations by Completing the Square . . . . . . . . . . . . . . 219–227

10.6 Solve Quadratic Equations by the Quadratic Formula . . . . . . . . . . . . . . . 228–234

10.7 Interpret the Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235–240

10.8 Compare Linear, Exponential, and Quadratic Models . . . . . . . . . . . . . . 241–251

Chapter Review Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

Chapter 11 Radicals and Geometry Connections Chapter Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253–256

11.1 Graph Square Root Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257–266

11.2 Simplify Radical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267–276

11.3 Solve Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277–283

11.4 Apply the Pythagorean Theorem and Its Converse . . . . . . . . . . . . . . . . . 284–289

11.5 Apply the Distance and Midpoint Formulas . . . . . . . . . . . . . . . . . . . . . . 290–296


Chapter 12 Rational Equations and Functions Chapter Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299–302

12.1 Model Inverse Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303–312

12.2 Graph Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313–321

12.3 Divide Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322–331

12.4 Simplify Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332–338

12.5 Multiply and Divide Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . 339–347

12.6 Add and Subtract Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 348–354

12.7 Solve Rational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355–361


Gridded Response Answer Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

Resource Book Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A1–A63


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Algebra 1Chapter Resource Book v

ContentsDescriptions of ResourcesThis Chapter Resource Book is organized by lessons within the chapter in order to make your planning easier. The following materials are provided:

Family Letter This guide helps families contribute to student success by providing an overview of the chapter along with questions and activities for families to work on together.

Graphing Calculator Activities with Keystrokes Keystrokes for two models of calculators are provided for each Graphing Calculator Activity in the Student Edition.

Activity Support Masters These blackline masters make it easier for students to record their work on selected activities in the Student Edition.

Practice A, B, and C These exercises offer additional practice for the material in each lesson, including application problems. There are three levels of practice for each lesson: A (basic), B (average), and C (advanced).

Review for Mastery These two pages provide additional instruction, worked-out examples, and practice exercises covering the key concepts and vocabulary in each lesson.

Problem Solving Workshops These blackline masters provide extra problem solving opportunities in addition to the workshops given in the textbook. There are three types of workshops: Alternative Methods, Worked-Out Examples, and Mixed Problem Solving.

Challenge Practice These exercises offer challenging practice on the mathematics of each lesson.

Chapter Review Game This worksheet offers fun practice at the end of the chapter and provides an alternative way to review the chapter content in preparation for the Chapter Test.

Gridded Response Answer Sheet This page provides 12 answer grids for the teacher to copy and distribute as needed for use with the Gridded Response questions in the Problem Solving Workshops.



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1Algebra 1

Chapter 7 Resource Book

Chapter Overview One way you can help your student succeed in Chapter 7 is by discussing the lesson goals in the chart below. When a lesson is completed, ask your student the following questions. “What were the goals of the lesson? What new words and formulas did you learn? How can you apply the ideas of the lesson to your life?”

Lesson Title Lesson Goals Key Applications

7.1: Solve Linear Systems by Graphing

Graph and solve systems of linear equations.

• Rental Business

• Television

• Fitness

7.2: Solve Linear Systems by Substitution

Solve systems of linear equations by substitution.

• Websites

• Antifreeze

• Fundraising

7.3: Solve Linear Systems by Adding or Subtracting

Solve linear systems using elimination.

• Kayaking

• Rowing

• Cellular Phones

7.4: Solve Linear Systems by Multiplying First

Solve linear systems by multiplying fi rst.

• Book Sale

• Music

• Farm Products

7.5: Solve Special Types of Linear Systems

Identify the number of solutions of a linear system.

• Art

• Recreation

• Photography

7.6: Solve Systems of Linear Inequalities

Solve systems of linear inequalities in two variables.

• Baseball

• Competition Scores

• Fish

Key Ideas for Chapter 7

In Chapter 7, you will apply the key ideas listed in the Chapter Opener (see page 437) and reviewed in the Chapter Summary (see page 489).

1. Solving linear systems by graphing

2. Solving linear systems using algebra

3. Solving systems of linear inequalities

CHAPTER

7 Family LetterFor use with Chapter 7 C

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2Algebra 1Chapter 7 Resource Book

Family Letter continuedFor use with Chapter 7

Key Ideas Your student can demonstrate understanding of key concepts by working through the following exercises with you.

Lesson Exercise

7.1 Solve the system by graphing. Check the solution.

x 1 2y 5 8

2x 2 y 5 6

7.2 Each day you either carpool to school, which takes 18 minutes, or ride the bus, which takes 35 minutes. After 20 days of school you have spent 598 minutes getting to school. How many days did you carpool? How many days did you ride the bus?

7.3 Solve the system using elimination. Check the solution.

3x 1 5y 5 9

5y 5 3x 1 21

7.4 You and a friend are playing in a basketball tournament. You buy 4 sports drinks and 5 power bars for $13. Your friend buys 3 sports drinks and 2 power bars for $7.65. How much did each sports drink cost? How much did each power bar cost?

7.5 Tell whether the linear system has one solution, no solution, or infi nitely many solutions. Explain.

(a) 3x 2 y 5 9 (b) 26x 1 8y 5 12

6x 2 2y 5 10 9x 2 12y 5 218

7.6 Graph the system of inequalities. x 1 y ≥ 1

x 2 y ≤ 4

y > 2

Home Involvement Activity

Directions Write systems of inequalities for triangular shaded regions that would be located solely within each of the four quadrants.

Answers7.1:

x

y

1

3

5

121

215

; (4, 2) 7.2: 14 bus rides, 6 carpool rides 7.6:

x

y6

622

22

7.3: (22, 3) 7.4: $1.75, $1.20

7.5: (a) no solution; same slope, different y-intercept

(b) many solutions; same slope and y-intercept

CHAPTER

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3Algebra 1


Vistazo al capítulo Una manera en que puede ayudar a su hijo a tener éxito en el Capítulo 7 es hablar sobre los objetivos de la lección en la tabla a continuación. Cuando se termina una lección, pregúntele a su hijo lo siguiente: “¿Cuáles fueron los objetivos de la lección? ¿Qué palabras y fórmulas nuevas aprendiste? ¿Cómo puedes aplicar a tu vida las ideas de la lección?”

Título de la lección Objetivos de la lección Aplicaciones clave

7.1: Resolver sistemas lineales con gráfi cas

Grafi car y resolver sistemas de ecuaciones lineales

• Negocio de alquiler

• Televisión

• Salud

7.2: Resolver sistemas lineales con la sustitución

Resolver sistemas de ecuaciones lineales con la sustitución

• Sitios web

• Anticongelante

• Recaudación de fondos

7.3: Resolver sistemas lineales con la suma o la resta

Resolver sistemas lineales usando la eliminación

• Hacer kayak

• Remar

• Teléfonos celulares

7.4: Resolver sistemas lineales multiplicando primero

Resolver sistemas lineales multiplicando primero

• Venta de libros

• Música

• Productos de la granja

7.5: Resolver tipos especiales de sistemas lineales

Identifi car la cantidad de soluciones de un sistema lineal

• Arte

• Recreo

• Fotografía

7.6: Resolver sistemas de desigualdades lineales

Resolver sistemas de desigualdades lineales con dos variables

• Béisbol

• Puntajes de concursos

• Peces

Ideas clave para el Capítulo 7

En el Capítulo 7, aplicarás las ideas clave enumeradas en la Presentación del capítulo (ver la página 437) y revisadas en el Resumen del capítulo (ver la página 489).

1. Resolver sistemas lineales con gráfi cas

2. Resolver sistemas lineales usando álgebra

3. Resolver sistemas lineales de desigualdades lineales

CAPÍTULO

7 Carta para la familiaUsar con el Capítulo 7 C

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Carta para la familia continúaUsar con el Capítulo 7

Ideas clave Su hijo puede demostrar la comprensión de las ideas clave al hacer los siguientes ejercicios con usted.

Lección Ejercicio

7.1 Resuelve el sistema al grafi car. Comprueba la solución.

x 1 2y 5 8

2x 2 y 5 6

7.2 Cada día o vas en carro a la escuela, que toma 18 minutos, o vas en autobús, que toma 35 minutos. Después de 20 días escolares, has pasado 598 minutos en llegar a la escuela. ¿Cuántos días fuiste en carro? ¿Cuántos días tomaste el autobús?

7.3 Resuelve el sistema usando la eliminación. Comprueba la solución.

3x 1 5y 5 9

5y 5 3x 1 21

7.4 Tú y un amigo juegan en un torneo de básquetbol. Compras 4 bebidas deportivas y 5 barras de nutrición por $13. Tu amigo compra 3 bebidas deportivas y 2 barras de nutrición por $7.65. ¿Cuánto costó cada bebida deportiva? ¿Cuánto costó cada barra de nutrición?

7.5 Indica si el sistema lineal tiene una solución, ninguna solución o muchas soluciones infi nitas. Explica.

(a) 3x 2 y 5 9 (b) 26x 1 8y 5 12

6x 2 2y 5 10 9x 2 12y 5 218

7.6 Grafi ca el sistema de desigualdades. x 1 y ≥ 1

x 2 y ≤ 4

y > 2

Actividad para la familia

Instrucciones Escribe sistemas de desigualdades para regiones triangulares sombreadas que se localizarían dentro de cada uno de los cuatro cuadrantes.

Respuestas7.1:

x

y

1

3

5

121

215

; (4, 2) 7.2: 14 días en autobús, 6 días en carro 7.6:

x

y6

622

22

7.3: (22, 3) 7.4: $1.75, $1.20

7.5: (a) ninguna solución; mismapendiente, diferenteintercepto en y

(b) muchas soluciones; mismapendiente y mismo intercepto en y

CAPÍTULO

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5Algebra 1


TI-83 Plus

Y= (�) ( 5 � 2 ) X,T,�,n � 3 ENTER ( 1 � 3 ) X,T,�,n � ( 5 � 3 ) ENTER ZOOM 6 2nd [CALC] 5 ENTER ENTER ENTER

Casio CFX-9850GC Plus

From the main menu, choose GRAPH.(�) ( 5 � 2 ) X, ,T� � 3 EXE

( 1 � 3 ) X, ,T� � ( 5 � 3 ) EXE SHIFT F3 F3 EXIT F6 SHIFT

F5 F5

Graphing Calculator Activity KeystrokesFor use with page 446

LESSON

7.1

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LESSON

7.1 Practice AFor use with pages 439–445

Tell whether the ordered pair is a solution of the linear system.

1. (0, 24); 2. (3, 3); 3. (1, 22);

x 1 y 5 24 x 1 2y 5 9 2x 2 3y 5 8 x 2 5y 5 20 4x 2 y 5 15 3x 1 2y 5 21

4. (24, 26); 5. (4, 21); 6. (2, 26); 23x 1 y 5 6 x 2 4y 5 8 4x 1 3y 5 210 22x 1 y 5 28 23x 1 5y 5 23 3x 1 2y 5 26

Match the linear system with its graph.

7. x 2 y 5 2 8. x 1 y 5 2 9. x 1 y 5 22 x 1 y 5 5 x 2 y 5 5 x 2 y 5 25

10. x 2 y 5 22 11. 2x 1 y 5 2 12. x 2 y 5 2 2x 2 y 5 5 x 1 y 5 5 2x 2 y 5 5

A.

x

y

1

121

23

3 5

B.

x

y

1

3

121

3 5

C.

x

y

1

3

23

D. xy

23 21

23

21

E.

x

y

1

3

1 3 521

F.

x

y

1

23 2121

23

Use the graph to solve the linear system. Check your solution.

13. 4x 1 3y 5 5 14. 2x 1 3y 5 9 15. 5x 2 y 5 24 2x 2 y 5 5 4x 2 y 5 8 22x 1 y 5 1

x

y

1

121

23

3 521

x

y

1

5

121

3 521

x

y

1 321

23

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7Algebra 1


Solve the linear system by graphing. Check your solution.

16. y 5 2x 1 6 17. y 5 22x 1 1 18. 4x 2 y 5 212

y 5 x 2 2 y 5 x 2 5 2x 2 y 5 3

x

y

1

3

5

12121

3 5

x

y

1

12121

23

25

3 5

x

y

3

9

3232923

29

9

19. y 5 x 20. y 5 22x 1 2 21. 3x 1 y 5 7

y 5 4x 2 9 y 5 x 1 5 22x 1 y 5 28

x

y

3

9

329 2323

29

9

x

y

1

3

5

12325 2121

x

y

2

6

2 626 2222

26

22. City Populations The graph shows the estimated populations

x

y

700

950

1000

1050

1100

1150

1200

1250

1 2 3 4 5 6Years since 1990

Po

pu

lati

on

(th

ou

san

ds)

Charlotte

Buffalo(in thousands of people) of the Buffalo, New York area and the Charlotte, North Carolina area. Use the graph to fi nd the year in which the populations of these two areas were the same. What was the population?

23. Juice You bought 15 one-gallon bottles of apple juice and

x14 1600

2

4

6

8

10

12

14

16

2 4 6 8 10 12Bottles of apple juice

Bo

ttle

s o

f o

ran

ge ju

ice

yorange juice for a school dance. The apple juice was on sale for $1.50 per gallon bottle. The orange juice was $2 per gallon bottle. You spent $26. Write algebraic models for the situation. Then graph the algebraic models. How many bottles of each type of juice did you buy?

LESSON

7.1 Practice A continuedFor use with pages 439–445

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LESSON

7.1 Practice BFor use with pages 439–445


1. (4, 1); 2. (22, 1); 3. (4, 23);

x 1 2y 5 6 5x 2 2y 5 212 23x 1 2y 5 218

3x 1 y 5 11 x 1 3y 5 1 6x 2 y 5 27

4. (24, 26); 5. (24, 3); 6. (22, 25);

3x 2 y 5 6 4x 1 3y 5 212 2x 1 y 5 23

2x 1 2y 5 8 x 1 2y 5 26 2x 1 3y 5 213

Use the graph to solve the linear system. Check your solution.

7. x 2 y 5 8 8. 5x 2 y 5 29 9. 2x 1 3y 5 2

x 1 y 5 22 y 1 2x 5 2 22x 1 y 5 6

x

y

121

23

25

3 5

x

y

1

5

23 21 3

x

y

3

21 1

10. 3x 2 2y 5 16 11. 2x 2 y 5 213 12. 6x 1 2y 5 8

5x 1 y 5 18 y 1 3x 5 212 23x 1 4y 5 16

x

y

121

23

25

3

x

y

1

5

3

2325 21

x

y

3

1

2123 1 3


13. y 5 3x 14. 2x 1 y 5 24 15. 23x 2 y 5 21

y 5 4x 2 1 x 2 y 5 28 2x 1 4y 5 216

x

y

1

3

12123 3

x

y

2

6

226

26

x

y1

12325 21

23

25

21

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9Algebra 1


16. 2x 1 2y 5 26 17. 26x 1 y 5 33 18. 29x 1 6y 5 26

25x 1 y 5 15 2x 2 8y 5 234 2x 2 3y 5 8

x

y

3

15

329215

x

y35

5215

x

y

1

123 21

25

21

19. 3x 1 2y 5 3 20. x 2 y 5 9 21. 6x 1 y 5 19

5x 1 y 5 29 3x 1 2y 5 2 5x 2 2y 5 24

x

y

9

929 23

29

x

y

6 102222

26

x

y

3

9

15

21

929 23

22. Hanging Flower Baskets You will be making hanging fl ower

3500

5

10

15

20

25

30

35

5 10 15 20 25 30Blooming annuals

No

n-b

loo

min

g a

nn

uals

y

x

baskets. The plants you have picked out are blooming annuals and non-blooming annuals. The blooming annuals cost $3.20 each and the non-blooming annuals cost $1.50 each. You bought a total of 24 plants for $49.60. Write a linear system of equations that you can use to fi nd how many of each type of plant you bought. Then graph the linear system and use the graph to fi nd how many of each type of plant you bought.

23. Baseball Outs In a game, 12 of a baseball team’s 27 outs

28 3200

4

8

12

16

20

24

28

32

4 8 12 16 20 24Outs made by infielders

Ou

ts m

ad

e b

y o

utf

ield

ers

y

x

were fl y balls. Twenty-fi ve percent of the outs made by infi elders and 100% of the outs made by outfi elders were fl y balls.

a. Write a linear system you can use to fi nd the number of outs made by infi elders and the number of outs made by outfi elders. (Hint: Write one equation for the total number of outs and another equation for the number of fl y ball outs.)

b. Graph your linear system.

c. How many outs were made by infi elders? How many outs were made by outfi elders?

LESSON

7.1 Practice B continuedFor use with pages 439–445

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LESSON

7.1 Practice CFor use with pages 439–445


1. (28, 4); 2. (7, 26); 3. (4, 26);

2x 1 4y 5 28 3x 1 2y 5 9 3x 1 y 5 26

3x 2 5y 5 3 24x 2 3y 5 210 2x 1 2y 5 8

4. (4, 22); 5. (23, 5);

21.5x 1 3.2y 5 11.5

4.1x 2 2y 5 222.3

6. (22.5, 2.5);

6x 2 8y 5 235

4x 1 2y 5 25

1 }

2 x 2

3 }

4 y 5

7 }

2

4x 1 3 }

8 y 5

61 }

4


7. 25x 1 8y 5 222 8. 210x 2 4y 5 64 9. 3x 2 7y 5 50

3y 2 2x 5 29 2x 1 2y 5 16 24x 1 2y 5 230

x

y

1

3

121

23

3 5 7

x

y

4

12

42421222024

212

x

y

121

23

25

27

3 5 7

10. 2 }

3 x 2

1 }

3 y 5 2

11 }

3 11.

2 } 5 x 1

3 } 5 y 5 2 12. 4x 2

1 }

3 y 5 2

19 }

3

x 1 1 }

2 y 5 2

1 }

2 2

2 }

3 x 1 y 5

2 }

3 2

2 }

3 x 1 y 5

23 }

3

x

y

2

6

10

2222621022

x

y

1

3

1 3 521

23

21

x

y

3

9

15

21

329215 23

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11Algebra 1


13. 1.8x 2 2.2y 5 24.2 14. 21.4x 1 6y 5 24.6 15. 3.2x 2 y 5 8.8

0.5x 1 3.2y 5 21.7 0.2x 1 y 5 0.2 5x 2 2.5y 5 10

x

y

1

3

5

7

1 3 521

x

y

2

6

10

2222621022

x

y

2

6

2 6 102222

26

16. Find the values for m and b so that the system y 5 3 }

4 x 2 2 and y 5 mx 1 b has (8, 4)

as a solution.

17. The graphs of the three lines given below form a triangle.

x

y

1

3

5

7

12121

23 3

Use a graph to fi nd the coordinates of the vertices of the triangle.

Line 1: 2x 1 y 5 7

Line 2: x 1 2y 5 2

Line 3: 2x 1 y 5 4

18. Investments A total of $45,000 is invested into two funds paying 5.5% and 6.5% annual interest. The combined annual interest is $2725. How much of the $45,000 is invested in each type of fund? (Hint: Write one equation for the amount invested in each fund and another for the interest earned.)

19. Umbrella Sales The table shows the number of automatic and

t700

5

10

15

20

25

30

35

1 2 3 4 5 6

Nu

mb

er

of

um

bre

llas

Years since 2000

ymanual opening umbrellas sold at a shop in 2000 and 2005. Use a linear model to represent the sales of each type of umbrella. Let t 5 0 correspond to 2000. Sketch the graphs and estimate when the number of automatic umbrellas sold equaled the number of manual umbrellas sold.

Year 2000 2005

Automatic 15 25

Manual 25 15

20. Credit Account With a minimum purchase of $100, you can open a credit account at a music store. The store is offering either $25 or 20% off your purchase if you open a credit account. You make a purchase of $135 and decide to open a credit account. Should you choose $25 or 20% off your purchase? Explain.

LESSON

7.1 Practice C continuedFor use with pages 439–445

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Graph and solve systems of linear equations.

VocabularyA system of linear equations, or simply a linear system, consists of two or more linear equations in the same variables.

A solution of a system of linear equations in two variables is an ordered pair that satisfi es each equation in the system.

GOAL

Check the intersection point

Use the graph to solve the system. Then

x

y

1

3

21

23

2123 3

check your solution algebraically.

2x 1 y 5 4 Equation 1

3x 2 5y 5 6 Equation 2

Solution

The lines appear to intersect at the point (2, 0).

CHECK Substitute 2 for x and 0 for y in each equation.

Equation 1 Equation 2 2x 1 y 5 4 3x 2 5y 5 6

2(2) 1 0 0 4 3(2) 2 5(0) 0 6

4 1 0 0 4 6 2 0 0 6

4 5 4 ✓ 6 5 6 ✓

Because the ordered pair (2, 0) is a solution of each equation, it is a solution of the system.

EXAMPLE 1

Use the graph-and-check method

Solve the linear system: x 2 3y 5 2 Equation 1

25x 1 y 5 4 Equation 2

STEP 1 Graph both equations. STEP 2 Estimate the point of the intersection. The two lines appear to intersect at (21, 21).

x

y

1

23

23 3

EXAMPLE 2

Review for MasteryFor use with pages 439–445

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13Algebra 1


STEP 3 Check whether (21, 21) is a solution by substituting 21 for x and 21 for y in each of the original equations.

Equation 1 Equation 2 x 2 3y 5 2 25x 1 y 5 4

21 2 3(21) 0 2 25(21) 1 (21) 0 4

21 1 3 0 2 5 2 1 0 4

2 5 2 ✓ 4 5 4 ✓

Because the ordered pair (21, 21) is a solution of each equation, it is a solution of the system.

Review for Mastery continuedFor use with pages 439–445

LESSON

7.1

Solve a multi-step problem

Delivery Service The Rosebud Flower Shop has a basic delivery charge of $5 plus a rate of $.25 per mile. The Beautiful Bouquets Shop has a basic delivery charge of $7 plus a rate of $.20 per mile. Determine the number of miles a delivery must be for the charges to be equal.

Solution

STEP 1 Write a linear system. Let x be the number of miles driven and y be the total cost of the delivery.

y 5 5 1 0.25x Equation for Rosebud Flower Shop

y 5 7 1 0.20x Equation for Beautiful Bouquets Shop

STEP 2 Graph both equations.

x

y

369

121518

20 30 40 50 601000

To

tal co

st

(do

llars

)

Miles driven

Delivery Service

STEP 3 Estimate the point of intersection. The two lines appear to intersect at (40, 15).

STEP 4 Check whether (40, 15) is a solution.

Equation 1 Equation 2 y 5 5 1 0.25x y 5 7 1 0.20x

15 0 5 1 0.25(40) 15 0 7 1 0.20(40)

15 5 15 ✓ 15 515 ✓

EXAMPLE 3

Exercises for Examples 1, 2, and 3

Solve the linear system by graphing.

1. 23x 1 y 5 4 2. x 1 1 }

2 y 5 4 3. 2x 2 6y 5 4

5x 2 2y 5 27 5x 1 2y 5 18 7x 2 4y 5 220

4. In Example 3, suppose Rosebud Flower Shop increases its basic charge to $10, and Beautiful Bouquets raises its basic charge to $13. Determine when the costs will be equal.

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Aerobics A fi tness club offers two aerobics classes. There are currently 28 people going to the afternoon class and attendance is increasing at a rate of 2 people per month. There are currently 16 people going to the night class and attendance is increasing at a rate of 4 people per month. Predict when the number of people in each class will be the same.

STEP 1 Read and Understand

What do you know? The number of people that go to each aerobic class and the increase each month.

What do you want to fi nd out? When each class has the same number of people.

STEP 2 Make a Plan Use what you know to write and solve a linear system.

STEP 3 Solve the Problem Let x be the number of months and y be the number of people in the class.

An equation that models the afternoon

x

y

7 8 900

10

20

30

40

50

1 2 3 4 5 6Number of months

Nu

mb

er

of

peo

ple

y 5 2x 1 28

y 5 4x 1 16

class is y 5 2x 1 28.

An equation that models the night class is y 5 4x 1 16.

Graph both equations. The point of intersection occurs at the point (6, 40).

After 6 months, both the afternoon class and the night class have the same number of people, 40.

STEP 4 Look Back Check whether (6, 40) is a solution.

y 5 2x 1 28 y 5 4x 1 16

40 0 2(6) 1 28 40 0 4(6) 1 16

40 5 40 ✓ 40 5 40 ✓

The answer is correct.

PROBLEM

1. Carpet Store A charges $4 per square foot for carpeting and $95 for installa-tion. Store B charges $6 per square foot for carpeting and $75 for installation. Find the square footage for which the total cost is the same for each store.

2. Football You are selling tickets to a football game. Student tickets cost $4 and general admission tickets cost $7. You sell 213 tickets and collect $1146. How many of each type of ticket did you sell?

3. What If? For the next football game, you sell 241 tickets and collect $1315. How many of each type of ticket did you sell?

4. Bowling Alley A charges $2.25 per game of bowling and $1.75 for shoe rentals. Alley B charges $2 per game of bowling and $2.75 for shoe rentals. Find the number of games for which the total cost is the same to bowl at each alley.

PRACTICE

LESSON

7.1 Problem Solving Workshop:Worked Out ExampleFor use with pages 439–445

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15Algebra 1


Tell whether the ordered pair is a solution of the system of linear equations.

1. 1 4 } 5 , 4 } 5 2 ; 2. 1 11

} 4 ,

9 }

4 2 ;

2x 1 3y 5 4 x 1 y 5 5

3x 1 2y 5 4 x 2 y 5 1 } 2

3. (4, 1); 4. 1 3a 1 2b }

b2 1 a2 , 3b 2 2a

} b2 1 a2 2 ;

x 1 2y 5 6

ax 1 by 5 3

2x 2 3y 5 4

bx 2 ay 5 2

In Exercises 5 and 6, use the table that shows the numbers of households in two cities in the years 1990 and 2000.

1990 2000

Bayside 100,000 105,000

Coal Flats 105,000 85,000

5. For each city, write a linear model to represent the number of households at time t, where t represents the number of years since 1990.

6. Use a graph to estimate when the two cities had the same number of households.

In Exercises 7–9, use the table that shows the annual number of spectators for three sports in a small town in the years 1950 and 2000.

1950 2000

Hockey 20,000 80,000

Soccer 0 100,000

Baseball 90,000 40,000

7. For each sport, write a linear model to represent the annual number of spectators at time t, where t represents the number of years since 1950.

8. Use a graph to estimate when the annual number of spectators of soccer overtook the annual number of spectators of hockey.

9. Use a graph to estimate when the annual number of spectators of soccer overtook the annual number of spectators of baseball.

Challenge PracticeFor use with pages 439–445

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LESSON


Solve for the indicated variable.

1. 9x 1 y 5 7; y 2. 3x 2 y 5 10; y 3. x 2 4y 5 1; x

4. 3x 1 6y 5 9; x 5. 2x 2 2y 5 8; y 6. 1 }

2 x 2 3y 5 7; x

Tell which equation you would use to isolate a variable. Explain your reasoning.

7. x 5 5y 2 8 8. 23x 1 2y 5 7 9. 4 1 8x 5 y

4x 1 3y 5 5 y 5 6x 1 1 6x 2 y 5 2

10. 2x 1 y 5 8 11. x 1 4y 5 22 12. 2x 5 4y 1 2

2y 2 3x 5 5 3x 2 y 5 1 25x 1 5y 5 13

Solve the linear system by using substitution.

13. x 5 1 2 y 14. x 5 4y 1 14 15. y 5 23x 2 1

y 5 2x 2 2 y 5 23x 1 3 4x 1 3y 5 2

16. y 5 22x 1 4 17. 4x 2 2y 5 14 18. x 1 2y 5 6

5y 2 2x 5 216 x 5 10 2 6y 27x 1 3y 5 28

19. 28x 1 3y 5 233 20. x 1 2y 5 11 21. 23x 1 y 5 8

5x 1 y 5 35 3x 2 4y 5 217 x 1 2y 5 25

22. x 1 y 5 3 23. x 2 y 5 0 24. 2x 1 2y 5 6

3x 2 4y 5 219 2x 1 4y 5 18 3x 2 5y 5 25

25. Driving Your brother and sister took turns driving on a 635-mile trip that took 11 hours to complete. Your brother drove at a constant speed of 60 miles per hour and your sister drove at a constant speed of 55 miles per hour. Let x be the number of miles your brother drove and let y be the number of miles your sister drove. Solve the linear system x 1 y 5 11 and 60x 1 55y 5 635 to fi nd the number of miles each of your siblings drove.

26. Fundraising A wilderness group is selling cans of nuts and boxes of microwaveable popcorn to raise money for a trip. A can of nuts sells for $4.50 and a box of microwaveable popcorn sells for $3. The group sells $252 in nuts and popcorn and they sell twice as many boxes of popcorn as cans of nuts.

a. Let x be the number of boxes of popcorn and let y be the number of cans of nuts sold. Write an equation that relates the number of boxes of popcorn sold to the number of cans of nuts sold.

b. Write an equation that gives the total amount of money made in terms of x and y.

c. How many boxes of popcorn did the group sell? How many cans of nuts did the group sell?

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17Algebra 1


Solve for the indicated variable.

1. 8x 1 4y 5 12; y 2. 3x 2 4y 5 12; y 3. 6x 2 4y 5 8; x


4. x 5 8y 2 3 5. 24x 1 5y 5 11 6. 9 2 3x 5 y

3x 2 4y 5 1 y 5 4x 2 1 3x 2 y 5 22


7. x 5 6 2 4y 8. 4x 1 3y 5 0 9. 2x 1 2y 5 26

2x 2 3y 5 1 2x 1 y 5 22 8x 1 y 5 31

10. 6x 2 y 5 235 11. 2x 1 3y 5 29 12. 3x 1 3y 5 218

5x 2 2y 5 235 8x 2 4y 5 32 4x 2 y 5 214

13. 2x 1 2y 5 6 14. 5x 1 2y 5 43 15. 4x 2 2y 5 24

23x 1 5y 5 233 26x 1 3y 5 230 7x 2 5y 5 219

16. 3x 1 2y 5 5 17. 4x 2 3y 5 28 18. 8x 1 8y 5 24

5x 2 9y 5 24 2x 1 3y 5 24 x 1 5y 5 11

19. Drum Sticks A drummer is stocking up on drum sticks and brushes. The wood sticks that he buys are $10.50 a pair and the brushes are $24 a pair. He ends up spending $90 on sticks and brushes and buys two times as many pairs of sticks as brushes. How many pairs of sticks and brushes did he buy?

20. Mowing and Shoveling Last year you mowed grass and shoveled snow for 12 households. You earned $225 for mowing a household’s lawn for the entire year and you earned $200 for shoveling a household’s walk and driveway for an entire year. You earned a total of $2600 last year.

a. Let x be the number of households you mowed for and let y be the number of households you shoveled for. Write an equation in x and y that shows the total number of households you worked for. Then write an equation in x and y that shows the total amount of money you earned.

b. How many households did you mow the lawn for and how many households did you shovel the walk and driveway for?

21. Dimensions of a Metal Sheet A rectangular hole 3 centimeters wide

4 cm 3 cm

x

y

and x centimeters long is cut in a rectangular sheet of metal that is 4 centimeters wide and y centimeters long. The length of the hole is 1 centimeter less than the length of the metal sheet. After the hole is cut, the area of the remaining metal sheet is 20 square centimeters. Find the length of the hole and the length of the metal sheet.

Practice BFor use with pages 447–453

LESSON

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LESSON



1. 6x 2 y 5 9 2. 22x 1 4y 5 10 3. 15 2 3x 5 2y

5x 2 3y 5 2 9y 5 5x 2 7 9x 2 3y 5 26


4. 13x 2 4y 5 38 5. 10x 2 20y 5 0 6. 3.5x 1 0.5y 5 14

x 2 6y 5 254 x 1 5y 5 228 y 2 x 5 4

7. 10x 1 y 5 285 8. 4x 2 3y 5 222 9. 4x 1 7y 5 8

0.1x 1 2.5y 5 11.6 0.2x 1 y 5 10.4 x 1 11y 5 76

10. 3x 1 2y 5 29 11. 5x 1 y 5 41 12. 210x 1 3y 5 21

2x 1 3y 5 4 3x 2 y 5 23 x 2 6y 5 15

13. 1 }

2 x 1

1 }

3 y 5

3 }

4 14. x 1

1 } 5 y 5 2

7 } 5 15. 6x 1 5y 5 2

7 }

3

x 2 1 }

4 y 5

13 }

16 23x 2 6y 5

3 }

2

1 }

3 x 2 y 5 2

5 }

9

16. Find the values of a and b so that the linear system shown has a solution of (4, 25).

ax 1 by 5 210 Equation 1


17. Painting and Cleaning During the spring and summer, you do a spring yard cleanup for households and you also paint houses. You earn $8 an hour doing the cleanups and $12 an hour painting. Last spring and summer, you worked a total of 400 hours and earned $3800. How many hours did you spend doing yard cleanups? How many hours did you spend painting?

18. Room Dimensions The area of the room shown is

12 ft

4 ft

8 ft

y ft x ft

224 square feet. The perimeter of the room is 64 feet. Find x and y.

19. Potting Soil You are creating a potting mix for your window boxes that is 20% peat moss and 80% potting soil. You add 100% potting soil to your mix that is currently 50% peat moss and 50% potting soil. You have 4 buckets of the mix that is half and half. Do you have enough of the half and half mix to make 8 buckets of the mix that is 20% peat moss and 80% potting soil? Explain.

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19Algebra 1


Solve systems of linear equations by substitution.GOAL

Use the substitution method

Solve the linear system: 2x 1 y 5 1 Equation 1

x 1 2y 5 5 Equation 2

Solution

STEP 1 Solve Equation 1 for y.

2x 1 y 5 1 Write original Equation 1.

y 5 22x 1 1 Subtract 2x from each side.

STEP 2 Substitute 22x 1 1 for y in Equation 2 and solve for x.

x 1 2y 5 5 Write Equation 2.

x 1 2(22x 1 1) 5 5 Substitute 22x 1 1 for y.

x 2 4x 1 2 5 5 Distributive property

23x 1 2 5 5 Simplify.

23x 5 3 Subtract 2 from each side.

x 5 21 Divide each side by 23.

STEP 3 Substitute 21 for x in the original Equation 1 to fi nd the value of y.

2x 1 y 5 1 Write original Equation 1.

2(21) 1 y 5 1 Substitute 21 for x.

22 1 y 5 1 Simplify.

y 5 3 Solve for y.

The solution is (21, 3).

CHECK Substitute 21 for x and 3 for y in each of the original equations.

Equation 1 Equation 2 2x 1 y 5 1 x 1 2y 5 5

2(21) 1 3 0 1 21 1 2(3) 0 5

1 5 1 ✓ 5 5 5 ✓

EXAMPLE 1


LESSON

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EXAMPLE 2


LESSON

7.2

Use the substitution method

Solve the linear system: 2x 1 5y 5 5 Equation 1

x 2 4y 5 9 Equation 2

Solution

STEP 1 Solve Equation 2 for x.

x 2 4y 5 9 Write original Equation 2.

x 5 4y 1 9 Revised Equation 2

STEP 2 Substitute 4y 1 9 for x in Equation 1 and solve for y.

2x 1 5y 5 5 Write Equation 1.

2(4y 1 9) 1 5y 5 5 Substitute 4y 1 9 for x.

8y 1 18 1 5y 5 5 Distributive property

13y 1 18 5 5 Simplify.

13y 5 213 Subtract 18 from each side.

y 5 21 Divide each side by 13.

STEP 3 Substitute 21 for y in the revised Equation 2 to fi nd the value of x.

x 5 4y 1 9 Revised Equation 2

x 5 4(21) 1 9 Substitute 21 for y.

x 5 5 Simplify.



Equation 1 Equation 2 2x 1 5y 5 5 x 2 4y 5 9

2(5) 1 5(21) 0 5 5 2 4(21) 0 9

5 5 5 ✓ 9 5 9 ✓

Exercises for Examples 1 and 2

Solve the linear system using the substitution method.

1. x 1 3y 5 210 2. 8x 1 5y 5 6 3. 6x 2 7y 5 22

7x 2 5y 5 34 5x 2 y 5 221 x 2 4y 5 22

4. 6x 1 y 5 26 5. x 1 3y 5 11

5x 2 2y 5 21 5x 1 6y 5 1

6. 3 }

2 x 1 y 5 8

4x 2 1 }

2 y 5 15

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21Algebra 1


Solve the linear system by using the substitution method.

1. 2x 1 y 5 1

3 }

2 x 1 y 5 6

2. 2 1 } 2 x 1

4 } 3 y 5 22

3 } 4 x 1

2 } 3 y 5

1 } 2

In Exercises 3 and 4, use the method shown in the following example to solve the system of equations.

Example: x2 1 y2 5 4

1 }

2 x2 1 3y2 5 8

Solution: Let u 5 x2 and v 5 y2. Using substitution, the system becomes

u 1 v 5 4

1 }

2 u 1 3v 5 8

.

Solving this system by substitution gives u 5 8 } 5 and v 5

12 } 5 .

Because u 5 x2 and v 5 y2, x 5 6 Î}

8 } 5 and y 5 6 Î}

12

} 5 . So, the solutions

are 1 2 Î}

8 } 5 , 2 Î}

12

} 5 2 , 1 2 Î}

8 } 5 , Î}

12

} 5 2 , 1 Î}

8 } 5 , 2 Î}

12

} 5 2 , and 1 Î}

8 } 5 , Î}

12

} 5 2 .

3. 2x2 1 4y2 5 11

x2 1 5y2 5 8

4. 3x 1 5y2 5 8

x 1 2y2 5 6


LESSON

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LESSON


Rewrite the linear system so that the like terms are arranged in columns.

1. 3x 2 y 5 23 2. 8x 5 y 1 1 3. 7x 2 4y 5 8

y 1 8x 5 11 3y 1 8x 5 7 4y 5 27x 1 9

4. 7x 2 y 5 13 5. 14 5 x 2 3y 6. 8x 1 1 5 4y

y 5 14x 2 3 x 1 10y 5 23 4y 1 3 5 14x

Describe the fi rst step you would use to solve the linear system.

7. x 1 4y 5 1 8. 2x 1 3y 5 21 9. 5x 1 y 5 8

6x 2 4y 5 23 3y 5 22x 1 3 x 1 y 5 26

10. 24x 2 4y 5 7 11. 6x 2 4y 5 5 12. 3x 5 y 2 9

4y 2 x 5 2 26x 2 5y 5 7 25x 1 y 5 8

Solve the linear system by using elimination.

13. 6x 2 y 5 5 14. x 1 4y 5 9 15. 5x 2 3y 5 214

3x 1 y 5 4 2x 2 2y 5 3 x 1 3y 5 2

16. 2x 1 y 5 7 17. 4x 1 3y 5 18 18. 25x 1 2y 5 22

x 1 y 5 1 4x 2 2y 5 8 3x 1 2y 5 210

19. 3x 5 y 1 5 20. x 2 4y 5 219 21. y 2 3 5 22x

2x 1 y 5 5 3y 2 15 5 x 2x 1 3y 5 13

22. 6x 2 3y 5 36 23. 24x 1 y 5 227 24. 9x 2 4y 5 255

5x 5 3y 1 30 2y 1 6x 5 43 3x 5 24y 2 21

25. Rollerblading One day, you are rollerblading on a trail while it is windy. You travel along the trail, turn around and come back to your starting point. On your way out on the trail, you are rollerblading against the wind. On your return trip, which is the same distance, you are rollerblading with the wind. You can only travel 3 miles an hour against the wind, which is blowing at a constant speed. You travel 8 miles an hour with the wind. Use the models below to write and solve a system of equations to fi nd the average speed when there is no wind and the speed of the wind.

Against the wind: Your speed with no wind 2 Speed of wind 5 Your speed

With the wind: Your speed with no wind 1 Speed of wind 5 Your speed

26. Car Wash A gas station has a car wash. If you get your gas tank fi lled, then you are charged a lower fl at fee x in dollars for a car wash plus y dollars per gallon for the gasoline. Two cars fi ll up with regular gasoline and both get a car wash. One car uses 8 gallons of gasoline and pays $22.80 for the gas and car wash and the other car uses 6 gallons of gasoline and pays $18.60 for the gas and car wash. Find the fee for the car wash and the cost of one gallon of regular gasoline.

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23Algebra 1


Rewrite the linear system so that the like terms are arranged in columns.

1. 8x 2 y 5 19 2. 4x 5 y 2 11 3. 9x 2 2y 5 5

y 1 3x 5 7 6y 1 4x 5 23 2y 5 211x 1 8


4. 22x 2 y 5 24 5. 25 5 x 2 7y 6. 3x 1 7 5 2y

y 5 6x 2 5 x 1 12y 5 28 22y 2 1 5 10x

7. x 1 9y 5 2 8. 4x 1 3y 5 26 9. 4x 1 y 5 210

14x 2 9y 5 24 3y 5 25x 1 1 x 1 y 5 214


10. x 1 5y 5 28 11. 7x 2 4y 5 230 12. 6x 1 y 5 39

2x 2 2y 5 213 3x 1 4y 5 10 22x 1 y 5 217

13. 3x 5 y 2 20 14. 2x 2 6y 5 210 15. x 2 3y 5 6

27x 2 y 5 40 4x 5 10 1 6y 22x 5 3y 1 33

16. 23x 5 y 2 20

2y 5 25x 1 4

17. x 2 1 }

2 y 5

11 } 2 18. 2

2 }

3 x 1 6y 5 38

2x 1 4y 5 26 x 2 6y 5 233

19. 3 }

2 x 1 y 5 2

5 }

2

4x 1 y 5 25

20. 7x 2 1 }

3 y 5 229 21.

1 }

2 x 2

3 }

2 y 5 2

29 } 2

2x 2 1 }

3 y 5 29 2

1 }

2 x 1 3y 5 33

22. Fishing Barge A fi shing barge leaves from a dock and moves upstream (against the current) at a rate of 3.8 miles per hour until it reaches its destination. After the people on the barge are done fi shing, the barge moves the same distance downstream (with the current) at a rate of 8 miles per hour until it returns to the dock. The speed of the current remains constant. Use the models below to write and solve a system of equations to fi nd the average speed of the barge in still water and the speed of the current.

Upstream: Speed of barge in still water 2 Speed of current 5 Speed of barge

Downstream: Speed of barge in still water 1 Speed of current 5 Speed of barge

23. Floor Sander Rental A rental company charges a fl at fee of x dollars for a fl oor sander rental plus y dollars per hour of the rental. One customer rents a fl oor sander for 4 hours and pays $63. Another customer rents a fl oor sander for 6 hours and pays $87.

a. Find the fl at fee and the cost per hour for the rental.

b. How much would it cost someone to rent a sander for 11 hours?

LESSON


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LESSON



1. 4x 2 y 5 221 2. 22x 1 5y 5 14 3. 2y 2 x 5 7

24x 1 7y 5 51 8x 1 5y 5 94 x 5 6y 2 28

4. 10y 2 2x 5 238 5. 8x 2 6y 5 212 6. 215x 1 4y 5 43

22x 5 8y 1 52 8x 1 4y 5 128 4y 5 23x 1 25

7. 6x 2 3y 5 54 8. 2y 2 3x 5 10 9. 9x 5 235 2 5y

6x 5 8y 2 36 7x 5 22y 2 50 5y 2 10x 5 250

10. 1.8x 2 4.2y 5 215.6 11. 27.4y 2 2.2x 5 47.2 12. 9.5x 2 7.4y 5 15.7

1.8x 1 7.5y 5 42.9 2.8y 5 2.2x 1 6.4 7.4y 2 4.2x 5 42.6

13. 2 }

3 x 1

1 }

3 y 5

2 }

3

1 }

3 y 1

1 }

3 x 5

5 }

3

14. 4.5x 1 0.5y 5 48.5 15. 3.2x 5 4.8y 1 8

2.5x 5 0.5y 1 14.5 6.4y 5 3.2x 2 19.2

16. For b Þ 0, what is the solution of the system 2x 1 by 5 22 and 4x 2 by 5 8?

17. Solve for x, y, and z in the system of equations below. Explain your steps.

x 1 3y 1 2z 5 9 Equation 1

2z 1 x 2 5y 5 27 Equation 2

6y 5 15 2 3x Equation 3

18. Car Rental A car rental company charges a daily rental fee plus a per mile fee over 150 miles. Two different people rent the same style of car for the same number of days. The total bill for one person’s rental is $207.50 for a 5-day rental and 180 miles. The total bill for the other person’s rental is $212.50 for a 5-day rental and 200 miles.

a. Write a linear system that you can use to fi nd the daily rental fee and the per mile fee over 150 miles. Explain how you got your linear system.

b. What is the daily rental fee? What is the fee per mile over 150 miles?

19. Greeting Cards Two friends are making their own greeting cards. They already have ink, but they will buy the stamps and cards. The table shows the numbers of stamps and packages of cards each person is buying. Another friend, George, wants to buy 3 stamps and 3 packages of cards. How much will it cost him? Explain.

Customer Stamps Packages of cards Total cost (dollars)

Stan 4 2 22.98

Leeza 7 2 32.73

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25Algebra 1


Solve linear systems by elimination.GOAL

Use addition to eliminate a variable



Solution

STEP 1 Add the equations to 2x 1 4y 5 2eliminate one variable. 4x 2 4y 5 16

STEP 2 Solve for x. 6x 5 18 x 5 3

STEP 3 Substitute 3 for x in either equation and solve for y.


2(3) 1 4y 5 2 Substitute 3 for x.

y 5 21 Solve for y.



Equation 1 Equation 2 2x 1 4y 5 2 4x 2 4y 5 16

2(3) 1 4(21) 0 2 4(3) 2 4(21) 0 16

2 5 2 ✓ 16 5 16 ✓

EXAMPLE 1

Use subtraction to eliminate a variable



Solution

STEP 1 Subtract the equations 7x 1 5y 5 18to eliminate one variable. 7x 2 3y 5 34

STEP 2 Solve for y. 8y 5 216 y 5 22

STEP 3 Substitute 22 for y in either equation and solve for x.


7x 1 5(22) 5 18 Substitute 22 for y.

x 5 4 Solve for x.


EXAMPLE 2


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Arrange like terms


13y 5 6x 1 8 Equation 2

Solution

STEP 1 Rewrite Equation 1 so that the like terms are arranged in columns.

6x 2 4y 5 10 6x 2 4y 5 10 13y 5 6x 1 8 26x 1 13y 5 8 STEP 2 Add the equations. 9y 5 18STEP 3 Solve for y. y 5 2

STEP 4 Substitute 2 for y in either equation and solve for x.



x 5 3 Solve for x.



Solve the linear system.

1. 5x 1 8y 5 36 2. 4x 1 5y 5 8 7x 2 8y 5 12 24x 2 3y 5 0

3. 9x 2 8y 5 7 4. 24x 1 7y 5 11 9x 1 2y 5 213 2x 1 7y 5 47

5. 9x 1 8y 5 230 6. 5y 5 4x 1 3 9x 5 4y 1 42 7x 5 36 2 5y

EXAMPLE 3


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27Algebra 1


Another Way to Solve Example 4 on page 458

Multiple Representations In Example 4 on page 458, you saw how to solve a problem about average speed using an inequality. You can also solve the problem by substitution.

Substitution You can solve the problem by substitution.

STEP 1 Write the system of equations from page 458.

Going upstream: x 2 y 5 4

Going downstream: x 1 y 5 6

STEP 2 Solve Equation 1 for x.

x 2 y 5 4 Write Equation 1.

x 5 y 1 4 Solve for x.

STEP 3 Substitute y 1 4 for x in Equation 2 and solve for y.

x 1 y 5 6 Write Equation 2.

y 1 4 1 y 5 6 Substitute y 1 4 for x.

y 5 1 Solve for y.

STEP 4 Substitute 1 for y in the revised Equation 1 to fi nd the value of x.

x 5 y 1 4 5 1 1 4 5 5

The average speed of the kayak in still water is 5 miles per hour, and the speed of the current is 1 mile per hour.

METHOD

1. Running Running into the wind, Calvin takes 56 minutes to run 7 miles. The return run takes 50 minutes. The wind speed remains constant during the trip. Find the average speed (in miles per hour) of Calvin in still air and the speed (in miles per hour) of the wind.

2. What If? Suppose in Exercise 1 it takes Calvin 70 minutes to run 7 miles into the wind and 50 minutes on the return run. Find the average speed of Calvin in still air and the speed of the wind.

3. Boating James and Bret take a boat out on a river. It takes them 15 minutes to travel 5 miles upstream (against the current). The return trip downstream (with the current) takes 10 minutes. The speed of the current remained constant during the trip. Find the average speed (in miles per hour) of the boat in still water and the speed of the current.

PRACTICE

LESSON

7.3 Problem Solving Workshop:Using Alternative MethodsFor use with pages 4562462

Kayaking During a kayaking trip, a kayaker travels 12 miles upstream (against the current) and 12 miles downstream (with the current), as shown on page 458. The speed of the current remained constant during the trip. Find the average speed of the kayak in still water and the speed of the current.

PROBLEM

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In Exercises 1–3, use the method shown in the following example to solve the system of equations.

Example: 3 1 1 } x 2 1 2 1 1 }

y 2 5 4

6 1 1 } x 2 2 2 1 1 }

y 2 5 5

Solution: Let u 5 1 }

x and v 5

1 }

y .

Using substitution, the system becomes

3u 1 2v 5 4

6u 2 2v 5 5.

Adding the equations results in the equation 9u 5 9.

So, u 5 1 5 1 } x and v 5

1 } 2 5

1 } y . So, x 5 1 and y 5 2.

1. 4 1 1 } x 2 1 7 1 1 }

y 2 5 3

24 1 1 } x 2 2 3 1 1 }

y 2 5 5

2. 4 1 1 1 1 } x 2 1 7y 5 3

24 1 1 1 1 } x 2 2 3y 5 5

3. 22(1 1 y3) 1 7 1 1 } x2 2 5 5

4(1 1 y3) 1 7 1 1 } x2 2 5 27

Solve the system for x and y in terms of a and b.

4. 3ax 1 2by 5 4

6ax 1 2by 5 7

5. ax 1 by 5 10

2ax 1 5by 5 13

6. 4ax 2 11y 5 b

2ax 1 2y 5 b


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29Algebra 1


LESSON


Match the linear system with an equivalent linear system.

1. 5x 2 2y 5 8 2. 7x 1 8y 5 3 3. 5x 1 2y 5 8

7x 1 8y 5 3 8x 2 2y 5 5 7x 1 8y 5 3

A. 220x 2 8y 5 232 B. 32x 2 8y 5 20 C. 20x 2 8y 5 32

7x 1 8y 5 3 7x 1 8y 5 3 7y 1 8y 5 3


4. x 1 y 5 4 5. 2x 1 6y 5 21 6. 3x 2 6y 5 21

3x 2 7y 5 10 24x 1 7y 5 8 x 1 y 5 4

7. 5x 2 2y 5 25 8. 23x 1 9y 5 13 9. 4x 2 y 5 7

10x 2 3x 5 7 7x 2 3y 5 14 10x 1 2y 5 8


10. x 1 y 5 3 11. 4x 1 y 5 28 12. 3x 2 y 5 10

22x 1 4y 5 6 3x 1 3y 5 3 2x 1 5y 5 35

13. 5x 2 4y 5 42 14. 2x 1 3y 5 210 15. 5x 1 6y 5 100

x 2 6y 5 24 24x 1 5y 5 22 2x 1 3y 5 46

16. 3x 2 5y 5 250 17. 26x 2 5y 5 243 18. 8x 2 6y 5 8

12x 1 2y 5 246 7x 1 15y 5 41 4x 1 5y 5 36

19. 4x 1 5y 5 100 20. 23x 1 11y 5 238 21. 5x 2 8y 5 235

3x 2 2y 5 6 2x 1 9y 5 240 27x 2 3y 5 222

22. Baseball Game Two families go to a baseball game. One family purchases two adult tickets and three youth tickets for $33. Another family purchases three adult tickets and two youth tickets for $37. Let x represent the cost in dollars of one adult ticket and let y represent the cost in dollars of one youth ticket. The linear system given by 2x 1 3y 5 33 and 3x 1 2y 5 37 represents this situation.

a. Solve the linear system to fi nd the cost of one adult and one youth ticket.

b. How much would it cost two adults and fi ve youths to attend the game?

23. Electricians Two different electrical businesses charge different rates. Business A charges $30 for a service call, plus an additional $45 per hour for labor. Business B charges $45 for a service call, plus an additional $40 per hour for labor.

a. Let x represent the number of hours of labor and let y represent the total charge in dollars. Write a linear system that you could use to fi nd the lengths of a service call for which both businesses charge the same amount.

b. Solve the linear system.

c. When will the businesses charge the same amount?

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1. 3x 2 4y 5 7 2. 9x 1 4y 5 13 3. 5x 1 7y 5 23 5x 1 8y 5 10 3x 1 5y 5 9 15x 1 4y 5 25

4. 7x 2 4y 5 6 5. 7x 1 9y 5 26 6. 9x 2 5y 5 14

3x 2 2y 5 215 25x 1 14y 5 11 26x 1 8y 5 13


7. x 1 3y 5 1 8. 23x 2 y 5 215 9. x 1 7y 5 237 25x 1 4y 5 224 8x 1 4y 5 48 2x 2 5y 5 21

10. 8x 2 4y 5 276 11. 23x 1 10y 5 23 12. 9x 2 4y 5 26 5x 1 2y 5 216 5x 1 2y 5 55 18x 1 7y 5 22

13. 4x 2 3y 5 16 14. 20x 1 10y 5 100 15. 3x 2 10y 5 225 16x 1 10y 5 240 25x 1 4y 5 53 5x 2 20y 5 255

16. 23x 2 4y 5 27 17. 2x 1 7y 5 2 18. 3x 2 5y 5 216 5x 2 6y 5 27 5x 2 2y 5 83 2x 2 3y 5 28

19. Hockey Game Two families go to a hockey game. One family purchases two adult tickets and four youth tickets for $28. Another family purchases four adult tickets and fi ve youth tickets for $45.50. Let x represent the cost in dollars of one adult ticket and let y represent the cost in dollars of one youth ticket.

a. Write a linear system that represents this situation.

b. Solve the linear system to fi nd the cost of one adult and one youth ticket.

c. How much would it cost two adults and fi ve youths to attend the game?

20. Travel Agency A travel agency offers two Chicago outings. Plan A includes hotel accommodations for three nights and two pairs of baseball tickets worth a total of $557. Plan B includes hotel accommodations for fi ve nights and four pairs of baseball tickets worth a total of $974. Let x represent the cost in dollars of one night’s hotel accommodations and let y represent the cost in dollars of one pair of baseball tickets.

a. Write a linear system you could use to fi nd the cost of one night’s hotel accommodations and the cost of one pair of baseball tickets.

b. Solve the linear system to fi nd the cost of one night’s hotel accommodations and the cost of one pair of baseball tickets.

21. Highway Project There are fi fteen workers employed on a highway project, some at $180 per day and some at $155 per day. The daily payroll is $2400. Let x represent the number of $180 per day workers and let y represent the number of $155 per day workers. Write and solve a linear system to fi nd the number of workers employed at each wage.

LESSON


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31Algebra 1


LESSON



1. 23x 1 5y 5 28 2. 2x 1 7y 5 213 3. 4x 1 7y 5 243

9x 1 4y 5 68 23x 1 14y 5 25 23x 1 6y 5 269

4. 8x 2 6y 5 2140 5. 4x 1 9y 5 253 6. 26x 1 12y 5 48

3x 1 5y 5 20 26x 2 4y 5 32 27x 1 18y 5 84

7. 3x 1 9y 5 27 8. 28x 1 5y 5 6 9. 10x 2 8y 5 28

14x 1 6y 5 18 6x 2 3y 5 6 12x 1 5y 5 92

10. 6x 2 11y 5 293 11. 215x 1 4y 5 22 12. 9x 2 8y 5 23

15x 1 13y 5 132 13x 2 10y 5 244 14x 2 12y 5 26

Solve the linear system by using any algebraic method.

13. 0.4x 1 0.1y 5 0.7 14. 4x 2 3y 5 7 15. 1.5x 1 2.6y 5 212.7

x 2 y 5 3 1.5x 1 y 5 9 24.5x 1 0.3y 5 21.9

16. x 1 y 5 7 17. 4x 1 y 5 2 7 }

4 18.

2 }

3 x 2

1 }

4 y 5 2

11 } 3

1 }

3 x 1

3 } 5 y 5

16 }

15

1 }

4 x 2

1 }

4 y 5

5 }

4 5x 2 2y 5 23

19. Find the values of a and b so that the linear system has a solution of (2, 4).


bx 2 ay 5 26 Equation 2

20. Lift Tickets Two families go skiing on a Saturday. One family purchases two adult lift tickets and four youth lift tickets for $166. Another family purchases four adult lift tickets and fi ve youth lift tickets for $263. Let x represent the cost in dollars of one adult lift ticket and let y represent the cost in dollars of one youth lift ticket.

a. Write a linear system that represents this situation.

b. Solve the linear system to fi nd the cost of one adult and one youth lift ticket.

c. How much would it cost two adults and fi ve youths to ski for a day?

21. Asian Cuisine A group of your friends goes to a restaurant that features different Asian foods. There are eight people in your group. Some of the group order the Thai special for $14.25 and the rest of the group order the Szechwan special for $13.95. If the total bill was $113.10, how many people ordered each dinner?

22. Getting to School You walk 1.75 miles to school at an average speed r (in miles per hour). On the way back home, you are walking with a friend and your average speed

is 3 }

4 r. The round trip took a total of 90 minutes. Find the average speed for each leg

of your trip.

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Solve linear systems by multiplying fi rst.GOAL

Multiply one equation, then add



Solution

STEP 1 Multiply Equation 1 by 22 so that the coeffi cients of y are opposites.

3x 2 2y 5 24 3 (22) 26x 1 4y 5 8 7x 2 4y 5 26 7x 2 4y 5 26STEP 2 Add the equations. x 5 2

STEP 3 Substitute 2 for x in either equation and solve for y.


3(2) 2 2y 5 24 Substitute 2 for x.

y 5 5 Solve for y.



Equation 1 Equation 2 3x 2 2y 5 24 7x 2 4y 5 26

3(2) 2 2(5) 0 24 7(2) 2 4(5) 0 26

24 5 24 ✓ 26 5 26 ✓

Exercises for Example 1

Solve the linear system using elimination.

1. 15x 1 4y 5 25 5x 2 3y 5 30

2. 5x 1 3y 5 18 9y 5 27x 1 6

3. 4x 5 7y 1 14 14y 5 3x 1 7

EXAMPLE 1

Review for MasteryFor use with pages 463– 469

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33Algebra 1


Multiply both equations, then add


7y 5 3x 1 19 Equation 2

Solution

STEP 1 Arrange the equations so that like terms are in columns.


23x 1 7y 5 19 Rewrite Equation 2.

STEP 2 Multiply Equation 1 by 3 and Equation 2 by 5 so that the coeffi cients of x in the equations are the least common multiple of 5 and 3, or 15.

5x 1 2y 5 218 3 3 15x 1 6y 5 254

23x 1 7y 5 19 3 5 215x 1 35y 5 95STEP 3 Add the equations. 41y 5 41STEP 4 Solve for y. y 5 1

STEP 5 Substitute 1 for y in either of the original equations and solve for x.



x 5 24 Solve for x.



Equation 1 Equation 2 5x 1 2y 5 218 7y 5 3x 1 19

5(24) 1 2(1) 0 218 7(1) 0 3(24) 1 19

218 5 218 ✓ 7 5 7 ✓


Solve the linear system using elimination.

4. 9x 1 5y 5 33

12x 2 7y 5 3

5. 3x 1 7y 5 20

5x 5 24y 1 41

6. 9y 5 10x 1 4

12x 5 5y 1 30

EXAMPLE 2

Review for Mastery continuedFor use with pages 463– 469

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LESSONS

7.1–7.4 Problem Solving Workshop:Mixed Problem SolvingFor use with pages 4392469

1. Multi-Step Problem You are selling tickets to a high school play. Student tickets cost $5 and general admission tickets cost $8. You sell 556 tickets and collect $3797.

a. Write a system of linear equations that represent the situation.

b. How many of each type of ticket did you sell?

2. Multi-Step Problem Biking into the wind on a fl at path, a bicyclist takes 5 hours to travel 30 miles. The return bike takes 3 hours. The wind speed remains constant during the trip.

a. Find the bicyclist’s average speed for each leg of the trip.

b. Write a system of linear equations that represent the situation.

c. What is the bicyclist’s average speed in still air? What is the speed of the wind?

3. Multi-Step Problem A total of $30,000 is invested in two accounts paying 3% and 4% annual interest. The combined annual interest is $1020.

a. Write a system of linear equations that represent the situation. (Hint: Write one equation for the amount invested in each account and another for the interest earned.)

b. How much of the $30,000 is invested in each account?

4. Gridded Response A bag contains dimes and nickels. There are 18 coins in the bag. The value of the coins is $1.25. How many nickels are in the bag?

5. Open-Ended Describe a real-world problem that can be modeled by a linear system. Then graph and solve the system and interpret the solution in the context of the problem.

6. Short Response At a grocery store, a customer pays a total of $11.10 for 1.6 pounds of chicken and 2 pounds of fi sh. Another customer pays a total of $12.15 for 2.4 pounds of chicken and 1.8 pounds of fi sh. How much do 2 pounds of chicken and 2 pounds of fi sh cost? Explain.

7. Open-Ended Find values for m and b so that the system y 5 2x 2 5 and y 5 mx 1 b has (6, 7) as a solution.

8. Gridded Response During one day, two cars are sold at a car dealership. The two customers each arrange payment plans with the salesperson. The graph shows the amount y of money (in dollars) paid for the car after x months. After how many months will each customer have paid the same amount?

x

y

00

2000

4000

6000

1 2 3 4 5 6Months since purchase

Am

ou

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paid

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llars

)

9. Extended Response A chemist needs 900 milliliters of a 40% acid solution for a chemistry experiment. The chemist combines x milliliters of a 20% acid solution and y milliliters of a 70% acid solution to make the 40% acid solution.

a. Write a linear system that represents the situation.

b. How many milliliters of the 20% acid solution and the 70% acid solution are combined to make the 40% acid solution?

c. The chemist also needs 900 millili-ters of a 45% acid solution. Does the chemist need more of the 20% acid solution than the 70% acid solution to make this new mix? Explain.

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35Algebra 1


In Exercises 1–3, use the following information.

Terry has a summer job mowing lawns in a neighborhood that has only two different lot sizes. After the fi rst day of work, Terry’s boss observed that Terry mowed 1 small lawn and 2 large lawns in 5 hours. After the second day of work, Terry’s boss observed that Terry mowed 3 small lawns and 3 large lawns in 8 hours.

1. Write a linear system to model this situation, where x represents the number of small lawns mowed and y represents the number of large lawns mowed.

2. Solve the linear system written in Exercise 1. What does the solution represent?

3. If Terry mows 2 large lawns in a 9-hour day, how many small lawns will he be able to mow?


Greyson has a paper delivery route which he completes by riding his bicycle. The drop-off station where he picks up his papers for delivery is located in the neighborhood where he delivers papers. When riding between his house and the drop-off station, Greyson averages 10 miles per hour. On Monday through Saturday the paper is a small daily and Greyson

averages 1 }

2 mile per hour while making his deliveries. When delivering the small daily it

takes Greyson 4 hours and 18 minutes, from the moment he leaves his house to the moment he returns in order to complete his route. On Sundays, the paper is much larger

and he averages 1 }

3 mile per hour while making his deliveries, which adds an additional

2 hours to the time it takes to complete his route.

4. Write a linear system to model this situation, where x represents the miles from Greyson’s house to the drop-off station and y represents the length of the route (in miles).

5. What is the distance from Greyson’s house to the drop-off station?

6. What is the length of the paper route?

Challenge Practice For use with pages 463–469

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LESSON


Identify the slope of the linear equation.

1. y 2 3x 5 8 2. 4x 1 2y 5 6 3. 9x 2 3y 5 15

Match the linear system with its graph. Then use the graph to tell whether the linear system has one solution, no solution, or infi nitely many solutions.

4. 23x 1 y 5 2 5. x 2 y 5 5 6. 4x 1 y 5 2

26x 1 2y 5 4 x 1 y 5 5 24x 2 y 5 1

A.

x

y

1 321

3

5

23

B.

x

y

1 321

23 21

C.

x

y

1 3 521

23

3

1

Graph the linear system. Then use the graph to tell whether the linear system has one solution, no solution, or infi nitely many solutions.

7. x 1 y 5 24 8. y 2 2x 5 3 9. 2x 1 2y 5 4

y 5 2x 1 1 x 1 y 5 2 y 5 2x 1 2

x

y

1

3

1212321

23

3

x

y

1

3

1212321

23

3

x

y

1

3

1212321

23

3

10. 3x 2 y 5 1 11. 4x 1 2y 5 8 12. 2x 2 4y 5 4

2x 1 y 5 22 3x 2 y 5 3 x 1 2 5 2y

x

y

1

3

1212321

23

3

x

y

1

3

1212321

23

3

x

y

1

3

1212321

23

3

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37Algebra 1


LESSON


Solve the linear system by using substitution or elimination.

13. 25x 1 5y 5 210 14. 4x 2 4y 5 218 15.

3x 2 3y 5 5 7x 2 7y 5 24

16. 24x 1 3y 5 1 17. 4x 2 y 5 2 18. 2x 1 4y 5 1

3x 2 4y 5 1 212x 1 3y 5 0 6x 1 12y 5 3

Without solving the linear system, tell whether the linear system has one solution, no solution, or infi nitely many solutions.

19. y 5 1 }

2 x 1 3 20. y 5 6x 1 4 21.

y 5 22x 1 3 y 5 26x 2 10

22. y 2 3x 5 8 23. 3y 1 6x 5 8 24.

3x 1 y 5 8 2x 1 y 5 210

25. 4x 2 6y 5 21 26. 2 2 }

3 x 1 y 5 2 27.

2 3 }

2 x 1 y 5

1 }

4 26x 1 3y 5 6

28. Water Park A water park charges a fee for admission to the park and a fee to rent a tube for the day. One admission to the water park costs x dollars and a tube rental for the wave pool costs y dollars. A group pays $263.25 for admission for 15 people and 8 tube rentals. Another group pays $358 for admission for 20 people and 13 tube rentals. Is there enough information to determine the cost of one admission to the water park? Explain.

29. Movie Tickets The table below shows the ticket sales at a small theater on a Thursday night and a Friday night.

DayNumber of

adult ticketsNumber of

children’s ticketsTotal sales

(dollars)

Thursday 45 10 425

Friday 225 50 2125

a. Let x represent the cost (in dollars) of one adult ticket and let y represent the cost (in dollars) of one children’s ticket. Write a linear system that models the situation.


c. Can you determine how much each kind of ticket costs? Why or why not?

2x 2 5y 5 0

5 }

2 x 2 y 5 0

y 5 3x 2 5

y 5 6 }

2 x2 5

4x 1 3y 5 9

3 }

4 x 1 y 5 3

9x 2 15y 5 15

x 1 3 } 5 y 5 1

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LESSON



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

3y 5 12x 2 9 2x 1 y 5 5 22x 1 y 5 23

A.

x

y

21

3

1

21

B.

x

y

321

21

C.

x

y

3121

23

2123

1


4. 26x 1 2y 5 22 5. 2y 2 x 5 24 6. 4x 2 y 5 2

23x 1 y 5 2 2x 1 y 5 3 2x 1 3y 5 9

x

y

3

123 3

x

y

1

3

12121

23

5

x

y

1

1212321

3

7. x 1 2y 5 3 8. 9. 2x 2 y 5 4

2x 1 2y 5 22 22x 1 y 5 24

x

y

1

3

3

23

521

x

y

1

3

121 3 5

x

y1

1212321

23

3

3x 1 y 5 4

x 1 1 }

3 y 5 2

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39Algebra 1



10. 3x 2 2y 5 24 11. 3x 1 2y 5 4 12. x 1 y 5 50

x 1 2y 5 8 26x 2 4y 5 28 23x 1 2y 5 0

13. 2x 1 4y 5 23 14. 2x 1 3y 5 9 15. 2x 1 y 5 6

23x 1 2y 5 1 2x 1 y 5 10 2x 1 y 5 27


16. 26x 1 6y 5 24 17. y 1 2x 5 8 }

3 18. 4x 1 3y 5 9

2x 2 2y 5 5 2x 1 y 5 210 3 }

4 x 1 y 5 3

19. 4x 2 6y 5 21 20. 2 2 } 3 x 1 y 5 2 21. 9x 2 15y 5 15

2 3 }

2 x 1 y 5

1 }

4 26x 1 3y 5 6 x 1

3 } 5 y 5 1

22. 23x 1 4y 5 2 23. 3x 1 y 5 4 24. 24x 1 3y 5 2

2y 5 3 }

2 x 1 1 x 1

1 }

3 y 5 2 4 2 6y 5 28x

25. Golf Clubs A sporting goods store stocks a “better” set of golf clubs in both left- handed and right-handed sets. The set of left-handed golf clubs sells for x dollars and the set of right-handed golf clubs sells for y dollars. In one month, the store sells 2 sets of left-handed golf clubs and 12 sets of right-handed golf clubs for a total of $1859.30. The next month, the store sells 2 sets of left-handed golf clubs and 22 sets of right-handed golf clubs for a total of $3158.80. Is there enough information to determine the cost of each kind of set? Explain.

26. Comedy Tickets The table below shows the ticket sales at an all-ages comedy club on a Friday night and a Saturday night.

DayNumber of

adult ticketsNumber of

student ticketsTotal sales (dollars)

Friday 30 20 910

Saturday 45 30 1365

a. Let x represent the cost (in dollars) of one adult ticket and let y represent the cost (in dollars) of one student ticket. Write a linear system that models the situation.


c. Can you determine how much each kind of ticket costs? Why or why not?

LESSON


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LESSON



1. 6x 1 4y 5 25 2. 3x 1 4y 5 12 3. y 5 3 } 5 x 1 5

3x 1 2y 5 2 5 } 2 24x 1 3y 5 29 23x 1 5y 5 210

A.

x

y

121

3

1

21

B.

x

y

1 321

1

21

C.

x

y

1

23

1

2123


4. 4y 5 3x 1 20 5. 3x 1 2y 5 8 6. 3y 2 4x 5 6

4y 1 12 5 5x 22x 1 3y 5 6 y 5 4 }

3 x 1 2

x

y

2

6

22222

26

6 10

x

y

1

3

5

1212321

3

x

y

1

3

1212321

23

3

7. 3x 1 4y 5 224 8. 2x 1 3y 5 21 9. 25x 1 2y 5 3

1 }

3 y 1

1 }

4 x 5 1 22x 1 3y 5 1 4y 2 10x 5 8

x

y

2

2222622

26

210

6

x

y

1

3

1212321

23

3

x

y

1

3

1212321

23

3

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41Algebra 1



10. 2x 1 2y 5 24 11. 4x 1 3y 5 2 12. x 1 8y 5 16

23x 1 4y 5 4 2x 1 3 }

2 y 5 1 23x 1 8y 5 28

13. 22x 1 5y 5 210 14. 22x 1 3y 5 2 1 } 2 15. 2y 2 10x 5 28

5y 2 2x 5 5 3x 1 2y 5 4 2y 2 x 5 4


16. 4y 5 12x 2 1 17. x 1 4y 5 3 18. 22x 1 3y 5 4

–12x 1 3y 5 21 1 }

2 x 1 2y 5 4 3x 2 2y 5 5

19. 5y 2 4x 5 3 20. y 2 1 }

4 x 5 22 21. 3y 1 5x 5 1

10y 5 8x 1 6 x 2 2y 5 8 25x 2 3y 5 1

22. 2y 2 x 5 3 23. 23x 1 4y 5 24 24. 4y 5 25x 1 3

2x 1 y 5 6 4x 1 3y 5 2 2y 1 5 }

2 x 5

3 }

2

25. Restaurant Sales The table below shows the number of each of the specials that has been sold on a Friday night and a Saturday night.

DayNumber of

vegetarian specialsNumber of

chicken specialsTotal sales

(dollars)

Friday 28 44 964.40

Saturday 21 33 723.30

a. Let x represent the cost (in dollars) of the vegetarian special and let y represent the cost (in dollars) of the chicken special. Write a linear system that models the situation.


c. Can you determine how much each kind of special costs? Why or why not?

26. Retail Prices Two employees at a store are given the task of putting price tags on items. One person starts pricing items at a rate of 10 items per minute. The second person starts 10 minutes after the fi rst person and prices items at a rate of 8 items per minute.

a. Let y be the number of items priced x minutes after the fi rst person starts pricing. Write a linear system that models the situation.


c. Does the solution of the linear system make sense in the context of the problem? Explain.

LESSON


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Identify the number of solutions of a linear system.

VocabularyA linear system with no solution is called an inconsistent system.

A linear system with infi nitely many solutions is called a dependent system.

GOAL

A linear system with no solution

Show that the linear system has no solution.



Solution

Method 1 Graphing

x

y

1

3

2121

3

Graph the linear system.

The lines are parallel because they have the same slope but different y-intercepts. Parallel lines do not intersect, so the system has no solution.

Method 2 Elimination

Add the equations. 25x 1 4y 5 16 5x 2 4y 5 8

0 5 24 This is a false statement.

The variables are eliminated and you are left with a false statement regardless of the values of x and y. This tells you that the system has no solution.

EXAMPLE 1


LESSON

7.5

A linear system with infi nitely many solutions

Show that the linear system has infi nitely many solutions.

y 5 2 }

3 x 1 5 Equation 1


Solution

Method 1 Graphing

x

y

1

3

7

2121

23 31

Graph the linear system.

The equations represent the same line, so any point on the line is a solution. So, the linear system has infi nitely many solutions.

EXAMPLE 2

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43Algebra 1


Identify the number of solutions


a. 7x 2 2y 5 9 Equation 1 b. 3x 1 y 5 210 Equation 1

7x 2 2y 5 21 Equation 2 26x 2 2y 5 20 Equation 2

Solution

a. y 5 7 }

2 x 2

9 }

2 Write Equation 1 in slope-intercept form.

y 5 7 }

2 x 1

1 }

2 Write Equation 2 in slope-intercept form.

Because the lines have the same slope but different y-intercepts, the system has no solution.

b. y 5 23x 2 10 Write Equation 1 in slope-intercept form.

y 5 23x 2 10 Write Equation 2 in slope-intercept form.

The lines have the same slope and y-intercept, so the system has infi nitely many solutions.



3. x 2 3y 5 7 4. 2x 1 3y 5 17 5. 24x 1 y 5 5

4x 5 12y 1 28 3x 1 2y 5 14 28x 2 14y 5 228

EXAMPLE 3

Method 2 Substitution

Substitute 2 }

3 x 1 5 for y in Equation 2 and solve for x.


22x 1 3 1 2 } 3 x 1 5 2 5 15 Substitute

2 }

3 x 1 5 for y.

22x 1 2x 1 15 5 15 Distributive property

15 5 15 Simplify.

The variables are eliminated and you are left with a statement that is true regardless of the values of x and y. This tells you the system has infi nitely many solutions.


Tell whether the linear system has no solution or infi nitely many solutions.

1. 215x 1 3y 5 6 2. 24x 1 y 5 5

y 5 5x 1 2 y 5 4x 1 3


LESSON

7.5

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LESSON

7.5

In Exercises 1–3, use the linear system.

ax 1 1 } 4 y 5 7

1 }

3 x 1

1 } 6 y 5 3

1. For what values of a does the system have no solution?

2. For what values of a does the system have infi nitely many solutions?

3. For what values of a does the system have exactly one solution?

In Exercises 4 and 5, suppose a, b, and c are non-zero constants. Use the linear system.

ax 1 by 5 3

cax 1 cby 5 12

4. Does the number of solutions depend on the values of a, b, and c?

5. Describe the number of solutions in each possible case.

In Exercises 6–9, suppose a1, a2, b1, b2, c1, and c2 are non-zero constants. Use the linear system.

a1x 1 b

1y 5 c

1

a2x 1 b

2y 5 c

2

6. Solve for x and y in terms of a1, a

2, b

1, b

2, c

1, and c

2.

7. State the relationship between the values of a1, a

2, b

1, b

2, c

1, and c

2 that will

guarantee there is exactly one solution.


2, b

1, b

2, c

1, and c

2 that will

guarantee there is no solution.


2, b

1, b

2, c

1, and c

2 that will

guarantee there are infi nitely many solutions.

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45Algebra 1


TI-83 PlusExample 1

Y= X,T,�,n � 2

ENTER ENTER 3 X,T,�,n �

1 ENTER

ENTER ENTER GRAPH

Press one of the arrow keys to place the cursor on the screen. Use the arrow keys to move the cursor to points in the graph of the system.

Example 2

Y= (�) 4 � 3 X,T,�,n � 1

ENTER

ENTER X,T,�,n � 5

ENTER ENTER ENTER

GRAPH

To identify a solution, use the cursor to locate a point in the graph of the system, or simply identify a solution visually.


Example 1From the main menu, choose GRAPH.

F3 F6 F1 X, ,T� � 2 EXE F3 F6 F4

3 X,T,�,n � 1 EXE F6

Press F1 [Trace] to place the cursor on the screen. Use the arrow keys to move the cursor to points in the graph of the system.

Example 2From the main menu, choose GRAPH.

F3 F6 F1 (�) 4 b ca 3 X, ,T� � 1 EXE

F3 F6 F4 X, ,T� � 5 EXE F6

To identify a solution, press F1 [Trace] and use the cursor to locate a point in the graph of the system, or simply identify a solution visually.

Graphing Calculator Activity KeystrokesFor use with pages 485–486

LESSON

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LESSON


Tell whether the ordered pair is a solution of the system of inequalities.

1. (2, 1) 2. (23, 2) 3. (0, 21)

x

y

1 3 521

3

x

y

121

212325

1

3

x

y

1 32123

1

4. (22, 0) 5. (2, 4) 6. (22, 3)

x

y

12325

3

23

x

y

1 3

3

1

21

x

y

3121

23

21

1

3

Match the system of inequalities with its graph.

7. x 1 y ≥ 4 8. x 1 y ≤ 4 9. x 2 y ≥ 4

x < 2 x < 22 y > 2

10. y 1 x ≤ 4 11. x 2 y ≤ 4 12. y 1 x ≥ 4

y < 2 x > 22 y < 22

A.

x

y

6 10222

22

6 B.

x

y

6226

6

10

2

C.

x

y

6226 22

6

26

22

D.

x

y

626 22

6

2

22

E.

x

y

6 102

26

22

6

2

F.

x

y

6226

2

22

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47Algebra 1


LESSON


Graph the system of inequalities.

13. x > 21 14. y > 23 15. x ≥ 2

x < 4 y ≤ 0 y > 0

x

y

1

3

1212321

23

3

x

y

1

3

1212321

23

3

x

y

1

3

1212321

23

3

16. x < 1 17. x > 0 18. y ≤ 3

y ≤ 22 y ≤ x y > 2x

x

y

1

3

1212321

23

3

x

y

1

3

1212321

23

3

x

y

1

3

1212321

23

3

19. Ordering Cups You work at an Italian ice shop during

x00

1

2

3

4

5

6

7

8

9

1 2 3 4 5 6 7 8 9Boxes of 5-ounce cups

Bo

xes o

f 8-o

un

ce c

up

s

ythe summer. You need to order 5-ounce and 8-ounce cups. The storage room will only hold 10 more boxes of cups. A box of 5-ounce cups costs $15 and a box of 8-ounce cups costs $18. A maximum of $90 is budgeted for cups.

a. Let x represent the number of boxes of 5-ounce cups and let y represent the number of boxes of 8-ounce cups. Write a system of linear inequalities for the number of cups that can be bought.

b. Graph the system of inequalities.

c. Identify two possible combinations of cups you can buy.

20. Studying You need at least 4 hours to do your science and

x00

1

2

3

4

5

6

1 2 3 4 5 6Hours spent on science

Ho

urs

sp

en

t o

n h

isto

ry

yhistory homework. It is 1:00 P.M. on Sunday and your friend wants you to go to the movies at 7:00 P.M.

a. How much time do you have between now and 7:00 P.M. to do your homework?

b. Let x represent the number of hours spent on science homework and let y represent the number of hours spent on history homework. Write and graph a system of linear inequalities that shows the number of hours you can work on each subject if you go to the movies.

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LESSON



1. (3, 0) 2. (2, 2) 3. (22, 2)

x

y

3

23

21

1

x

y

31 521

1

5

3

21

x

y

321

2123

1


4. 1 }

2 x 1 y ≥ 3 5. y 2

1 }

2 x ≤ 3 6. y ≤ 1 }

2 x 1 3

x > 21 x < 21 x > 21

A.

x

y

1 323

1

B.

x

y

23

1

1 3

C.

x

y

23

1

1 3


7. x > 21 8. y ≥ 2 9. x 1 y > 1

x < 1 y < 3 x ≤ y

x

y

323

3

1

23

21

x

y

3123 21

1

23

21

x

y

3123 21

3

23

10. x ≥ y 1 2 11. y ≥ 2 12. x ≤ 2y

2x 1 y < 4 x 1 y ≤ 23 2x 2 y < 4

x

y

3121

3

1

21

x

y

212325

3

1

21

x

y

3123 21

1

23

21

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49Algebra 1


LESSON


Write a system of inequalities for the shaded region.

13.

x

y

3123 21

23

21

14.

x

y

212325

1

21

25

15.

x

y

3123

3

23

21

16.

x

y

312123

3

1

21

17.

x

y

3123

3

23

18.

x

y

312123

3

5

1

21

19. Cookout You are planning a cookout. You fi gure that you will need at least 5 packages of hot dogs and hamburgers. A package of hot dogs costs $1.90 and a package of hamburgers costs $5.20. You can spend a maximum of $20 on the hot dogs and hamburgers.

a. Let x represent the number of packages of hot dogs and let y represent the number of packages of hamburgers. Write a system of linear inequalities for the number of packages of each that can be bought.


c. Identify two possible combinations of packages of hot dogs and hamburgers you can buy.

20. Chores You need at least 4 hours to do your chores, which are cleaning out the garage and weeding the fl ower beds around your house. It is 1:30 P.M. on Sunday and your friend wants you to go to the movies at 7:00 P.M.

a. How much time do you have between now and 7:00 P.M. to do your chores?

b. Let x represent the number of hours spent cleaning out the garage and let y represent the number of hours spent on weeding the fl ower beds. Write and graph a system of linear inequalities that shows the number of hours you can work on each chore if you go to the movies.

c. Identify two possible combinations of time you can spend on each chore.

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LESSON



1. (0, 1) 2. (0, 21) 3. (1, 4)

x

y

31

21

23 x

y

31

21

x

y

3 5121

3

1

21


4. 3x 1 2y ≥ 4 5. 3x 1 2y ≥ 24 6. 3x 2 2y ≤ 4

y > 4 2 x x 1 y < 4 x 1 y < 4

A.

x

y

42421222024

20

12

B.

x

y

6222

6

2

C.

x

y

2226

6

10

22


7. x ≥ 22 8. x < 0 9. 3x 1 y < 0

y ≤ 5 y > 21 4x 2 y ≤ 1

x

y

12123

1

3

5

21

x

y

3123 21

3

1

23

21

x

y

3123 21

3

1

23

21

10. x ≥ 0, y ≥ 0 11. x > 4, x < 8 12. y > 22, x ≥ 0

2x 1 y < 3 y ≥ 2x 1 1 y ≥ 3x

x

y

3121

3

1

21

x

y

12424

12

20

4

24

x

y

312123

3

1

21

23

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51Algebra 1


LESSON


Write a system of inequalities for the shaded region.

13.

x

y

1 323

1

3

14.

x

y

312123

3

1

23

15.

x

y

3121

3

1

21

16.

x

y

312123

3

21

23

17.

x

y

1 32123

3

1

21

18.

x

y

6222

2

19. School Play The tickets for a school play cost $8 for adults and

x00

200

400

600

100

300

500

200 400 600Adult tickets

Stu

den

t ti

ckets

y$5 for students. The auditorium in which the play is being held can hold at most 525 people. The organizers of the school play must make at least $3000 to cover the costs of the set construction, costumes, and programs.

a. Write a system of linear inequalities for the number of each type of ticket sold.


c. If the organizers sell out and sell twice as many student tickets as adult tickets, can they reach their goal? Explain how you got your answer.

20. Exercise You exercise 15 hours per week by swimming and

1400

2

4

6

8

10

12

14

2 4 6 8 10 12

Sw

imm

ing

Running

y

x

running. You want to spend at least twice the amount of time swimming as running.

a. Write a system consisting of an equation and an inequality that describes the situation.

b. Draw a graph to show the possible combinations of hours that you could exercise.

c. Interpret the graph in the context of the problem.

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Solve systems of linear inequalities in two variables.

VocabularyA system of linear inequalities in two variables, or simply a system of inequalities, consists of two or more linear inequalities in the same variables.

A solution of a system of linear inequalities is an ordered pair that is a solution of each inequality in the system.

The graph of a system of linear inequalities is the graph of all solutions of the system.

GOAL

Graph a system of two linear inequalities


y < 1 }

2 x 1 2 Inequality 1

x

y

1

3

5

7

2121

23 1

y ≥ 22x 1 5 Inequality 2

Solution

Graph both inequalities in the same coordinate plane. The graph of the system is the intersection of the two half-planes, which is shown as the shaded region.

CHECK Choose a point in the shaded region, such as (2, 2). To check this solution, substitute 2 for x and 2 for y into each inequality.

Inequality 1 Inequality 2

y < 1 } 2 x 1 2 y ≥ 22x 1 5

2 <? 1 } 2 (2) 1 2 2 ≥? 22(2) 1 5

2 < 3 ✓ 2 ≥ 1 ✓

EXAMPLE 1


LESSON

7.6

Graph a system of three linear inequalities


y ≤ 5 Inequality 1

x < 4 Inequality 2

y ≥ 22x 1 2 Inequality 3

Solution

x

y

2

6

2222

26

26 6Graph all three inequalities in the same coordinate plane. The graph of the system is the triangular region shown.

EXAMPLE 2

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53Algebra 1



LESSON

7.6


Graph the system of linear inequalities.

1. y > 3x 2 2 2. x > 22 3. y > 2

y ≤ 2 } 3 x 1 1 y > 23 y < 8

y ≤ 3 } 4 x 1 2 y ≥ 4x 2 1

Write a system of linear inequalities

Write a system of inequalities for the shaded

x

y

1

3

5

2121

1 5

region.

Solution

Inequality 1 One boundary for the shaded region has a slope of 24 and a y-intercept of 5. So, its equation is y 5 24x 1 5. Because the shaded region is below the solid line, the inequality is y ≤ 24x 1 5.

Inequality 2 Another boundary line for the shaded region has a slope of 3 } 5 and a

y-intercept of 22. So, its equation is y 5 3 } 5 x 2 2. Because the shaded region is

above the dashed line, the inequality is y > 3 } 5 x 2 2.

The system of inequalities for the shaded region is: y ≤ 24x 1 5 Inequality 1

y > 3 } 5 x 2 2 Inequality 2


Write a system of inequalities that defi nes the shaded region.

4.

x

y

1

3

2123 31

5.

x

y6

6 102

EXAMPLE 3

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LESSONS

7.5–7.6 Problem Solving Workshop:Mixed Problem SolvingFor use with pages 471–484

1. Multi-Step Problem Stacy can read 32 pages per hour. Anthony starts 15 minutes after Stacy and can read 28 pages per hour.

a. Let y be the number of pages read x hours after Stacy began reading. Write a linear system that models the situation.


c. Does the solution of the linear system make sense in the context of the problem? Explain.

2. Multi-Step Problem A restaurant offers two different meals each evening and has at least 260 customers. For Friday night, the restaurant offers salmon and lemon chicken. The restaurant expects that more people will order the chicken than the salmon. The salmon costs $6 per serving and the chicken costs $4 per serving. The restaurant has a budget of at most $1600 for meat for Friday night.

a. Let x be the number of customers who ordered salmon and let y be the number of customers who ordered lemon chicken. Write a system of linear inequalities that models the situation.


c. Use the graph to determine whether 120 orders of salmon and 160 orders of chicken can be ordered.

3. Open-Ended Write a linear system so that it has no solution and one of the equations is 5x 2 4y 5 26.

4. Gridded Response What is the area (in square units) of the garden defi ned by the system of inequalities below?

y ≥ 0 x ≥ 0 y ≤ 4 x 1 y ≤ 8

5. Short Response During a sale at a clothing store, all shirts are priced the same and all shorts are priced the same. Lucy buys 6 shirts and 3 shorts for $78. The next day, while the sale is still in progress, Lucy goes back and buys 2 shirts and 1 pair of shorts for $26. Is there enough information to determine the cost of 1 shirt? Explain.

6. Extended Response During the summer, you want to earn at least $120 per week. You earn $9 per hour babysitting and you earn $6 per hour working at a grocery store. You can work at most 25 hours per week.

a. Write and graph a system of linear inequalities that models the situation.

b. You work 7 hours per week babysitting and 8 hours per week at the grocery store. Will you earn at least $120 per week? Explain.

c. You are scheduled to work 12 hours babysitting. What is the range of hours you can work at the grocery store to earn at least $120 per week?

7. Short Response Is it possible to fi nd a value for c so that the linear system below has no solution? Explain.


y 5 3 } 7 x 1 c Equation 2

8. Extended Response Martin decides to make a walkway in his backyard. He spends $94 on 6 large bricks and 8 small bricks. Then he decides to make another walkway using the same kinds of bricks. He spends $188 for 12 large bricks and 16 small bricks.

a. Write a system of linear equations that models the situation.

b. Is there enough information given to determine the cost of one brick of each type? Explain.

c. A large brick costs $4 more than a small brick. What is the cost of one brick of each type?

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55Algebra 1



1. y ≥ ⏐x⏐

y ≤ 6 2 ⏐x⏐

2. ⏐x⏐ ≤ 2

⏐y⏐ ≤ 2


Your school club decides to hold a fundraiser by selling trail mix, and you are in charge of making the mix. You plan to offer two mixes, Country Blend and Premium Mix, each

sold in one pound bags. Each pound of Country Blend consists of 1 }

2 pound of toasted

oats, 1 }

4 pound of peanuts, and

1 }

4 pound of raisins. Each pound of Premium Mix consists

of 1 }

4 pound of toasted oats,

1 }

4 pound of peanuts, and

1 }

2 pound of raisins. You have available

to use at most 40 pounds of oats, 22 pounds of peanuts, and 35 pounds of raisins.

3. Model the situation above by letting x represent the number of pounds of Country Blend and y represent the number of pounds of Premium Mix. Your algebraic model should be a system of fi ve inequalities. (Remember that you cannot make a negative number of pounds of trail mix.)

4. Graph the system of inequalities from Exercise 3.

5. You sell the trail mix for $5 per pound for Country Blend and $7 per pound for Premium Mix. How many bags of each type of mix should you make in order to maximize your income? (Hint: the maximum income must occur at one of the vertices of the graph.)

6. Using the answer from Exercise 5, what will be your club’s income if all the bags of mix are sold?


LESSON

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CHAPTER

7 Chapter Review GameFor use after Chapter 7

Magic Square

Solve each linear system in the table using any method. Place the indicated coordinate on the line given in the box. When the puzzle is completed correctly, the sum of each row, column, and diagonal should be the same. Place the sum of each row, column, and diagonal on the given lines next to the square.

Diagonal: _______

2x 2 y 5 0

2x 1 y 5 4

y-coordinate

______

3x 1 y 5 4

22x 1 y 5 21

x-coordinate

______

y 5 2 1 }

2 x 1 8

y 5 2x 2 7

x-coordinate

______

3 }

2 x 1 2y 5 12

1 }

4 x 1 y 5 4

y-coordinate

______

4x 1 3y 5 8

x 2 2y 5 13

x-coordinate

______

y 5 x 1 4

y 5 5x 2 8

y-coordinate

______

7x 2 y 5 225

2x 1 5y 5 14

y-coordinate

______

y 5 22x 1 21

y 5 1 } 3 x 1 7

y-coordinate

______

3x 1 2y 5 8

3x 2 4y 5 2

x-coordinate

______

Column 1: _______ Column 2: _______ Column 3: _______ Diagonal: _______

Row 1: _______

Row 2: _______

Row 3: _______

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57Algebra 1


CHAPTER

8



8.1: Apply Exponent Properties Involving Products

Use properties of exponents involving products.

• Bees

• Ice Cream Composition

• Coastal Landslide

8.2: Apply Exponent Properties Involving Quotients

Use properties of exponents involving quotients.

• Fractal Tree

• Astronomy

• Space Travel

8.3: Defi ne and Use Zero and Negative Exponents

Focus on Operations

Use zero and negative exponents.

Use fractional exponents.

• Mass

• Botany

• Medicine

8.4: Use Scientifi c Notation Read and write numbers in scientifi c notation.

• Blood Vessels

• Insect Lengths

• Agriculture

8.5: Write and Graph Exponential Growth Functions

Write and graph exponential growth models.

• Collector Car

• Compound Interest

• Investments

8.6: Write and Graph Exponential Decay Functions

Focus on Functions

Write and graph exponential decay functions.

Identify, graph, and write geometric sequences.

• Forestry

• Cell Phones

• Guitars



1. Applying properties of exponents to simplify expressions

2. Working with numbers in scientifi c notation

3. Writing and graphing exponential functions

Family LetterFor use with Chapter 8 C

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Family Letter continuedFor use with Chapter 8


Lesson Exercise

8.1 A farming corporation plants 103 seeds per acre of land. The corporation plants 105 acres. Use order of magnitude to fi nd the number of seeds that were planted.

8.2 A city has 1000 gas pumps. During the past year, 94,750,000 gallons of gas were sold in the city. Use order of magnitude to fi nd the approximate number of gallons sold per gas pump.

8.3 Simplify the expression. Write your answer using only positive exponents.

(a) (23x)5 • (23)27 (b) (22x4y24z)23

(c) 1 }

(4x)22 (d) (5x)23 p y4

} 2x6y26

Focus on Operations

Evaluate the expression 1002 • 10023/2.

8.4 Evaluate the expression (4.3 3 106)(2.1 3 1022).(a) Write the answer in scientifi c notation.

(b) Write the answer in standard form.

8.5 You inherited a stamp collection valued at $400 when you were 10 years old. The value of the collection increases at a rate of 4.3% per year. How much will it be worth when you turn 18? Round your answer to the nearest dollar.

8.6 Find the value of a $20,000 boat after 5 years if the boat depreciates 8% per year. Round your answer to the nearest dollar.

Focus on Functions

Tell whether the sequence is arithmetic or geometric. Then write the next term of the sequence. 7, 14, 21, 28, 35, 42, ...


Directions Investigate fi ve different banks or credit unions to learn their interest rates and how money is compounded (monthly, yearly) in their savings accounts. Then fi nd the balance for each account after one, fi ve, and ten years with principal amounts of $500, $2000, and $10,000. Analyze your fi ndings to determine the best account for short-term and long-term investments.

Answers8.1: 108 seeds 8.2: about 105 gallons 8.3: (a)

x5 }

9 (b) y12

} 28x12z3 (c) 16x2 (d)

y10

} 2125x9

Focus on Operations: 10 8.4: (a) 9.03 3 104 (b) 90,300 8.5: $560 8.6: $13,182 Focus on Functions: arithmetic; 49

CHAPTER

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59Algebra 1


CAPÍTULO

8



8.1: Aplicar propiedades de exponentes con productos

Usar propiedades de exponentes con productos

• Abejas

• Redacción del helado

• Derrumbamiento de tierras costales

8.2: Aplicar propiedades de exponentes con cocientes

Usar propiedades de exponentes con cocientes

• Árbol de factores

• Astronomía

• Viaje espacial

8.3: Defi nir y usar cero y exponentes negativos

Enfoque en las operaciones

Usar cero y exponentes negativos

Usar exponentes fraccionales

• Masa

• Botánica

• Medicina

8.4: Usar la notación científi ca Leer y escribir números usando la notación científi ca

• Vasos sanguíneos

• Longitudes de insectos

• Agricultura

8.5: Escribir y grafi car funciones de crecimiento exponencial

Escribir y grafi car modelos de crecimiento exponencial

• Carro coleccionable

• Interés compuesto

• Inversiones

8.6: Escribir y grafi car funciones de decrecimiento exponencial

Enfoque en las funciones

Escribir y grafi car modelos de decrecimiento exponencial

Identifi car, grafi car y escribir secuencias geométricas

• Silvicultura


• Guitarras



1. Aplicar propiedades de exponentes para simplifi car expresiones

2. Trabajar con números en notación científi ca

3. Escribir y grafi car funciones exponenciales

Carta para la familiaUsar con el Capítulo 8 C

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Carta para la familia continúaUsar con el Capítulo 8


Lección Ejercicio

8.1 Una compañía agrícola siembra 103 semillas por acre de tierra. La compañía siembra 105 acres. Usa el orden de magnitud para hallar el número de semillas que se sembraron.

8.2 Una ciudad tiene 1000 bombas de gasolina. Durante el año pasado, 94,750,000 galones de gasolina se vendieron en la ciudad. Usa el orden de magnitud para hallar el número aproximado de galones vendidos por bomba de gasolina.

8.3 Simplifi ca la expresión. Escribe tu respuesta usando solo exponentes positivos.

(a) (23x)5 • (23)27 (b) (22x4y24z)23

(c) 1 }

(4x)22 (d) (5x)23 p y4

} 2x6y26


Evalúa la expresión 1002 • 10023/2.

8.4 Evalúa la expresión (4.3 3 106)(2.1 3 1022).(a) Escribe la respuesta usando la notación científi ca.

(b) Escribe la respuesta en forma usual.

8.5 Heredaste una colección de sellos con un valor de $400 cuando tenías 10 años. El valor de la colección aumenta a una tasa de 4.3% por año. ¿Qué será su valor cuando cumples los 18 años? Redondea tu respuesta al dólar más próximo.

8.6 Halla el valor de un barco de $20,000 después de 5 años si se deprecia 8% por año. Redondea tu respuesta al dólar más próximo.


Indica si la secuencia es aritmética o geométrica. Luego escribe el término que sigue en la secuencia. 7, 14, 21, 28, 35, 42, ...


Instrucciones Investiga cinco bancos o cooperativos de crédito diferentes para saber sus tasas de interés y cómo se compone el dinero (mensualmente, anualmente) en sus cuentas de ahorros. Luego halla el saldo de cada cuenta después de un, cinco y diez años con cantidades principales de $500, $2000 y $10,000. Analiza tus hallazgos para determinar la mejor cuenta para inversiones de corto plazo y de largo plazo.

Respuestas8.1: 108 semillas 8.2: aproximadamente 105 galones 8.3: (a)

x5 }

9 (b) y12

} 28x12z3

(c) 16x2 (d) y10

} 2125x9

Enfoque en las operaciones: 10 8.4: (a) 9.03 3 104 (b) 90,300 8.5: $560 8.6: $13,182 Enfoque en las funciones: aritmética; 49

CAPÍTULO

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61Algebra 1


ExpressionExpression as

repeated multiplicationNumber

of factorsSimplifi edexpression

74 p 75 (7 p 7 p 7 p 7) p (7 p 7 p 7 p 7 p 7) 9 79

(24)2 p (24)3 [(24) p (24)] p [(24) p (24) p (24)]

x1 p x5

Expression Expanded expressionExpression as repeated

multiplication

Numberof factors

Simplifi edexpression

(53)2 (53) p (53) (5 p 5 p 5) p (5 p 5 p 5) 6 56

F (26)2 G 4 F (26)2 G p F (26)2 G p F (26)2 G p F (26)2 G

(a3)3

Activity Support MasterFor use with page 504

LESSON

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LESSON


Name the property that is demonstrated by the example.

1. (2x)3 5 23 p x3 5 8x3 2. x4 p x5 5 x415 5 x9 3. ( y3)2 5 y3p2 5 y6

Fill in the blanks.

4. (z3)5 5 z3 5 5. (5x)4 5 5 p x

5 z 5 x

6. 33 p 31 5 33 1 7. (24y2)3 5 (24) ( y )

5 3 5 y

8. (x2y4)3 5 ( x ) ( y ) 9. x2(x3y)2 5 x ( x ) y

5 x y 5 x x y

5 x y

Simplify the expression. Write your answer using exponents.

10. 82 p 85 11. 52 p 54 12. 7 p 78

13. (24)5 14. (63)7 15. (42)9

16. (13 p 18)2 17. (21 p 25)5 18. (7 p 154)6

Simplify the expression.

19. x3 p x 20. y2 p y6 21. z10 p z3

22. (m4)7 23. (b9)2 24. ( p5)3

25. (3n)3 26. (2x)5 27. (xy)6

28. State Populations The table below shows the populations of selected states in 1870. Write the order of magnitude of each of the populations.

State Wisconsin Nebraska New Jersey Oregon

Population 1,054,670 122,993 906,096 90,923

29. U.S. National Parks Hot Springs National Park in Arkansas covers an area of about 101 square miles. Kenai Fjords National Park in Arkansas covers an area that is about 102 times the area of Hot Springs National Park. Find the approximate area of Kenai Fjords National Park. Write your answer using exponents.

30. Mining In 2000, Canada mined approximately 104 metric tons of uranium. The amount of metric tons of zinc mined in Canada in 2000 was approximately 102 times this amount. About how many metric tons of zinc were mined in Canada in 2000?

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63Algebra 1



1. 54 p 58 2. (24)7 p (24)3

3. (210)5 p (210)2 4. 82 p 84 p 8

5. 25 p 2 p 24 6. (35)2

7. (93)7 8. (152)4

9. [(24)5]9 10. (13 p 19)4

11. (48 p 27)6 12. (135 p 8)5


13. x5 p x2 14. y3 p y p y4

15. a10 p a2 p a6 16. (z5)5

17. (b7)2 18. [(b 1 1)2]3

19. (23x)4 20. 2(3x)4

21. (2ab)5 22. (2x3y)6

23. (3m7)4 p m3 24. 4p2 p (3p5)2

Find the missing exponent.

25. x6 p x? 5 x12 26. (x4)? 5 x12 27. (3z?)3 5 27z18

28. Newspaper Circulation In 1996, the newspaper circulation in the country of Algeria was approximately 103 times the newspaper circulation in the country of Mauritania. The newspaper circulation in Mauritania was 103. What was the newspaper circulation in Algeria?

29. Metric System The metric system has names for very large weights.

a. One gigaton is 102 times the weight of a hectaton. One hectaton is 102 ton. Write one gigaton in tons.

b. One teraton is 109 times the weight of a kiloton. One kiloton is 103 ton. Write one teraton in tons.

c. One exaton is 106 times the weight of a teraton. Use your answer to part (b) to write one exaton in tons.

30. Wall Mural You are designing a wall mural that will be composed of squares of different sizes. One of the requirements of your design is that the side length of each square is itself a perfect square.

a. If you represent the side length of a square as x2, write an expression for the area of a mural square.

b. Find the area of a mural square when x 5 5.

c. Find the area of a mural square when x 5 10.

LESSON


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LESSON



1. (29)10 p (29)4 2. 103 p 105 p 10 3. (27) p (27)3 p (27)4

4. (48)7 5. (113)9 6. [(26)6]3

7. (20 p 31)5 8. (125 p 8)8 9. [(216) p 26]6


10. x4 p x p x7 11. [(c 1 5)3]6

12. (24c7)3 13. 2(4c7)3

14. (5x8y5)4 15. (210a7b)5

16. (5p3)3 p 2p4 17. 10m4 p (2m5)6

18. (6x3)2(24x5)3 19. 2(4n4)3(212n5)

20. 1 1 } 3 z4 2 3(3z2)4 21. (210c)3(22c2)5

Find the missing exponent.

22. (5d 4)? 5 625d16 23. (2a4)? p 3a5 5 96a25 24. 5a6 p (10a5)? 5 5000a21

25. Write three expressions that involve products of powers, powers of powers, or powers of products and are equivalent to 24x12.

26. Personal Computers In 2001, there were 103 personal computers in use in Samoa. The number of personal computers in use in Bahrain in 2001 was 10 times the number used in Samoa. The number of personal computers in use in Australia in 2001 was 10 times the number used in Bahrain. How many personal computers were in use in Australia in 2001? Explain how you got your answer.

27. Bananas In 1999, Venezuela produced approximately 106 metric tons of bananas. This is 102 times the number of bananas produced in Samoa in 1999. How many metric tons of bananas were produced in Samoa in 1999? Explain how you got your answer.

28. Storage Cubes You are designing open storage cubes that will hang on the walls of your room. These cubes will be artistic as well as functional. One of the requirements of your design is that the side length of the cube be a perfect square.

a. If you represent the side length of a cube as x2, write an expression for the volume of a wall cube.

b. Find the volume of a wall cube when x 5 5.

c. Find the volume of a wall cube when x 5 10.

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65Algebra 1



LESSON

8.1

Use the power of a power property


a. (33)6 5 33 p 6

5 318

b. [(212)7]6 5 (212)7 p 6

5 (212)42

c. (d5)2 5 d 5 p 2

5 d10

d. [(x 2 3)3]4 5 (x 2 3)3 p 4

5 (x 2 3)12

Use properties of exponents involving products.

VocabularyThe order of magnitude of a quantity can be defi ned as the power of 10 nearest the quantity.

GOAL

Use the product of powers property


a. 26 p 28 5 26 1 8

5 214

b. (23)7 p (23) 5 (23)7 p (23)1

5 (23)711

5 (23)8

c. (27)3 p (27) p (27)4 5 (27)3 p (27)1 p (27)4

5 (27)3 1 1 1 4

5 (27)8

d. m p m5 p m6 5 m1 1 5 1 6

5 m12



1. 83 p 811 2. 6 p 63

3. y3 p y6 p y2 4. (210)2 p (210) p (210)5

EXAMPLE 1

EXAMPLE 2

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Use all three properties

Simplify (23y5)3 • 2y2.

Solution

(23y5)3 p 2y2 5 (23)3 p ( y5)3 p 2y2 Power of a product property

5 227 p y15 p 2y2 Power of a power property

5 254y17 Product of powers property



9. (5 p 18)6 10. 2(11p)3

11. (23x2y5)2 12. (2m)3 p (m4)5

EXAMPLE 4

Use the power of a product property


a. (16 p 21)4 5 164 p 214

b. (6mn)3 5 (6 p m p n)3

5 63m3n3

5 216m3n3

c. (25p)3 5 (25 p p)3

5 (25)3 p p3

5 2125p3

d. 2(2q)4 5 2(2 p q)4

5 2(24 p q4) 5 216q4

EXAMPLE 3


LESSON

8.1



5. (133)10 6. [(28)7]3

7. ( f 8)2 8. [(w 1 8)9]2

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67Algebra 1



LESSON

8.1

In Exercises 1–5, simplify the expression, if possible. Write your answer as a power.

1. ax/3a3

2. (a2b)5y p (ab2)2y

3. (x1/2 p y1/4)2

4. [(xy)(x3y5)]2

5. (x 1 2)2a 1 1 p (x 1 2)3a 2 5


You are constructing a storage bin to hold bird seed. You decide the length, width, and height of the bin will each have a length of a feet.

6. Write an expression that gives the volume of the storage bin in terms of a.

7. Suppose the length and width of the storage bin are doubled. By what factor would the height of the bin have to change so that the volume of the bin remains the same?

8. Suppose the length of the original storage bin is tripled and the width of the storage bin is halved. By what factor would the height of the bin have to change so that the volume of the bin is doubled?

9. Suppose the length, width, and height of the bin each have 1 foot added to them. Write an expression for the volume of the storage bin.

10. An exam has 10 true-false questions and 10 multiple choice questions. Each multiple choice question has 6 possible answers. Assuming a student guesses at each question on the exam, write an exponential expression for the number of different ways it is possible to answer the 20 questions.

11. Using the fact that 6 5 2 p 3, write the expression from Exercise 10 as powers of 2 and 3.

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LESSON


Name the property demonstrated by the example.

1. x5

} x3 5 x523 5 x2 2. 1 a }

b 2

4 5

a4 }

b4 3. 2m8

} m6 5 2m826 5 m2

Fill in the blanks.

4. 38

} 35 5 38 5 5. 1 3 }

4 2 4 5

3 }

4 6.

86 }

84 p 82 5 8

} 8

5 3 5 8


7. 47

} 43 8.

910 }

97 9. 36

} 31

10. (25)4

} (25)3 11.

(27)5

} (27)1 12. 1 1 }

4 2

5

13. 1 5 } 3 2

7 14. 1 2 } 7 2

9 15. 45 p 1 }

42


16. 1 }

y5 p y11 17. z3 p 1 }

z2

18. 1 }

m4 p m8 19. 1 x }

y 2

3

20. 1 a } b 2

13 21. 1 1 }

z 2

9

22. Internet Users The table shows the numbers of Internet users in selected countries in 2001.

Country Albania Jamaica Marshall Islands Romania

Internet Users 104 105 103 106

a. How many times greater is the number of users from Romania than the number of users from the Marshall Islands?

b. How many times greater is the number of users from Albania than the number of users from the Marshall Islands?

c. How many times greater is the number of users from Jamaica than the number of users from the Marshall Islands?

d. How many times greater is the number of users from Romania than the number of users from Albania?

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69Algebra 1



1. 614

} 68 2.

145 }

144 3. (25)7

} (25)2

4. 125 p 123

} 124 5.

817 }

83 p 87 6. 1 3 } 4 2 5

7. 1 2 1 } 5 2 6 8. 38 p 1 }

31 9. 1 1 } 4 2 5 p 413


10. 1 }

y9 p y15 11. z16 p 1 }

z7 12. 1 a } b 2

8

13. 1 2 6 } z 2

3 14. 1 a3 }

2b5 2 4 15. 1 3x4 }

y6 2 5

16. 1 m4 }

5n9 2 3 17. 1 3x7 }

2y12 2 4 18. 1 2m5 }

3n9 2 5

19. Area The area of New Zealand is 104,454 square miles and the area of Saint Kitts and Nevis, islands in the Caribbean Sea, is 104 square miles. Use order of magnitude to estimate how many times greater New Zealand’s area is than Saint Kitts and Nevis’ area.

20. Cell Phone Subscribers The table below shows the approximate number of cell phone subscribers in selected countries in 2001.

Country Algeria Dominican Republic Poland Solomon Islands

Number of subscribers 105 106 107 103

a. How many times greater is the number of cell phone subscribers in Poland than in the Solomon Islands?

b. How many times greater is the number of cell phone subscribers in the Dominican Republic than in the Solomon Islands?

21. Glass Vase You are taking a glass-blowing class and have created a vase in the shape of a sphere. The vase will have a hole in the top so you can put fl owers in it

and it will sit on a stand. The radius of your vase is 21

} 2 inches. Use the formula

V 5 4 }

3 πr3 to write an expression for the volume of your vase.

LESSON


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LESSON



1. 152 p 159

} 156 2.

613 }

64 p 65 3. 1 2 8 }

9 2 7

4. 813 p 1 } 86 5. 1 1 } 5 2 7 p 517 6. 108 p 1 2

1 }

10 2 3


7. 1 2 a }

b 2

7 8. 1 3x6

} y9 2 4 9. 1 m

7 }

2n10 2 6

10. 1 4a2 }

5b3 2 3 11. 1 7x3 }

8y7 2 2 12. 1 3x5 }

10y2 2 3 p 5 } x4

13. 1 }

4x5 p 1 2x2 }

y3 2 5 14. 3y3

} 5 p 1 10x7 }

9y8 2 2 15. 1 2 6 }

x 2 3 p 1 x

4 }

3y7 2 5

16. Find the values of x and y if you know that bx

} by 5 b5 and

bx12 }

b2y 5 b4. Explain how you found your answer.

17. U.S. Postal Service In 2004, the U.S. Postal Service handled 97,926,396 pieces of fi rst class mail and 848,633 pieces of priority mail. Use order of magnitude to estimate how many times greater a volume of fi rst class mail the U.S. Postal Service handled than the volume of priority mail.

18. Large Numbers Very large numbers are named differently in the American and British systems. In the American system, one quintillion is the name for the number 1018. In the British system, one quintillion is the name for the number 1030. How many times larger is one quintillion in the British system than in the American system?

19. Lawn Ornaments You have learned how to make lightweight plant containers using a mixture of peat, sand, and cement. You are going to take these skills and make

lawn ornaments in the shapes of spheres. Use the formula for volume V 5 4 }

3 πr3

to write an expression for the volume of each sphere shown.

ft32

ft34

ft12

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71Algebra 1


Use properties of exponents involving quotients.GOAL

Use the quotient of powers property


a. 713

} 78 5 71328 b.

(21)6

} (21)2 5 (21)622

5 75 5 (21)4

c. 23 p 29

} 24 5

212 }

24 d. 1 }

y7 p y18 5

y18

} y7

5 212 2 4 5 y18 2 7

5 28 5 y11



1. 1215

} 126 2.

(28)20

} (28)16

3. 136 p 138

} 139 4.

1 }

w16 p w21

EXAMPLE 1


LESSON

8.2

Use the power of a quotient property


a. 1 m } n 2

5 5

m5 }

n5

b. 1 3 } p 2

3 5

33 }

p3 5 27

} p3



5. 1 b } c 2

7 6. 1 2

3 }

w 2

4

EXAMPLE 2

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Solve a real world problem Distances The distance from Earth to the nearest galaxy is about 1022 meters. The distance from Earth to the North Star is about 1019 meters. How many times farther from Earth is the nearest galaxy than the North Star?

Solution

Distance to the nearest galaxy

}}} Distance to the North Star

5 1022

} 1019 5 1022219 5 103

The nearest galaxy is about 103 times farther than the North Star.

Exercise for Example 4 9. The distance from the sun to Saturn is 1012 meters. The distance from the sun

to Venus is 1011 meters. How much further is Saturn than Venus from the sun?

EXAMPLE 4

Use the properties of exponents


a. 1 2x3 }

5y2 2 2 5 (2x3)2

} (5y2)2

Power of a quotient property

5 22(x3)2

} 52( y2)2

Power of a product property

5 4x6

} 25y4 Power of a power property

b. 1 3k3 }

4l5 2 2 p l

2 }

6k2 5 32(k3)2

} 42(l5)2

p l3 }

6k2 Power of a quotient property

5 9k6

} 16l10 p

l3 }

6k2 Power of a power property

5 9k6l3

} 96l10k2 Multiply fractions.

5 3k 4

} 32l7 Quotient of a powers property



7. 1 3s5 }

t4 2 3 8.

1 }

3m4 p 1 3m2n }

n2 2 3

EXAMPLE 3


LESSON

8.2

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73Algebra 1


LESSON

8.2 Challenge PracticeFor use with pages 511–517

1. Solve for the value of a if ax

} a2y

5 a3 and x 5 a5y.

2. Solve for the value of b if (b 1 1)2

} b2 5

4(b 2 1)2

} b2 .

3. Solve for the values of x and y if cxcy

} cxy 5 c and cy 2 1 5 c3.

4. Solve for the value of c if 2c 1 4 5 3b2 and b6 5 c3.

5. Solve for the value of y if d3x

} d3y 5 d3x 2 y.


A common formula used to compute annual salary raises is

Salary 5 Starting Salary p (1 1 r)n

where r is the rate of annual raise and n is the number of years of employment.

Example:

Find the salary of an employee who has worked for 2 years and whose starting salary was $25,000 at a company that gives annual raises at a rate of r 5 0.1.

Solution:

New Salary 5 $25,000(1 1 0.1)2

5 $25,000(1.21)

5 $30,250

Suppose a company gives annual raises at a rate of r 5 0.05.

6. What is the salary of an employee whose starting salary was $40,000 per year and has worked at the company for 10 years?



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LESSON


Match the equivalent expressions.

1. 1 2 } 3 2 22

2. 222 p 322 3. 1 3 } 2 2

22

A. 1 }

36 B.

4 }

9 C.

9 }

4

Evaluate the expression.

4. 523 5. 822 6. 225

7. (23)24 8. (29)21 9. 60

10. (25)0 11. 1 1 } 2 2 0 12. 1 1 }

6 2 22

13. 1 3 } 4 2

21 14. 1 2 } 5 2 23

15. 022

Simplify the expression. Write your answer using only positive exponents.

16. x25 17. m29 18. 6y23

19. 8a210 20. (3b)24 21. x3y22

22. x24y3 23. a21b22 24. 2x23y1

25. Finger Thickness Your friend tells you that her fi nger is 1 4 } 3 2

21 inch thick. Evaluate

the expression that represents the thickness of your friend’s fi nger.

26. Floor Tile The minimum recommended width of the space between 6-inch by 6-inch tiles is 222 inch and the maximum recommended width is 221 inch. Simplify the expressions for the minimum and maximum widths of the space between the 6-inch by 6-inch fl oor tiles.

27. Hole Punch Your hole punch makes holes in your paper that have a diameter of 421 inch.

a. Write an expression for the area of one punched hole. Use the formula for the area of a circle A 5 πr2.

b. Your hole punch makes three holes in a page. Write an expression for the total area punched out of one sheet of paper.

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75Algebra 1



1. 325 2. 1023 3. (22)26

4. 50 5. (26)0 6. 1 4 } 3 2 0

7. 1 5 } 8 2 22 8. 1 7 }

4 2 3 9. 025

10. 1022 p 1023 11. 426 p 43 12. 1 }

524


13. x27 14. 6y24 15. (2b)25

16. (23m)24 17. a2b24 18. 3x22y25

19. (4x24y2)23 20. (8mn3)0 21. c23

} d25

22. x2

} y24 23.

x26 }

4y5 24. 1 }

3x23y27

25. Paper A sheet of 67-pound paper has a thickness of 10021 inch.

a. Write and evaluate an expression for the total thickness of 5 sheets of

67-pound paper.

b. Write and evaluate an expression for the total thickness of 23 sheets of

67-pound paper.

26. Frogs A frog egg currently has a radius of 521 centimeter. Write an expression for the volume of the frog egg. Use the formula for the volume of a sphere

V 5 4 }

3 πr3.

27. Metric System The metric system has names for very small lengths.

a. One micrometer is 103 times the length of one nanometer. One nanometer is 1029 meter. Write one micrometer in meters.

b. One femtometer is 103 times the length of one attometer. One attometer is 10218 meter. Write one femtometer in meters.

c. One centimeter is 1010 times the length of one picometer. One picometer is 10212 meter. Write one centimeter in meters.

LESSON


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LESSON



1. 324 p 321 2. 924 p 98 3. (521)4

4. 1 }

1025 5. 526

} 529 6.

8210 }

828

7. 15 1 3 } 5 2 21 8. 32 1 224

} 23 2 9. 4 2 2 p 1 7

} 120 2


10. (4x23y4)22 11. 1 }

9x24y28 12. 1 }

6x4y210

13. 1 }

(4x25)22 14.

8 }

(22d2)24 15.

(2x)24y8

} 2x5y23

16. x26y4

} (23x2)24y21

17. 20x3y24

} (2x24y21)2

18. (4x24y7)2

} 24x26y2

Tell whether the statement is true or false for all nonzero values of a and b. If it is false, give a counterexample.

19. a25

} a26 5

1 }

a 20.

b21 }

a21 5 a }

b 21.

1 }

a21 1 b21 5 a 1 b

22. Guitar The world’s smallest guitar is only 1026 meter tall. An average guitar is about 100 meter tall. How many times taller is an average guitar than the world’s smallest guitar?

23. Knitting Needles A size 1 knitting needle has a diameter of about 421 centimeter and a size 8 knitting needle has a diameter of about 221 centimeter.

a. How many times larger is the diameter of a size 8 needle than the diameter of a size 1 needle?

b. Suppose that each needle is 14 inches long. Write expressions for the approximate volume of each size of knitting needle. Use the formula for the volume of a cylinder V 5 πr2h.

c. How many times larger is the approximate volume of a size 8 needle than the approximate volume of a size 1 needle?

d. Are your approximations in part (b) overestimates or underestimates? Explain your reasoning.

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77Algebra 1


Use zero and negative exponents.GOAL

Use defi nition of zero and negative exponents


a. 423 5 1 }

43 Defi nition of negative exponents

5 1 }

64 Evaluate exponent.

b. 150 5 1 Defi nition of zero exponent

c. 1 3 } 2 2

23 5

1 }

1 3 } 2 2

3 Defi nition of negative exponents

5 1 }

1 27 }

8 2

Evaluate exponents.

5 8 }

27 Simplify.



1. 1 2 1 }

2 2

0 2. (25)24

3. 1 }

622 4. 1 5 } 2 2

23

EXAMPLE 1


LESSON

8.3

Evaluate exponential expressions


a. 1316 p 13214 5 1316 2 14 Product of powers property

5 132 Subtract exponents.

5 169 Evaluate power.

b. [(22)24]2 5 (22)24 p 2 Power of a power property

5 (22)28 Multiply exponents.

5 1 }

(22)8 Defi nition of negative exponents

5 1 }

256 Evaluate power.

EXAMPLE 2

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5. 825

} 825 6.

1 }

922

7. (24)7 p (24)29 8. 102

} 1023


LESSON

8.3

Use properties of exponents


a. (3m22n3)3 5 33 p (m22)3 p (n3)3 Power of a product property

5 27 p m26 p n9 Power of a power property

5 27n9

} m6 Defi nition of negative exponents

b. (25st)2t24

} 210s3t28

5 (25st)2t8

} 210s3t4 Defi nition of negative exponents

5 (25s2t2)t8

} 210s3t4

Power of a product property

5 25s2t10

} 210s3t4 Product of powers property

5 5t6

} 22s

Quotient of powers property



9. (5x2y23z)4

10. 4m22np3

} 12m2n25p

11. (2r2t)23rst4

} 6r6s23

EXAMPLE 3

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79Algebra 1


LESSONS


1. Multi-Step Problem A department store sells plastic cubical containers that can be used to store food.

a. One of the containers has a side length

of 3 3 }

4 inches. Find the container’s

volume by writing the side length as an improper fraction and substituting the length into the formula for the volume of a cube.

b. Identify the property of exponents you used to fi nd the volume in part (a).

2. Multi-Step Problem There are about 103 white corpuscles in 1 cubic millimeter of blood.

a. Copy and complete the table by fi nding the number of white corpuscles for the given amounts of blood (in cubic millimeters).

Blood (cubic millimeters)

Number of white corpuscles

10 ?

100 ?

1000 ?

10,000 ?

100,000 ?

b. A particular sample of blood is 95,000 cubic millimeters. Use order of magnitude to write an expression you can use to fi nd the approximate number of white corpuscles in the sample of blood. Simplify the expression. Verify your answer using the table.

3. Short Response A carrot seed has a mass of about 1024 gram and is 103 times less massive than a sweet corn seed. A student says that a sweet corn seed has a mass of about 1 gram. Is the student correct? Explain.

4. Open-Ended The table shows units of measurement of length and their equivalents in meters.

Name of unit Length (meters)

Terameter 1012

Kilometer 103

Centimeter 1022

Micrometer 1026

a. Use the table to write a conversion problem that can be solved by applying a property of exponents involving products.

b. Use the table to write a conversion problem that can be solved by applying a property of exponents involving quotients.

5. Gridded Response The mass of a grain of sand is about 1023 gram. About how many grains of sand are in a bag of sand that weighs 2.8 grams?

6. Extended Response For an experiment, a scientist dropped about 1024 cubic inch of olive oil into a container of water to see how the oil would spread out over the surface of the water. The scientist found that the oil spread until it covered an area of about 1022 square inch.

a. About how thick was the layer of oil that spread out across the water? Check your answer using unit analysis.

b. The water has a surface area of 102 square inches. If the oil spreads to the same thickness as in part (a), how many cubic inches of olive oil would be needed to cover the entire surface of the water?

c. Explain how you could fi nd the amount of oil needed to cover a container of water with a surface area of 10x square inches.

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LESSON


In Exercises 1–5, a and b are real numbers such that a > 0 and b > 0. Tell whether the statement is always true, sometimes true, or never true. If it is sometimes true, give a pair of values for which it is true and a pair of values for which it is false.

1. a23

} b24 5

b4

} a3

2. (a 1 b)22 5 a22 1 b22

3. (a2 1 b2)1/2 5 a 1 b

4. (a2 1 b2)2 5 a4 1 2a2b2 1 b4

5. a 1 b 5 a2 1 b2

6. Determine which positive values of a make a23 > a24 a true statement.

In Exercises 7–10, evaluate the given expression for the given values of a.

7. [(a 1 1)22]3

} [(a 2 1)23]2

; a 5 0

8. [(a2 1 3)a 2 2]3

}} [(a 2 1)2]4

; a 5 2

9. [(a 1 2)a]a 2 1 2 2a 1 1; a 5 0

10. (aa)22a

} (a 1 1)2a ; a 5 2

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81Algebra 1


FOCUS ON

8.3 Practice For use with pages 525–526


1. 6251/2 2. 16921/2 3. 45/2

4. 923/2 5. 19621/2 6. 493/2

7. 1251/3 8. 34321/3 9. (227)2/3

10. 6424/3 11. (264)1/3 12. 1 }

824/3

13. 253/2 • 251/2 14. 21621/3

} 2162/3 15.

1 }

3622 • 3623/2

16. (264)2/3 • (264)21/3 17. 813/2 4 811/2

} 8121/2 18. (28)5/3 •

(28)1/3

} (28)2/3

19. Reasoning Show that the product of the cube root of a and a can be written as a4/3 using an argument similar to the one given for square roots on page 525.

20. Challenge Evaluate the expression 644/3 • 23 }

163/2 ÷ 256.

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Use fractional exponents.

In lesson 2.7, you learned to write the square root of a number using a radical sign. You can also write a square root of a number using exponents. For a nonnegative number a, Ï}

a 5 a1/2. You can work

with exponents of 1 }

2 and multiples of

1 }

2 just as you work with integer

exponents.

GOAL

Evaluate expressions involving square roots

a. 811/2 5 Ï}

81 b. 10021/2 5 1 }

1001/2

5 9 5 1 }

Ï}

100

5 1 }

10

c. 363/2 5 36(1/2)?3 d. 925/2 5 9(1/2) • (25)

5 (361/2)3 5 (91/2)25

5 1 Ï}

36 2 3 5 1 Ï}

9 2 25

5 63 5 325

5 216 5 1 }

35

5 1 }

243



1. 14421/2 2. 93/2

3. 6423/2 4. 251/2

EXAMPLE 1


FOCUS ON

8.3

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83Algebra 1



FOCUS ON

8.3

Evaluate expressions involving cube roots

a. 81/3 5 3 Ï}

8 b. 6421/3 5 1 }

641/3

5 3 Ï}

23 5 1 }

3 Ï}

64

5 2

5 1 }

4

c. 1254/3 5 125(1/3) • 4 d. 2722/3 5 27(1/3) • (22)

5 (1251/3)4 5 (271/3)22

5 1 3 Ï}

125 2 4 = 1 3 Ï}

27 2 22

5 54 5 322

5 625 5 1 }

32

5 1 }

9



5. 21621/3 6. 274/3 7. 642/3 8. 12524/3

EXAMPLE 2

Use properties of exponents

a. 723/2 • 77/2 5 7(23/2) 1 (7/2) b. 57/3 • 5

} 5

1/3 5 5(7/3) 1 1

} 5

1/3

= 74/2 57/3 • 5

} 5

1/3 5 510/3

} 51/3

= 72 57/3 • 5

} 5

1/3 5 5(10/3) 2 (1/3)

= 49 57/3 • 5

} 5

1/3 5 53

57/3 • 5

} 5

1/3 5 125



9. 1621/2 • 162 10. 273 • 2728/3 11. 87/3 • 84/3

} 82 12. 423/2 •

41/2 }

423/2

EXAMPLE 3

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TI-83 Plus

( 1.1 � 10 ^ (�) 8 ) ( 1.4 �

10 ^ 21 ) ENTER


From the main menu, choose RUN.

( 1.1 � 10 ^ (�) 8 ) ( 1.4 �

10 ^ 21 ) EXE


LESSON

8.4

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85Algebra 1


LESSON


Match the equivalent numbers.

1. 0.004 2. 0.04 3. 4000

A. 40 3 1023 B. 4 3 103 C. 4 3 1023

Write the number in scientifi c notation.

4. 6.4 5. 85.2 6. 0.25

7. 0.104 8. 540 9. 9124.5

10. 0.0095 11. 630,000 12. 0.03

13. 23,960 14. 0.0457 15. 0.000045

Write the number in standard form.

16. 5.2 3 104 17. 9.1 3 108 18. 6.25 3 105

19. 6.05 3 102 20. 8.125 3 106 21. 1.113 3 1010

22. 4.7 3 1023 23. 1.6 3 1028 24. 4.45 3 1026

25. 9.24 3 1024 26. 7.1123 3 1023 27. 2.0123 3 1025

Order the numbers from least to greatest.

28. 21,000; 4.5 3 103; 15,625; 3 3 104

29. 0.0006; 7.8 3 1026; 0.0012; 2.15 3 102

30. 1.765; 1.3 3 1022; 0.0125; 6.15 3 1021

31. Body Makeup The table below shows the amounts (in pounds) of some elements that are in the body of a 150-pound person. Complete the table.

Element Oxygen Chlorine Cobalt Magnesium Sodium Hydrogen

Weight in decimal form

97.5 ? 0.00024 ? 0.165 ?

Weight in scientifi c notation

? 3 3 1021 ? 6 3 1022 ? 1.5 3 101

32. Internet Users In 2003, there were about 5.8078 3 108 people using the Internet in the world and about 1.6575 3 108 of these people were in the United States. What percent of Internet users in 2003 were in the United States? Round your answer to the nearest tenth of a percent.

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1. 10.4 2. 6751 3. 0.54

4. 0.000103 5. 415,620 6. 0.08104

7. 3,412,000 8. 525.5 9. 104.25

10. 0.0000456 11. 0.000000207 12. 23,551


13. 15.8 3 104 14. 3.21 3 108 15. 450.21 3 107

16. 8.1045 3 105 17. 17.22 3 106 18. 1.012 3 102

19. 8.12 3 1024 20. 4.014 3 1027 21. 8.1025 3 1023

22. 3.12056 3 1029 23. 1.211 3 1022 24. 7.00135 3 1025


25. 1.3759 3 104; 14,205; 9.287 3 103; 3.0214 3 104

26. 0.16; 2.5 3 1023; 1.04 3 1023; 0.0985

27. 8.79 3 102; 1146; 1.0085 3 103; 1023

28. 1.2 3 1025; 0.001023; 1.045 3 1023; 0.01036

Evaluate the expression. Write your answer in scientifi c notation.

29. (6 3 108)(5 3 1022) 30. 4.5 3 1025

} 9 3 1022 31. (2 3 1025)5

32. Pixels The images on a computer screen are made up of more than 5000 pixels, or dots, per square inch. How many pixels are on a computer screen that measures 108 square inches? Write your answer in scientifi c notation.

33. Oregon Oregon has an area of approximately 2.52 3 105 square kilometers. In 2000, the population of Oregon was approximately 3.42 3 106 people. How many people were there per square kilometer in Oregon in 2000?

34. Uranus’ Moons The table below shows the masses in kilograms of some of Uranus’ moons.

Moon Miranda Titania Ariel Oberon Umbriel

Mass (kg) 6.6 3 1019 3.52 3 1021 13.5 3 1020 30.1 3 1020 11.7 3 1020

a. Write the moons in order of largest mass to smallest mass.

b. How many times larger is the moon of largest mass than the moon of smallest mass?

LESSON


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87Algebra 1


LESSON



1. 0.0015 2. 30,400 3. 0.0000046

4. 9,120,006 5. 24.5 6. 0.1256

7. 705 8. 100,456 9. 0.000000501


10. 1.325 3 105 11. 7.05123 3 108 12. 8.15 3 1028

13. 9.044 3 1022 14. 5.1 3 103 15. 3.1112 3 1010

16. 8.1101 3 1025 17. 7.7 3 1027 18. 6.25 3 107


19. 758.4; 7.208 3 103; 72,165; 7.914 3 103

20. 1.305 3 1023; 0.000526; 2.018 3 1023; 0.00205

21. 0.000316; 3.28 3 1024; 3.016 3 1024; 0.003028


22. (5.7 3 103)(2.2 3 1026) 23. 6.5 3 1027

} 1.3 3 1023 24. (3 3 1029)5

25. California California has an area of approximately 4.11 3 105 square kilometers. In 2000, the population of California was approximately 3.39 3 107 people. How many people were there per square kilometer in California in 2000?

26. Helium Atom A proton and a neutron each weigh 1.67 3 10224 gram. An electron weighs 9.11 3 10228 gram. One helium atom contains 2 protons, 2 neutrons, and 2 electrons. Find the mass of one helium atom.

27. Saturn’s Moons The table below shows the masses in kilograms of some of Saturn’s moons.

Moon Mimas Calypso Tethys Dione Phoebe

Mass (kg) 3.75 3 1019 4 3 1015 6.27 3 1020 11 3 1020 4 3 1017

a. Write the moons in order of largest mass to smallest mass.

b. How many times larger is the moon of largest mass than the moon of smallest mass?

c. There are approximately 2.2 pounds in one kilogram. Write each mass in pounds.

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Read and write numbers in scientifi c notation.

VocabularyA number is written in scientifi c notation when it is of the form c 3 10n where 1 ≤ c < 10 and n is an integer.

GOAL

Write numbers in scientifi c notation


a. 397,000,000 b. 0.000712

Solution

a. 397,000,000 5 3.97 3 108 Move decimal point 8 places to the left. Exponent is 8.

b. 0.000712 5 7.12 3 1024 Move decimal point 4 places to the right. Exponent is 24.

EXAMPLE 1


LESSON

8.4

Write numbers in standard form


a. 3.02 3 104 b. 9.131 3 1023

Solution

a. 3.02 3 104 5 30,200 Exponent is 4. Move decimal point 4 places to the right.

b. 9.131 3 1023 5 0.009131 Exponent is 23. Move decimal point 3 places to the left.



1. 0.0000079 2. 1,356,000


3. 1.012 3 103 4. 3.7 3 1025

EXAMPLE 2

Order numbers in scientifi c notation

Order 5.2 3 107, 910,000, and 13,200,000 from least to greatest.

Solution

STEP 1 Write each number in scientifi c notation, if necessary.

9,100,000 5 9.1 3 106 13,200,000 5 1.32 3 107

EXAMPLE 3

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89Algebra 1


Compute with numbers in scientifi c notation


a. (3.2 3 103)(4.7 3 104)

5 (3.2 p 4.7) 3 (103 p 104) Commutative and associative properties

5 15.04 3 107 Product of powers property

5 (1.504 3 101) 3 107 Write 15.04 in scientifi c notation.

5 1.504 3 (101 3 107) Associative property

5 1.504 3 108 Product of powers property

b. (3.8 3 1024)2 5 3.82 3 (1024)2 Power of a product property

5 14.44 3 1028 Power of a power property


5 1.444 3 1027 Associative property and product of powers property

c. 2.6 3 106

} 6.5 3 1022 5

2.6 }

6.5 3

106 }

1022 Product rule for fractions

5 0.4 3 108 Quotient of powers property


5 4.0 3 107 Associative property and product of powers property


5. Order 361,000, 2.1 3 106, and 2.8 3 105 from least to greatest.


6. 7.2 3 1023

} 1.8 3 106 7. (9.1 3 107)(2.3 3 1025)

8. (2.9 3 106)2

EXAMPLE 4

STEP 2 Order the numbers. First order the numbers with different powers of 10. Then order the numbers with the same power of 10.

Because 106 < 107, you know that 9.1 3 106 is less than both 1.32 3 107 and 5.2 3 107. Because 1.32 < 5.2, you know that 1.32 3 107 is less than 5.2 3 107.

So, 9.1 3 106 < 1.32 3 107 < 5.2 3 107.

STEP 3 Write the original numbers in order from least to greatest.

9,100,000, 13,200,000, 5.2 3 107


LESSON

8.4

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LESSON


In Exercises 1–5, evaluate the expression without using a calculator. Write your answer in scientifi c notation.

1. (1.2 3 1023) p (1.2 3 105)

2. (2.5 3 106) p (1 3 108)

}} 5 3 103

3. (3 3 106) 1 (5 3 105)

4. 6(4 3 1022) 1 4

5. 2.2(2 3 104) 1 1.2(2 3 105)

}}} (7.1 3 1022) 1 (2.13 3 1021)

6. The population of Earth in the year 2000 was estimated to be 6 3 109 people. The population of the U.S. in the year 2000 was estimated to be 3 3 108 people. What proportion of the world’s population in the year 2000 resided in the U.S.?

7. The population of the People’s Republic of China in the year 2000 was estimated to be 1.3 3 109 people. The population of the Republic of China (Taiwan) in the year 2000 was estimated to be 2.6 3 107 people. What was the proportion of the popula-tion of the Republic of China to the People’s Republic of China?

8. In the year 2002 there were approximately 9.6 3 105 dogs registered with the American Kennel Club (AKC) and 2.3 3 104 of those dogs were Rottweilers. What proportion of the dogs registered to the AKC in the year 2002 were Rottweilers?

In Exercises 9 and 10, convert the decimal expressions to scientifi c notation and then simplify the expression. Write your answer in decimal form.

9. (0.0000032) p (2000000)

}} (8 3 103) p (8 3 1024)

10. (0.0000012) 2 (0.000002)

}} 16

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91Algebra 1


EXCEL

Select cell A1.

Years since 1984, t TAB Value, C (dollars) ENTER

Select cell A2.

0 TAB 11000 ENTER

Select cell A3.

5A2 + 1 TAB 5B2*1.069 ENTER

Select cells A3–A22. From the Edit menu, choose Fill. From the Fill submenu, choose Down. Select cells B2 and B3. From the Format menu, choose Cells. Select the Number tab. In the Category list, choose Number. For Decimal places, use the up and down arrows to set the number of decimal places to 2. Click OK. Select cells B3–B22. From the Edit menu, choose Fill. From the Fill submenu, choose Down.

Spreadsheet Activity KeystrokesFor use with pages 544 and 545

CHAPTER

8.5

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LESSON


Write a rule for the function.

1. x 21 0 1 2 3

y 1 } 3 1 3 9 27

2. x 21 0 1 2 3

y 1 } 5 1 5 25 125

Match the function with its graph.

3. y 5 5x 4. y 5 (2.5)x 5. y 5 (1.5)x

A.

x

y

1 321

3

5

23 21

B.

x

y

1 321

3

5

23 21

C.

x

y

1 321

3

5

23 21

Graph the function and identify its domain and range.

6. y 5 4x 7. y 5 10x 8. y 5 6x

x

y

1

3

5

1212321

3

x

y

2

6

10

1212322

3

x

y

1

3

5

1212321

3

9. y 5 (3.5)x 10. y 5 (1.4)x 11. y 5 (2.2)x

x

y

1

3

5

1212321

3

x

y

1

3

5

1212321

3

x

y

1

3

5

1212321

3

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93Algebra 1


12. y 5 1 7 } 3 2

x 13. y 5 1 5 }

2 2

x 14. y 5 1 7 }

4 2

x

x

y

1

3

5

1212321

3

x

y

1

3

5

1212321

3

x

y

1

3

5

1212321

3

Graph the function. Compare the graph with the graph of y 5 4x.

15. y 5 24x 16. y 5 3 p 4x 17. y 5 1 }

4 p 4x

x

y

1

3

123 2121

23

3

x

y

1

3

1212321

23

3

x

y

1

3

1212321

23

3

In the growth model, identify the growth rate, the growth factor, and the initial amount.

18. y 5 3(1 1 0.05)t 19. y 5 2(1 1 0.25)t 20. y 5 0.1(1.75)t

21. Investments You deposit $200 in a savings account that earns 3% interest compounded yearly. Find the balance in the account after the given amounts of time.

a. 1 year

b. 2 years

c. 5 years

22. Grade Point Average From Chad’s freshman year to his senior year, his grade point average has increased by approximately the same percentage each year. Chad’s grade point average in year t can be modeled by

y 5 2 1 5 } 4 2 t

where t 5 0 corresponds to Chad’s freshman year. Complete the table showing Chad’s grade point average throughout his high school career.

Year, t 0 1 2 3

Grade point average ? ? ? ?

LESSON


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LESSON



1. x 22 21 0 1 2

y 1 } 121

1 }

11 1 11 121

2. x 21 0 1 2 3

y 1 } 8

1 }

4

1 }

2 1 2


3. y 5 12x 4. y 5 (1.75)x 5. y 5 (3.1)x

x

y

2

6

10

1212322

3

x

3

1212321

23

3

y

x

3

1212321

23

3

y

6. y 5 1 9 } 2 2

x 7. y 5 25x 8. y 5 2 1 3 }

2 2

x

x

1

3

5

1212321

3

y

x

1

12321

23

25

3

y

x

1

1

23

25

3

y

9. y 5 5 p 2x 10. y 5 2 p 1 4 } 3 2

x 11. y 5 23 p 2x

x

6

2

10

123 2122

3

y

x

1

3

5

1212321

3

y

x

1

12121

23

25

3

y

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95Algebra 1



12. y 5 2 p 6x 13. y 5 26x 14. y 5 1 }

2 p 6x

x

y

2

6

10

1212322

3

x

1

12321

23

25

3

y

x

1

3

5

1212321

3

y

15. y 5 23 p 6x 16. y 5 2 1 }

4 p 6 x 17. y 5 2

3 }

2 p 6x

x

3

1212323

29

215

3

y

x

1

1212321

23

25

3

y

x

2

1212322

26

210

3

y

18. Investments You deposit $500 in a savings account that earns 2.5% interest compounded yearly. Find the balance in the account after the given amounts of time.

a. 1 year

b. 5 years

c. 20 years

19. College Tuition From 1995 to 2005, the tuition at a

t

y

7 8 9 1000

4,000

8,000

12,000

16,000

1 2 3 4 5 6Years since 1995

Tu

itio

n (

do

llars

)

(0, 8000)

college increased by about 7% per year. Use the graph to write an exponential growth function that models the tuition over time.

20. Profi t A business had $10,000 profi t in 2000. Then the profi t increased by 8% each year for the next 10 years.

a. Write a function that models the profi t in dollars over time.

b. Use the function to predict the profi t in 2009.

LESSON


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LESSON



1. x 22 21 0 1 2

y 2 1 }

16 2

1 } 4 21 24 216

2. x 21 0 1 2 3

y 5 } 2 5 10 20 40


3. y 5 15x 4. y 5 (2.25)x 5. y 5 (5.2)x

x

y

3

9

15

12123

23 3

x

y

1

3

12121

23

23 3

x

y

1

3

5

12121

23 3

6. y 5 1 9 } 8 2

x 7. y 5 27x 8. y 5 2 1 5 }

2 2

x

x

y

1

3

12121

23

23 3

x

y

12121

23

25

27

23 3

x

y

1

3

12121

23

23 3

9. y 5 3 p 6x 10. y 5 4 p 1 3 } 2 2

x 11. y 5 22 p 4x

x

y

3

9

15

2312123 3

x

y

2

6

10

12122

23 3

x

y2

12122

26

210

23 3

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97Algebra 1



12. y 5 2 p 5x 13. y 5 25x 14. y 5 1 }

2 p 5x

x

y

2

6

10

12122

23 3

x

y

1

12121

23

25

23 3

x

y

1

3

12121

23

23 3

15. y 5 23 p 5x 16. y 5 2 1 }

2 p 5x 17. y 5 2

3 }

4 p 5x

x

y3

12123

29

215

23 3

x

y

1

3

12121

23

23 3

x

y

1

3

12121

23

23 3

18. Investments You deposit $375 in a savings account that earns 2.75% interest compounded yearly. Find the interest earned by the account after the given amounts of time. Explain how you got your answers.

a. 1 year

b. 5 years

c. 20 years

19. Population A town had a population of 65,000 in 2000. Then the population increased by 2.5% each year for the next 5 years.

a. Write a function that models the population over time.

b. Use the function to predict the population in 2004.

20. Internet Users The number of students who have applied for Internet privileges at school has doubled each month.

a. What is the percent of increase each month?

b. Ten students had applied for Internet privileges initially. Write a function that models the number of students applying for Internet privileges over time.

c. How many students will have applied for Internet privileges in 4 months?

LESSON


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Write and graph exponential growth models.

VocabularyAn exponential function is a function of the form y 5 abx where a Þ 0, b > 0, and b Þ 1.

When a > 0 and b > 1, the function y 5 abx represents exponential growth.

Compound interest is interest earned on both an initial investment and on previously earned interest.

GOAL

Write a function rule


Solution 11 11 11 11

x 22 21 0 1 2

y 2 } 5 2 10 50 250

3 5 3 5 3 5 3 5

STEP 1 Tell whether the function is exponential. Here, the y-values are multiplied by 5 for each increase of 1 in x, so the table represents an exponential function of the form y 5 a p bx where b 5 5.

STEP 2 Find the value of a by fi nding the value of y when x 5 0. When x 5 0, y 5 ab0 5 a p 1 5 a. The value of y when x 5 0 is 10, so a 5 10.

STEP 3 Write the function rule. A rule for the function is y 5 10 p 5x.

EXAMPLE 1


LESSON

8.5

Graph an exponential function

Graph the function y 5 5 • 3x. Identify its domain and range.

Solution x 22 21 0 1 2

y 5 }

9

5 }

3 5 15 45

STEP 1 Make a table by choosing a few values for x and fi nding the values of y. The domain is all real numbers.

STEP 2 Plot the points.

2123 1 3

15

25

35

45

x

y

STEP 3 Draw a smooth curve through the points. From either the table or the graph, you can see that the range is all positive real numbers.

EXAMPLE 2

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99Algebra 1



LESSON

8.5

Compare graphs of exponential functions

Graph y 5 2 1 } 2 p 4x and y 5 2 • 4x. Compare each graph with the graph of

y 5 4x.

Solution

To graph each function, make a table of values, plot the points, and draw a smooth curve through the points.

x y 5 4x y 5 2 1 } 2 p 4x y 5 2 p 4x

22 1 }

16 2

1 } 32

1 }

8

21 1 }

4 2

1 }

8

1 }

2

0 1 2 1 }

2 2

1 4 22 8

2 16 28 32

Because the y-values for y 5 2 1 } 2 p 4x are 2 1 }

2 times the corresponding y-values for

y 5 4x, the graph of y 5 2 1 } 2 p 4x is a vertical shrink and a refl ection in the x-axis of

the graph of y 5 4x.

Because the y-values for y 5 2 p 4x are 2 times the corresponding y-values for y 5 4x, the graph of y 5 2 p 4x is a vertical stretch of the graph of y 5 4x.

Exercises for Examples 1, 2, and 3 1. Write a rule for the function.

x 22 21 0 1 2

y 1 3 9 27 81

2. Graph y 5 4 p 3x and identify its domain and range.

3. Graph y 5 25 p 6x. Compare the graph with the graph of y 5 6x.

EXAMPLE 3

123 321

23

3

x

y

y 5 4x

y 5 2(4)x

y 5 2 (4)x12

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Savings You put $125 in a savings account that earns 3% annual interest compounded yearly. You do not make any deposits or withdrawals. How much will your investment be worth in 4 years?


What do you know? The amount deposited, the annual interest, and the years

What do you want to fi nd out? How much is in the account after 4 years?

STEP 2 Make a Plan Use what you know to write and solve an exponential growth model.

STEP 3 Solve the Problem Write and solve an exponential growth model.

y 5 a(1 1 r)t Write exponential growth model.

5 125(1 1 0.03)4 Substitute 125 for a, 0.03 for r, and 4 for t.

5 125(1.03)4 Simplify.

ø 140.69 Use a calculator.

You will have $140.69 in 4 years.

STEP 4 Look Back Use the simple interest formula to estimate the amount of interest earned.

I 5 Prt Write simple interest formula.

5 (125)(0.03)(4) 5 15 Substitute 125 for P, 0.03 for r, and 4 for t.

The compounded interest is slightly more than $15. So, the answer is correct.

PROBLEM

1. Internet In 1996, consumer spending per person per year for the Internet was $13.24. The spending increased by about 36% per person per year from 1996 to 2007. Predict the spending per person per year on the Internet in 2007.

2. Error Analysis Describe and correct the error made in solving Exercise 1.

y 5 13.24(0.36)x

5 13.24(0.36)11

ø 0.10

The consumer spending per person per year for the Internet increased by $.10 from 1996 to 2007. The spending in 2007 was $13.34.

3. Population In 1960, the population of the United States was 179,323,175. By 2000, the population was 281,423,231. Write an exponential model for the U.S. population from 1960 to 2000. Use the model to predict the U.S. population in 2010.

4. Pond When a stone is dropped into a pond, the initial 1-foot radius of the ripple increases at a rate of about 50% per second. Find the radius of the initial ripple 5 seconds after the stone is dropped.

5. What If? Suppose a larger stone is dropped into the pond and the initial 1-foot radius of the ripple increases at a rate of about 75% per second. Find the radius of the initial ripple 5 seconds after the stone is dropped.

PRACTICE

LESSON


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101Algebra 1


LESSON


In Exercises 1–5, fi nd an exponential function of the form f(x) 5 abx that passes through the given points.

1. (0, 1), (2, 9), (4, 81)

2. (0, 3), (1, 6), 1 21, 3 }

2 2

3. 1 0, 1 }

2 2 , 1 21,

1 }

10 2 , 1 3,

125 }

2 2

4. 1 0, 1 }

9 2 , 1 1,

1 }

3 2 , (2, 1)

5. 1 0, 3 }

2 2 , (1, 3), (3, 12)

In Exercises 6–10, use the properties of exponents to write both functions so that each has the same constant raised to a power, then determine which function has the greater value when x 5 1.

6. f (x) 5 3 p 28x

g(x) 5 3 p 46x

7. f (x) 5 2 p 42x 2 1

g(x) 5 5 p 16x 1 2

8. f (x) 5 25x 1 1

g(x) 5 1 1 } 5 2 22x

9. f (x) 5 6 p 16x

g(x) 5 1 } 2 p 64x

10. f (x) 5 1000 p (2.25)5x

g(x) 5 2000 p (1.5)3x

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LESSON


Tell whether the table represents an exponential function. If so, write a rule for the function.

1. x 22 21 0 1 2

y 100 10 1 1 }

10

1 }

100

2. x 21 0 1 2 3

y 25 23 21 1 3


3. y 5 1 1 } 2 2

x 4. y 5 2x 5. y 5 2 1 1 }

2 2

x

A.

x

y3

12121

23

23 3

B.

x

y3

21

23

23 3

1

C.

x

y3

12121

23

23 3


6. y 5 1 1 } 6 2

x 7. y 5 1 2 } 5 2

x 8. y 5 1 3 }

8 2

x

x

y

1

3

5

12121

23 3

x

y

1

3

5

12121

23 3

x

y

1

3

5

12121

23 3

9. y 5 (0.4)x 10. y 5 (0.7)x 11. y 5 (0.2)x

x

y

1

3

5

12121

23 3

x

y

1

3

5

12121

23 3

x

y

1

3

5

12121

23 3

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103Algebra 1


Graph the function. Compare the graph with the graph of y 5 1 1 } 3 2 x.

12. y 5 2 p 1 1 } 3 2

x 13. y 5 2 1 1 }

3 2

x 14. y 5

1 }

3 p 1 1 }

3 2

x

x

y

1

3

5

12121

23 3

x

y

1

3

12121

23

23 3

x

y

1

3

12121

23

23 3

Tell whether the graph represents exponential growth or exponential decay.

15.

x

y3

12121

23

23 3

16.

x

y

3

1

12121

23

17.

x

y

1

12121

3

18.

x

y

3

25 7522521

275

19.

x

y

3

1 32121

23

20.

x

y

3

25 7522521

275

21. Car Value You buy a used car for $12,000. It depreciates at the rate of 15% per year. Find the value of the car after the given number of years.

a. 1 year

b. 3 years

c. 5 years

22. Declining Employment A business had 4000 employees in 2000. Each year for the next 5 years, the number of employees decreased by 2%.

a. Write a function that models the number of employees over time.

b. Use the function to predict the number of employees in 2004. Round to the nearest whole number.

LESSON


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LESSON



1. x 22 21 0 1 2

y 25 5 1 1 } 5

1 }

25

2. x 21 0 1 2 3

y 1 4 7 10 13


3. y 5 1 1 } 12

2 x

4. y 5 1 7 } 8 2

x 5. y 5 2 1 1 } 8 2

x

x

6

10

1212322

3

y

x

3

5

2222621

6

y

x12123 3

y2

6. y 5 2 p 1 1 } 5 2 x

7. y 5 2 p (0.25)x 8. y 5 20.5 p (0.3)x

x

6

2

10

1212322

3

y

x

1

5

7

12123 3

y

x

0.5

3212320.5

y

Graph the function. Compare the graph with the graph of y 5 1 1 } 8 2

x.

9. y 5 2 p 1 1 } 8 2

x 10. y 5 2 1 1 } 8 2

x 11. y 5

1 }

4 p 1 1 }

8 2

x

x

20

1212324

3

y

x

2

12123 3

y

x

3

5

1212321

3

y

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105Algebra 1


Decide whether the given statement is always, sometimes, or never true. Justify your answer.

12. For a positive real number b other than 1, the graphs of y 5 bx and y 5 2bx are refl ections in the y-axis.

13. For a positive real number b other than 1, the graphs of y 5 bx and y 5 1 1 } b 2

x

intersect.

14. For a nonzero number a and a positive real number b, the graphs of y 5 abx and

y 5 1 } a • bx are not identical.

Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function.

15.

x

y

1 321

1

5

23 21

(0, 3)(21, 4)

16.

x

y

2 624

4

12

26 22

(0, 2)

(26, 17) 17.

x

y

121

1

5

23 21

(0, 4)

(22, 1)

18. Computer Value You buy a computer for $3000. It depreciates at the rate of 20% per year. Find the value of the computer after the given number of years.

a. 1 year

b. 3 years

c. 5 years

19. Unemployment Rate In 2000, the unemployment rate

t

y

7 8 9 10 1100

4

5

6

7

1 2 3 4 5 6Months since January

Un

em

plo

ym

en

t

rate

(p

erc

en

t) (0, 7)of a city decreased by approximately 2.1% each month. In January, the unemployment rate was 7%.

a. Use the graph at the right to write a function that models the unemployment rate of the city over time.

b. What was the unemployment rate in December?

20. Indoor Water Park An indoor water park had a declining attendance from 2000 to 2005. The attendance in 2000 was 18,000. Each year for the next 5 years, the attendance decreased by 5.5%.

a. Write a function that models the attendance since 2000.

b. What was the attendance in 2005?

LESSON


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LESSON



1. x –2 –1 0 1 2

y 100 } 81

10

} 9 1

9 }

10

81 }

100

2. x 22 21 0 1 2

y 2 17

} 2 2 33

} 4 28 2 31

} 4 2 15

} 2


3. y 5 1 1 } 15

2 x 4. y 5 1 4 }

9 2

x 5. y 5 2 1 1 }

4 2

x

x

y

3

9

15

12123

23 3

x

y

1

3

12121

23

23 3

x

y

1

3

12121

23

23 3

6. y 5 4 p 1 1 } 9 2

x 7. y 5 3 p (0.25)x 8. y 5 20.2 p (0.3)x

x

y

6

18

30

12126

23 3

x

y

2

6

10

12122

23 3

x

y0.1

12120.1

20.3

20.5

23 3

Graph the function. Compare the graph with the graph of y 5 1 1 } 5 2

x.

9. y 5 5 p 1 1 } 5 2 x 10. y 5 2 1 1 } 5 2

x 11. y 5 2

1 } 5 p 1 1 } 5 2

x

x

y

5

15

25

12125

23 3

x

y

1

12121

23

25

23 3

x

y

1

3

12121

23

23 3

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107Algebra 1


Decide whether the given statement is always, sometimes, or never true. Justify your answer.

12. For a positive real number b other than 1, the graphs of y 5 bx and y 5 2bx are refl ections in the x-axis.

13. For a positive real number b other than 1, the graphs of y 5 bx and y 5 1 1 } b 2

x

have the same range.

14. For a positive real number b, the function y 5 2bx is an exponential growth function.


15.

x

y

1

3

5

12121

3 5

(0, 5)

(2, 0.8)

16.

x

y

1

5

12121

2325

(0, 3)(21, 2.4)

17.

x

y

2

10

12122

23 3

(21, 7.5) (1, 4.8)

18. Truck Value You buy a used truck for $15,000. It depreciates at a rate of 18% per year. Find how much the value of the truck depreciated after the given number of years have passed.

a. 1 year

b. 3 years

c. 5 years

19. Sleeping Behavior On average, as people grow older,

t

y

7 800

1

2

3

4

5

6

7

8

1 2 3 4 5 6Years since 2000

Ho

urs

of

sle

ep

(0, 8)they sleep fewer hours during the night. The amount of sleep that your great-aunt gets has decreased by 1.8% since 2000.

a. Use the graph at the right to write a function that models the number of hours your great-aunt sleeps each night over time.

b. How many hours of sleep did your aunt average a night in 2003?

20. Investment You invested $2000 into the stock market in 2000. Your investment increased 6% each year for fi ve years. Over the next fi ve years your investment decreased in value of 6% each year. Did you have the $2000 again at the end of ten years? Explain your reasoning.

LESSON


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Write and graph exponential decay functions.

VocabularyWhen a > 0 and 0 < b < 1, the function y 5 a p bx represents exponential decay.

GOAL

Write a function rule


11 11 11

x 21 0 1 2

y 2 1 } 3 22 212 272

3 6 3 6 3 6

The y-values are multiplied by 6 for each increase of 1 in x, so the table represents an exponential function of the form y 5 abx with b 5 6.

The value of y when x 5 0 is 22, so a 5 22.The table represents the exponential function y 5 22 p 6x.

Exercise for Example 1 1. Tell whether the table represents an

x 22 21 0 1 2

y 1 3 9 27 81

exponential function. If so, write a rule for the function.

EXAMPLE 1


LESSON

8.6

EXAMPLE 2 Graph an exponential function

Graph the function y 5 1 1 } 10

2 x. Identify its domain and range.

Solution

STEP 1 Make a table of values. The domain is

2123 1 3

1

5

7

9

x

y 5 110)x(

y

all real numbers.

x 21 0 1 2

y 10 1 1 }

10

1 }

100


STEP 3 Draw a smooth curve through the points. From either the table or the graph, you can see the range is all positive real numbers.

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109Algebra 1


Classify and write rules for functions


a.

2123 1 3

3

9

15

x

y

(1, 5)

(0, 15)

b.

2123 1 3

1

5

7

x

y

(0, 4)

(1, 8)

Solution

a. The graph represents exponential decay ( y 5 abx where 0 < b < 1). The y-intercept is 15, so a 5 15. Find the value of b by using the point (1, 5) and a 5 15.

y 5 abx Write function.

5 5 15 p b1 Substitute.

1 }

3 5 b Solve.

A function rule is y 5 15 p 1 1 } 3 2

x.

b. The graph represents exponential growth (y 5 abx where b > 1). The y-intercept is 4, so a 5 4. Find the value of b by using the point (1, 8) and a 5 4.

y 5 abx Write function.

8 5 4 p b1 Substitute.

2 5 b Solve.

A function rule is y 5 4 p 2x.

Exercises for Examples 2 and 3 2. Graph y 5 (0.7)x and identify its domain and range.

3. The graph of an exponential function passes through the points (0, 4) and 1 1, 1 }

2 2 .

Graph the function. Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function.

EXAMPLE 3


LESSON

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LESSONS


1. Multi-Step Problem The radius of Jupiter is about 71,492 kilometers. The radius of Callisto, one of Jupiter’s moons, is about 2400 kilometers.

a. Write each radius in scientifi c notation.

b. The surface area S of a sphere with radius r is given by S 5 4πr2. Assume Jupiter and Callisto are spheres. Find their surface areas. Write your answers in scientifi c notation.

c. What is the ratio of the surface area of Jupiter to the surface area of Callisto? What does the ratio tell you?

2. Multi-Step Problem The half-life of a pesticide is the time it takes for the pesticide to reduce to half of its original amount in soil. A certain pesticide has a half-life of about 45 days.

a. A yard is sprayed with 20 ounces of pesticide. Write a function that models the amount of the pesticide in the soil over time.

b. How much of the 20 ounces sprayed will be in the soil after 180 days?

3. Multi-Step Problem The graph shows the number of mobile phone subscribers in the world over time.

x

y

700

400

800

1200

1600

1 2 3 4 5 6Years since 1995

Nu

mb

er

of

su

bscri

bers

(milli

on

s)

(1, 145)(0, 91)

a. Does the graph represent exponential growth or exponential decay?

b. Write a function that models the number of mobile phone subscribers over time.

c. How many mobile phone subscribers were there in 1998?

4. Short Response In 2004, a family bought a boat for $7000. The boat depreciates (loses value) at a rate of 15% annually. In 2006, a person offers to buy the boat for $5500. Should the family sell the boat? Explain.

5. Gridded Response A new television costs $400. The value of the television decreases over time. The value V in dollars of the television after t years is given by the function V 5 400(0.86)t. What is the decay rate, written as a decimal, of the value of the television?

6. Open-Ended Write two numbers in scientifi c notation whose product is 5.4 3 107. Write two numbers in scientifi c notation whose quotient is 5.4 3 107.

7. Short Response The graph shows the value of a car over time.

14 16 1800

4,000

8,000

12,000

16,000

20,000

2 4 6 8 10 12Time (years)

Valu

e (

do

llars

) (0, 20,000)

(1, 18,800)

x

y

a. Write an equation for the function whose graph is shown.

b. At what rate is the car losing value? Explain.

8. Extended Response A skier is saving money to buy a new pair of ski boots. The skier puts $200 in a saving account that pays 4% annual interest compounded yearly.

a. Write a function that models the amount of money in the account over time.

b. Graph the function.

c. The skier wants a pair of ski boots that cost $234.99. Will there be enough in the account after 3 years to buy the ski boots? Explain.

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111Algebra 1


LESSON


In Exercises 1–5, fi nd an exponential function of the form f(x) 5 abx that passes through the given points.

1. 1 1, 3 }

2 2 , 1 2,

3 }

4 2 , 1 4,

3 }

16 2

2. 1 1, 2 }

3 2 , 1 3,

2 }

27 2 , 1 5,

2 }

243 2

3. (0, 4), 1 2, 36

} 25

2 , 1 3, 108

} 125

2

4. (1, 1), 1 2, 2 } 5 2 , 1 3,

4 }

25 2

5. 1 0, 7 }

3 2 , (1, 1), 1 2,

3 } 7 2

In Exercises 6–9, use the properties of exponents to write both functions so that each has the same constant raised to a power, then determine which function has the greater value when x 5 1.

6. f (x) 5 3 p 1 1 } 9 2

5x

g(x) 5 4 p 1 1 } 3 2 6x

7. f (x) 5 2 p 1 1 } 4 2

2x 2 1

g(x) 5 5 p 1 1 } 16

2 x 1 2

8. f (x) 5 1 1 } 5 2

x 1 1

g(x) 5 1 1 } 25

2 2x

9. f (x) 5 6 p 1 3 } 4 2

2x

g(x) 5 1 } 2 p 1 18

} 32

2 x

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FOCUS ON


Tell whether the sequence is arithmetic or geometric. Then graph the sequence.

1. 2, 4, 6, 8, ... 2. 64, 232, 16, 28, ... 3. 21, 23, 25, 27, ...

Write a rule for the nth term of the geometric sequence. Then graph the sequence, and identify the domain and the range.

4. 64, 16, 4, 1, ... 5. 1, 26, 36, 2216, ... 6. 3, 6, 12, 24, ...

7. 1, 1 }

4 ,

1 }

16 ,

1 }

64 , ... 8. 21,

1 }

2 , 2

1 } 4 ,

1 }

8 , ... 9. 281, 227, 29, 23, ...

10. Challenge A certain type of pea-plant germinates 6 seeds per generation. Write a rule for the nth term of the sequence in the table. Then graph the fi rst six terms of the sequence.

Number of generations, n 1 2 3 4

Number of new pea-plants, an 1 6 36 216

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113Algebra 1


Identify, graph, and write geometric sequences.

VocabularyIn a geometric sequence, the ratio of any term to the previous term is constant.

This constant ratio is called the common ratio and is denoted by r.

The General Rule for a Geometric Sequence is given by an 5 a1rn 2 1.

GOAL

Identify a geometric sequence

Tell whether the sequence is arithmetic or geometric. Then write the next term of the sequence.

a. 4, 8, 12, 16, 20, ... b. 486, 162, 54, 18, 6, ...

Solution

a. The fi rst term is a1 5 4. Find the ratios of consecutive terms:

a2

} a1 5

8 }

4 = 2

a3 } a2 5

12 }

8 5 1

1 }

2

a4 } a3 5

16 }

12 5 1

1 }

3

a5 } a4 5

20 }

16 5 1

1 }

4

Because ratios are not constant, the sequence is not geometric. To see if the sequence is arithmetic, fi nd the differences of consecutive terms.

a2 2 a1 5 8 2 4 = 4 a3 2 a2 5 12 2 8 = 4

a4 2 a3 5 16 2 12 = 4 a5 2 a4 5 20 2 16 = 4

The common difference is 4, so the sequence is arithmetic. The next term of the sequence is a6 5 a5 1 4 5 24.

b. The fi rst term is a1 5 486. Find the ratios of consecutive terms:

a2

} a1 5

162 }

486 5

1 }

3

a3 } a2 5

54 }

162 5

1 }

3

a4 } a3 5

18 }

54 5

1 }

3

a5 } a4 5

6 }

18 5

1 }

3

Because the ratios are constant, the sequence is geometric. The common ratio

is 1 }

3 . The next term of the sequence is a6 5 a5 •

1 }

3 5 2.


Tell whether the sequence is arithmetic or geometric. Then write the next term of the sequence.

1. 4, 20, 100, 500, ... 2. 0.5, 1.25, 2, 2.75, ...

3. 32, 16, 8, 4, ...

EXAMPLE 1


FOCUS ON

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Graph a geometric sequence To graph the sequence from part (b) of Example 1, let each term’s position number in the sequence be the x-value. The term is the corresponding y-value. Then make and plot the points.

Position, x 1 2 3 4 5

Term, y 486 162 54 18 6

x

y

O 1 2 3 4 5

240

320

400

480

80

160


Graph the sequence.

4. 4, 20, 100, 500, ... 5. 0.5, 1.25, 2, 2.75, ...

6. 32, 16, 8, 4, ...

EXAMPLE 2


FOCUS ON

8.6

Write a rule for a geometric sequence

Write a rule for the nth term of the geometric sequence in Example 1. Then fi nd a10.

Solution

To write a rule for the nth term of the sequence, substitute the values for a1 and r in the

general rule an 5 a1rn 2 1. Because a1 5 486 and r 5 1 }

3 , an 5 486 • 1 1 }

3 2 n 2 1. The 10th

term of the sequence is a10 5 486 • 1 1 } 3 2 10 2 1

= 2 }

81 .


Write a rule for the nth term of the geometric sequence. Then fi nd a10.

7. 1, 22, 4, 28, ... 8. 1, 1 }

3 ,

1 }

9 ,

1 }

27 , ...

9. 10, 20, 40, 80, ...

EXAMPLE 3

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115Algebra 1


Math History

Solve the following exercises. Find the answer at the right of the page. Place the letter associated with the correct answer on the line with the exercise number to answer the following question.

Who was the fi rst mathematician to use exponential notation the way we use it today?

Exercises Answers

1. Simplify: x3 p x5 (S) 1.495 3 1011 (B) 0

2. Write in scientifi c notation: 31,009,100 (R) x8 (N) 1

3. Simplify: (8x4y3)0 (E) 0.055 (D) 0.891

4. Simplify: 1 x2 } y 2 3 (L) 1.055 (E)

x6 }

y3

5. What is the decay factor in the model y 5 35(0.891)t? (A) 3.0 3 1025 (P) x15

6. Simplify: 2(3x2)4(2x)2 (T) 2 500

} x4y6

(C) 2x7y

7. Write in standard form: 9.87 3 1025 (K) 987,000

8. Simplify: 16x22y4

} (2x23y)3

(R) y14

} 16x6

9. Evaluate: 7.5 3 1023

} 2.5 3 102 (U)

8x7 } y

10. Simplify: (2x23y4)2 p (4y22)23 (E) 3.10091 3 107

11. Simplify: 1 2 5 }

x2 2 3 p 1 2x

} y3 2

2 (F)

x6 }

y

12. What is the growth rate in the model y 5 17(1.055)t? (S) 0.0000987

13. Evaluate: (6.5 3 106)(2.3 3 104) (E) 2324x10

1 2 3 4

5 6 7 8 9 10 11 12 13

CHAPTER


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AP

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R R

EV

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117Algebra 1


Chapter Overview �� !


9.1: Add and Subtract Polynomials

Add and subtract polynomials.

• Baseball Attendance

• Backpacking and Camping

• Car Costs

9.2: Multiply Polynomials Multiply polynomials. • Skateboarding

• Swimming Pool

• Sound Recordings

9.3: Find Special Products of Polynomials

Use special product patterns to multiply polynomials.

• Border Collies

• Pea Plants

• Football Statistics

9.4: Solve Polynomial Equations in Factored Form

Solve polynomial equations.

• Armadillo

• Spittlebug • Soccer

9.5: Factor x2 1 bx 1 c Factor trinomials of the form x2 1 bx 1 c.

• Banner Dimensions

• Card Design

• Construction

9.6: Factor ax2 1 bx 1 c Factor trinomials of the form ax2 1 bx 1 c.

• Discus • Diving

• Scrapbook Design

9.7: Factor Special Products Factor special products. • Falling Object

• Falling Brush

• Grasshopper

9.8: Factor Polynomials Completely

Factor polynomials completely.

• Terrarium • Carpentry

• Jumping Robot



1. Adding, subtracting, and multiplying polynomials

2. Factoring polynomials

3. Writing and solving polynomial equations to solve problems

CHAPTER


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Algebra 1Chapter 9 Resource Book118

CHAPTER

9 Family Letter continuedFor use with Chapter 9


Lesson Exercise

9.1 Find the sum or difference.

(a) (2x3 1 4x2 2 6x 2 8) 1 (x3 2 5x 1 4)(b) (3x2 2 4x 2 5) 2 (22x2 2 8x 1 7)

9.2 You frame a picture that has a length of 10 inches and a width of 8 inches with a border that is the same width on every side.

(a) Write a polynomial that represents the total area of the picture and border.

(b) Find the total area when the width of the border is 3 inches.

9.3 Find the product. (a) (3x 2 4)2 (b) (x 1 5y)(x 2 5y)

9.4 While lying on the ground, you throw a paper airplane straight up in the air with an initial vertical velocity of 20 feet per second. The airplane’s height h, t seconds after you throw it, can be modeled by h 5 216t2 1 20t. After how many seconds does it land on the ground?

9.5 Factor the trinomial. (a) x2 1 2x 2 35 (b) y2 2 11y 1 24

9.6 Solve the equation. (a) 2x2 1 9x 1 7 5 0 (b) 9y2 1 12y 2 12 5 0

9.7 A clothesline runs between two apartment buildings 144 feet in the air. A wet sock is dropped while being placed on the line. Use the vertical motion model to write an equation for the height h (in feet) of the sock as a function of the time t (in seconds) after it is dropped. After how many seconds does the sock land on the ground?

9.8 Factor the expression completely: 6x2y 1 45xy2 1 75y3.


Directions Measure the length and width of a rectangular-sized yard, to the nearest foot. Suppose you were going to put a rectangular shaped pool in the yard with a space x feet wide on all four sides. Find a model for the area of the pool. Write it as a quadratic trinomial. If x 5 7, what is the area of the pool the yard could hold?

Answers9.1: (a) 3x3 1 4x2 2 11x 2 4 (b) 5x2 1 4x 2 12 9.2: (a) x2 1 18x 1 80 (b) 143 in.2 9.3: (a) 9x2 2 24x 1 16 (b) x2 2 25y2 9.4: 1.25 seconds 9.5: (a) (x 1 7)(x 2 5)

(b) (y 2 8)(y 2 3) 9.6: (a) x 5 2 7 } 2 , 21 (b) y 5

2 } 3 , 22 9.7: h 5 216t2 1 144;

3 seconds 9.8: 3y(2x 1 5y)(x 1 5y)

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119Algebra 1


Vistazo al capítulo �� !�"#�$�� %�� &�#'�� %�� &�#�� &(


9.1: Sumar y restar polinomios Sumar y restar polinomios • Asistencia de béisbol• Ir de excursión y de

camping• Gastos de carro

9.2: Multiplicar polinomios Multiplicar polinomios • Patinaje • Piscina• Grabaciones de sonidos

9.3: Hallar productos especiales de polinomios

Usar patrones de productos especiales para multiplicar polinomios

• Pastor fronterizo• Plantas de guisantes• Estadística de fútbol

americano

9.4: Resolver ecuaciones de polinomios en forma de factores

Resolver ecuaciones de polinomios

• Armadillo• Insecto • Fútbol

9.5: Hallar factores de x2 1 bx 1 c

Hallar factores de trinomios en forma de x2 1 bx 1 c.

• Dimensiones de estandarte

• Diseño de tarjeta• Construcción

9.6: Hallar factores de ax2 1 bx 1 c

Hallar factores de trinomios en forma de ax2 1 bx 1 c.

• Disco • Zambullidas• Diseño de álbum de

recortes

9.7: Hallar factores de productos especiales

Hallar factores de productos especiales

• Objeto en caída• Cepillo en caída• Saltamontes

9.8: Hallar factores de polinomios completamente

Hallar factores de polinomios completamente

• Terrario • Carpintería• Robot de saltos



1. Sumar, restar y multiplicar polinomios

2. Hallar factores de polinomios

3. Escribir y resolver ecuaciones de polinomios para resolver problemas

CAPÍTULO


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Algebra 1Chapter 9 Resource Book120

CAPÍTULO

9 Carta para la familia continúaUsar con el Capítulo 9


Lección Ejercicio

9.1 Halla la suma o la diferencia.

(a) (2x3 1 4x2 2 6x 2 8) 1 (x3 2 5x 1 4)(b) (3x2 2 4x 2 5) 2 (22x2 2 8x 1 7)

9.2 Enmarcas un cuadro que tiene un largo de 10 pulgadas y un ancho de 8 pulgadas con un borde que tiene el mismo ancho en cada lado.

(a) Escribe un polinomio que represente el área total del cuadro y del borde.

(b) Halla el área total cuando el ancho del borde es 3 pulgadas.

9.3 Halla el producto. (a) (3x 2 4)2 (b) (x 1 5y)(x 2 5y)

9.4 Recostado en el suelo, tiras un avión de papel hacia arriba con una velocidad vertical inicial de 20 pies por segundo. La altura del avión h, t segundos después de tirarlo, se puede modelar por h 5 216t2 1 20t. ¿Después de cuántos segundos aterriza el avión?

9.5 Halla los factores del trinomio. (a) x2 1 2x 2 35 (b) y2 2 11y 1 24

9.6 Resuelve la ecuación. (a) 2x2 1 9x 1 7 5 0 (b) 9y2 1 12y 2 12 5 0

9.7 Un tendedero se extiende 114 pies en el aire entre dos edifi cios de apartamentos. Un calcetín mojado se cae del tendedero. Usa el modelo de moción vertical para escribir una ecuación para la altura h (en pies) del calcetín como una función del tiempo t (en segundos) después de caerse. ¿Después de cuántos segundos aterriza el calcetín?

9.8 Halla los factores de la expresión completamente: 6x2y 1 45xy2 1 75y3.


Instrucciones Mide el largo y el ancho de un patio trasero rectangular al pie más próximo. Supón que deseas poner una piscina rectangular en el patio con un es-pacio x pies de ancho en los cuatro lados. Halla un modelo para el área de la piscina. Escríbelo como un trinomio cuadrático. Si x 5 7, ¿qué es el área de la piscina que se puede poner en el patio?

Respuestas9.1: (a) 3x3 1 4x2 2 11x 2 4 (b) 5x2 1 4x 2 12 9.2: (a) x2 1 18x 1 80 (b) 143 pulg2 9.3: (a) 9x2 2 24x 1 16 (b) x2 2 25y2 9.4: 1.25 segundos 9.5: (a) (x 1 7)(x 2 5)

(b) (y 2 8)(y 2 3) 9.6: (a) x 5 2 7 } 2 , 21 (b) y 5

2 } 3 , 22 9.7: h 5 216t2 1 144;

3 segundos 9.8: 3y(2x 1 5y)(x 1 5y)

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121Algebra 1


TI-83 Plus

Part a.

Y= ( X,T,�,n x 2

� 2 X,T,�,n � 3 ) � ( 2 X,T,�,n x 2

� 4 X,T,�,n � 5 ) ENTER 3 X,T,�,n x 2

� 2 X,T,�,n � 2 ENTER

Use the arrow keys to move the cursor to the graph style icon in the fi rst column before y2. Press ENTER until you see the graph style thick.

WINDOW (�) 5 ENTER 5 ENTER 1ENTER (�) 5 ENTER 5 ENTER 1 ENTER GRAPH

Part b.

Y= CLEAR ( X,T,�,n MATH 3 � X,T,�,n � 1 ) � ( 5 X,T,�,n

MATH 3 � 2 X,T,�,n � 7 ) ENTER CLEAR (�) 4 X,T,�,n

MATH 3 � X,T,�,n � 6 ENTER

Use the arrow keys to move the cursor to the graph style icon in the fi rst column before y2. Press ENTER until you see the graph style thick.

WINDOW (�) 5 ENTER 5 ENTER 1ENTER (�) 10 ENTER 10 ENTER 1 ENTER GRAPH


Part a.

From the main menu, choose GRAPH.

( X, ,T� x 2

� 2 X, ,T� � 3 ) �

( 2 X, ,T� x 2

� 4 X, ,T� � 5 ) EXE 3 X, ,T� x 2

� 2 X, ,T� � 2 EXE F4 F2 EXIT SHIFT F3 (�) 5 EXE 5 EXE 1 EXE (�) 5 EXE 5 EXE

1 EXE EXIT F6

Part b.


( X, ,T� ^ 3 � X, ,T� � 1 ) �

( 5 X, ,T� ^ 3 � 2 X, ,T� � 7 ) EXE (�) 4 X, ,T� ^ 3 � X, ,T� � 6 EXE F4 F2 EXIT SHIFT F3 (�)

5 EXE 5 EXE 1 EXE (�) 10 EXE 10EXE 1 EXE EXIT F6


LESSON

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Write the polynomial so that the exponents decrease from left to right. Identify the degree and leading coeffi cient of the polynomial.

1. 8n6 2. 29z 1 1 3. 4 1 2x5

4. 18x 2 x2 1 2 5. 3y3 1 4y2 1 8 6. m 2 20m3 1 5

7. 28 1 10a4 2 3a7 8. 4z 1 z3 2 5z2 1 6z4 9. 8h3 2 6h4 1 h7

Tell whether the expression is a polynomial. If it is a polynomial, fi nd its degree and classify it by the number of its terms. Otherwise, tell why it is not a polynomial.

10. 6m2 11. 3x 12. y22 1 4

13. 3b2 2 2 14. 1 } 2 x2 2 2x 1 1 15. 6x3 2 1.4x

Find the sum or difference.

16. (6x 1 4) 1 (x 1 5) 17. (4m2 2 5) 1 (3m2 2 2)

18. (2y2 1 y 2 1) 1 (7y2 1 4y 2 3) 19. (3x2 1 5) 2 (x2 1 2)

20. (10a2 1 4a 2 5) 2 (3a2 1 2a 1 1) 21. (m2 2 3m 1 4) 2 (2m2 1 5m 1 1)

Write a polynomial that represents the perimeter of the fi gure.

22.

x 1 2 x 1 1

2x 1 1

23.

x 1 1

x 1 4

x 1 5

x 2 1

24. Library Books For 1995 through 2005, the number F of fi ction books (in ten thousands) and the number N of nonfi ction books (in ten thousands) borrowed from a library can be modeled by

F 5 0.01t2 1 0.08t 1 7 and N 5 0.004t2 1 0.05t 1 5

where t is the number of years since 1995. Find the total number B of books borrowed from the library in a year from 1995 to 2005.

25. Photograph Mat A mat in a frame has an opening

a � 3b � 4x � 2

xNot drawn to scale

4x

for a photograph as shown in the fi gure. Find the area of the mat if the area of the opening is given by A 5 πab. Leave your answer in terms of π.

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123Algebra 1


Write the polynomial so that the exponents decrease from left to right. Identify the degree and leading coeffi cient of the polynomial.

1. 4n5 2. 4x 2 2x2 1 3 3. 6y3 2 2y2 1 4y4 2 5


4. 10x 5. 26n2 2 n3 1 4 6. w23 1 5


7. (3z2 1 z 2 4) 1 (2z2 1 2z 2 3) 8. (8c2 2 4c 1 1) 1 (23c2 1 c 1 5)

9. (2x2 1 5x 2 1) 1 (x2 2 5x 1 7) 10. (10b2 2 3b 1 2) 2 (4b2 1 5b 1 1)

11. (24m2 1 3m 2 1) 2 (m 1 2) 12. (3m 1 4) 2 (2m2 2 6m 1 5)

Write a polynomial that represents the perimeter of the fi gure.

13. 3x

3x

2x 1 12x 1 1

14.

2x 2 1

4x 2 3

x 1 2

2x 1 1

15. Floor Plan The fi rst fl oor of a home has the fl oor plan shown. Find the area of the fi rst fl oor.

x2

x

4x

x 2 48

x2

16. Profi t For 1995 through 2005, the revenue R (in dollars) and the cost C (in dollars) of producing a product can be modeled by

R 5 1 } 4 t2 1

21 } 4 t 1 400 and C 5

1 } 12 t2 1

13 } 4 t 1 200

where t is the number of years since 1995. Write an equation for the profi t earned from 1995 to 2005. (Hint: Profi t 5 Revenue 2 Cost)

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1. 28 2. x2 2 5x 1 x21 3. 23b2 2 5 1 1 } 2 b


4. (3m3 1 2m 1 1) 1 (4m2 2 3m 1 1) 5. (24y2 1 y 1 5) 1 (4 2 3y 2 y2)

6. (24c 1 c3 1 8) 1 (c2 2 5c 2 3) 7. (23z 1 6) 2 (4z2 2 7z 2 8)

8. (14x4 2 3x2 1 2) 2 (3x3 1 4x2 1 5) 9. (5 2 x4 2 2x3) 2 (26x2 1 5x 1 5)

10. Find the sum f (x) 1 g(x) and the difference f (x) 2 g(x) for the functions f (x) 5 25x2 1 2x 2 1 and g(x) 5 6x3 1 2x2 2 5.


11. (10a2b2 2 7a2b) 1 (24a3b2 1 5a2b2 2 3a2b 1 5)

12. (6m2n 2 5mn2 2 8n 1 2m) 2 (6n2m 1 3m2n)

13. Mineral Production For 1997 through 2003, the amount P of peat produced (in thousand metric tons) and the amount L of perlite produced (in thousand metric tons) in the United States can be modeled by

P 5 3.09t4 2 36.74t3 + 121.38t2 2 77.65t 1 663.57

and

L 5 1.84t4 2 20.04t3 1 56.27t2 2 48.77t 1 703.94

where t is the number of years since 1997.

a. Write an equation that gives the total number T of thousand metric tons of peat and perlite produced as a function of the number of years since 1997.

b. Was more peat and perlite produced in 1997 or in 2003? Explain your answer.

14. Home Sales In 1997, the median sale price for a one-family home in the Northeast was about $187,443 and the median sale price for a one-family home in the Midwest was about $151,629. From 1997 through 2003, the median sale price for a one-family home in the Northeast increased by about $13,857 per year and the median sale price for a one-family home in the Midwest increased by about $5457 per year.

a. Write two equations that model the median sale prices of a one-family home in the Northeast and Midwest as functions of the number of years since 1997.

b. How much more did a home in the Northeast cost than a home in the Midwest in 1997 and 2003? What was the change in the sale price of each area from 1997 to 2003?

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125Algebra 1


Add and subtract polynomials.

VocabularyA monomial is a number, a variable, or the product of a number and one or more variables with whole number exponents.

The degree of a monomial is the sum of the exponents of the variables in the monomial.

A polynomial is a monomial or a sum of monomials, each called a term of the polynomial.

The degree of a polynomial is the greatest degree of its terms.

When a polynomial is written so that the exponents of a variable decrease from left to right, the coeffi cient of the fi rst term is called the leading coeffi cient.

A polynomial with two terms is called a binomial.

A polynomial with three terms is called a trinomial.

GOAL


LESSON

9.1

Rewrite a polynomial

Write 12x 3 2 15x 1 13x 5 so that the exponents decrease from left to right. Identify the degree and the leading coeffi cient of the polynomial.

Solution

Consider the degree of each of the polynomial’s terms.

Degree is 3. Degree is 1. Degree is 5.

12x3 2 15x 1 13x5

The polynomial can be rewritten as 13x5 1 12x3 2 15x. The greatest degree is 5, so the degree of the polynomial is 5, and the leading coeffi cient is 13.


Write the polynomial so that the exponents decrease from left to right. Identify the degree and the leading coeffi cient of the polynomial.

1. 9 2 2x2 2. 16 1 3y3 1 2y 3. 6z3 1 7z2 2 3z5

EXAMPLE 1

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Subtract polynomials

Find the difference.

a. (3x2 2 9x) 2 (2x2 2 5x 1 6) b. (11x2 1 6x 2 1) 2 (2x2 2 7x 1 5)

Solution

a. Vertical format: Align like terms in vertical columns.

3x2 2 9x 3x2 2 9x

2 (2x2 2 5x 1 6) 2 2x2 1 5x 2 6

_______________ _____________

x2 2 4x 2 6

b. Horizontal format: Group like terms and simplify.

(11x2 1 6x 2 1) 2 (2x2 2 7x 1 5) 5 11x2 1 6x 2 1 2 2x2 1 7x 2 5

5 (11x2 2 2x2) 1 (6x 1 7x) 1 (21 2 5)

5 9x2 1 13x 2 6



4. (2a2 1 7) 1 (7a2 1 4a 2 3)

5. (9b2 2 b 1 8) 1 (4b2 2 b 2 3)

6. (7c3 2 6c 1 4) 2 (9c3 2 5c2 2 c)

7. (d2 2 15d 1 10) 2 (212d2 1 8d 2 1)

EXAMPLE 3

Add polynomials

Find the sum.

a. (3x4 2 2x3 1 5x2) 1 (7x2 1 9x32 2x) b. (7x22 3x 1 6) 1 (9x2 1 6x2 11)

Solution

a. Vertical format: Align like terms in vertical columns.

3x4 2 2x3 1 5x2

1 9x3 1 7x2 2 2x ______________________

3x4 1 7x3 1 12x2

2 2x

b. Horizontal format: Use the associative and commutative properties to group like terms and simplify.

(7x2 2 3x 1 6) 1 (9x2 1 6x 2 11) 5 (7x2 1 9x2) 1 (23x 1 6x) 1 (6 2 11)

5 16x2 1 3x 2 5

EXAMPLE 2


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127Algebra 1


1. Hockey Attendance During the period 199922003, the attendance S and P (in thousands) at National Hockey League regular season and playoff games, respectively, can be modeled by

S 5 2359.93t2 1 2272.61t 1 17,084.14

P 5 214t3 1 72t2 2 23t 1 1475.6

where t is the number of years since 1999. About how many people attended National Hockey League games in 2003?

2. Salaries During the period 1999–2003, the average salaries B and F (in thousands of dollars) for Major League Baseball and National Football League players, respectively, can be modeled by

B 5 236.57t2 1 339.29t 1 1602.86

F 5 219.58t3 1 117.14t2 2 17.49t 1 707.8

where t is the number of years since 1999. About how much more was the average baseball salary than the average football salary in 2003?

College Basketball Attendance During the period 1999–2003, the attendance M and W (in thousands) at men’s and women’s NCAA basketball games, respectively, can be modeled by

M 5 73.3t3 2 372.4t2 1 722.2t 1 28,524.4 and W 5 40.3t3 2 208.6t2 1 727.7t 1 8035.7

where t is the number of years since 1999. About how many people attended NCAA basketball games in 2003?


What do you know? The equations that model the attendance for men’s and women’s NCAA basketball games from 1999–2003.

What do you want to fi nd out? The attendance of NCAA basketball games in 2003.

STEP 2 Make a Plan Use what you know to add the two equations.

STEP 3 Solve the Problem Add the models for the attendance to men’s and women’s games to fi nd a model for A, the total attendance (in thousands).

A 5 (73.3t3 2 372.4t2 1 722.2t 1 28,524.4) 1 (40.3t3 2 208.6t2 1 727.7t 1 8035.7)

5 (73.3t3 1 40.3t3) 1 (2372.4t2 2 208.6t2) 1 (722.2t 1 727.7t) 1 (28,524.4 1 8035.7)

5 113.6t3 2 581t2 1 1449.9t 1 36,560.1

Substitute 4 for t in the model, because 2003 is 4 years after 1999.

A 5 113.6(4)3 2 581(4)2 1 1449.9(4) 1 36,560.1 5 40,334.1

About 40,334,100 people attended NCAA basketball games in 2003.

STEP 4 Look Back Substitute 4 into each attendance equation and then add to fi nd the total attendance in 2003.When you substitute 4 into the men’s attendance, you obtain 30,146. When you substitute 4 into the women’s attendance, you obtain 10,188.1. When you add the men’s and women’s attendance, you get 40,334,100 people.

PRACTICE

PROBLEM

LESSON

9.1 Problem Solving Workshop:Worked Out ExampleFor use with pages 5722578

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Suppose you have x number of quarters, x 1 4 number of dimes, 2x 1 1 number of nickels, and 3x 1 5 number of pennies. For each combination of coins, determine whether the number of coins is even, odd, or can’t be determined from the given information.

1. The total number of quarters and dimes

2. The total number of quarters and nickels

3. The total number of quarters and pennies

4. The total number of dimes and pennies

5. The total number of dimes, nickels, and pennies

In Exercises 6–12, simplify the given expression. Assume x is positive.

6. (2x 1 1)[(3x2 2 2x 1 5) 1 (2x2 1 4x 2 3) 2 (5x2 1 2x 1 2)]

7. (2x 1 1)[(3x2 2 2x 1 5) 1 (2x2 1 4x 2 3) 2 (5x2 1 2x 1 2)]

8. x3x 1 5 p x22x 2 2 p x2x 2 2

9. 3xx p x22x p xx

10. 32x2 2 5x 1 1 p 322x2 1 5x 1 3

}} 26x 2 1 p 226x 1 3

11. 52x2 2 3x 2 4

} 52x2 2 3x 2 6

12. 3x2 2 2

} 3x2 2 5

2 2x4 1 3

} 2x4

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129Algebra 1


Find the product.

1. x(3x2 2 2x 1 1) 2. 2y(3y3 1 y2 2 4) 3. 23m(m2 1 4m 2 1)

4. d2(4d2 2 3d 1 1) 5. 2w3(w2 1 3w) 6. 2a2(a2 1 3a 2 1)

Use a table to fi nd the product.

7. (x 1 1)(x 2 4) 8. (y 1 6)(y 1 2) 9. (a 2 5)(a 2 3)

10. (2m 1 1)(m 1 3) 11. (3z 1 4)(z 2 5) 12. (d 1 6)(3d 2 1)

Use a vertical or a horizontal format to fi nd the product.

13. (y 1 8)(y 2 3) 14. (n 1 5)(n 1 6) 15. (3x 2 2)(x 1 5)

16. (4a 1 1)(2a 2 1) 17. (w 1 1)(w2 1 2w 1 1) 18. (m 2 2)(m2 2 2m 1 3)

Use the FOIL pattern to fi nd the product.

19. (y 2 3)(8y 1 1) 20. (5b 2 1)(3b 1 2) 21. (2d 2 4)(3d 2 1)

22. (3x 1 1)(2x 1 2) 23. (6x 2 2)(x 1 4) 24. (2s 2 5)(s 1 3)

25. (8c 1 2)(5c 2 7) 26. (8p 2 3)(2p 2 5) 27. (14t 2 2)(t 1 2)

28. Volume You have come up with a plan for building

(4x 1 8) in.

(3x 1 6) in.

24 in.a wooden box to hold all of your sports equipment as shown.

a. Write a polynomial that represents the volume of the box.

b. Find the volume of the box when x 5 10.

29. National Park System During the period 1990–2002, the number A of acres (in thousands) making up the national park system in the United States and the percent P (in decimal form) of this amount that is parks can be modeled by

A 5 211t 1 76,226

and

P 5 20.0008t2 1 0.009t 1 0.6


a. Find the values of A and P for t 5 0. What does the product A p P mean for t 5 0 in the context of this problem?

b. Write an equation that models the number of acres (in thousands) that are just parks as a function of the number of years since 1990.

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Find the product.

1. x2(6x2 2 3x 2 1) 2. 25a3(4a4 2 3a 1 1) 3. 4d2(22d3 1 5d2 2 6d 1 2)

4. (3x 1 1)(2x 2 5) 5. (2y 1 3)(y 2 5) 6. (6a 2 3)(4a 2 1)

7. (b 2 8)(5b 2 2) 8. (8m 1 7)(2m 1 3) 9. (2p 1 2)(3p2 1 1)

10. (2z 2 7)(–z 1 3) 11. (23d 1 10)(2d 2 1) 12. (n 1 1)(n2 1 4n 1 5)

13. (w 2 3)(w2 1 8w 1 1) 14. (2s 1 5)(s2 1 3s 2 1) 15. (x2 2 4xy 1 y2)(5xy)


16. a(3a 1 1) 1 (a 1 1)(a 2 1)

17. (x 1 2)(x 1 5) 2 x(4x 2 1)

18. (m 1 7)(m 2 3) 1 (m 2 4)(m 1 5)

Write a polynomial for the area of the shaded region.

19.

3x

x 5

20.

2

x

x

4

21. Flower Bed You are designing a rectangular fl ower bed

x ft

x ft6 ft

5 ft

that you will border using brick pavers. The width of the border around the bed will be the same on every side, as shown.

a. Write a polynomial that represents the total area of the fl ower bed and the border.

b. Find the total area of the fl ower bed and border when the width of the border is 1.5 feet.

22. School Enrollment During the period 1995–2002, the number S of students (in thousands) enrolled in school in the U.S. and the percent P (in decimal form) of this amount that are between 7 and 13 years old can be modeled by

S 5 32.6t3 2 376.45t2 1 1624.2t 1 66,939

and

P 5 0.000005t4 2 0.0003t3 1 0.003t2 2 0.007t 1 0.4


a. Find the values of S and P for t 5 0. What does the product S p P mean for t 5 0 in the context of this problem?

b. Write an equation that models the number of students (in thousands) that are between 7 and 13 years old as a function of the number of years since 1995.

c. How many students between 7 and 13 years old were enrolled in 1995?

LESSON


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131Algebra 1


LESSON


Find the product.

1. 28y3(2y4 2 5y2 1 3) 2. (b 1 3)(3b2 2 2b 1 1) 3. (6w 2 3)(4 2 3w)

4. (9m3 1 1)(4m2 2 1) 5. (2x2 1 5x 2 2)(x 1 3) 6. (8n2 2 1)(3n2 2 4n 1 5)

7. (3p4 2 5)(2p2 1 4) 8. (28r3 1 2)(6r2 2 1) 9. (25z2 2 3)(22z2 1 9)

10. xy(x2 1 2y) 11. 23x(2xy 1 5y) 12. y2(x2y 1 y2x)

13. (x 2 y)(5x 1 6y) 14. (xy2 1 70)(3x 1 2y) 15. (x2 2 4xy 1 y2)(5xy)


16. (7n 1 1)(3n 1 5) 1 (4n 2 2)(3n 1 1) 17. 5w2(3w3 2 2w 1 1) 1 w4(w2 2 2w 1 3)

Write a polynomial for the area of the shaded region.

18.

x 1 4

x 1 3

19.

8

12

x 1 1

2x

20. Car Production During the period 1995–2002, the number of cars C (in thousands) produced in the U.S. and the average price P (in dollars) spent on one of these cars can be modeled by

C 5 2198.02t 1 6320.49 and P 5 1.67t4 2 22.28t3 1 44.84t2 1 531.16t 1 16,860


a. Write an equation that models the total amount spent (in thousands of dollars) on new cars in the U.S. by consumers as a function of the number of years since 1995.

b. How much money was spent in the U.S. on new cars by consumers in 1995?

21. Sporting Goods Equipment During the period 1990–2002, the amount of money E (in millions of dollars) spent on sporting goods equipment in the U.S. and the percent P (in decimal form) of this amount that is spent on exercise equipment can be modeled by

E 5 25.56t4 1 149.93t3 2 1314.65t2 1 4396.75t 1 14,439.09

and P 5 20.00002t4 2 0.0005t3 1 0.0028t2 1 0.001t 1 0.126


a. Find the values of E and P for t 5 0. What does the product E p P mean for t = 0 in the context of this problem?

b. Write an equation that models the amount spent (in millions of dollars) on exercise equipment as a function of the number of years since 1990.

c. How much money was spent in the U.S. on exercise equipment in 1990?

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Multiply polynomials.GOAL

Multiply a monomial and a polynomial

Find the product 5x4(2x3 2 3x2 1 x 2 6).

Solution

5x4(2x3 2 3x2 1 x 2 6) Write product.

5 5x4(2x3) 2 5x4(3x2) 1 5x4(x) 2 5x4(6) Distributive property

5 10x7 2 15x6 1 5x5 2 30x4 Product of powers property


Find the product.

1. 3x2(7x2 2 2x 1 3) 2. 4x5(3x3 2 2x2 2 8x 1 9)

EXAMPLE 1

Multiply polynomials horizontally

Find the product (9x2 2 x 1 6)(5x 2 2).

Solution

(9x2 2 x 1 6)(5x 2 2) Write product.

5 9x2(5x 2 2) 2 x(5x 2 2) 1 6(5x 2 2) Distributive property

5 45x3 2 18x2 2 5x2 1 2x 1 30x 2 12 Distributive property

5 45x3 2 23x2 1 32x 2 12 Combine like terms.


LESSON

9.2

EXAMPLE 3

Multiply polynomials vertically

Find the product (5m2 2 2m 1 3)(2m 1 7).

Solution

STEP 1 Multiply by 7. STEP 2 Multiply by 2m. STEP 3 Add products.

5m2 2 2m 1 3 5m2 2 2m 1 3 5m2 2 2m 1 33 2m 1 7 3 2m 1 7 3 2m 1 7

35m2 2 14m 1 21 35m2 2 14m 1 21 35m2 2 14m 1 21 10m3 2 4m2 1 6m 10m3 2 4m2 1 6m

10m3 1 31m2 2 8m 1 21

EXAMPLE 2

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133Algebra 1


Standardized Test Practice

The dimensions of a rectangle are 3x 2 1 and x 1 5. Which expression represents the area of the rectangle?

A 3x2 1 16x 2 5 B 3x2 1 14x 2 4 C 3x2 1 14x 2 5 D 4x 1 4

Solution

Area 5 length p width Formula for area of a rectangle

5 (3x 2 1)(x 1 5) Substitute for length and width.

5 (3x)(x) 1 (3x)(5) 1 (21)(x) 1 (21)(5) Use FOIL pattern.

5 3x2 1 15x 1 (2x) 1 (25) Multiply.

5 3x2 1 14x 2 5 Combine like terms.

The correct answer is C.

Exercise for Example 5 9. The dimensions of a rectangle are y 1 9 and 2y 2 3. Write an expression for

the area of the rectangle.

EXAMPLE 5

Multiply binomials using FOIL pattern

Find the product (2x 2 1)(7x 1 6).

Solution

(2x 2 1)(7x 1 6) Write product.

5 (2x)(7x) 1 (2x)(6) 1 (21)(7x) 1 (21)(6) Write product of terms.

5 14x2 1 12x 1 (27x) 1 (26) Multiply.

5 14x2 1 5x 2 6 Combine like terms.


Find the product.

3. (m2 1 6m 1 4)(3m 2 1) 4. (2n 1 7)(3n 1 4)

5. (2p2 2 p 1 6)( p 1 7) 6. (6q2 2 5q 2 4)(2q 2 3)

7. (5t 1 9)(3t 2 8) 8. (8s 2 7)(9s 2 7)

EXAMPLE 4


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In Exercises 1–5, fi nd the product and simplify.

1. (x3 1 2x)(x4 1 x2)

2. (3y 2 y3)(y4 1 y)

3. (2x3 1 2y)(x4 1 2y3)

4. x3(x5 1 4x3)(2x4 1 3x2)

5. (x2 1 1)(x 1 2)(x2 1 2)

In Exercises 6–10, simplify the expression and write the result as a polynomial in standard form.

6. x(x2 1 2x) 2 x2(x 1 2)

7. (x 1 1)(x 1 1) 2 (x 2 1)(x 2 1)

8. (x2 1 1)(x2 1 1) 2 (x2 2 1)(x2 2 1)

9. (2x2 1 3x 2 1)(x 2 1) 2 2x(x 1 1)

10. (x 1 3)(2x2 1 2) 1 2(x 1 1)(x 2 2) 1 3


A ship storage compartment is being designed to carry trailers, each of which has dimensions 50 feet long by 9 feet tall by 8 feet wide. It is decided that the storage container will have dimensions 50x 1 150 feet long by 9x tall by 8x 1 16 feet wide.

11. Write an expression for the volume of the storage compartment in terms of x.

12. Simplify the expression found in Exercise 11 and write it as a polynomial in standard form.

13. If x is 4, how many trailers will fi t inside the storage compartment?

LESSON


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135Algebra 1


Find the missing term.

1. (a 2 b)2 5 a2 2 ? 1 b2 2. (m 1 n)2 5 m2 1 ? 1 n2

3. (x 2 1)2 5 x2 2 ? 1 1 4. (x 1 5)2 5 x2 1 ? 1 25

5. (x 2 y)(x 1 y) 5 x2 2 ? 6. (x 2 3)(x 1 3) 5 x2 2 ?

Match the product with its polynomial.

7. (2x 1 3)(2x 2 3) 8. (2x 1 3)2 9. (2x 2 3)2

A. 4x2 1 12x 1 9 B. 4x2 2 12x 1 9 C. 4x2 2 9

Find the product of the square of the binomial.

10. (x 1 4)2 11. (m 2 8)2 12. (a 1 10)2

13. (p 2 12)2 14. (2y 1 1)2 15. (3y 2 1)2

16. (10r 2 1)2 17. (4n 1 2)2 18. (3c 2 2)2

Find the product of the sum and difference.

19. (z 1 5)(z 2 5) 20. (b 2 2)(b 1 2) 21. (n 2 8)(n 1 8)

22. (a 1 10)(a 2 10) 23. (2x 1 1)(2x 2 1) 24. (5m 2 1)(5m 1 1)

25. (4d 1 1)(4d 2 1) 26. (3p 1 2)(3p 2 2) 27. (2r 2 3)(2r 1 3)

Describe how you can use mental math to fi nd the product.

28. 13 p 7 29. 24 p 36 30. 51 p 69

31. Total Profi t For 1995 through 2005, the number N of units (in thousands) produced by a manufacturing plant can be modeled by N 5 3t 1 2 and the profi t per unit P (in dollars) can be modeled by P 5 3t 2 2 where t is the number of years since 1995. Write a polynomial that models the total profi t T (in thousands of dollars).

32. Eye Color In humans, the brown eye gene B is dominant Mother

Fath

er

B

B

b

b

BB Bb

bB bb

and the blue eye gene b is recessive. This means that humans whose eye genes are BB, Bb, or bB have brown eyes and those with bb have blue eyes. The Punnett square at the right shows the results of eye colors for children of parents who each have one B gene and one b gene.

a. Write a polynomial that models the possible gene combinations of a child.

b. What percent of the possible gene combinations results in a child with blue eyes?

LESSON

9.3 Practice AFor use with pages 5872592

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Find the product of the square of the binomial.

1. (x 2 9)2 2. (m 1 11)2 3. (5s 1 2)2

4. (3m 1 7)2 5. (4p 2 5)2 6. (7a 2 6)2

7. (10z 2 3)2 8. (2x 1 y)2 9. (3y 2 x)2

Find the product of the sum and difference.

10. (a 2 9)(a 1 9) 11. (z 2 20)(z 1 20) 12. (5r 1 1)(5r 2 1)

13. (6m 1 10)(6m 2 10) 14. (7p 2 2)(7p 1 2) 15. (9c 2 1)(9c 1 1)

16. (4x 1 3)(4x 2 3) 17. (4 2 w)(4 1 w) 18. (5 2 2y)(5 1 2y)


19. 15 p 25 20. 43 p 57 21. 182

Perform the indicated operation using the functions f (x) 5 4x 1 0.5 and g(x) 5 4x 2 0.5.

22. f(x) p g(x) 23. ( f(x))2 24. (g(x))2

25. Pea Plants In pea plants, the gene S is for spherical seed shape, and the gene s is for wrinkled seed shape. Any gene combination with an S results in a spherical seed shape. Suppose two pea plants have the same gene combination Ss.

a. Make a Punnett square that shows the possible gene combinations of an offspring pea plant and the resulting seed shape.

b. Write a polynomial that models the possible gene combinations of an offspring pea plant.

c. What percent of the possible gene combinations of the offspring results in a wrinkled seed shape?

26. Basketball Statistics You are on the basketball team Made Missed

Missed

Madeand you want to fi gure out some statistics about foul shots. The area model shows the possible outcomes of two attempted foul shots.

a. What percent of the two possible outcomes of two attempted foul shots results in you making at least one foul shot? Explain how you found your answer using the table.

b. Show how you could use a polynomial to model the possible results of two attempted foul shots.

LESSON

9.3 Practice BFor use with pages 5872592

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137Algebra 1


LESSON

9.3 Practice CFor use with pages 5872592

Find the product.

1. (8x 2 5)2 2. (4p 1 4)2 3. (10m 2 11)2

4. (11s 2 10)2 5. (20b 2 15)2 6. (m 1 4n)2

7. (r 2 8s)2 8. (10a 1 3b)2 9. (2x 2 4y)2

10. (8p 2 3)(8p 1 3) 11. (11t 1 4)(11t 2 4) 12. (7n 2 5)(7n 1 5)

13. (9z 1 12)(9z 2 12) 14. (15 2 w)(15 1 w) 15. (6 2 5p)(6 1 5p)

16. (20 2 3m)(20 1 3m) 17. (10a 2 5b)(10a 1 5b) 18. (4x 2 3y)(4x 1 3y)


19. 36 p 44 20. 232 21. 492

Perform the indicated operation using the functions f(x) 5 9x 2 0.5 and g(x) 5 9x 1 0.5.

22. f(x) p g(x) 23. (f(x) 1 g(x))2 24. (f(x) 2 g(x))2

25. Write two binomials that have the product x2 2 144. Explain how you found your answer.

26. Write a pattern for the cube of a binomial (a 2 b)3. Justify.

27. Soccer Statistics You are on the soccer team and you want Made Missed

Missed

Madeto fi gure out some statistics about attempted goals. The area model shows the possible outcomes of two attempted goals.

a. What percent of the two possible outcomes of two attempted goals results in you making at least one goal? Explain how you found your answer using the table.

b. Show how you could use a polynomial to model the possible results of two attempted goals.

28. Greenhouse You are drawing up a plan to build a greenhouse

8 ftin the shape of a rectangular prism. The height of the greenhouse is constant at 8 feet tall. You have 144 feet of material to form the base of the greenhouse into a square with a side length of 12 feet. You want to change the dimensions of the enclosed region. For every 1 foot you increase the width, you must decrease the length by 1 foot. Write a polynomial that gives the volume of the prism after you increase the width by x feet and decrease the length by x feet. Explain why any change in dimensions results in a volume less than that of the original prism.

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Use special product patterns to multiply polynomials.

Square of a Binomial PatternAlgebra Example

(a 1 b)2 5 a2 1 2ab 1 b2 (x 1 3)2 5 x2 1 6x 1 9

(a 2 b)2 5 a2 2 2ab 1 b2 (3x 2 2)2 5 9x2 2 12x 1 4

GOAL

Use the square of a binomial pattern

Find the product.

a. (7x 1 2)2 b. (6x 2 5y)2

Solution

a. (7x 1 2)2 5 (7x)2 1 2(7x)(2) 1 22 Square of a binomial pattern

5 49x2 1 28x 1 4 Simplify.

b. (6x 2 5y)2 5 (6x)2 2 2(6x)(5y) 1 (5y)2 Square of a binomial pattern

5 36x2 2 60xy 1 25y2 Simplify.


Find the product.

1. (y 1 9)2

2. (3z 1 7)2

3. (2w 2 3)2

4. (10r 2 3s)2

EXAMPLE 1

Sum and Difference PatternAlgebra Example

(a 1 b)(a 2 b) 5 a2 2 b2 (x 1 5)(x 2 5) 5 x2 2 25


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139Algebra 1


Use special products and mental math

Use special products to fi nd the product of 37 p 43.

Solution

Notice that 37 is 3 less than 40 while 43 is 3 more than 40.

37 p 43 5 (40 2 3)(40 1 3) Write as a product of difference and sum.

5 402 2 32 Sum and difference pattern

5 1600 2 9 Evaluate powers.

5 1591 Simplify.


Describe how you can use special products to fi nd the product.

9. 552

10. 31 p 49

EXAMPLE 3

Use the sum and difference pattern

Find the product.

a. (m 1 9)(m 2 9) b. (4n 2 3)(4n 1 3)

Solution

a. (m 1 9)(m 2 9) 5 m2 2 92 Sum and difference pattern

5 m2 2 81 Simplify.

b. (4n 2 3)(4n 1 3) 5 (4n)2 2 32 Sum and difference pattern

5 16n2 2 9 Simplify.


Find the product.

5. (g 1 11)(g 2 11)

6. (7f 2 1)(7f 1 1)

7. (2h 1 9)(2h 2 9)

8. (6k 2 8)(6k 1 8)

EXAMPLE 2


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In Exercises 1–5, simplify by multiplying and then adding and subtracting. Write the result as a polynomial in standard form.

1. (2x 1 3)2 1 (2x 2 3)2

2. (2x2 1 1)2 1 (x2 1 2)2

3. (ax 1 by)2 1 (ax 2 by)2

4. (ax2 1 by2)2 1 (ax2 2 by2)2

5. (x 1 5)2 2 (x 2 25)(x 1 1)

6. Show that (a 2 b 1 c)2 5 a2 1 b2 1 c2 2 2ab 1 2ac 2 2bc.

In Exercises 7 and 8, use the result from Exercise 6 to fi nd the product.

7. (3x 2 2y 1 5z)2

8. (ax 2 by 1 cz)2

In Exercises 9–12, assume x is a positive integer.

9. Find an expression for the product of three consecutive even integers, with 2x as the smallest of the three integers. Write the result as a polynomial in standard form.

10. Explain why the result from Exercise 9 is an even number.

11. Find an expression for the product of three consecutive odd integers, with 2x 1 1 as the smallest of the three integers. Write the result as a polynomial in standard form.

12. Explain why the result from Exercise 11 is an odd number.

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141Algebra 1


Match the equation with its solutions.

1. (x 1 4)(x 1 5) 5 0 A. 25 and 4

2. (x 2 4)(x 1 5) 5 0 B. 25 and 24

3. (x 2 5)(x 2 4) 5 0 C. 4 and 5

Solve the equation.

4. (x 1 6)(x 1 2) 5 0 5. (p 2 5)(p 1 3) 5 0 6. (b 2 7)(b 2 10) 5 0

7. (m 2 8)(m 1 1) 5 0 8. (a 2 9)(a 1 9) 5 0 9. (y 1 15)(y 1 12) 5 0

10. (c 2 25)(c 1 50) 5 0 11. (2z 2 2)(z 1 3) 5 0 12. (2n 2 6)(n 2 2) 5 0

Factor out the greatest common monomial factor.

13. 4m 2 2 14. 5x 2 10 15. 6y 1 15

16. 8x 1 8y 17. 7a 2 7b 18. 2a 1 10b

19. 9m 2 18n 20. 15p 2 3q 21. 12x 1 4y

22. 2c2 1 4c 23. 9m3 1 m2 24. 2w2 1 4w

Match the equation with its solutions.

25. 4a2 1 a 5 0 A. 0 and 4

26. a2 1 4a 5 0 B. 0 and 24

27. a2 2 4a 5 0 C. 0 and 2 1 } 4

Solve the equation.

28. a2 1 8a 5 0 29. n2 2 7n 5 0 30. 2w2 1 2w 5 0

31. 3p2 2 3p 5 0 32. 4c2 2 8c 5 0 33. 5x2 1 10x 5 0

34. Hot Air Balloon An object is dropped from a hot-air balloon 1296 feet above the ground. The height of the object is given by

h 5 216(t 2 9)(t 1 9)

where the height h is measured in feet, and the time t is measured in seconds. After how many seconds will the object hit the ground?

35. Kickball A kickball is kicked upward with an initial vertical velocity of 3.2 meters per second. The height of the ball is given by

h 5 29.8t2 1 3.2t

where the height h is measured in meters, and the time t is measured in seconds. After how many seconds does the ball land?

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Solve the equation.

1. (x 1 14)(x 2 3) 5 0 2. (m 2 12)(m 1 5) 5 0 3. (p 1 15)(p 1 24) 5 0

4. (n 2 8)(n 2 9) 5 0 5. (d 1 8) 1 d 2 1 }

2 2 5 0 6. 1 c 1

3 } 4 2 (c 2 6) 5 0

7. (2z 2 8)(z 1 5) 5 0 8. (y 2 3)(5y 1 10) 5 0 9. (6b 2 4)(b 2 8) 5 0

10. (8x 1 4)(6x 2 3) 5 0 11. (3x 1 9)(6x 2 3) 5 0 12. (4x 1 5)(4x 2 5) 5 0


13. 10x 2 10y 14. 8x2 1 20y 15. 18a2 2 6b

16. 4x2 2 4x 17. r2 1 2rs 18. 2m2 1 6mn

19. 5p2q 1 10q 20. 9a5 1 a3 21. 6w3 2 14w2

Solve the equation.

22. m2 2 10m 5 0 23. b2 1 14b 5 0 24. 5w2 2 5w 5 0

25. 24k2 1 24k 5 0 26. 8r2 2 24r 5 0 27. 9p2 1 18p 5 0

28. 6n2 2 15n 5 0 29. 28y2 2 10y 5 0 30. 210b2 1 25b 5 0

31. 8c2 5 4c 32. 30r2 5 215r 33. 224y2 5 9y

34. Diving Board A diver jumps from a diving board that is 24 feet above the water. The height of the diver is given by

h 5 216(t 2 1.5)(t 1 1)

where the height h is measured in feet, and the time t is measured in seconds. When will the diver hit the water? Can you see a quick way to fi nd the answer? Explain.

35. Dog To catch a frisbee, a dog leaps into the air with an initial velocity of 14 feet per second.

a. Write a model for the height of the dog above the ground.

b. After how many seconds does the dog land on the ground?

36. Desktop Areas You have two components to the desktop w

w

7 ft

3 ft

where you do your homework that fi t together into an L shape. The two components have the same area.

a. Write an equation that relates the areas of the desktop components.

b. Find the value of w.

c. What is the combined area of the desktop components?

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143Algebra 1


LESSON


Solve the equation.

1. (x 1 3) 1 x 2 2 } 5 2 5 0 2. 1 m 2

5 } 2 2 1 m 1

3 } 2 2 5 0 3. (4b 1 16)(b 2 6) 5 0

4. (7a 2 14)(a 1 8) 5 0 5. (2y 1 3)(y 2 9) 5 0 6. (5z 2 8)(3z 1 2) 5 0

7. (9w 2 2)(7w 2 3) 5 0 8. (8 2 2c)(5c 1 1) 5 0 9. (9 2 8r)(10 2 4r) 5 0


10. 9x2 2 21y 11. 4m3 1 24m 12. 10p2q 2 5pq2

13. 6x3y 1 9y2 14. 35a2b2 2 5ab 15. 12m2n 2 8mn2

16. w4 2 2w3 1 w 17. 23p4 1 15p2 1 6p 18. 8r5 2 20r4 2 12r2

Solve the equation.

19. 12a2 2 9a 5 0 20. 18x2 1 12x 5 0 21. 6z2 2 8z 5 0

22. 20p2 5 224p 23. 228m2 5 14m 24. 230r2 5 225r

25. 100m2 5 26m 26. 15y 2 50y2 5 0 27. 26w 1 34w2 5 0

Find the zeros of the function.

28. f (x) 5 228x2 1 7x 29. f (x) 5 29x2 1 4x 30. f (x) 5 5x2 2 3x

31. Fish A fi sh jumps out of the water while swimming. The height h (in feet) of the fi sh can be modeled by h 5 216t2 1 3.5t where t is the time (in seconds) since the fi sh jumped out of the water.

a. Find the zeros of the function. Explain what the zeros mean in this situation.

b. What is a reasonable domain for the function? Explain your answer.

32. Storage Structure The cross section of a wooden storage

x

y

10

30

50

70

525215 15

Center ofstructurestructure can be modeled by the polynomial function

y 5 2 3 } 80 (2x 2 40)(2x 1 40)

where x and y are measured in feet, and the center of the structure is where x 5 0.

a. Explain how to use the algebraic model to fi nd the width of the structure.

b. Use the model to fi nd the structure’s width. Show your work

c. Use the model to fi nd the coordinates of the center of the structure. Show your work.

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Solve polynomial equations.

VocabularyThe zero-product property is used to solve an equation when one side is zero and the other side is a product of polynomial factors. The solutions of such an equation are also called roots.

The height of a projectile can be described by the vertical motion model: h 5 216t2 1 vt 1 s, where t is the time (in seconds) the object has been in the air, v is the initial vertical velocity (in feet per second), and s is the initial height (in feet).

GOAL

Use the zero-product property

Solve (x 2 3)(x 1 6) 5 0.

Solution

(x 2 3)(x 1 6) 5 0 Write original equation.

x 2 3 5 0 or x 1 6 5 0 Zero-product property

x 5 3 or x 5 26 Solve for x.

The roots of the equation are 3 and 26.

CHECK Substitute each root into the original equation to check.

(3 2 3)(3 1 6) 0 0 (26 2 3)(26 1 6) 0 0

0 p 9 0 0 29 p 0 0 0

0 5 0 ✓ 0 5 0 ✓


Solve the equation.

1. (m 2 7)(m 2 9) 5 0 2. (5n 1 10)(4n 1 12) 5 0

EXAMPLE 1

Solve an equation by factoring

Solve 6x2 1 12x 5 0.

6x2 1 12x 5 0 Write original equation.

6x(x 1 2) 5 0 Factor left side.

6x 5 0 or x 1 2 5 0 Zero-product property



Review for MasteryFor use with pages 593– 598

LESSON

9.4

EXAMPLE 2

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145Algebra 1


Solve a multi-step problem Jump Rope A child jumping rope leaves the ground at an initial vertical velocity of 8 feet per second. After how many seconds does the child land on the ground?

Solution

STEP 1 Write a model for the height above the ground.

h 5 216t2 1 vt 1 s Vertical motion model

h 5 216t2 1 8t 1 0 Substitute 8 for v and 0 for s.

h 5 216t2 1 8t Simplify.

STEP 2 Substitute 0 for h. When the child lands, the child’s height above the ground is 0 feet. Solve for t.

0 5 216t2 1 8t Substitute 0 for h.

0 5 8t(22t 1 1) Factor right side.

8t 5 0 or 22t 1 1 5 0 Zero-product property

t 5 0 or t 5 1 }

2 Solve for t.

The child lands on the ground 1 }

2 second after the child jumps.

Exercise for Example 4 6. In Example 4, suppose the initial velocity is 10 feet per second. After how

many seconds will the child land on the ground?

EXAMPLE 4

Solve an equation by factoring

Solve 9y2 5 21y.

9y2 5 21y Write original equation.

9y2 2 21y 5 0 Subtract 21y from each side.

3y(3y 2 7) 5 0 Factor left side.

3y 5 0 or 3y 2 7 5 0 Zero-product property

y 5 0 or y 5 7 }

3 Solve for y.

The roots of the equation are 0 and 7 }

3 .


Solve the equation.

3. q2 1 16q 5 0 4. 4k2 2 8k 5 0 5. 12h2 5 36h

EXAMPLE 3

Review for Mastery continuedFor use with pages 593– 598

LESSON

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LESSONS

9.1–9.4

1. Multi-Step Problem You are making a scrapbook out of pictures that are 7 inches long and 5 inches wide. You want to place a border of equal width on each edge.

a. Write a polynomial that represents the total area of the picture with the border.

b. Find the total area of the picture with border when the width of the border is 2 inches.

2. Multi-Step Problem During the period 199722002, the sporting goods sales S (in millions of dollars) and the percent P (in decimal form) of sporting goods sales that are for exercise equipment can be modeled by

S 5 1990.5t 1 67,530

P 5 0.0022t 1 0.0436


a. Write an equation that models the sales (in millions of dollars) of exercise equipment as a function of the number of years since 1997.

b. Find the amount of exercise equipment sales in 2001.

3. Open-Ended In fl owers, the gene P is for purple coloring and the gene w is for white coloring. Any gene combination with a P results in purple coloring.

a. Suppose one fl ower has the gene combination Pw. Choose a color gene combination for another fl ower. Create a Punnett square to show the possible gene combinations of an offspring fl ower.

b. What percent of the possible gene combinations of the offspring result in purple coloring?

c. Show how you could use a polynomial to model the possible color gene combinations of the offspring.

4. Gridded Response During the period 1997–2003, the total number N (in thousands) of mechanics employed by the airline industry can be modeled by

N 5 21.16t2 1 5.51t 1 65.34

where t is the number of years since 1997. What is the degree of the polynomial that represents N?

5. Short Response The height h (in feet) of a kangaroo’s jump can be modeled by h 5 216t2 1 18t where t is the time (in seconds) since the kangaroo jumped off of the ground. Find the zeros of the function. Explain what the zeros mean in this situation.

6. Short Response On Brian’s fi rst vertical jump, he has an initial vertical velocity of 40 inches per second. On his second vertical jump, Brian has an initial vertical velocity of 35 inches per second. For which jump is Brian in the air for more time? Justify your answer.

7. Extended Response During the period 1999–2003, the retail sales F (in millions of dollars) for fl ower gardening and the retail sales V (in millions of dollars) for vegetable gardening can be modeled by

F 5 93.4t3 2 642.5t2 1 837.6t 1 3956.5

V 5 50.9t3 2 198.6t2 2 317.1t 1 2602.8


a. Write an equation that models the total retail sales S (in millions of dollars) of fl ower gardening and vegetable gardening as a function of the number of years since 1999.

b. Find the total retail sales in these types of gardening in 1999 and 2003.

c. What was the average rate of change in total retail sales from 1999 to 2003? Explain how you found this rate.

Problem Solving Workshop:Mixed Problem SolvingFor use with pages 5722598

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147Algebra 1


In Exercises 1–5, fi nd a polynomial that has these given roots. Write the result as a polynomial with x as the variable, in both factored form and standard form.

1. 1, 2, 3

2. 21, 0, 1

3. 0, 0, 1, 1

4. 0, 1 } 2 , 2

5. 21, 2 2 } 3 , 23

6. A rectangular pool whose long side is twice as long as its narrow side is being built. There will be a paved border around all sides of the pool that is 5 feet wide around three sides and 10 feet wide around one of the narrow ends to accommodate a diving platform. The total area of the pool and the border is 1650 square feet. Write an equation for the area of the pool and border where x represents the length of the short side of the pool.

7. In Exercise 6, fi nd the length of the sides of the pool.

8. Consider the equation x3 2 xy2 5 0. What are the possible values of x and y that make the equation hold true?

9. Consider the equation x4 2 x2y2 5 0. What are the possible values of x and y that make the equation hold true?

10. Consider the equation (x2 1 y2)(x2 2 y2) 5 0. What are the possible values of x and y that make the equation hold true?

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Match the trinomial with its correct factorization.

1. x2 2 4x 2 12 2. x2 2 x 2 12 3. x2 1 4x 2 12

A. (x 1 6)(x 2 2) B. (x 2 6)(x 1 2) C. (x 1 3)(x 2 4)

Factor the trinomial.

4. x2 1 6x 1 5 5. a2 1 10a 1 21 6. w2 1 8w 1 15

7. p2 2 3p 2 10 8. c2 1 10c 2 11 9. y2 1 5y 2 14

10. n2 2 4n 1 3 11. b2 2 5b 1 6 12. r2 2 12r 1 35

13. z2 1 7z 1 12 14. s2 2 3s 2 18 15. d2 2 5d 2 24

Solve the equation.

16. x2 1 5x 1 4 5 0 17. d2 1 7d 1 10 5 0 18. p2 1 9p 1 14 5 0

19. w2 2 12w 1 11 5 0 20. n2 2 n 2 6 5 0 21. a2 2 12a 1 35 5 0

22. y2 2 4y 2 5 5 0 23. m2 1 2m 2 15 5 0 24. b2 1 6b 2 7 5 0

Match the equivalent equations.

25. s(s 2 6) 5 28 26. s(s 2 2) 5 8 27. s(s 1 2) 5 8

A. s2 2 2s 2 8 5 0 B. s2 1 2s 2 8 5 0 C. s2 2 6s 1 8 5 0

Solve the equation.

28. w(w 1 1) 5 12 29. x(x 2 3) 5 10 30. m(m 2 5) 5 6

31. b(b 1 4) 5 21 32. p(p 1 5) 5 36 33. r(r 2 3) 5 4

34. Boardwalk A boardwalk is being built along two sides

x ft

80 ft

120 ftx ft

of a beach area. The beach area is rectangular with a width of 80 feet and a length of 120 feet. The boardwalk will have the same width on each side of the beach area. If the combined area of the beach and the boardwalk is 16,500 square feet, then the area can be modeled by (x 1 80)(x 1 120) 5 16,500. How wide should the boardwalk be?

35. Note Board Design You are designing a note board

1.5 ft(x 1 1) ft

x ftCorkboardDry

eraseboard

that is made of corkboard and dry erase board. The area of the corkboard is 6 square feet.

a. Write an equation for the area of the corkboard.

b. Find the dimensions of the corkboard.

c. Find the area of the dry erase board.

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149Algebra 1



1. x2 1 8x 1 7 2. b2 2 7b 1 10 3. w2 2 12w 2 13

4. p2 1 10p 1 25 5. m2 2 10m 1 24 6. y2 2 5y 2 24

7. a2 1 13a 1 36 8. n2 1 2n 2 48 9. z2 2 14z 1 40

Solve the equation.

10. y2 1 17y 1 72 5 0 11. a2 2 9a 2 36 5 0 12. w2 2 13w 1 42 5 0

13. m2 2 5m 2 14 5 0 14. x2 1 11x 1 24 5 0 15. n2 2 12n 1 27 5 0

16. d2 1 5d 2 50 5 0 17. p2 1 16p 1 48 5 0 18. z2 2 z 2 30 5 0

Find the zeros of the polynomial function.

19. f (x) 5 x2 2 5x 2 36 20. g(x) 5 x2 1 8x 2 20 21. h(x) 5 x2 2 11x 1 24

22. f (x) 5 x2 1 11x 1 28 23. g(x) 5 x2 1 11x 2 12 24. h(x) 5 x2 1 3x 2 18

Solve the equation.

25. x(x 1 17) 5 260 26. p(p 2 4) 5 32 27. w(w 1 8) 5 215

28. n(n 1 6) 5 7 29. s2 2 3(s 1 2) 5 4 30. d2 1 18(d 1 4) 5 29

31. Patio Area A community center is building a patio area 100 ftx ft

x ft

50 ft

along two sides of its pool. The pool is rectangular with a width of 50 feet and a length of 100 feet. The patio area will have the same width on each side of the pool.

a. Write a polynomial that represents the combined area of the pool and the patio area.

b. The combined area of the pool and patio area should be 8400 square feet. How wide should the patio area be?

32. Area Rug You are creating your own area rug from a x in.

3 in.

4 in.

x in.Area rug

square piece of remnant carpeting. You plan on cutting 4 inches from the length and 3 inches from the width. The area of the resulting area rug is 1056 square inches.

a. Write a polynomial that represents the area of your area rug.

b. What is the perimeter of the original piece of remnant carpeting?

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LESSON



1. x2 2 x 2 56 2. m2 1 14m 1 48 3. y2 2 15y 1 54

4. p2 1 12p 1 20 5. w2 2 14w 1 45 6. x2 1 2x 2 24

Solve the equation.

7. n2 2 11n 2 60 5 0 8. z2 1 22z 1 121 5 0 9. c2 2 24c 1 144 5 0

10. x2 1 5x 2 500 5 0 11. b2 1 b 2 132 5 0 12. m2 1 17m 1 72 5 0

13. r2 2 4r 2 60 5 0 14. p2 2 6p 2 72 5 0 15. y2 2 16y 1 64 5 0


16. f (x) 5 x2 1 30x 1 225 17. h(x) 5 x2 2 5x 2 150 18. g(x) 5 x2 2 13x 1 30

19. g(x) 5 x2 2 10x 2 600 20. f (x) 5 x2 1 16x 1 28 21. f (x) 5 x2 1 13x 1 40

Solve the equation.

22. x(x 2 4) 5 21 23. b(b 1 2) 5 24 24. n(n 2 11) 5 224

25. x2 1 13(x 1 2) 5 210 26. x2 2 10(x 1 2) 5 4 27. y(y 2 15) 5 256

28. x2 1 2 1 1 } 2 x 2 10 2 5 0 29. x(x 1 17) 5 242 30. c(c 2 11) 5 218

31. Zoo Exhibit A zoo is building a walkway along two sides of 400 ftx ft

x ft

200 ft

an exhibit. The exhibit is rectangular with a width of 400 feet and a length of 200 feet. The walkway will have the same width on each side of the exhibit.

a. Write a polynomial that represents the combined area of the exhibit and the walkway.

b. The combined area of the exhibit and walkway should be 95,625 square feet. How wide should the walkway be?

c. If concrete costs $15 per square foot, how much will it cost to pave the walkway?

32. Fish Pond A rectangular fi sh pond is positioned in the center

(a 1 25) ft(a 1 5) ft (a 1 5) ft

(a 1 15) ft

a ft

a ft

of a rectangular grassy area, as shown. The area of the pond is 2000 square feet.

a. Use the dimensions given in the diagram to fi nd the dimensions of the pond.

b. The combined area of the pond and the surrounding grassy area is 9900 square feet. Find the length and width of the grassy area.

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151Algebra 1



LESSON

9.5

EXAMPLE 2

Factor when b and c are positive

Factor x2 1 10x 1 24.

Solution

Find two positive factors of 24 whose sum is 10. Make an organized list.

Factors of 24 Sum of factors

24, 1 24 1 1 5 25

12, 2 12 1 2 5 14

8, 3 8 1 3 5 11

6, 4 6 1 4 5 10

The factors 6 and 4 have a sum of 10, so they are the correct values of p and q.

x2 1 10x 1 24 5 (x 1 6)(x 1 4)

CHECK (x 1 6)(x 1 4) 5 x2 1 4x 1 6x 1 24 Multiply binomials.

5 x2 1 10x 1 24 ✓ Simplify.

EXAMPLE 1

✗

✗

✗

correct sum

Factor when b is negative and c is positive

Factor w2 2 10w 1 9.

Solution

Because b is negative and c is positive, p and q must be negative.


29, 21 29 1 (21) 5 210

23, 23 23 1 (23) 5 26


w2 2 10w 1 9 5 (x 2 9)(x 2 1)



1. x2 1 10x 1 16 2. y2 1 6y 1 5 3. z2 2 7z 1 12

correct sum

✗

Factor trinomials of the form x2 1 bx 1 c. GOAL

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Solve a polynomial equation

Solve the equation h2 2 4h 5 21.

Solution

h2 2 4h 5 21 Write original equation.

h2 2 4h 2 21 5 0 Subtract 21 from each side.

(h 1 3)(h 2 7) 5 0 Factor left side.

h 1 3 5 0 or h 2 7 5 0 Zero-product property

h 5 23 or h 5 7 Solve for h.


Exercise for Example 4

7. Solve the equation x2 1 30 5 11x.

EXAMPLE 4


LESSON

9.5

Factor when b is positive and c is negative

Factor k2 1 6x 2 7.

Solution

Because c is negative, p and q must have different signs.


27, 1 27 1 1 5 26

7, 21 7 1 (21) 5 6


k2 1 6k 2 7 5 (x 1 7)(x 2 1)



4. x2 2 10x 2 11

5. y2 1 2y 2 63

6. z2 2 5z 2 36

EXAMPLE 3

✗

correct sum

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153Algebra 1


LESSON


In Exercises 1–5, use the given factor formula and the substitution method to factor the expression.

x2 1 (a 1 b)x 1 ab 5 (x 1 a)(x 1 b)

Example: y 1 y1/2 2 6

Solution: Let x 5 y1/2. Then x2 5 y and the expression y 1 y1/2 2 6 becomes x2 1 x 2 6. Now factor this expression using the given factor formula.

x2 1 x 2 6 5 (x 1 3)(x 2 2)

Finally, replace x with y1/2.

(x 1 3)(x 2 2) 5 (y1/2 1 3)(y1/2 2 2)

1. y2/3 1 6y1/3 1 8

2. y4 2 y2 2 12

3. 1 }

y2 2 8 }

y 2 9

4. 5 Ï}

y2 1 16 5 Ï}

y 1 48

5. Ï}

y 1 12 4 Ï}

y 1 11

In Exercises 6–10, use substitution to factor, then solve for x.

6. x4 2 3x2 2 4 5 0

7. x4 2 13x2 1 36 5 0

8. 1 }

x2 2 1 }

x 2 12 5 0

9. x 2 Ï}

x 2 6 5 0

10. x4 2 16x2 1 48 5 0

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1. 4x2 2 2x 2 2 A. (4x 1 1)(x 2 2)

2. 4x2 2 7x 2 2 B. (2x 1 1)(2x 2 2)

3. 4x2 1 7x 2 2 C. (4x 2 1)(x 1 2)


4. 2x2 2 2x 1 15 5. 2m2 1 3m 2 2 6. 2p2 1 5p 1 14

7. 2w2 1 7w 1 3 8. 3y2 1 5y 1 2 9. 2b2 1 b 2 1

10. 3n2 2 3 11. 5a2 1 13a 2 6 12. 2z2 1 9z 2 5

13. 7d2 2 15d 1 2 14. 2r2 2 12r 1 10 15. 6s2 2 13s 1 2

Solve the equation.

16. 2x2 1 7x 2 15 5 0 17. 3n2 1 13n 1 4 5 0 18. 4b2 1 2b 2 2 5 0

19. 2m2 1 5m 2 3 5 0 20. 3p2 1 11p 2 4 5 0 21. 3y2 1 11y 1 10 5 0

22. 4r2 1 8r 1 3 5 0 23. 9w2 1 3w 2 2 5 0 24. 5a2 2 8a 2 4 5 0

25. 3c2 1 19c 2 14 5 0 26. 8z2 1 6z 1 1 5 0 27. 12d2 1 14d 2 6 5 0


28. f (x) 5 2x2 2 4x 1 5 29. g(x) 5 3x2 2 13x 2 10 30. h(x) 5 22x2 1 9x 1 5

31. g(x) 5 2x2 1 5x 2 6 32. f (x) 5 4x2 2 9x 1 2 33. g(x) 5 22x2 2 9x 1 18

34. h(x) 5 2x2 1 7x 2 4 35. h(x) 5 6x2 1 3x 2 9 36. f (x) 5 24x2 2 9x 2 2

37. Ball Toss A ball is tossed into the air from a height of 8 feet with an initial velocity of 8 feet per second. Find the time t (in seconds) it takes for the object to reach the ground by solving the equation 216t2 1 8t 1 8 5 0.

38. Wallpaper You trimmed a large strip of wallpaper from a scrap

(x 2 6) in.

4x in.

x in.

(4x 2 15) in.

to fi t into the corner of a wall you are wallpapering. You trimmed 15 inches from the length and 6 inches from the width. The area of the resulting strip of wallpaper is 684 square inches.

a. If the length of the original strip of wallpaper is four times the original width, write a polynomial that represents the area of the trimmed strip of wallpaper.

b. What are the dimensions of the original scrap of wallpaper?

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155Algebra 1



1. 2x2 2 3x 1 28 2. 2p2 1 8p 2 12 3. 2m2 2 13m 2 40

4. 2y2 1 15y 1 7 5. 3a2 2 13a 1 4 6. 5d2 2 18d 2 8

7. 6c2 1 7c 1 2 8. 10n2 2 26n 1 12 9. 12w2 1 8w 2 15

10. 22b2 2 5b 1 12 11. 23r2 2 17r 2 10 12. 24s2 1 6s 1 4

Solve the equation.

13. 2x2 1 x 1 20 5 0 14. 2m2 2 10m 2 16 5 0 15. 2p2 1 13p 2 42 5 0

16. 2c2 2 11c 1 5 5 0 17. 2y2 1 y 2 10 5 0 18. 16r2 1 18r 1 5 5 0

19. 3w2 1 19w 1 6 5 0 20. 12n2 2 11n 1 2 5 0 21. 15a2 2 2a 2 8 5 0

22. 22x2 2 9x 2 4 5 0 23. 23s2 2 s 1 10 5 0 24. 8d2 2 6d 2 5 5 0


25. f (x) 5 2x2 1 6x 1 27 26. f (x) 5 6x2 1 45x 2 24 27. f (x) 5 23x2 2 14x 1 24

28. f (x) 5 22x2 1 2x 1 4 29. f (x) 5 3x2 2 17x 1 20 30. f (x) 5 8x2 1 53x 2 21

31. f (x) 5 4x2 1 29x 1 30 32. f (x) 5 22x2 2 17x 1 30 33. f (x) 5 10x2 1 5x 2 5

34. Summer Business Your weekly revenue R (in dollars) from your tie-dye T-shirt business can be modeled by

R 5 22t2 1 87t 1 90

where t represents the number of weeks since the fi rst week you started selling T-shirts. How much did you make your fi rst week?

35. Cliff Diving A cliff diver jumps from a ledge 96 feet above the ocean with an initial upward velocity of 16 feet per second. How long will it take until the diver enters the water?

36. Wall Mirror You plan on making a wall hanging that contains two

x in.

x in.

4 in.

2 in.2 in.

2x in.

2x in.

small mirrors as shown.

a. Write a polynomial that represents the area of the wall hanging.

b. The area of the wall hanging will be 480 square inches. Find the length and width of the mirrors you will use.

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LESSON



1. 2x2 2 11x 1 180 2. 22m2 1 19m 2 24 3. 23p2 1 26p 1 40

4. 8r2 1 26r 1 15 5. 14b2 1 38b 2 12 6. 10y2 2 36y 1 18

Solve the equation.

7. 232x2 2 28x 1 15 5 0 8. 28n2 2 16n 2 6 5 0 9. 215s2 1 4s 1 4 5 0

10. 26p2 2 17p 2 5 5 0 11. 63m2 2 31m 2 10 5 0 12. 40r2 2 42r 1 9 5 0

13. 16a2 2 2a 2 3 5 0 14. 215d2 2 2d 1 8 5 0 15. 26y2 1 32y 2 10 5 0


16. f (x) 5 216x2 1 50x 2 25 17. h(x) 5 220x2 1 44x 2 21 18. h(x) 5 20x2 1 18x 2 44

19. g(x) 5 236x2 2 30x 2 6 20. f (x) 5 12x2 1 8x 2 15 21. g(x) 5 21x2 1 14x 2 7

Multiply each side of the equation by an appropriate power of 10 to obtain integer coeffi cients. Then solve the equation.

22. 0.2x2 2 0.3x 2 3.5 5 0 23. r2 1 0.6r 2 0.4 5 0 24. 0.8m2 1 m 2 0.3 5 0

25. 20.5x2 1 1.2x 5 0.4 26. 1.2(p2 1 1) 5 2.5p 27. 20.36n2 1 0.6n 2 0.25 5 0

28. Baseball A baseball player releases a baseball at a height of 7 feet with an initial velocity of 54 feet per second. How long will it take the ball to reach the ground?

29. Rocket Launch A miniature rocket is launched off a roof 20 feet above the ground with an initial velocity of 22 feet per second. How much time will elapse before the rocket reaches the ground?

30. Frog Jump A frog jumps from the ground into the air with an initial vertical velocity of 8 feet per second.

a. Write an equation that gives the frog’s height (in feet) as a function of the time (in seconds) since it left the ground.

b. After how many seconds is the frog 12 inches above the ground?

c. Does the frog go any higher than 12 inches? Explain your reasoning using your answer from part (b).

d. Suppose the frog now jumps from 4 feet above the ground with the same initial vertical velocity. Write an equation that gives the frog’s height (in feet) as a function of the time (in seconds) since it left the ground.

e. Should the frog reach the ground in the same time in both jumps? Explain why or why not.

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157Algebra 1


Factor trinomials of the form ax2 1 bx 1 c.GOAL

Factor when b is negative and c is positive

Factor 5n2 2 12n 1 7.

Solution

Because b is negative and c is positive, both factors of c must be negative. Make a table to organize your work.

You must consider the order of the factors of 7, because the x-terms of the possible factorization are different.

Factorsof 5

Factorsof 7

Possiblefactorization

Middle termwhen multiplied

1, 5 21, 27 (n 2 1)(5n 2 7) 25n 2 7n 5 212n

1, 5 27, 21 (n 2 7)(5n 2 1) 2n 2 35n 5 236n

5n2 2 12n 1 7 5 (n 2 1)(5n 2 7)

EXAMPLE 1

correct

✗


LESSON

9.6

EXAMPLE 2 Factor when b is negative and c is negative

Factor 3m2 2 5m 2 22.

Solution

Because b is negative and c is negative, p and q must have different signs.

Factorsof 3

Factorsof 22



1, 3 1, 222 (m 1 1)(3m 2 22) 3m 2 22m 5 219m

1, 3 21, 22 (m 2 1)(3m 1 22) 22m 2 3m 5 19m

1, 3 2, 211 (m 1 2)(3m 2 11) 211m 1 6m 5 25m

1, 3 211, 2 (m 2 11)(3m 1 2) 2m 2 33m 5 231m

3m2 2 5m 2 22 5 (m 1 2)(3m 2 11)



1. 7a2 2 50a 1 7 2. 4b2 2 8b 2 5 3. 6c2 1 5c 2 14

✗

✗

correct

✗

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LESSON

9.6

Factor when a is negative

Factor 22x2 1 9x 2 9.

Solution

STEP 1 Factor 21 from each term of the trinomial.

22x2 1 9x 2 9 5 2(2x2 2 9x 1 9)

STEP 2 Factor the trinomial 2x2 2 9x 1 9. Because b is negative and c is positive, both factors of c must be negative. Use a table to organize information about the factors of a and c.

Factorsof 2

Factorsof 9



1, 2 21, 29 (x 2 1)(2x 2 9) 29x 2 2x 5 211x

1, 2 29, 21 (x 2 9)(2x 2 1) 2x 2 18x 5 219x

1, 2 23, 23 (x 2 3)(2x 2 3) 23x 2 6x 5 29x

22x2 1 9x 2 9 5 2(x 2 3)(2x 2 3)



4. 23r2 2 7r 2 4

5. 23s2 1 8s 1 16

6. 28t2 1 6t 2 1

EXAMPLE 3

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159Algebra 1



Multiple Representations In Example 4 on page 613, you saw how to solve a problem about a discus by factoring a quadratic equation. You can also solve the problem by using a graph.

Discus An athlete throws a discus from an initial height of 6 feet and with an initial vertical velocity of 46 feet per second. Write an equation that gives the height (in feet) of the discus as a function of the time (in seconds) since it left the athlete’s hand. After how many seconds does the discus hit the ground?

PROBLEM

Using a Graph You can solve the problem by using a graph.

STEP 1 Use the vertical motion model to write an equation for the height h (in feet) of the discus. In this case, v 5 46 and s 5 6.

h 5 216t2 1 vt 1 s Vertical motion model

h 5 216t2 1 46t 1 6 Substitute 46 for v and 6 for s.

STEP 2 Graph the equation for the height of the discus

ZeroX=3 Y=0

using a graphing calculator. Graph y1 5 216x2 1 46x 1 6. Because you are looking for when the discus hits the ground, you need to fi nd the time when the height is 0.

STEP 3 Find the zeros of the graph by using the zero feature on your calculator. You only need to consider positive values of x because a negative solution does not make sense in this situation. There is a zero at (3, 0). The discus hits the ground after 3 seconds.

METHOD

1. Cliff Diving A cliff diver jumps from a ledge 88 feet above the ocean with an initial upward velocity of 12 feet per second. How long will it take until the diver enters the water?

2. Error Analysis Describe and correct the error made in Exercise 1.

216t2 2 12t 1 88 5 0

24(4t2 1 3t 2 22) 5 0

24(4t 1 11)(t 2 2) 5 0

The cliff diver enters the water after 2 seconds.

3. Tennis A tennis ball is hit when it is 6 feet off the ground with an initial upward velocity of 20 feet per second. How long does it take for the tennis ball to hit the ground?

4. Football You throw a football from a height of 6 feet into the air with an initial vertical velocity of 12 feet per second. The football is caught at a height of 2 feet. After how many seconds is the football caught?

5. What If? Suppose in Exercise 4 the football is thrown with an initial vertical velocity of 30 feet per second. After how many seconds is the football caught?

PRACTICE

Problem Solving Workshop:Using Alternative MethodsFor use with pages 6102617

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LESSON


In Exercises 1–5, use the substitution method to factor the expression.

Example: 3y 1 11y1/2 2 4

Solution: Let x 5 y1/2. Then x2 5 y and the expression 3y 1 11y1/2 2 4 becomes 3x2 1 11x 2 4. Now factor this expression.

3x2 1 11x 2 4 5 (3x 2 1)(x 1 4)

Finally, replace x with y1/2.

(3x 2 1)(x 1 4) 5 (3y1/2 2 1)(y1/2 1 4)

1. 4y2/3 1 12y1/3 1 5

2. 8y4 2 10y2 2 3

3. 9 }

y2 2 12

} y 2 5

4. 7 3 Ï}

y2 1 36 3 Ï}

y 1 5

5. 28 4 Ï}

y 1 8 Ï}

y 1 6


6. 6x6 1 x3 2 2 5 0

7. 9x4 2 12x2 2 5 5 0

8. 5 }

x2 1 28

} x 1 15 5 0

9. 3x 2 Ï}

x 2 14 5 0

10. 5x4 1 21x2 2 20 5 0

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161Algebra 1



1. x2 2 25 2. x2 1 10x 1 25 3. x2 2 10x 1 25

A. (x 1 5)2 B. (x 2 5)(x 1 5) C. (x 2 5)2

Factor the difference of two squares.

4. x2 2 1 5. b2 2 81 6. m2 2 100

7. p2 2 225 8. 4y2 2 1 9. 16n2 2 25

10. 9w2 2 100 11. 64z2 2 36 12. 49d2 2 25

13. 4r2 2 121 14. 9s2 2 144 15. c2 2 625

Factor the perfect square trinomial.

16. x2 1 6x 1 9 17. b2 1 10b 1 25 18. w2 2 12w 1 36

19. m2 2 8m 1 16 20. r2 2 20r 1 100 21. z2 1 16z 1 64

22. s2 1 22s 1 121 23. x2 2 16x 1 64 24. 4c2 1 4c 1 1

25. 16d2 1 8d 1 1 26. 9y2 2 6y 1 1 27. 9p2 2 12p 1 4

Solve the equation.

28. x2 2 9 5 0 29. p2 1 14p 1 49 5 0 30. d2 2 10d 1 25 5 0

31. 25m2 2 1 5 0 32. r2 2 2r 1 1 5 0 33. n2 1 20n 1 100 5 0

34. 4y2 2 9 5 0 35. 36x2 2 64 5 0 36. w2 1 4w 1 4 5 0

37. Washers Washers are available in many different sizes.

x

y

a. Write and factor an expression for the area of one side of the washer. Leave your answer in terms of π.

b. Find the area of one side of the washer when x 5 8 centimeters and y 5 3 centimeters.

38. Cherry Tree A cherry falls from a tree branch that is 9 feet above the ground.

a. How far above the ground is the cherry after 0.2 second?

b. After how many seconds does the cherry reach the ground?

39. Wind Chime A wind chime falls from a roof that is 10 feet above the ground.

a. How far above the ground is the wind chime after 0.5 second?

b. After how many seconds does the wind chime reach the ground?

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Factor the polynomial.

1. x2 2 36 2. 25p2 2 144 3. 4b2 2 100

4. 36m2 2 81 5. 22x2 1 32 6. 24r2 1 100s2

7. y2 1 24y 1 144 8. 9c2 1 24c 1 16 9. 25w2 2 20w 1 4

10. 16n2 2 56n 1 49 11. 218a2 2 12a 2 2 12. 20z2 2 140z 1 245

Solve the equation.

13. x2 1 14x 1 49 5 0 14. 8w2 5 50 15. 64p2 2 16p 1 1 5 0

16. 8a2 2 72 5 0 17. 3m2 1 30m 1 75 5 0 18. 24y2 1 32y 2 64 5 0

19. 25x2 1 125 5 0 20. 27r2 1 140r 2 700 5 0 21. 24w2 2 24w 1 6 5 0

22. 18n2 1 60n 1 50 5 0 23. 25

} 2 x2 1 15x 1

9 } 2 5 0 24. 4x2 5

9 } 16

Find the value of x in the geometric shape.

25. Area 5 144π cm2 26. Area 5 225 in.2

(x 1 4) cm

(4x 1 3) in.

27. Measuring Tape A measuring tape drops from a roof that is 16 feet above the ground. After how many seconds does the measuring tape land on the ground?

28. Playground A curved ladder that children can climb on can be modeled by the equation

y 5 2 1 } 20 x2 1 x

where x and y are measured in feet.

a. Make a table of values that shows the height of the ladder for x 5 0, 5, 10, 15, and 20 feet from the left end.

b. For what additional values of x does the equation make sense? Explain.

c. Plot the ordered pairs in the table from part (a) as

00

1

2

3

4

5

5 10 15 20Distance from left end (feet)

Heig

ht

(feet)

x

ypoints in the coordinate plane. Connect the points with a smooth curve.

d. At approximately what distance from the left end does the ladder reach a height of 5 feet? Check your answer algebraically.

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163Algebra 1


LESSON



1. 25x2 2 81 2. 225p2 2 100 3. 121w2 2 625

4. 36m2 2 64 5. 9 } 16

r2 2 1 } 16 6. 81x2 2 49y2

7. 23y2 2 48y 2 192 8. 4n2 2 40n 1 100 9. 12z2 1 12z 1 3

10. 24a2 2 120ab 1 150b2 11. 218s2 2 48st 2 32t2 12. 5z2 1 2z 1 1 } 5

Solve the equation.

13. 25m2 2 64 5 0 14. 2p2 1 36p 1 162 5 0 15. 216r2 1 196 5 0

16. 3w2 2 60w 1 300 5 0 17. 36x2 2 132x 1 121 5 0 18. 225a2 2 120a 1 16 5 0

19. 275y2 2 90y 2 27 5 0 20. 196n2 2 224n 1 64 5 0 21. 160z2 5 640

22. 0.9r2 2 4.8r 1 6.4 5 0 23. 25

} 2 b2 1 5b 1

1 } 2 5 0 24. 296d2 1 144d 2 54 5 0

Determine the value(s) of k that make the expression a perfect square trinomial.

25. 81x2 1 kx 1 25 26. 100x2 1 kx 1 49 27. 25x2 2 60x 1 k

28. kx2 1 72x 1 81 29. 4x2 2 12x 1 k 30. 49x2 1 kxy 1 4y2

31. Squirrel A squirrel jumps straight up with an initial vertical velocity of 16 feet per second. How many times does the squirrel reach a height of 4 feet? Explain your answer.

32. Foot Bridge A foot bridge that spans a small creek can be modeled by the equation

y 5 2 3 } 800 x2 1

3 } 10 x

where x and y are measured in feet.

a. Make a table of values that shows the height of the bridge for x 5 0, 20, 40, 60, and 80 feet from the left end.

b. For what additional values of x does the equation make sense? Explain.

c. Plot the ordered pairs in the table from part (a)

00

1

2

3

4

5

6

20 40 60 8010 30 50 70Distance from left end (feet)

Heig

ht

(feet)

x

yas points in the coordinate plane. Connect the points with a smooth curve.

d. At approximately what distance from the left end does the bridge reach a height of 6 feet? Check your answer algebraically.

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Factor special products.

VocabularyThe pattern for fi nding the square of a binomial gives you the pattern for factoring trinomials of the form a2 1 2ab 1 b2 and a2 2 2ab 1 b2. These are called perfect square trinomials.

GOAL

Factor the difference of squares


a. r2 2 81 5 r2 2 92 Write as a2 2 b2.

5 (r 2 9)(r 1 9) Difference of two squares pattern

b. 9s2 2 4t2 5 (3s)2 2 (2t)2 Write as a2 2 b2.

5 (3s 2 2t)(3s 1 2t) Difference of two squares pattern

c. 80 2 125q2 5 5(16 2 25q2) Factor out common factor.

5 5[42 2 (5q)2] Write 16 2 25q2 as a2 2 b2.

5 5(2 2 5q)(2 1 5q) Difference of two squares pattern



1. m2 2 121

2. 9n2 2 64

3. 3y2 2 147z2

EXAMPLE 1

Review for MasteryFor use with pages 6182623

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165Algebra 1




4. m2 2 1 }

2 m 1

1 }

16 5. 16r2 1 40rs 1 25s2 6. 36x2 2 36x 1 9


Solve the equation q2 2 100 5 0.

Solution

q2 2 100 5 0 Write original equation.

q2 2 102 5 0 Write left side as a2 2 b2.

(q 1 10)(q 2 10) 5 0 Difference of two squares pattern

q 1 10 5 0 or q 2 10 5 0 Zero-product property

q 5 210 or q 5 10 Solve for q.



Solve the equation.

7. r2 2 10r 1 25 5 0 8. 16m2 2 81 5 0

EXAMPLE 3


LESSON

9.7

Factor perfect square trinomials


a. x2 1 14x 1 49 5 x2 1 2(x)(7) 1 72 Write as a2 1 2ab 1 b2.

5 (x 1 7)2 Perfect square trinomial pattern

b. 144y2 2 120y 1 25 5 (12y)2 2 2(12y)(5) 1 52 Write as a2 2 2ab 1 b2.

5 (12y 2 5)2 Perfect square trinomial pattern

c. 150z2 2 60z 1 6 5 6(25z2 2 10z 1 1) Factor out common factor.

5 6[(5z)2 2 2(5z)(1) 1 12] Write 25z2 2 10z 1 1 as a2 2 2ab 1 b2.

5 6(5z 2 1)2 Perfect square trinomial pattern

EXAMPLE 2

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In Exercises 1–3, factor the expression.

1. x2 2 6xy 1 9y2

2. 4x2 2 20xy 1 25y2

3. 25x2y2 1 40xy 116

In Exercises 4 and 5, use the substitution method to factor the expression.

Example: 16(y 1 3)2 2 40(y 1 3) 1 25

Solution: Let x 5 y 1 3. Then the expression 16(y 1 3)2 2 40(y 1 3) 1 25 becomes 16x2 2 40x 1 25. Now factor this expression.

16x2 2 40x 1 25 5 (4x 2 5)2

Finally, replace x with (y 1 3).

(4x 2 5)2 5 [4(y 1 3) 2 5]2 5 (4y 1 7)2

4. 4(x 2 7)2 2 24(x 2 7) 1 36

5. 25(x 1 3)2 2 20(x 1 3) 1 4


6. (x 2 5)4 2 10(x 2 5)2 1 25 5 0

7. 4(2x 2 7)6 2 28(2x 2 7)3 1 49 5 0

8. 25(x 1 2)2 1 30(x 1 2) 1 9 5 0

9. 16

} x2 1

56 }

x 1 49 5 0

10. 9 }

(x 1 1)2 1 12 }

x 1 1 1 4 5 0

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167Algebra 1



1. 2x(x 1 5) 2 (x 1 5) 2. 2x(x 1 5) 1 (x 1 5) 3. 2x(x 2 5) 2 (x 2 5)

A. (2x 1 1)(x 1 5) B. (2x 2 1)(x 2 5) C. (2x 2 1)(x 1 5)

Factor the expression.

4. x(x 1 4) 1 (x 1 4) 5. b(b 1 3) 2 (b 1 3) 6. 2m(m 1 1) 1 (m 1 1)

7. 5r(r 1 2) 2 (r 1 2) 8. w(w 1 6) 1 3(w 1 6) 9. y(y 1 4) 2 6(y 1 4)

10. n(n 2 3) 2 7(n 2 3) 11. 3z(z 2 4) 1 8(z 2 4) 12. 2p(p 1 5) 2 3(p 1 5)

Factor the polynomial by grouping.

13. x2 1 x 1 3x 1 3 14. x2 2 x 1 2x 2 2 15. x2 1 8x 2 x 2 8

16. x3 2 5x2 1 2x 2 10 17. x3 2 4x2 2 6x 1 24 18. x3 1 3x2 1 5x 1 15

19. x3 2 x2 1 7x 2 7 20. x3 1 3x2 2 3x 2 9 21. x3 1 3x2 2 x 2 3

Determine whether the polynomial has been completely factored.

22. x4 1 x3 23. x2 1 1 24. 2x2 1 4

Factor the polynomial completely.

25. x5 2 x3 26. 4a4 2 25a2 27. 5y6 2 125y4

Solve the equation.

28. x3 1 x2 2 25x 2 25 5 0 29. x3 1 x2 2 16x 2 16 5 0 30. x3 2 x2 2 4x 1 4 5 0

31. x3 2 x2 2 9x 1 9 5 0 32. z3 2 4z 5 0 33. c4 2 64c2 5 0

34. Metal Plate You have a metal plate that you have drilled a hole into. The entire area enclosed by the metal plate is given by 5x2 1 12x 1 10 and the area of the hole is given by x2 1 2. Write an expression for the area in factored form of the plate that is left after the hole is drilled.

35. Storage Container A plastic storage container in the shape of a cylinder has a height of 8 inches and a volume of 72π cubic inches.

a. Write an equation for the volume of the storage container.

b. What is the radius of the storage container?

36. Tennis Ball For a science experiment, you toss a tennis ball from a height of 32 feet with an initial upward velocity of 16 feet per second. How long will it take the tennis ball to reach the ground?

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1. 4x(x 1 5) 2 3(x 1 5) 2. 12(a 2 3) 2 2a(a 2 3)

3. w2(w 1 8) 2 5(w 1 8) 4. 2b2(b 1 6) 1 3(b 1 6)

5. y(15 1 x) 2 (x 1 15) 6. 3x(4 1 y) 2 6(4 1 y)

Factor the polynomial by grouping.

7. x3 1 x2 1 5x 1 5 8. y3 2 14y2 1 y 2 14

9. m3 2 6m2 1 2m 2 12 10. p3 1 9p2 1 4p 1 36

11. t3 1 12t2 2 2t 2 24 12. 3n3 2 3n2 1 n 2 1


13. 7x3 1 28x2 14. 4m3 2 16m 15. 216p3 2 2p

16. 48r3 2 30r2 17. 15y 2 60y2 18. 18xy 2 24x2

19. 5m2 1 20m 1 40 20. 6x2 1 6x 2 120 21. 4z3 2 4z2 2 8z

22. 9x3 1 36x2 1 36 23. x3 1 x2 1 5x 1 5 24. d3 1 4d2 1 5d 1 20

Solve the equation.

25. 3x2 1 18x 1 24 5 0 26. 10x2 5 250 27. 4m2 2 28m 1 49 5 0

28. 12x2 1 18x 1 6 5 0 29. 18x2 2 48x 1 32 5 0 30. 218x2 2 60x 2 50 5 0

31. Countertop A countertop will have a hole drilled in it to hold a cylindrical container that will function as a utensil holder. The area of the entire countertop is given by 5x2 1 12x 1 7. The area of the hole is given by x2 1 2x 1 1. Write an expression for the area in factored form of the countertop that is left after the hole is drilled.

32. Film Canister A fi lm canister in the shape of a cylinder has a height of 8 centimeters and a volume of 32π cubic centimeters.

a. Write an equation for the volume of the fi lm canister.

b. What is the radius of the fi lm canister?

33. Badminton You hit a badminton birdie upward with a racket from a height of 4 feet with an initial velocity of 12 feet per second.

a. Write an equation that models this situation.

b. How high is the birdie at 0.1 second?

c. How high is the birdie at 0.25 second?

d. How long will it take the birdie to reach the ground?

Practice BFor use with pages 6242631

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169Algebra 1



1. 13a 2 26a2 2. 30xy 2 45x2 3. 22m2 2 16m 2 14

4. 14p2 2 35p 1 21 5. r3 1 10r2 1 25r 6. 5b4 1 40b3 1 80b2

7. 4n5 1 4n4 2 120n3 8. 7c3 2 28c2 1 28c 9. 210t2 2 5t 1 75

10. x2 1 9x 2 xy 2 9y 11. x3 1 5x2 2 8x 2 40 12. 9x2 2 64y2

13. 3x5y 2 243x3y 14. 8r3s4 2 72rs4 15. 25x3y 2 100x2y

Solve the equation.

16. 5x3 1 20x2 1 15x 5 0 17. 219x2 1 76 5 0 18. 218p3 2 21p2 1 15p 5 0

19. 48p2 2 675 5 0 20. 14x3 2 68x2 2 10x 5 0 21. 23n4 2 36n3 2 108n2 5 0

22. 20t4 1 28t3 5 24t2 23. 64t 5 12t2 1 45 24. 900x2 5 625

25. 16m4 2 81m2 5 0 26. 16x 1 280 5 8x2 27. 2r2 1 392 5 56r

28. 75a3 1 90a2 1 27a 5 0 29. 2p2 5 12p 1 54 30. 81x3 5 100x

31. Use factoring by grouping to show that a trinomial of the form a2 2 2ab 1 b2 can be factored as (a 2 b)2. Justify your steps.

32. Work Bench You are drilling holes into your work bench that will hold caddies for some of your gardening equipment. The area of the entire work bench before the holes are drilled is given by 24x2 1 5x. The area of one hole is given by 3x2 1 x 1 3. Write an expression for the area in factored form of the work bench that is left after the holes are drilled.

33. Poster Tube A poster tube in the shape of a cylinder has a height of 2 feet and

a volume of 1 }

2 π cubic feet.

a. Write an equation for the volume of the poster tube.

b. What is the radius of the poster tube?

34. Moon On the moon, the vertical motion model is given by h 5 2 16

} 6 t2 1 vt 1 s

where h is the height (in feet), v is the initial velocity (in feet per second), t is the time (in seconds), and s is the initial height (in feet). On the moon, an astronaut

tosses a baseball from a height of 64 feet with an initial upward velocity of 23 2 }

3 feet

per second. How long does it take the ball to reach the ground?

Practice CFor use with pages 6242631

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Factor polynomials completely.

VocabularyFactoring a common monomial from pairs of terms, then looking for a common binomial factor is called factor by grouping.

A polynomial of two or more terms is prime if it cannot be written as the product of polynomials of lesser degree using only integer coeffi cients and constants, and if the only common factors of its terms are 1 and –1.

A polynomial is factored completely if it is written as a monomial or as the product of a monomial (possibly 1 or 21) and one or more prime polynomials.

GOAL

Factor out a common binomial


a. 5x2(x 2 2) 2 3(x 2 2) b. 7y(5 2 y) 1 3( y 2 5)

Solution

a. 5x2(x 2 2) 2 3(x 2 2) 5 (x 2 2)(5x2 2 3)

b. The binomials 5 2 y and y 2 5 are opposites. Factor 21 from 5 2 y to obtain a common binomial factor.

7y(5 2 y) 1 3( y 2 5) 5 27y( y 2 5) 1 3( y 2 5) Factor 21 from (5 2 y).

5 ( y 2 5)(27y 1 3) Distributive property

EXAMPLE 1

Factor by grouping


a. m3 1 7m2 2 2m 2 14 b. n3 1 30 1 6n2 1 5n

Solution

a. m3 1 7m2 2 2m 2 14 5 (m3 1 7m2) 1 (22m 2 14) Group terms.

5 m2(m 1 7) 2 2(m 1 7) Factor each group.

5 (m 1 7)(m2 2 2) Distributive property

b. n3 1 30 1 6n2 1 5n 5 n3 1 6n2 1 5n 1 30 Rearrange terms.

5 (n3 1 6n2) 1 (5n 1 30) Group terms.

5 n2(n 1 6) 1 5(n 1 6) Factor each group.

5 (n 1 6)(n2 1 5) Distributive property


LESSON

9.8

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171Algebra 1



Solve the equation 7x3 1 14x2 5 105x.

Solution

7x3 1 14x2 5 105x Write original equation.

7x3 1 14x2 2 105x 5 0 Subtract 105x from each side.

7x(x2 1 2x 2 15) 5 0 Factor out 7x.

7x(x 1 5)(x 2 3) 5 0 Factor the trinomial.

7x 5 0 or x 1 5 5 0 or x 2 3 5 0 Zero-product property

x 5 0 or x 5 25 or x 5 3 Solve for x.

The roots of the equation are 0, 25, and 3.


Solve the equation.

4. 2c3 1 8c2 2 42c 5 0

5. 4x3 1 48x2 1 144x 5 0

6. 5r3 1 15r 5 20r2

EXAMPLE 3



1. 11x(x 2 8) 1 3(x 2 8)

2. 9x3 1 9x2 2 7x 2 7

3. 10x3 1 21y 2 35x2 2 6xy


LESSON

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LESSONS

9.5–9.8

1. Multi-Step Problem The length of a box is 12 inches more than its height. The width of the box is 3 inches less than its height.

a. Draw a diagram of the box. Label its dimensions in terms of the height h.

b. Write a polynomial that represents the volume of the box.

c. The box has a volume of 324 cubic inches. What are the length, width, and height of the box?

2. Open-Ended Describe a situation that can be modeled using the vertical motion model h 5 216t2 1 24t. Then fi nd the value of t when h 5 0. Explain what this value of t means in this situation.

3. Multi-Step Problem A block of wood has the dimensions shown.

(x 1 3) in.(x 2 5) in.

x in.

a. Write a polynomial that represents the surface area of the wood.

b. The wood has a surface area of 384 square inches. What are the length, width, and height of the block?

4. Short Response The shape of an underpass for cars can be modeled by the graph of the equation y 5 20.4x(x 2 14) where x and y are measured in feet. On a coordinate plane, the ground is represented by the x-axis. How wide is the underpass at its base? Explain how you found your answer.

5. Extended Response You hit a baseball straight up into the air. The baseball is hit with an initial vertical velocity of 60 feet per second when it is 4 feet off the ground.

a. Write an equation that gives the height (in feet) of the baseball as a function of the time (in seconds) since it was hit.

b. After how many seconds does the ball reach a height of 54 feet?

c. Does the ball reach a height of 54 feet more than once? Justify your answer.

6. Gridded Response While standing on a ladder, you drop a paintbrush from a height of 9 feet. After how many seconds does the paintbrush land on the ground?

7. Extended Response You want to make a box with no lid out of a 9 inch by 13 inch piece of cardboard. You cut out squares of the same size from each corner. Then you fold up the sides and tape them together.

x in.

9 in.

13 in.

x in.

a. Write a polynomial that represents the volume of the box.

b. Find the volume of the box for cut out square side lengths of 1 inch, 2 inches, 3 inches, and 4 inches. Which cut out side square length gives the largest volume?

c. Could a box be formed using cut out squares with side lengths of 5 inches? Explain why or why not.

Problem Solving Workshop:Mixed Problem SolvingFor use with pages 6002631

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173Algebra 1


In Exercises 1–5, factor the expression completely.

1. 8(y 1 3)3 1 22(y 1 3)2 1 15(y 1 3)

2. (y 2 1)4 2 16

3. (9x2 2 12x 1 4) 2 9

4. 21x2 1 15x 114x 1 10

5. 2y5 2 32y

In Exercises 6–10, factor completely to solve for x.

6. (x 1 3)2 1 3(x 1 3) 5 10

7. x5 5 81x

8. 8x2 1 14x 1 21 5 212x

9. 2x2 2 5x 1 30 5 12x

10. 1 }

x3 2 6 }

x2 52 9 } x

In Exercises 11 and 12, use the following information.

A roller coaster has a velocity v (in miles per hours) described by the polynomial v(t) 5 210t4 1 100t2 2 90 for times from t 5 1 to t 5 3 minutes.

11. Find the velocity of the roller coaster when t 5 2 minutes.

12. For what times on the interval from t 5 1 to t 5 3 minutes does the roller coaster have a velocity of 0?

LESSON


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Crossword Puzzle

Use the clues at the bottom of the page to fi ll in the correct vocabulary word from Chapter 9 in the crossword puzzle.

1

4

7

8

6

11

14

13

12

109

5

32

Across Down

2. x2 2 4x 1 4 is an example of a _______ 1. Using the distributive property to factor square trinomial. polynomials with four terms is called factoring by _______.

4. Solutions of an equation 3. 4x2 2 2x 1 1

7. A monomial or sum of monomials 5. Use this to multiply binomials

9. The number 6 in the polynomial 6x2 2 24 6. Writing a polynomial as a product of is called the _______ coeffi cient. other polynomials

11. 2x 8. A polynomial with two terms

13. Sum of the exponents of the variables 10. A polynomial that cannot be factored in a monomial using integer coeffi cients

14. An object propelled into the air but has 12. The height of a projectile can be described no power to keep itself in the air by the _______ motion model.

CHAPTER


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175Algebra 1




10.1: Graph y 5 ax2 1 c Graph simple quadratic functions.

• Solar Energy • Astronomy

• Sailing

10.2: Graph y 5 ax2 1 bx 1 c

Focus on Functions

Graph general quadratic functions.

Graph quadratic functions in intercept form.

• Suspension Bridges

• Spiders

• Architecture

10.3: Solve Quadratic Equations by Graphing

Solve quadratic equations by graphing.

• Sports • Soccer

• Diving

10.4: Use Square Roots to Solve Quadratic Equations

Solve a quadratic equation by fi nding square roots.

• Sports Event • Gemology

• Internet Usage

10.5: Solve Quadratic Equations by Completing the Square

Focus on Functions

Solve quadratic equations by completing the square.

Graph quadratic functions in vertex form.

• Crafts

• Landscaping

• Snowboarding

10.6: Solve Quadratic Equations by the Quadratic Formula

Solve quadratic equations using the quadratic for-mula.

• Film Production

• Advertising

• Cell Phones

10.7: Interpret the Discriminant Use the value of the discriminant.

• Fountains • Biology

• Food

10.8: Compare Linear, Exponential, and Quadratic Models

Compare linear, exponen-tial, and quadratic models.

• Cycling

• Lizards

• Nautilus



1. Graphing quadratic functions

2. Solving quadratic functions

3. Comparing linear, exponential, and quadratic models

CHAPTER


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CHAPTER



Lesson Exercise

10.1 What is the vertex of the graph of the function y 5 2 1 } 2 x2 1 5?

10.2 Find the axis of symmetry and the vertex of the graph of the function y 5 22x2 1 6x 2 4.

Focus on Functions

Graph y 5 2(x 1 1)(x 2 3). Label the vertex, axis of symmetry, and x-intercepts.

10.3 Find the zeros of f (x) 5 2x2 2 2x 1 8.

10.4 Solve the equation 2x2 2 25 5 103 by using square roots. Round your solution to the nearest hundredth, if necessary.

10.5 Solve the equation x2 1 16x 1 20 5 0 by completing the square. Round your solution to the nearest hundredth, if necessary.

Focus on Functions

Write y 5 x2 2 14x 1 48 in vertex form. Then graph the function. Label the vertex and axis of symmetry.

10.6 Use the quadratic formula to solve the equation 5x2 1 12x 1 4 5 0.

10.7 Tell whether the equation 8x2 2 8x 5 22 has two, one, or no solution.

10.8 Tell whether the table of values represents x 23 22 21 0 1 2

y 4 1 0 1 4 9a linear function, an exponential function, or a quadratic function.


Directions Write a function that gives the surface area (in square feet) of a room in your home. Investigate how much it would cost to paint fi ve of the surfaces and carpet (or tile) the sixth. (Remember to subtract the area of doors and windows.)

Answers10.1: (0, 5) 10.2: axis of symmetry: x 5

3 }

2 ; vertex: 1 3 }

2 , 1 }

2 2 Focus on Functions:

x

y

O1

1(1, 0)(3, 0)

(1, 4)

x 1

10.3: 2, 24 10.4: 8, 8 10.5: 21.37 and 214.63 Focus on

Functions: 10.6: 22 and 2 2 }

5 10.7: one solution 10.8: quadratic; y 5 (x 1 1)2

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177Algebra 1




10.1: Grafi car y 5 ax2 1 c Grafi car funciones cuadráticas simples

• Energía solar • Astronomía• Navegación

10.2: Grafi car y 5 ax2 1 bx 1 c


Grafi car funciones cuadráticas generales

Grafi car funciones cuadráticas en forma de intersección

• Puentes colgantes• Arañas• Arquitectura

10.3: Resolver ecuaciones cuadráticas al grafi car

Resolver ecuaciones cuadráticas al grafi car

• Deportes • Fútbol• Zambullidas

10.4: Usar raíces cuadradas para resolver ecuaciones cuadráticas

Resolver ecuaciones cuadráticas al elevar al cuadrado

• Evento deportivo • Joyería• Uso de Internet

10.5: Resolver ecuaciones cuadráticas al elevar al cuadrado


Grafi car funciones cuadráticas en forma de vértice

Grafi car funciones cuadráticas en forma de vértice

• Manualidades • Jardinería• Hacer snowboard

10.6: Resolver ecuaciones cuadráticas por la fórmula cuadrática

Resolver ecuaciones cuadráticas usando la fórmula cuadrática

• Rodaje de película• Publicidad• Teléfonos celulares

10.7: Interpretar el discriminante

Usar el valor del discriminante

• Fuentes • Biología• Comida

10.8: Comparar modelos lineales, exponenciales y cuadráticos

Comparar modelos lineales, exponenciales y cuadráticos

• Ciclismo• Lagartijas• Sepia



1. Grafi car funciones cuadráticas

2. Resolver funciones cuadráticas

3. Comparar modelos lineales, exponenciales y cuadráticos

CAPÍTULO


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CAPÍTULO



Lección Ejercicio

10.1 ¿Qué es el vértice de la gráfi ca de la función y 5 2 1 } 2 x2 1 5?

10.2 Halla el eje de simetría y el vértice de la gráfi ca de la función y 5 22x2 1 6x 2 4.


Grafi ca y 5 2(x 1 1)(x 2 3). Rotula el vértice, el eje de simetría y los interceptos en x.

10.3 Halla los ceros de f (x) 5 2x2 2 2x 1 8.

10.4 Resuelve la ecuación 2x2 2 25 5 103 al usar raíces cuadradas. Redondea tu solución al centésimo más próximo, si es necesario.

10.5 Resuelve la ecuación x2 1 16x 1 20 5 0 en forma de vértice. Luego grafi ca la función. Rotula el vértice y el eje de simetría.


Escribe y 5 x2 2 14x 1 48 en forma de vértice. Luego grafi ca la función. Rotula el vértice y el eje de simetría.

10.6 Usa la fórmula cuadrática para resolver la ecuación 5x2 1 12x 1 4 5 0.

10.7 Indica si la ecuación 8x2 2 8x 5 22 tiene una, dos o ninguna solución.

10.8 Indica si la tabla de valores representa x 23 22 21 0 1 2

y 4 1 0 1 4 9una función lineal, una función exponencial o una función cuadrática.


Instrucciones Escribe una función que da el área de la superfi cie (en pies cuadrados) de una habitación en tu casa. Investiga cuánto costaría pintar cinco de las superfi cies y ponerle alfombra (o baldosas) a la sexta superfi cie. (Recuérdate de restar el área de puertas y ventanas.)

Respuestas10.1: (0, 5) 10.2: eje de simetría: x 5

3 }

2 ; vértice: 1 3 }

2 , 1 }

2 2 Enfoque en las funciones:

x

y

O1

1(1, 0)(3, 0)

(1, 4)

x 1

10.3: 2, 24 10.4: 8, 8 10.5: 21.37 y 214.63

Enfoque en las funciones: 10.6: 22 y 2 2 }

5 10.7: una solución 10.8: cuadrática; y 5 (x 1 1)2

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179Algebra 1


Use the quadratic function to complete the table of values.

1. y 5 5x2 2. y 5 24x2

x 22 21 0 1 2

y ? ? ? ? ?

x 22 21 0 1 2

y ? ? ? ? ?

3. y 5 x2 1 6 4. y 5 x2 2 8

x 22 21 0 1 2

y ? ? ? ? ?

x 22 21 0 1 2

y ? ? ? ? ?


5. y 5 2 1 } 2 x2 6. y 5 2x2 7. y 5

1 } 4 x2

A.

x

y

1

3

12121

23 3

B.

x

y

1

3

12121

23 3

C.

x

y

1

21

23

23 3

Graph the function and identify its domain and range. Compare the graph with the graph of y 5 x2.

8. y 5 5x2 9. y 5 2 1 } 3 x2 10. y 5 26x2

x

y

1

3

5

12121

23 3

x

y

1

12121

23

23 3

x

y

12121

23

25

23 3

Identify the vertex and axis of symmetry of the graph.

11.

x

y

2

6

10

12123 3

12.x

y

12121

23

23 3

13.

x

y

0.5

123 3

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14. y 5 x2 2 3 15. y 5 3x2 2 1 16. y 5 2x2 1 3

A.

x

y

123 3

1

21

B.

x

y

1

12121

23 3

C.

x

y

1

3

12123 3


17. y 5 x2 2 5 18. y 5 x2 1 7 19. y 5 2x2 2 3

x

y1

12121

23

25

23 3

x

y

2

6

10

12122

23 3

x

y

1

3

12121

23

23 3

Complete the statement.

20. The graph of y 5 x2 1 5 can be obtained from the graph of y 5 x2 by shifting the graph of y 5 x2 ? .

21. The graph of y 5 10x2 can be obtained from the graph of y 5 x2 by ? the graph of y 5 x2 by a factor of ? .

22. Pot Rack A cross section of the pot rack shown can be modeled

x

y

2

6

22226 6

by the graph of the function y 5 20.08x2 1 8 where x and y are measured in inches.

a. Find the domain of the function in this situation.

b. Find the range of the function in this situation.

23. Drawer Handle A cross section of the drawer handle shown

x

y

1

3

1212325 3 5

can be modeled by the graph of the function y 5 2 1 } 18 x2 1 2

where x and y are measured in centimeters.



LESSON

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181Algebra 1


LESSON



1. y 5 9x2 2. y 5 25x2

x 22 21 0 1 2

y ? ? ? ? ?

x 22 21 0 1 2

y ? ? ? ? ?

3. y 5 5 }

2 x2 1 1 4. y 5 2

1 } 8 x2 2 2

x 24 22 0 2 4

y ? ? ? ? ?

x 216 28 0 8 16

y ? ? ? ? ?

5. y 5 24x2 1 3 6. y 5 6x2 2 5

x 22 21 0 1 2

y ? ? ? ? ?

x 22 21 0 1 2

y ? ? ? ? ?


7. y 5 24x2 1 3 8. y 5 3x2 1 4 9. y 5 1 } 3 x2 2 4

10. y 5 1 } 4 x2 2 3 11. y 5 23x2 1 4 12. y 5 4x2 1 3

A.

x

y

1

3

12123 3

B.

x

y1

12121

25

C.

x

y

1

32321

D.

x

y1

12121

23

25

E.

x

y

1

3

12123 3

F.

x

y

1

323

23

Describe how you can use the graph of y 5 x2 to graph the given function.

13. y 5 x2 2 8 14. y 5 2x2 1 4 15. y 5 2x2 1 3

16. y 5 25x2 1 1 17. y 5 1 } 2 x2 2 2 18. y 5 2

3 } 4 x2 1 5

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19. y 5 x2 1 9 20. y 5 2 1 } 5 x2 21. y 5 2

3 } 2 x2

x

y

3

9329 2323

x

y

1

3

32321

23

x

y

1

3

1 32123

23

22. y 5 x2 2 3.5 23. y 5 2x2 2 9 24. y 5 25x2 1 2

x

y1

1 3212321

25

x

y

3

9

9329 23

29

x

y3

1 32123

25. Serving Plate The top view of a freeform serving plate you made in

t

y

4

12

241224212

a ceramics class is shown in the graph. One edge of the plate can be

modeled by the graph of the function y 5 2 5 } 81 x2 1 20 where x and y

are measured in inches.



26. Roof Shingle A roof shingle is dropped from a rooftop that is

00

20

40

60

80

100

1 2Time (seconds)

Heig

ht

(feet)

y

t

100 feet above the ground. The height y (in feet) of the dropped roof shingle is given by the function y 5 216t2 1 100 where t is the time (in seconds) since the shingle is dropped.

a. Graph the function.

b. Identify the domain and range of the function in this situation.

c. Use the graph to estimate the shingle’s height at 1 second.

d. Use the graph to estimate when the shingle is at a height of 50 feet.

e. Use the graph to estimate when the shingle is at a height of 0 feet.

LESSON

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183Algebra 1


LESSON



1. y 5 10x2 2 4 2. y 5 21.5x2 1 3

x 22 21 0 1 2

y ? ? ? ? ?

x 22 21 0 1 2

y ? ? ? ? ?


3. y 5 1 } 6 x2 1 2 4. y 5 24x2 2 3 5. y 5 9x2 2

7 } 2

x

y

1

3

12121

23

23 3

x

y2

12122

26

210

23 3

x

y

2

6

12122

23 3

6. y 5 3 } 5 x2 1

1 } 5 7. y 5 2

1 } 2 x2 1 4 8. y 5 6x2 1

3 } 4

x

y

1

3

5

12121

23 3

x

y

1

3

12121

23 3

x

y

6

18

30

12126

23 3

9. y 5 4x2 2 2 } 3 10. y 5 22x2 2

1 } 2 11. y 5 25x2 1 15

x

y

3

9

15

12123

23 3

x

y2

12122

26

210

23 3

x

y

5

15

12125

215

23 3

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Tell how you can obtain the graph of g from the graph of f byusing transformations.

12. f (x) 5 x2 1 6 13. f (x) 5 2x2 1 14 14. f (x) 5 2 1 } 2 x2 2 3

g(x) 5 x2 2 2 g(x) 5 2x2 1 9

g(x) 5 2 1 } 2 x2 2 7

15. f (x) 5 3x2 2 5 16. f (x) 5 3x2 17. f (x) 5 8x2

g(x) 5 3x2 1 11 g(x) 5 9x2 g(x) 5 4x2

Write a function of the form y 5 ax2 1 c whose graph passes through the two given points. Then graph the function.

18. (0, 6), (2, 10) 19. (0, 1), (21, 0) 20. (0, 24), (23, 5)

x

y

2

6

10

12123 3

x

y

1

12121

23

23 3

x

y

2

6

12122

26

23 3

21. Nylon Rope The breaking weight w (in pounds) of a nylon

00

20,00040,00060,00080,000

100,000

0.5 1.0 1.5 2.0Diameter (inches)

Wei

gh

t (p

ou

nd

s)

w

d

rope can be modeled by the function w 5 22,210d2 where d is the diameter (in inches) of the rope.


b. Use the graph to estimate the diameter of a nylon rope that has a breaking weight of 50,000 pounds.

22. Foam Ball A foam ball is dropped from a

0048

121620

0.4 0.8Time (seconds)

Dis

tan

ce (

feet

)

y

t

0048

121620


Hei

gh

t (f

eet)

y

t

deck that is 20 feet above the ground.

a. The distance y (in feet) that the ball falls is given by the function y 5 16t2 where t is the time (in seconds) since the ball wasdropped. Graph the function.

b. The height y (in feet) of the dropped ball is given by the function y 5 216t2 1 20 where t is the time (in seconds) since the ball was dropped. Graph the function.

c. How are the graphs from part (a) and part (b) related? Explain how you can use each graph to fi nd the number of seconds after which the ball has dropped 8 feet.

LESSON

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185Algebra 1


Graph simple quadratic functions.

VocabularyA quadratic function is a nonlinear function that can be written in the standard form y 5 ax2 1 bx 1 c where a Þ 0.

Every quadratic function has a U-shaped graph called a parabola.

The most basic quadratic function in the family of quadratic functions, called the parent quadratic function, is y 5 x2.

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

The line that passes through the vertex and divides the parabola into two symmetric parts is called the axis of symmetry.

GOAL

Graph y 5 ax2 when a > 1

Graph y 5 26x2. Compare the graph with the graph of y 5 x2.

Solution

STEP 1 Make a table of values for y 5 26x2.

x

9

3

29

215

221

12123 3

y

y 5 26x2

y 5 x2

x 22 21 0 1 2

y 224 26 0 26 224

STEP 2 Plot the points from the table.

STEP 3 Draw a smooth curve through the points.

STEP 4 Compare the graphs of y 5 26x2 and y 5 x2.Both graphs have the same vertex, (0, 0), and the same axis of symmetry, x 5 0. However, the graph of y 5 26x2 is narrower than the graph of y 5 x2 and it opens down. This is because the graph of y 5 26x2 is a vertical stretch (by a factor of 6) of the graph of y 5 x2 and a refl ection in the x-axis of the graph of y 5 x2.

EXAMPLE 1


LESSON

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Graph y 5 ax2 when⏐a⏐< 1

Graph y 5 2 } 5 x2. Compare the graph with the graph of y 5 x2.

STEP 1 Make a table of values for y 5 2 } 5 x2.

x

35

25

525215 15

y

y 5 x2 y 5 x225

x 210 25 0 5 10

y 40 10 0 10 40



STEP 4 Compare the graphs of y 5 2 } 5 x2 and y 5 x2.

Both graphs have the same vertex, (0, 0), and the same axis of symmetry,

x 5 0. Both graphs open upward. However, the graph of y 5 2 } 5 x2 is wider

than the graph of y 5 x2.

This is because the graph of y 5 2 } 5 x2 is a vertical shrink 1 by a factor of

2 } 5 2

of the graph of y 5 x2.

EXAMPLE 2

Graph y 5 ax2 1 c

Graph y 5 3x2 2 1. Compare the graph with the graph of y 5 x2.

STEP 1 Make a table of values for y 5 3x2 2 1.

x

10

6

2

12123 3

y

22y 5 3x2 2 1

y 5 x2

x 22 21 0 1 2

y 11 2 21 2 11



STEP 4 Compare the graphs of y 5 3x2 2 1 and y 5 x2. Both graphs open up and have the same axis of symmetry, x 5 0.

However, the graph of y 5 3x2 2 1 is narrower and has a lower vertex than the graph of y 5 x2. This is because the graph of y 5 3x2 2 1 is a vertical stretch (by a factor of 3) and a vertical translation (1 unit down) of the graph of y 5 x2.


Graph the function. Compare the graph with the graph of y 5 x2.

1. y 5 28x2 2. y 5 1 } 7 x2 3. y 5 2

1 } 3 x2

4. y 5 x2 2 3 5. y 5 1 }

4 x2 1 2 6. y 5 2

1 } 2 x2 2 1

EXAMPLE 3

Review for Mastery continuedFor use with pages 6482654

LESSON

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187Algebra 1


In Exercises 1–5, write the function of the form y 5 ax2 1 c whose graph passes through the given points.

1. (0, 4), (21, 7), (1, 7)

2. (1, 21), (21, 21), (3, 217)

3. (1, 26), (2, 6), (3, 26)

4. (21, 4), (2, 1), (3, 24)

5. 1 1, 3 }

2 2 , (0, 2), (22, 0)


Einstein’s famous formula E 5 mc2 relates mass m (in kilograms) to the energy E (in joules) contained within the mass. The constant c is equal to the speed of light in a vacuum (in meters per second), c ø 3.1 3 108 meters per second.

6. What is the mass (in kilograms) of an object containing 9.61 3 1016 joules of energy?

7. The average automobile uses 5 3 1010 joules of energy per year. What is the mass represented by this energy?

8. Suppose Einstein’s formula holds true in an alternate universe where the speed of light is not the same as in our universe. If an experiment is conducted in which 1 kilogram of mass is equivalent to 1 3 1020 joules of energy, then what is the speed of light in the alternate universe?

9. The average home uses 1 3 108 joules of energy per year. What is the mass represented by this energy?

10. Suppose Einstein’s formula holds true in an alternate universe where the speed of light is not the same as in our universe. If the speed of light in the alternate universe is 4 3 105 meters per second, then how much mass would be needed to produce 5 3 1011 joules of energy?

LESSON

10.1 Challenge PracticeFor use with pages 6482654

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Activity Support MasterFor use with pages 655–656

Equation graphed y-intercept x-intercept(s) Axis of symmetry

y 5 2x2 0 0 x 5 0

y 5 2x2 2 4x 0 0, 2 x 5 1

Equation graphed y-intercept x-intercept(s) Axis of symmetry

y 5 2x2 2 4x 0 0, 2 x 5 1

y 5 2x2 2 4x 2 6 26 21, 3 x 5 1

LESSON

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189Algebra 1


Identify the values of a, b, and c in the quadratic function.

1. y 5 7x2 1 2x 1 11 2. y 5 3x2 2 5x 1 1 3. y 5 4x2 1 2x 2 2

4. y 5 23x2 1 9x 1 4 5. y 5 1 } 2 x2 2 x 2 5 6. y 5 2x2 1 7x 2 6

Tell whether the graph opens upward or downward. Then fi nd the axis of symmetry of the graph of the function.

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

10. y 5 x2 2 4x 1 5 11. y 5 2x2 1 4x 2 5 12. y 5 2x2 1 8x 1 3

13. y 5 x2 1 3x 2 6 14. y 5 2x2 1 7x 2 2 15. y 5 3x2 1 6x 1 10

Find the vertex of the graph of the function.

16. y 5 x2 1 5 17. y 5 2x2 1 3 18. y 5 x2 1 10x 1 3

19. y 5 2x2 1 4x 2 2 20. y 5 3x2 1 6x 1 1 21. y 5 22x2 1 8x 2 3

22. y 5 10x2 2 10x 1 7 23. y 5 x2 1 x 1 3 24. y 5 x2 2 x 1 1


25. y 5 x2 2 6x 1 8 26. y 5 2x2 1 12x 2 5

x 1 2 3 4 5

y ? ? ? ? ?

x 4 5 6 7 8

y ? ? ? ? ?

27. y 5 7x2 1 14x 1 2 28. y 5 22x2 2 4x 1 1

x 23 22 21 0 1

y ? ? ? ? ?

x 23 22 21 0 1

y ? ? ? ? ?


29. y 5 8x2 1 2x 1 3 30. y 5 2x2 1 8x 1 1 31. y 5 1 } 2 x2 1 8x 1 5

A.

x

y4

424220

B. x

y

121

27

2325

C.

x

y

1

12123 3

LESSON


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Graph the function. Label the vertex and axis of symmetry.

32. y 5 2x2 2 6 33. y 5 x2 1 7 34. y 5 x2 1 2x 1 5

x

y2

12122

26

210

23 3

x

y

2

6

10

12122

23 3

x

y

1

3

5

7

1212325 3

35. y 5 x2 2 8x 1 1 36. y 5 22x2 1 x 2 3 37. y 5 2x2 2 4x 1 3

x

y

2 6 102222

26

210

214

x

y

12121

23

25

27

23 3

x

y

1

3

5

7

1212325

Tell whether the function has a minimum value or a maximum value. Then fi nd the minimum or maximum value.

38. f (x) 5 x2 2 7 39. f (x) 5 2x2 1 9 40. f (x) 5 2x2 1 4x

41. Greenhouse The dome of the greenhouse shown can be modeled by the graph of the function y 5 20.15625x2 1 2.5x where x and y are measured in feet. What is the height h at the highest point of the dome as shown in the diagram?

x

y

2

6

10

2 6 10 14

h

42. Fencing A parabola forms the top of a fencing panel as shown. This parabola can be modeled by the graph of the function y 5 0.03125x2 2 0.25x 1 4 where x and y are measured in feet and y represents the number of feet the parabola is above the ground. How far above the ground is the lowest point of the parabola formed by the fence?

LESSON


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191Algebra 1


LESSON


Identify the values of a, b, and c in the quadratic function.

1. y 5 6x2 1 3x 1 5 2. y 5 3 } 2 x2 2 x 1 8 3. y 5 7x2 2 3x 2 1

4. y 5 22x2 1 9x 5. y 5 3 } 4 x2 2 10 6. y 5 28x2 1 3x 2 7

Tell whether the graph opens upward or downward. Then fi nd the axis of symmetry and vertex of the graph of the function.

7. y 5 x2 2 5 8. y 5 2x2 1 9 9. y 5 22x2 1 6x 1 7

10. y 5 3x2 2 12x 1 1 11. y 5 3x2 1 6x 2 2 12. y 5 22x2 1 7x 2 21

13. y 5 1 } 2 x2 1 5x 2 4 14. y 5 2

1 } 4 x2 2 24 15. y 5 23x2 1 9x 2 8

16. y 5 3x2 2 2x 1 3 17. y 5 22x2 1 7x 1 1 18. y 5 3x2 1 2x 2 5

Find the vertex of the graph of the function. Make a table of values using x-values to the left and right of the vertex.

19. y 5 x2 2 10x 1 3 20. y 5 2x2 1 6x 2 2

x ? ? ? ? ?

y ? ? ? ? ?

x ? ? ? ? ?

y ? ? ? ? ?

21. y 5 1 } 2 x2 2 x 1 7 22. y 5

1 } 3 x2 2 2x 1 3

x ? ? ? ? ?

y ? ? ? ? ?

x ? ? ? ? ?

y ? ? ? ? ?


23. y 5 2x2 2 10 24. y 5 2x2 1 3 25. y 5 22x2 1 2x 1 1

x

y5

5 152521525

x

y

1

5

1 3212321

x

y

1

3

1 32123

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26. y 5 5x2 1 2x 27. y 5 22x2 1 x 2 4 28. y 5 x2 2 8x 1 5

x

y

1

3

1 32123

x

y

1 3212321

23

x

y2

2 6 102222

26

210

29. y 5 2 1 } 2 x2 2 8x 1 3 30. y 5

1 } 4 x2 1 3x 2 1 31. y 5 2

3 } 4 x2 2 2x 1 2

x

y

10230

10

x

y2

22210

210

x

y

1

2521

23


32. f (x) 5 8x2 2 40 33. f (x) 5 25x2 1 10x 2 2 34. f (x) 5 8x2 2 4x 1 4

35. Storage Building The storage building shown can be modeled by the graph of the function y 5 20.12x2 1 2.4x where x and y are measured in feet. What is the height h at the highest point of the building as shown in the diagram?

x

y

2

6

10

2 6 10 14 18

h

36. Velvet Rope A parabola is formed by a piece of velvet rope found around a museum display as shown. This parabola can be

modeled by the graph of the function y 5 4 } 225 x2 2

16 } 15 x 1 40

where x and y are measured in inches and y represents the number of inches the parabola is above the ground. How far above the ground is the lowest point on the rope?

LESSON


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193Algebra 1


LESSON


Tell whether the graph opens upward or downward. Then fi nd the axis of symmetry and vertex of the graph of the function.

1. y 5 23x2 1 3x 1 5 2. y 5 5 } 2 x2 2 2x 1 1 3. y 5 8x2 2 2x 1 3

4. y 5 29x2 1 9x 5. y 5 2 } 3 x2 2 9 6. y 5 25x2 1 2x 2 3

7. y 5 1 } 8 x2 2 2x 8. y 5 2

1 } 5 x2 1 7 9. y 5 26x2 1 12x 1 5

10. y 5 4x2 2 12x 1 8 11. y 5 5x2 1 10x 2 3 12. y 5 26x2 1 8x 2 10

Find the vertex of the graph of the function. Make a table of values using x-values to the left and right of the vertex.

13. y 5 1 } 4 x2 2 2x 1 5 14. y 5 2

5 } 2 x2 1 10x 2 1

x ? ? ? ? ?

y ? ? ? ? ?

x ? ? ? ? ?

y ? ? ? ? ?


15. y 5 2x2 2 15 16. y 5 6x2 1 8 17. y 5 24x2 1 4x 1 3

x

y5

52525

215

225

215 15

x

y

2

6

10

14

22226 6

x

y

1

3

12121

23

23 3

18. y 5 2x2 1 20 19. y 5 7x2 2 14x 1 6 20. y 5 23x2 1 18x 2 4

x

y

5

15

52525

215

215 15

x

y

1

3

5

12123 321

x

y

4

12

20

42424

212 12

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21. y 5 2 7 } 2 x2 1 21x 2 5 22. y 5

1 } 4 x2 2 2x 1 10 23. y 5 6x2 2 12x 1 13

x

y

6

18

30

62626

218 18

x

y

2

6

10

14

22226 6 10

x

y

2

6

10

14

22226 6

24. y 5 5 } 3 x2 2 15x 1 2 25. y 5

7 } 4 x2 1 35x 2 4 26. y 5 2

2 } 5 x2 2 20x 1 5

x

y6

22226

218

230

26 6 10

x

y30

525230

290

2150

215 15

x

y

50

150

250

50250250

2150 150


27. f (x) 5 9x2 2 36 28. f (x) 5 2 3 } 4 x2 1 18x 2 7 29. f (x) 5

5 } 4 x2 2 10x 1 3

30. Lamps A lighting company offers two models of small lamps, both of which contain a refl ector in the shape of a parabola. The shape of the refl ector in lamp A can be modeled by the function y 5 20.16x2 1 25 and the shape of the refl ector in lamp B can be modeled by the function y 5 20.2x2 1 20 where x and y are measured in millimeters.

a. Find the maximum value of each function, which gives the height of the refl ector.

b. How much taller is the refl ector for lamp A than the refl ector for lamp B?

31. Window An artist designs a window in a house to be in the shape of a parabola as shown. The top part of the window can be modeled by the function y 5 21.875x2 1 7.5x and the bottom part of the window can be modeled by the function y 5 1.5 where x represents the width of the window (in feet) and y represents the height of the window (in feet) above the ground. How tall is the window? Explain how you got your answer.

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195Algebra 1


Graph general quadratic functions.

VocabularyFor y 5 ax2 1 bx 1 c, the y-coordinate of the vertex is the minimum value of the function if a > 0 and the maximum value of the function if a < 0.

GOAL

Find the axis of symmetry and the vertex

Consider the function y 5 3x2 2 18x 1 11.

a. Find the axis of symmetry of the graph of the function.

b. Find the vertex of the graph of the function.

Solution

a. For the function y 5 3x2 2 18x 1 11, a 5 3 and b 5 218.

x 5 2 b

} 2a 5 2 (218)

} 2(3)

5 3 Substitute 3 for a and 218 for b. Then simplify.

The axis of symmetry is x 5 3.

b. The x-coordinate of the vertex is 2 b } 2a , or 3. To fi nd the y-coordinate,

substitute 3 for x in the function and fi nd y.

y 5 3(3)2 218(3) 1 11 5 216 Substitute 3 for x. Then simplify.

The vertex is (3, 216).

EXAMPLE 1

EXAMPLE 2 Find the minimum or maximum value

Tell whether the function f (x) 5 x2 1 14x 2 3 has a minimum value or a maximum value. Then fi nd the minimum or maximum value.

Solution

Because a 5 1 and 1 > 0, the parabola opens up and the function has a minimum value. To fi nd the minimum value, fi nd the vertex.

x 5 2 b } 2a 5 2

14 }

2(1) 5 27 The x-coordinate is 2

b } 2a .

f (27) 5 (27)2 1 14(27) 2 3 5 252 Substitute 27 for x. Then simplify.

The minimum value of the function is f (x) 5 252.


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Graph y 5 ax2 1 bx 1 c

Graph y 5 1 } 5 x2 2 2x 1 3.

Solution

STEP 1 Determine whether the parabola opens up or down. Because a > 0, the parabola opens up.

STEP 2 Find and draw the axis of symmetry:

x 5 2 b } 2a 5 2

(22) }

2 1 1 } 5 2 5 5.

x

10

14

622 10 14

y

(0, 3)

(1, 1.2)

(5, 22)

(10, 3)

(9, 1.2)

x 5 5

STEP 3 Find and plot the vertex.

The x-coordinate of the vertex is 2 b } 2a ,

or 5. To fi nd the y-coordinate, substitute 5 for x in the function and simplify.

y 5 1 } 5 (5)2 2 2(5) 1 3 5 22

So, the vertex is (5, 22).

STEP 4 Plot two points. Choose two x-values less than the x-coordinate of the vertex. Then fi nd the corresponding y-values.

x 0 1

y 3 1.2

STEP 5 Refl ect the points plotted in Step 4 in the axis of symmetry.

STEP 6 Draw a parabola through the plotted points.


4. Graph the function f (x) 5 x2 2 4x 1 7. Label the vertex and axis of symmetry.

EXAMPLE 3


Find the axis of symmetry and the vertex of the graph of the function.

1. y 5 5x2 1 20x 1 9 2. y 5 1 }

3 x2 2 4x 2 19

3. Tell whether the function f (x) 5 1 }

2 x2 2 8x 1 13 has a minimum value or a

maximum value. Then fi nd the minimum value or maximum value.


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197Algebra 1


1. Suspension Bridges The cables between the two towers of the Golden Gate Bridge in California form a parabola that can be modeled by the graph of y 5 0.00012x2 2 0.505x 1 746 where x and y are measured in feet. What is the height of the cable above the water at its lowest point?

2. Baseball You hit a baseball whose path can be modeled by the graph of y 5 216x2 1 40x 1 3 where x is the time (in seconds) since the ball was hit and y is the height (in feet) of the baseball. What is the maximum height of the baseball?

3. Tunnel The shape of a tunnel for cars can be modeled by the graph of the equation y 5 20.5x2 1 4x where x and y are measured in feet. On a coordinate plane, the ground is represented by the x-axis. How wide is the tunnel at its base?

4. Sprinkler A sprinkler ejects water at an angle of 35° with the ground. The path of the water can be modeled by the equation y 5 20.06x2 1 0.7x 1 0.5 where x and y are measured in feet. What is the maximum height of the water?

Basketball You throw a basketball whose path can be modeled by the graph of y 5 216x2 1 19x 1 6 where x is the time (in seconds) and y is the height (in feet) of the basketball. What is the maximum height of the basketball?


What do you know?

The equation that models the path of a basketball

What do you want to fi nd out?

The maximum height of the basketball

STEP 2 Make a Plan Use what you know to fi nd the vertex of the parabola.

STEP 3 Solve the Problem The highest point of the basketball is at the vertex of the parabola. Find the x-coordinate of the vertex. Use a 5 216 and b 5 19.

x 5 2 b } 2a 5 2

19 }

2(216) ø 0.59 Use a calculator.

Substitute 0.59 for x in the equation to fi nd the y-coordinate of the vertex.

y ø 216(0.59)2 1 19(0.59) 1 6 ø 11.64

The basketball reaches a maximum height of about 11.64 feet.

STEP 4 Look Back By graphing the function, it

002468

1012

0.4 0.8 1.2Time (seconds)

Hei

gh

t (f

eet)

y

x

appears that the maximum occurs after about 0.6 second and at a height between 11 and 12 feet. The answer seems reasonable.

PROBLEM

PRACTICE

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In Exercises 1–5, write the function of the form y 5 ax2 1 bx 1 c whose graph passes through the three given points.

1. (0, 1), (1, 0), (2, 3)

2. (1, 2), (0, 4), (21, 4)

3. (21, 6), (1, 2), (3, 6)

4. (2, 0), (1, 1), (0, 4)

5. (1, 12), (2, 9), (3, 0)

In Exercises 6–10, use the given information to write a function of the form f (x) 5 ax2 1 bx 1 c.

6. f (x) has an axis of symmetry at x 5 3 } 2 , x-intercepts at x 5 1 and x 5 2, and a

y-intercept at y 5 2.

7. f (x) has an axis of symmetry at x 5 3 } 4 , x-intercepts at x 5 21 and x 5

5 } 2 ,

and a y-intercept at y 5 5.

8. f (x) has an axis of symmetry at x 5 2 5 } 4 , x-intercepts at x 5 2

7 } 2 and x 5 1,


9. f (x) has an axis of symmetry at x 5 5 } 12 , x-intercepts at x 5

1 } 3 and x 5

1 } 2 ,


10. f (x) has an axis of symmetry at x 5 19

} 6 , x-intercepts at x 5 1 } 3 and x 5 6,


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199Algebra 1


FOCUS ON

10.2 PracticeFor use with pages 663–664

Graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts. Identify the domain and range of the function.

1. y 5 (x 1 2)(x 2 4) 2. y 5 2(x 1 1)(x 2 3) 3. y 5 2(x 1 5)(x 1 1)

4. y 5 24(x 2 1)(x 2 3) 5. y 5 (x 1 4)2 6. y 5 5x2 2 45

7. y 5 2x2 + 4x 2 4 8. y 5 2x2 + 6x 2 8 9. y 5 25x2 + 10x 1 40

10. Follow the steps below to write an equation of the parabola shown.

a. Find the x-intercepts.

b. Use the values of p and q and the coordinates of the vertex to fi nd the value of a in the equation y 5 a(x 2 p)(x 2 q).

c. Write a quadratic equation in intercept form.

11. Challenge A baseball is thrown into the air. The path of a baseball is parabolic. The ball reaches a height of 25 feet before it starts to descend and lands 50 feet from the point where it was thrown. What is the equation, in intercept form, which models the path of the baseball? Assume the baseball was thrown at (0, 0).

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Graph quadratic functions in intercept form.

VocabularyThe intercept form of a quadratic function is y 5 a(x 2 p)(x 2 q) where a Þ 0 and p and q are the x-intercepts. The axis of symmetry is halfway between (p, 0) and (q, 0). The parabola opens up if a . 0 and opens down if a , 0.

GOAL

Graph a quadratic function in intercept form

Graph y 5 2(x 2 1)(x 1 3).

Solution

STEP 1 Identify and plot the x-intercepts. Because p 5 1 and q 5 23, the x-intercepts occur at the points (1, 0) and (23, 0).

STEP 2 Find and draw the axis of symmetry:

x 5 p 1 q

} 2 5

1 1 (23) }

2 5 21

x

y

O 1(1, 0)( 3, 0)

( 1, 8)


The x-coordinate of the vertex is 21.

To fi nd the y-coordinate of the vertex, substitute 21 for x and simplify.

y 5 2(21 2 1)(21 1 3) 5 28


STEP 4 Draw a parabola through the vertex and the points where the x-intercepts occur.



1. y 5 22(x 1 3)(x 2 3) 2. y = 4(x 1 2)(x 2 4)

EXAMPLE 1


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201Algebra 1


Graph a quadratic function

Graph y 5 3x2 2 3.

Solution

STEP 1 Rewrite the quadratic function in intercept form.

y 5 3x2 2 3 Write original function.

5 3(x2 2 1) Factor out common factor.

5 3(x 1 1)(x 2 1) Difference of two squares pattern

STEP 2 Identify and plot the x-intercepts. Because p 5 21 and q 5 1, the x-intercepts occur at the points (21, 0) and (1, 0).

STEP 3 Find and draw the axis of symmetry.

x 5 p 1 q

} 2 5

21 1 1 }

2 5 0


The x-coordinate of the vertex is 0.

The y-coordinate of the vertex is:

y 5 3(0)2 2 3 5 23


STEP 5 Draw a parabola through the vertex and the points where the x-intercepts occur.



3. y 5 3x2 2 12 4. y 5 23x2 1 12x

EXAMPLE 2


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TI-83 Plus

Example 1

Y= (�) 2 X,T,�,n x 2 � 6 X,T,�,n

� 7 WINDOW (�) 6 ENTER

10 ENTER 1 ENTER (�) 20 ENTER

20 ENTER 2 ENTER 2nd [CALC] 4

(�) 4 ENTER 1 ENTER (�) 1 ENTER

Example 2

Y= CLEAR 3 X,T,�,n x 2 � 2

X,T,�,n � 4 WINDOW (�) 5

ENTER 5 ENTER 1 ENTER (�) 10

ENTER 10 ENTER 1 2nd [CALC] 2

(�) 2 ENTER (�) 1 ENTER (�) 1.5

ENTER 2nd [CALC] 2 0 ENTER 2

ENTER .5 ENTER


Example 1


(�) 2 X, ,T� x 2 � 6 X, ,T� � 7

EXE SHIFT F3 (�) 6 EXE 10 EXE 1

EXE (�) 20 EXE 20 EXE 2 EXE

EXIT F6 SHIFT F5 F2

Example 2


3 X, ,T� x 2 � 2 X, ,T� � 4 EXE

SHIFT F3 (�) 5 EXE 5 EXE 1

EXE (�) 10 EXE 10 EXE 1 EXE

EXIT F6 SHIFT F5 F1

Graphing Calculator Activity KeystrokesFor use with pages 672 and 673

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203Algebra 1


Write the equation in standard form.

1. x2 1 3x 5 212 2. x2 2 8x 5 14 3. x2 5 9x 2 1

4. x2 5 6 2 10x 5. 14 2 x2 5 3x 6. 1 }

2 x2 5 23x 2 7

Determine whether the given value is a solution of the equation.

7. x2 1 36 5 0; 26 8. 100 2 x2 5 0; 210 9. 0 5 x2 1 6x 1 5; 21

10. x2 2 5x 1 6 5 0; 2 11. 2x2 1 4x 2 4 5 0; 4 12. 0 5 2x2 1 8x 1 3; 8

Use the graph to fi nd the solutions of the given equation.

13. x2 1 5 5 0 14. 2x2 1 4 5 0 15. x2 1 4x 1 3 5 0

x

y

1

3

12123 3

x

y

1

3

5

12123 3

x

y

3

121

25

16. x2 2 16 5 0 17. x2 2 2 5 0 18. x2 1 2x 2 8 5 0

x

y

4

212 12

x

y1

12123 321

25

x

y

2

2222

26

26

Solve the equation by graphing.

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

x

y

3

9

15

12123 3

x

y

1

3

5

12123 3

x

y

2

6

22222

26

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22. x2 2 2x 2 15 5 0 23. 22x2 1 x 2 3 5 0 24. 2x2 2 2x 1 3 5 0

x

y3

32323

29

215

29 9

x

y3

12123

29

215

23 3

x

y

1

3

12121

23

Find the zeros of the function by graphing the function.

25. f (x) 5 x2 2 25 26. f (x) 5 2x2 1 9 27. f (x) 5 2x2 1 4x

x

y5

52525

215

225

215 15

x

y

3

9

32323

29

29 9

x

y

1

3

12121

23

23 3

28. f (x) 5 x2 2 4x 2 12 29. f (x) 5 2x2 2 3x 1 40 30. f (x) 5 3x2 2 30x

x

y4

22224

212

220

6

x

y

10

30

50

222210

26

x

y

222212

236

260

6 10

31. Plate Cover A plate cover made of netting has a cross section

002468

1012

4 8 12 162 6 10 14Width (inches)

Hei

gh

t (i

nch

es)

y

x

in the shape of a parabola. The cross section can be modeled by the function y 5 20.1875x2 1 3x where x is the width of the cover (in inches) and y is the height of the cover (in inches).


b. Find the domain and range of the function in this situation.

c. How wide is the cover?

d. How tall is the cover?

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205Algebra 1


LESSON



1. x2 2 2x 1 15 5 0; 3 2. x2 2 4x 2 12 5 0; 2 3. 2x2 2 5x 2 6 5 0; 3

4. x2 1 3x 2 4 5 0; 1 5. 2x2 1 9x 2 5 5 0; 22 6. 3x2 2 5x 2 2 5 0; 2

Use the graph to fi nd the solutions of the given equation.

7. x2 1 8x 1 16 5 0 8. 2x2 1 36 5 0 9. x2 1 5x 2 24 5 0

x

y

12

20

42421224

x

y

6

18218

x

y

2222626

218

6

10. x2 1 11x 1 30 5 0 11. x2 2 25 5 0 12. x2 1 7 5 0

x

y

2

6

10

226210 22

x

y

15215

225

5

x

y

2

6

2 62226


13. 2x2 2 6x 5 0 14. 2x2 5 2 15. x2 2 7x 1 10 5 0

x

y

22222

x

y

1

3

323

23

x

y

2

6

10

62222

16. x2 5 10x 17. x2 2 6x 1 9 5 0 18. 2x2 1 9x 5 18

x

y5

5 152525

215

225

x

y

2

6

10

2 6222622

x

y3

9232923

29

215

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Find the zeros of the function by graphing.

19. f (x) 5 2x2 2 5x 2 10 20. f (x) 5 x2 1 12x 1 36 21. f (x) 5 2x2 1 24x

x

y2

2 6222622

210

x

y

30

6 1826218

x

y

6 1826218212

22. f (x) 5 x2 2 49 23. f (x) 5 2x2 1 1 24. f (x) 5 3x2 1 12x

x

y10

2 622210

230

x

y3

32321

23

x

y

2 6222622

25. Stunt Double A movie stunt double jumps from the top of a

00

10

20

30

40

50

0.5 1.0 1.5 2.0Time (seconds)

Heig

ht

(feet)

h

t

building 50 feet above the ground onto a pad on the ground below. The stunt double jumps with an initial vertical velocity of 10 feet per second.

a. Write and graph a function that models the height h (in feet) of the stunt double t seconds after she jumps.

b. How long does it take the stunt double to reach the ground?

26. Wastebasket You throw a wad of used paper towards a

00

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.1 0.2 0.3 0.4Time (seconds)

Heig

ht

(feet)

h

t

wastebasket from a height of about 1.3 feet above the fl oor with an initial vertical velocity of 3 feet per second.

a. Write and graph a function that models the height h (in feet)of the paper t seconds after it is thrown.

b. If you miss the wastebasket and the paper hits the fl oor, how long does it take for the ball of paper to reach the fl oor?

c. If the ball of paper hits the rim of the wastebasket one-half foot above the ground, how long was the ball in the air?

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207Algebra 1


LESSON



1. x2 5 4 2. x2 1 3x 5 4 3. 2x2 2 14x 2 49 5 0

x

y

1

12121

23

23 3

x

y

2

6

22222

26

26 6

x

y5

2225

215

225

26210

4. 2x2 1 6x 1 16 5 0 5. x2 1 10x 1 25 5 0 6. x2 1 8x 1 15 5 0

x

y

4

12

20

222 6

x

y

5

15

25

52525

215 15

x

y

3

9

15

2123

2325

7. x2 1 2 5 0 8. x2 5 4x 1 12 9. 2x2 1 25 5 0

x

y

1

3

12121

23

23 3

x

y

4

22224

212

26 6

x

y

5

15

25

52525

215 15

Find the zeros of the function by graphing.

10. f (x) 5 2x2 2 8x 2 10 11. f (x) 5 23x2 2 6x 1 24 12. f (x) 5 4x2 2 4x 2 8

x

y

4

12124

212

3 5

x

y

6

18

30

12126

23

x

y

2

12122

26

23 3

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Approximate the zeros of the function to the nearest tenth by graphing.

13. f (x) 5 22x2 1 5x 1 1 14. f (x) 5 3x2 2 5 15. f (x) 5 4x2 2 3x 2 4

x

y

1

3

12121

23 3

x

y1

12121

23

25

23 3

x

y

1

12121

23

23 3

Use the given surface area S of the cylinder to fi nd the radius r to the nearest tenth. (Use 3.14 for π.)

16. S 5 301 in.2 17. S 5 58 ft2 18. S 5 1356 cm2

6 in.

r

3 ft

r

12 cm

r

19. Jumping A cat jumps from a countertop 30 inches above the fl oor.

00

0.51.01.52.02.5


Hei

gh

t (f

eet)

h

t

It jumps with an initial vertical velocity of 5 feet per second.

a. Write and graph a function that models the height h (in feet) of the cat t seconds after it jumps. Explain how you got your model.

b. How far above the ground is the cat after one half of a second?

c. How long does it take the cat to reach the ground?

20. Basketball A basketball player throws a ball towards a hoop at a

00

10203040

1 2 3Time (seconds)

Hei

gh

t (f

eet)

h

t

height of 6 feet with an initial vertical velocity of 50 feet per second.

a. Write and graph a function that models the height h (in feet) of the ball t seconds after it is thrown.

b. If the player misses the hoop completely and the ball lands on the ground, how long was the ball in the air?

c. If an opposing player catches the ball at a height of 5 feet, how long was the ball in the air? Explain your reasoning.

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209Algebra 1


Solve quadratic equations by graphing.

VocabularyA quadratic equation is an equation that can be written in the standard form ax2 1 bx 1 c 5 0 where a Þ 0 and a is called the leading coeffi cient.

GOAL

Solve a quadratic equation having two solutions

Solve x2 1 5x 5 14 by graphing.

Solution

STEP 1 Write the equation in standard form. 22

22

26

210

214

x

yx 5 27

x 5 2

x2 1 5x 5 14 Write original equation.

x2 1 5x 2 14 5 0 Subtract 14 from each side.

STEP 2 Graph the function y 5 x2 1 5x 2 14.

The x-intercepts are 27 and 2.

The solutions of the equation x2 1 5x 5 14 are 27 and 2.

CHECK You can check 27 and 2 in the original equation.

x2 1 5x 5 14 x2 1 5x 5 14 Write original equation.

(27)2 1 5(27) 0 14 (2)2 1 5(2) 0 14 Substitute for x.

14 5 14 ✓ 14 5 14 ✓ Simplify. Each solution checks.

EXAMPLE 1


LESSON

10.3

Solve a quadratic equation having one solution

Solve x2 1 25 5 10x by graphing.

Solution

STEP 1 Write the equation in standard form.

10

6

2

x

y

x 5 5

2 6 10

x2 1 25 5 10x Write original equation.

x2 2 10x 1 25 5 0 Subtract 10x from each side.


The x-intercept is 5.

The solution of the equation x2 1 25 5 10x is 5.

EXAMPLE 2

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Find the zeros of a quadratic function

Find the zeros of f (x) 5 x2 2 10x 1 24.

Solution

Graph the function f (x) 5 x2 2 10x 1 24.

10

6

2

x

y

x 5 4 10

x 5 6

The x-intercepts are 4 and 6.

The zeros of the function are 4 and 6.


Find the zeros of the function.

4. f (x) 5 x2 2 4

5. f (x) 5 x2 1 5x 2 14

EXAMPLE 4

Solve a quadratic equation having no solution

Solve x2 1 11 5 5x by graphing.

Solution


10

6

2

x

y

222 6

x2 1 11 5 5x Write original equation.

x2 2 5x 1 11 5 0 Subtract 5x from each side.


The graph has no x-intercepts.

The equation x2 1 11 5 5x has no solution.



1. x2 5 2x 1 15

2. x2 1 4 5 24x

3. x2 1 6x 5 24

EXAMPLE 3


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211Algebra 1


In Exercises 1–5, graph each quadratic function on the same coordinate system and use the graph to identify the points of intersection.

1. y 5 3x2 1 1

y 5 2x2 1 5

2. y 5 1 } 2 x2 2 1

y 5 2 1 } 2 x2 1 8

3. y 5 2x2 2 1 } 2

y 5 x2 1 7 } 2

4. y 5 2x2 1 4x 1 3

y 5 x2 1 x 1 3

5. y 5 2x2 1 3x 1 1

y 5 22x2 2 3x 1 1


A batter hits a baseball in such a way that its path is described by the quadratic function

y 5 20.00126875x2 1 0.5x 1 3.

A fence of varying height surrounds the baseball fi eld. Given the information in the exercise, determine whether the ball goes over the fence, hits the fence, or hits the ground before reaching the fence.

6. The fence is 380 feet away from the batter, and the fence is 10 feet high.



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1. Ï}

49 2. Ï}

225 3. Ï}

100

Isolate the variable in the equation.

4. 9x2 2 18 5 0 5. 4x2 2 12 5 0 6. 10x2 2 40 5 0

Solve the equation.

7. x2 5 36 8. x2 2 9 5 0 9. 5x2 5 20

10. 5x2 2 45 5 0 11. 2x2 2 18 5 0 12. 3x2 2 12x 5 0

Evaluate the expression. Round your answer to the nearest hundredth.

13. Ï}

5 14. Ï}

10 15. Ï}

12

Solve the equation. Round the solutions to the nearest hundredth.

16. x2 5 8 17. x2 2 3 5 0 18. 7x2 2 14 5 0

Use the given area A of the circle to fi nd the radius r or the diameter d of the circle. Round the answer to the nearest hundredth, if necessary.

19. A = 25π m2 20. A = 121π in.2 21. A = 23π cm2

r

r

d

22. Boat Racing The maximum speed s (in knots or nautical miles per hour) that some

kinds of boats can travel can be modeled by s2 5 16

} 9 x where x is the length of the

water line in feet. Find the maximum speed of a sailboat with a 20-foot water line. Round your answer to the nearest hundredth.

23. Stockpile You can fi nd the radius r (in inches) of a cylindrical air compressor

receiver tank by using the formula c 5 1 } 73.53 hr2 where h is the height of the tank

(in inches) and c is the capacity of the tank (in gallons). Find the tank radius of each tank in the table. Round your answers to the nearest inch.

Tank Height (in.) Radius (in.) Capacity (in.3)

A 24 ? 12

B 36 ? 24

C 48 ? 65

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213Algebra 1


Solve the equation.

1. 6x2 2 24 5 0 2. 8x2 2 128 5 0 3. x2 2 13 5 23

4. 3x2 2 60 5 87 5. 2x2 2 33 5 17 6. 5x2 2 200 5 205

7. 4x2 2 125 5 225 8. 7x2 2 50 5 13 9. 1 }

2 x2 2

1 } 2 5 0


10. x2 1 15 5 23 11. x2 2 16 5 213 12. 12 2 x2 5 17

13. 3x2 2 8 5 7 14. 9 2 x2 5 9 15. 4 1 5x2 5 34

16. 48 5 14 1 2x2 17. 8x2 5 50 18. 3x2 1 23 5 18

19. (x 2 3)2 5 5 20. (x 1 2)2 5 10 21. 3(x 2 4)2 5 18

Use the given area A of the circle to fi nd the radius r or the diameter d of the circle. Round the answer to the nearest hundredth, if necessary.

22. A 5 169π m2 23. A 5 38π in.2 24. A 5 45π cm2

r

r

d

25. Flower Seed A manufacturer is making a cylindrical can that will hold

6 in.

and dispense fl ower seeds through small holes in the top of the can. The manufacturer wants the can to have a volume of 42 cubic inches and be 6 inches tall. What should the diameter of the can be? (Hint: Use the formula for volume, V = πr2h, where V is the volume, r is the radius, and h is the height.) Round your answer to the nearest inch.

26. Stockpile You can fi nd the diameter D (in feet) of a conical pile of sand, dirt, etc. by using the formula V 5 0.2618hD2 where h is the height of the pile (in feet) and V is the volume of the pile (in cubic feet). Find the diameter of each stockpile in the table. Round your answers to the nearest foot.

Stockpile Height (ft) Diameter (ft) Volume (ft3)

A 10 ? 68

B 15 ? 230

C 20 ? 545

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LESSON


Solve the equation.

1. 4x2 2 29 5 7 2. 2x2 2 50 5 48 3. 5x2 2 120 5 240

4. 1 }

2 x2 2 2 5 0 5.

1 }

3 x2 2 8 5 4 6. 0.1x2 2 6.4 5 0


7. 4x2 2 8 5 12 8. 7x2 2 43 5 34 9. 2x2 1 7 5 1

10. 3x2 1 23 5 74 11. 6x2 2 27 5 9 12. 5(x 2 8)2 5 15

13. 4(x 1 9)2 5 24 14. 1 }

2 (x 2 4)2 5 7 15.

3 }

4 (x 1 7)2 5 9

16. 2 } 5 (x 2 4)2 5 16 17. 7x2 2 34 5 2x2 1 16 18. 24 5 3(x2 1 7)

19. 9x2 1 3 5 4(3x2 2 6) 20. 1 x 2 4 } 5 2

2 5 36 21. (16x2 2 8)2 5 81

Solve the equation without graphing.

22. x2 1 6x 1 9 5 16 23. x2 2 4x 1 4 5 100 24. x2 2 10x 1 25 5 121

25. 2x2 2 28x 1 98 5 72 26. 23x2 1 6x 2 3 5 227 27. 1 }

2 x2 1 4x 1 8 5 8

28. Plant Food A manufacturer is making a cylindrical canister that will

18 cm

hold granulated plant food. The manufacturer wants the canister to have a volume of 2036 cubic centimeters and be 18 centimeters tall. What should the diameter of the canister be? (Hint: Use the formula for volume, V = πr2h, where V is the volume, r is the radius, and h is the height.) Round your answer to the nearest centimeter.

29. Speed To estimate the speed s (in feet per second) of a car involved in an accident,

investigators use the formula s 5 11

} 2 Ï}

3 }

4 l where l represents the length (in feet)

of tire skid marks on the pavement. After an accident, an investigator measures skid marks that are 180 feet long. Approximately how fast was the car traveling?

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215Algebra 1


Solve a quadratic equation by fi nding square roots.GOAL

Solve quadratic equations

Solve the equation.

a. x2 2 7 5 9 b. 11y2 5 11 c. z2 1 13 5 5

Solution

a. x2 2 7 5 9 Write original equation.

x2 5 16 Add 7 to each side.

x 5 6 Ï}

16 Take square roots of each side.

5 64 Simplify.

The solutions are 24 and 4.

b. 11y2 5 11 Write original equation.

y2 5 1 Divide each side by 11.

y 5 6 Ï}


5 61 Simplify.


c. z2 1 13 5 5 Write original equation.

z2 5 28 Subtract 13 from each side.

Negative real numbers do not have real square roots. So, there is no solution.

EXAMPLE 1


LESSON

10.4

Take square roots of a fraction

Solve 9m2 5 169.

Solution

9m2 5 169 Write original equation.

m2 5 169

} 9 Divide each side by 9.

m 5 6 Ï}

169

} 9 Take square roots of each side.

m 5 6 13

} 3 Simplify.

The solutions are 2 13

} 3 and 13

} 3 .

EXAMPLE 2

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Solve a quadratic equation

Solve 3(x 1 3)2 5 39. Round the solutions to the nearest hundredth.

Solution

3(x 1 3)2 5 39 Write original equation.

(x 1 3)2 5 13 Divide each side by 3.

x 1 3 5 ± Ï}


x 5 23 ± Ï}

13 Subtract 3 from each side.

The solutions are 23 1 Ï}

13 ø 0.61 and 23 2 Ï}

13 ø 26.61.


Solve the equation.

10. 5(x 2 1)2 5 40 11. 2( y 1 4)2 5 18 12. 4(z 2 5)2 5 32

EXAMPLE 4

Approximate solutions of a quadratic equation

Solve 2x2 1 5 5 15. Round the solutions to the nearest hundredth.

Solution

2x2 1 5 5 15 Write original equation.

2x2 5 10 Subtract 5 from each side.

x2 5 5 Divide each side by 2.

x 5 ± Ï}


x ≈ ±2.24 Use a calculator. Round to the nearest hundredth.

The solutions are about 22.24 and about 2.24.


Solve the equation.

1. w2 2 9 5 0 2. 4r2 2 7 5 9 3. 5s2 1 13 5 9

4. 36x2 5 121 5. 16m2 1 81 5 81 6. 4q2 2 225 5 0


7. 7x2 2 8 5 13 8. 26y2 1 15 5 215 9. 4z2 1 7 5 12

EXAMPLE 3


LESSON

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217Algebra 1


LESSONS

10.1–10.4

1. Multi-Step Problem The research and development expenditures for a company from 1991 to 2003 can be modeled by the function y 5 2x2 2 12x 1 3600 where y is the expenditure (in thousands of dollars) and x is the number of years since 1991.

a. In what year was the research and development expenditure the least?

b. What was the lowest research and development expenditure?

2. Multi-Step Problem Use the rectangle below.

(8 2 x) in.

3x in.

a. Find the value of x that gives the greatest possible area of the rectangle.

b. What is the greatest possible area of the rectangle?

3. Short Response For the period 1998–2001, the number of oil spills O in U.S. water can be modeled by the function O 5 2256t2 1 519t 1 8305 where t is the number of years since 1998. Did the greatest number of oil spills occur in 1999? Explain.

4. Open-Ended Write an equation that models the height of an object being dropped as a function of time. Use the equation to determine the time it takes the object to hit the ground.

5. Gridded Response The skid distance D (in feet) a car travels after applying the

brakes is given by D 5 S2

} 30f

where S is the

speed of the car (in miles per hour) at the time of applying the brakes and f is the drag factor of the road surface. A car skids for 75 feet on a road surface that has a drag factor of 0.9. Find the speed (in miles per hour) when the brakes were applied.

6. Extended Response You throw a football twice into the air.

a. For your fi rst throw, the ball is released 6 feet above the ground with an initial vertical velocity of 25 feet per second. Use the vertical motion model to write an equation for the height h (in feet) of the football as a function of time t (in seconds).

b. For your second throw, the ball is released 5.5 feet above the ground with an initial vertical velocity of 30 feet per second. Use the vertical motion model to write an equation for the height h (in feet) of the football as a function of time t (in seconds).

c. If no one catches either throw, for which of your throws is the ball in the air longer? Explain.

7. Gridded Response

9 ft

r ft

The volume of the cylinder is 144π cubic feet. What is the radius of the cylinder, in feet?

8. Extended Response Students are selling T-shirts to raise money for a class trip. Last year, when the students charged $8 per T-shirt, they sold 100 T-shirts. The students want to increase the cost per T-shirt. They estimate that they will lose 5 sales for each $1 increase in the cost per T-shirt. The revenue R (in dollars) generated by selling the T-shirts is given by the function R 5 (8 1 n)(100 2 5n) where n is the number of $1 increases.

a. Write the function in standard form.

b. Find the maximum value of the function.

c. At what price should the T-shirts be sold to generate the most revenue? Explain your reasoning.

Problem Solving Workshop:Mixed Problem SolvingFor use with pages 648–680 LE

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In Exercises 1–5, solve the equation by writing the left side of the equation as a perfect square, then use square roots to solve the problem.

1. x2 1 6x 1 9 5 81

2. 4x2 1 20x 1 25 5 16

3. 1 } 4 x2 1 2x 1 4 5 0

4. 36x2 1 12x 1 1 5 4

5. 49x2 1 112x 1 64 5 25


A NASA mission plans to send a probe to a moon of a distant planet in our solar system. The probe will orbit the moon at a height of 100 kilometers above the moon’s surface, then fall out of orbit to the surface of the moon. Once the probe begins to fall to the surface of

the moon, its height is modeled by the equation h 5 2 1 } 4 t2 1 100, where t is the time in

minutes and h is the height in kliometers.

6. Once the probe begins to fall, how many minutes pass until the probe hits the surface of the moon?

7. A NASA scientist needs to know how many minutes pass between the time the probe falls out of orbit until the probe is 64 kilometers above the surface of the moon. Find the number of minutes to answer the scientist’s question.

8. Suppose that once the probe reaches a height of 64 kilometers above the surface of the moon it fi res a rocket to temporarily stop the descent and then releases a parachute. Once the parachute is released, the height of the probe is modeled by the

equation h 5 2 1 }

16 t2 1 64. Find the number of minutes between the release of the

parachute and the probe striking the surface of the moon.

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219Algebra 1


Match the expression with the value of c that makes the expression a perfect square trinomial.

1. x2 1 8x 1 c 2. x2 1 16x 1 c 3. x2 1 4x 1 c

A. 4 B. 16 C. 64

Write the expression as a square of a binomial.

4. x2 1 2x 1 1 5. x2 2 14x 1 49 6. x2 1 18x 1 81

7. x2 2 4x 1 4 8. x2 1 22x 1 121 9. x2 2 24x 1 144

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as a square of a binomial.

10. x2 2 10x 1 c 11. x2 2 8x 1 c 12. x2 2 6x 1 c

13. x2 1 22x 1 c 14. x2 2 12x 1 c 15. x2 1 20x 1 c

16. x2 2 30x 1 c 17. x2 1 26x 1 c 18. x2 1 40x 1 c

19. x2 1 3x 1 c 20. x2 1 11x 1 c 21. x2 2 7x 1 c

Solve the equation by completing the square. Round your solutions to the nearest hundredth, if necessary.

22. x2 1 6x 5 2 23. x2 1 10x 5 1 24. x2 2 4x 5 3

25. Flight of an Arrow An arrow is shot into the air with an upward velocity of 64 feet per second from a hill 32 feet high. The height h of the arrow (in feet) can be found by using the model h 5 216t2 1 64t 1 32 where t is the time (in seconds).

a. Write an equation that you can use to fi nd when the arrow will be 64 feet above the ground.

b. When will the arrow be 64 feet above the ground? Round your answer(s) to the nearest hundredth.

c. Write and solve an equation that you can use to fi nd when the arrow will be 32 feet above the ground.

26. Tile Floor You are tiling a fl oor so that it has marble in the center

12 ft

15 ft

x

x

x

xand ceramic tile around the border as shown. The ceramic tile border has a uniform width x (in feet). You have enough money in your budget to purchase marble to cover 28 square feet.

a. Solve the equation 28 5 (12 2 2x)(15 2 2x) to fi nd the width of the border.

b. How many square feet of ceramic tile will you need for the project? Explain how you found your answer.

LESSON


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1. x2 1 12x 1 c 2. x2 1 50x 1 c 3. x2 2 26x 1 c

4. x2 2 18x 1 c 5. x2 1 13x 1 c 6. x2 2 9x 1 c

7. x2 2 11x 1 c 8. x2 1 1 } 2 x 1 c 9. x2 2

6 } 5 x 1 c


10. x2 1 6x 5 1 11. x2 1 4x 5 13 12. x2 2 10x 5 15

13. x2 1 8x 5 10 14. x2 2 2x 2 7 5 0 15. x2 2 12x 2 21 5 0

16. x2 1 3x 2 2 5 0 17. x2 1 5x 2 3 5 0 18. x2 2 x 5 1

Find the value of x. Round your answer to the nearest hundredth, if necessary.

19. Area of triangle 5 30 ft2 20. Area of rectangle 5 140 in.2

x ft

(x 1 4) ft

2x in.

(3x 2 1) in.

21. Colorado The state of Colorado is almost perfectly rectangular, with its north border 111 miles longer than its west border. If the state encompasses 104,000 square miles, estimate the dimensions of Colorado. Round your answer to the nearest mile.

22. Baseball After a baseball is hit, the height h (in feet) of the ball above the ground t seconds after it is hit can be approximated by the equation h 5 216t2 1 64t 1 3. Determine how long it will take for the ball to hit the ground. Round your answer to the nearest hundredth.

23. Fenced-In Yard You have 60 feet of fencing to fence in part of

ww

l

Houseyour backyard for your dog. You want to make sure that your dog has 400 square feet of space to run around in. The back of your house will be used as one side of the enclosure as shown.

a. Write equations in terms of l and w for the amount of fencing and the area of the enclosure.

b. Use substitution to solve the system of equations from part (a). What are the possible lengths and widths of the enclosure?

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221Algebra 1


LESSON



1. x2 1 3.6x 1 c 2. x2 2 1 } 4 x 1 c 3. x2 1

2 } 3 x 1 c


4. x2 2 3x 5 7 } 4 5. x2 1 11x 5 2

15 } 4 6. x2 2

1 } 3 x 5 8

7. x2 2 9x 2 8 5 0 8. x2 2 5x 1 1 5 0 9. x2 1 7x 1 3 } 4 5 0

10. 2x2 2 10x 2 16 5 0 11. 2x2 1 36x 1 12 5 0 12. 3x2 2 42x 1 30 5 0

13. 2x2 1 18x 1 5 5 3 14. 3x2 2 15x 2 10 5 9 15. 4x2 1 4x 2 9 5 0

Find the value of x. Round your answer to the nearest hundredth, if necessary.

16. Area of triangle 5 52 ft2 17. Area of rectangle 5 180 in.2

(x 1 6) ft

(x 1 5) ft

2x in.

(2x 1 3) in.

18. The product of two consecutive negative integers is 240. Find the integers.

19. Stopping Distance A car with good tire tread can stop in less distance than a car with poor tire tread. The formula for the stopping distance d (in feet) of a car with good tread on dry cement is approximated by d 5 0.04v2 1 0.5v where v is the speed of the car (in miles per hour). If the driver must be able to stop within 80 feet, what is the maximum safe speed of the car? Round your answer to the nearest mile per hour.

20. Day Care A day care center has 100 feet of fencing to fence in part

ww

l

Buildingof its land for a safe play area for the children. The people that run the center fi gure that they will need 1000 square feet of space for the play area. One side of the day care building will be used as one side of the play area as shown.

a. Write equations for the length of the fencing and the area of the play area.

b. Use substitution to solve the system of equations from part (a). What are the possible lengths and widths of the play area?

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Solve quadratic equations by completing the square.

VocabularyFor an expression of the form x2 1 bx, you can add a constant c to the expression so that the expression x2 1 bx 1 c is a perfect square trinomial. This process is called completing the square.

GOAL

Complete the square

Find the value of c that makes the expression x2 1 7x 1 c a perfect square trinomial. Then write the expression as the square of a binomial.

Solution

STEP 1 Find the value of c. For the expression to be a perfect square trinomial, c needs to be the square of half the coeffi cient of x.

c 5 1 7 } 2 2 25

49 }

4 Find the square of half the coeffi cient of x.

STEP 2 Write the expression as a perfect square trinomial. Then write the expression as the square of a binomial.

x2 1 7x 1 c 5 x2 1 7x 1 49

} 4 Substitute

49 }

4 for c.

5 1 x 1 7 }

2 2 2 Square of a binomial.

EXAMPLE 1


LESSON

10.5


Solve x2 1 14x 5 213 by completing the square.

Solution

x2 1 14x 5 213 Write original equation.

x2 1 14x 1 (7)2 5 213 1 72 Add 1 14 }

2 2

2, or 72, to each side.

(x 1 7)2 5 213 1 72 Write left side as the square of a binomial.

(x 1 7)2 5 36 Simplify the right side.

x 1 7 5 6 6 Take square roots of each side.

x 5 27 6 6 Subtract 7 from each side.

The solutions of the equation are 27 1 6 5 21 and 27 2 6 5 213.

EXAMPLE 2

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223Algebra 1


Solve a quadratic equation in standard form

Solve 3x2 1 18x 2 9 5 0 by completing the square. Round your solutions to the nearest hundredth.

Solution

3x2 1 18x 2 9 5 0 Write original equation.

3x2 1 18x 5 9 Add 9 to each side.

x2 1 6x 5 3 Divide each side by 3.

x2 1 6x 1 32 5 3 1 32 Add 1 6 } 2 2 2, or 32, to each side.

(x 1 3)2 5 12 Write left side as the square of a binomial.

x 1 3 5 Ï}


x 5 23 6 Ï}

12 Subtract 3 from each side.

The solutions are 23 1 Ï}

12 ø 0.46 and 23 2 Ï}

12 ø 26.46.


Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.

1. x2 2 9x 1 c

2. x2 1 11x 1 c

3. x2 2 16x 1 c

Solve the equation by completing the square. Round your solutions to the nearest hundredth if necessary.

4. q2 2 8q 5 7

5. r2 1 12r 5 23

6. 2s2 2 28s 1 8 5 0

EXAMPLE 3


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1. The product of two consecutive positive even integers is 224. Find the integers.

2. The product of two consecutive positive odd integers is 143. Find the integers.

3. The product of two consecutive positive integers is equal to eleven times the sum of the two integers plus 35. Find the integers.

4. The sum of the squares of two consecutive positive integers is 421. Find the integers.

5. The sum of the squares of a positive integer and fi ve more than twice the integer is equal to 1810. Find the integer.

In Exercises 6–9, complete the square to solve for x.

6. x2 1 bx 1 5 5 12

7. x2 2 5x 1 c 5 3

8. x2 1 bx 1 c 5 0

9. ax2 1 bx 1 c 5 0

10. You are planning a vegetable garden and you lay out a rectangular design 10 feet wide by 20 feet long. After laying out the design you decide you want a larger garden and decide to increase the length of the garden by a length of 2x feet and increase the width by a length of x feet. You have enough dirt to cover an area of 600 square feet, and you want to make the garden as large as possible. What are the dimensions of the fi nished garden? Round your answer to the nearest foot.

11. The path of a rocket shot into the air is modeled by the equation h 5 225t2 1 50t 1 4 where h is the height (in feet) of the rocket above the ground t seconds after it is launched. Find the number of seconds after launch it takes for the rocket to touch back down to the ground. Round your answer to the nearest hundredth second.

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225Algebra 1


FOCUS ON

10.5 PracticeFor use with pages 691–692

Graph the quadratic function. Label the vertex and axis of symmetry.

1. y 5 (x 1 1)2 2 3 2. y 5 3(x 1 2)2 2 1 3. y 5 2(x 2 2)2 1 4

4. y 5 22(x 1 4)2 1 2 5. y 5 1 }

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

3 } 2 (x 2 2)2 1 2

Write the function in vertex form, then graph the function. Label the vertex and axis of symmetry.

7. y 5 2x2 2 12x 1 2 8. y 5 24x2 2 2x 1 16 9. y 5 1 }

2 x2 2 2x 2 1

x

y

1

4

O

10. Write an equation in vertex form of the parabola

x

y

O 1

1

( , )12

12

( , 1)12

( , ) 3 2

1 2

shown. Use the coordinates of the vertex and the coordinates of a point on the graph to write the equation.

11. Challenge The path of a soccer ball is parabolic. The ball reaches a height of 12 feet before it starts to descend and lands 32 feet from the point where it was kicked. What is the equation, in vertex form, which models the path of the soccer ball? Assume the ball was kicked at (0, 0).

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Graph quadratic functions in vertex form.

VocabularyThe vertex form of a quadratic function is y 5 a(x 2 h)2 1 k where a Þ 0. The vertex of the graph is (h, k) and the axis of symmetry is x 5 h. The parabola opens up if a . 0 and opens down if a , 0.

The graph of y 5 a(x 2 h)2 1 k is the graph of y 5 ax2 translated h units horizontally and k units vertically.

GOAL

Graph a quadratic function in vertex form

Graph y 5 2(x 2 1)2 2 2.

Solution

STEP 1 Identify the values of a, h, and k: a 5 2, h 5 1, and k 5 22.Because a . 0, the parabola opens up.

STEP 2 Draw the axis of symmetry, x 5 1.

x

y

O

1

(1, 2)

x 1

1

STEP 3 Plot the vertex (h, k) 5 (1, 22).

STEP 4 Plot four points. Evaluate the function for two x-values less than the x-coordinate of the vertex.

x 5 0: y 5 2(0 2 1)2 2 2 5 0

x 5 21: y 5 2(21 2 1)2 2 2 5 6

Plot the points (0, 0) and (–1, 6) and their refl ections (2, 0) and (3, 6), in the axis of symmetry.



Graph the quadratic function. Label the vertex and axis of symmetry.

1. y 5 3(x 1 1)2 2 5 2. y = 22(x 2 3)2 1 1

EXAMPLE 1


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227Algebra 1


Graph a quadratic function

Graph y 5 x2 2 4x 1 1.

Solution

STEP 1 Write the function in vertex form by completing the square.

y 5 x2 2 4x 1 1 Write original function.

y 1 5 x2 2 4x 1 1 1 Prepare to complete the square.

y 1 4 5 (x2 2 4x 1 4) 1 1 Add 1 – 4 }

2 2 2 5 (–2)2 5 4 to each side.

y 1 4 5 (x 2 2)2 1 1 Write x2 2 4x 1 4 as a square of a binomial.

y 5 (x 2 2)2 2 3 Subtract 4 from each side.

STEP 2 Identify the values of a, h, and k: a 5 1, h 5 2,

x

y

O 1

1

(2, 3)

x 2k 5 23. Because a . 0, the parabola opens up.

STEP 3 Draw the axis of symmetry, x 5 2.

STEP 4 Plot the vertex (h, k) 5 (2, 23).

STEP 5 Plot four more points. Evaluate the function for two x-values less than the x-coordinate of the vertex.

x 5 1: y 5 (1 2 2)2 2 3 5 22

x 5 0: y 5 (0 2 2)2 2 3 5 1

Plot the points (1, 22) and (0, 1) and their refl ections (3, 22) and (4, 1), in the axis of symmetry.



Write the function in vertex form, then graph the function. Label the vertex and the axis of symmetry.

3. y 5 22x2 2 8x 2 7 4. y 5 2x2 1 4x 2 1

EXAMPLE 2


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Identify the values of a, b, and c in the quadratic equation.

1. 5x2 1 7x 1 1 5 0 2. 2x2 2 6x 1 11 5 0

3. 2x2 1 17x 2 23 5 0 4. 10x2 2 8x 2 13 5 0

5. 23x2 1 x 2 2 5 0 6. 5x2 2 18x 2 3 5 0

Match the quadratic equation with the formula that gives its solution(s).

7. 2x2 1 x 2 4 5 0 8. 4x2 2 x 1 2 5 0 9. 2x2 1 4x 1 2 5 0

A. x 5 24 6 Ï

}}

42 2 4(21)(2) }}

2(21) B. x 5

21 6 Ï}}

12 2 4(2)(24) }}

2(2) C. x 5

2(21) 6 Ï}}

(21)2 2 4(4)(2) }}}

2(4)

Use the quadratic formula to solve the equation. Round your solutions to the nearest hundredth, if necessary.

10. x2 1 6x 2 10 5 0 11. x2 2 4x 2 9 5 0

12. 5x2 1 2x 2 3 5 0 13. x2 1 8x 1 2 5 0

14. x2 1 10x 1 1 5 0 15. 2x2 2 3x 1 5 5 0

16. 3x2 1 5x 2 2 5 0 17. 6x2 2 2x 1 5 5 0

18. 2x2 2 8x 1 3 5 0 19. 2x2 1 4x 2 16 5 0

20. 23x2 1 7x 2 2 5 0 21. 5x2 2 2x 1 1 5 0

22. Nuts For the period 1990–2002, the amount of shelled nuts y (in millions of pounds) imported into the United States can be modeled by the function y 5 1.55x2 2 5.1x 1 197 where x is the number of years since 1990.

a. Write and solve an equation that you can use to approximate the year in which 300 million pounds of nuts were imported.

b. Write and solve an equation that you can use to approximate the year in which 237 million pounds of nuts were imported.

23. Soybeans For the period 1995–2003, the number of acres y (in millions) of soybeans harvested in the United States can be modeled by the function y 5 20.31x2 1 3.8x 1 61.6 where x is the number of years since 1995.

a. Write and solve an equation that you can use to approximate the year(s) in which 73 million acres of soybeans were harvested.

b. Graph the function on a graphing calculator. Use the trace feature to fi nd the year in which 73 million acres of soybeans were harvested. Use the graph to check your answer from part (a).

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229Algebra 1



1. x2 1 7x 2 80 5 0 2. 3x2 2 x 2 16 5 0

3. 8x2 2 2x 2 30 5 0 4. x2 1 4x 1 1 5 0

5. 2x2 1 x 1 12 5 0 6. 23x2 2 4x 1 10 5 0

7. 5x2 1 30x 1 32 5 0 8. x2 1 6x 2 100 5 0

9. 4x2 2 x 2 20 5 0 10. 5x2 1 x 2 9 5 0

11. 6x2 1 7x 2 3 5 0 12. 10x2 2 7x 1 5 5 0

Tell which method(s) you would use to solve the quadratic equation. Explain your choice(s).

13. 6x2 2 216 5 0 14. 8x2 5 56 15. 5x2 2 10x 5 0

16. x2 1 8x 1 7 5 0 17. x2 2 6x 1 1 5 0 18. 29x2 1 10x 5 5

Solve the quadratic equation using any method. Round your solutions to the nearest hundredth, if necessary.

19. 210x2 5 250 20. x2 2 16x 5 264 21. x2 1 3x 2 8 5 0

22. x2 5 14x 2 49 23. x2 1 6x 5 14 24. 25x2 1 x 5 13

25. Pasta For the period 1990–2003, the amount of biscuits, pasta, and noodles y (in thousands of metric tons) imported into the United States can be modeled by the function y 5 1.36x2 1 27.8x 1 304 where x is the number of years since 1990.

a. Write and solve an equation that you can use to approximate the year in which 500 thousand metric tons of biscuits, pasta, and noodles were imported.

b. Write and solve an equation that you can use to approximate the year in which 575 thousand metric tons of biscuits, pasta, and noodles were imported.

26. Eggs For the period 1997–2003, the number of eggs y (in billions) produced in the United States can be modeled by the function y 5 20.27x2 1 3.3x 1 77 where x is the number of years since 1997.

a. Write and solve an equation that you can use to approximate the year(s) in which 80 billion eggs were produced.

b. Graph the function on a graphing calculator. Use the trace feature to fi nd the year when 80 billion eggs were produced. Use the graph to check your answer from part (a).

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LESSON



1. 15x2 1 8x 1 1 5 0 2. 4x2 2 6x 1 2 5 0

3. 9x2 1 9x 2 1 5 0 4. x2 2 6x 5 15

5. 4x2 2 3 5 10x 6. 2x2 1 6x 1 5 5 7

7. 8x2 5 5x2 1 9x 1 3 8. 212 5 x2 2 14x 1 30

9. 5x2 2 10x 2 16 5 4x 10. 10x2 1 10 5 8 2 6x

11. 6x2 2 5x 5 3 2 5x2 12. 22x2 2 x 1 4 5 2x 1 3

Tell which method(s) you would use to solve the quadratic equation. Explain your choice(s).

13. 13x2 2 26x 5 0 14. 2x2 2 9x 1 5 5 0 15. x2 2 8x 1 1 5 0

Solve the quadratic equation using any method. Round your solutions to the nearest hundredth, if necessary.

16. 23x2 5 218 17. x2 2 5x 1 10 5 0 18. x2 1 3x 2 1 5 0

19. x2 5 9x 2 81 20. x2 1 6x 5 10 21. 25x2 1 x 5 13

22. 10x2 2 4 5 6x2 1 5 23. 2x2 2 18 5 x2 1 12x 24. (x 1 9)2 5 64

25. Books For the period 1990–2002, the amount of money y (in billions of dollars) spent in the United States on books and maps can be modeled by the function y 5 0.0178x2 1 1.5x 1 16 where x is the number of years since 1990.

a. Find the year in which 20 billion dollars were spent on books and maps.

b. Find the year in which 32 billion dollars were spent on books and maps.

c. Graph the function on a graphing calculator. Use the trace feature to check your answers from parts (a) and (b).

26. Spectator Sports For the period 1990–2002, the amount of money y (in billions of dollars) spent in the United States on admissions to spectator sports can be modeled by the function y 5 0.0284x2 1 0.388x 1 5 where x is the number of years since 1990.

a. Find the year in which 7 billion dollars were spent.

b. Graph the function on a graphing calculator. Use the trace feature to fi nd the year in which 7 billion dollars were spent. Use the graph to check your answer from part (a).

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231Algebra 1


Solve quadratic equations using the quadratic formula.

VocabularyBy completing the square for the quadratic equation ax2 1 bx 1 c 5 0,

you can develop a formula, x 5 2b 6 Ï

}

b2 2 4ac }}

2a , that gives the

solutions of any quadratic equation in standard form. This formula is called the quadratic formula.

GOAL


Solve 5x2 2 3 5 4x.

Solution

5x2 2 3 5 4x Write original equation.

5x2 2 4x 2 3 5 0 Write in standard form.

x 5 2b 6 Ï

}

b2 2 4ac }}

2a Quadratic formula

5 2(24) 6 Ï

}}

(24)2 2 4(5)(23) }}}

2(5) Substitute values in the quadratic formula:

a 5 5, b 5 24, and c 5 23.

5 4 6 Ï

}

76 }

10 Simplify.

The solutions are 4 1 Ï

}

76 }

10 ø 1.27 and

4 2 Ï}

76 }

10 ø 20.47.



1. x2 2 12x 2 14 5 0

2. 5y2 2 7 5 11y

3. 9z2 1 3z 5 5

EXAMPLE 1


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Choose a solution method

Tell what method you would use to solve the quadratic equation. Explain your choice(s).

a. 3x2 1 13x 5 11 b. x2 1 8x 5 7 c. 4x2 2 25 5 0

Solution

a. The quadratic equation cannot be factored easily, and completing the square will result in many fractions. So, the equation can be solved using the quadratic formula.

b. The quadratic equation can be solved by completing the square because the equation can be rewritten in the form ax2 1 bx 1 c 5 0 where a 5 1 and b is an even number.

c. The quadratic equation can be solved using square roots because the equation can be written in the form x2 5 d.

Exercises for Examples 2 and 3 4. In Example 2, fi nd the year when $18,000 was invested.

Tell what method you would use to solve the quadratic equation. Explain your choice(s).

5. x2 1 11x 5 0 6. 23x2 1 19x 5 27 7. 4x2 1 16x 5 12

EXAMPLE 3

Use the quadratic formula Retirement Savings For the period 1995–2005, the amount of dollars invested in an individual’s retirement account can be modeled by the function y 5 30x2 2 24x 1 15,500 where x is the number of years since 1995. In what year was $17,000 invested?

Solution

y 5 30x2 2 24x 1 15,500 Write function.

17,000 5 30x2 2 24x 1 15,500 Substitute 17,000 for y.

0 5 30x2 2 24x 2 1500 Write in standard form.

x 5 2(224) 6 Ï

}}

(224)2 2 4(30)(21500) }}}

2(30) Substitute values in the quadratic formula:

a 5 30, b 5 224, and c 5 21500.

5 24 6 Ï

}

180,576 }}

60 Simplify.

The solutions are 24 1 Ï

}

180,576 }}

60 ø 7 and

24 2 Ï}

180,576 }}

60 ø 27.

The year when $17,000 is invested is about 7 years after 1995, or 2002.

EXAMPLE 2


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233Algebra 1


Problem Solving Workshop:Using Alternative MethodsFor use with pages 693–698


Multiple Representations In Example 3 on page 694, you saw how to solve a problem about fi lms produced in the world from 1971–2001 by using the quadratic formula. You can also solve the problem by using a graph.

Using a Graph You can solve the problem by using a graph.

STEP 1 Graph the equation for the number of fi lms

5 10 15 20 25N

um

ber

of

film

s0

Years since 1971

30

X=11.91734Intersection

Y=4200

10,000

8,000

6,000

4,000

2,000

0

produced in the world using a graphing calculator. Graph y1 5 10x2 2 94x 1 3900. Because you are looking for when the number of fi lms produced is 4200, graph y2 5 4200 and fi nd the intersection between the graphs. You only need to consider x-values between 0 and 30 because that is the interval for the equation.

STEP 2 Find the intersection of the graphs by using the intersect feature on your calculator. You only need to consider positive values of x because a negative solution does not make sense in this situation. The intersection occurs at (11.91734, 4200). There were 4200 fi lms produced about 12 years after 1971, or in 1983.

1. Cassettes For the period 1998–2003, the number y of cassettes (in millions) in manufacturers’ shipments can be modeled by the function y 5 4.3x2 2 50.4x 1 162 where x is the number of years since 1998. In what year were 50 million cassettes shipped?

2. Error Analysis Describe and correct the error made in Exercise 1.

x 5 50.4 6 Ï

}}

(50.4)2 2 4(4.3)(162) }}}

2(4.3)

x 5 50.4 6 Ï

}

2246.24 }} 8.6

There was no time from 1998–2003 when 50 million cassettes were shipped.

3. Diving Board A person jumps off of a 6-foot high diving board with an initial velocity of 13 feet per second. How many seconds does it take the person to hit the water? Round your answer to the nearest tenth of a second.

4. Federal Aid For the period 1998–2003, the amount of money y (in billions of dollars) of federal aid grants to state and local governments can be modeled by the function y 5 1.71x2 1 19.14x 1 244.92 where x is the number of years since 1998. In what year was 290 billion dollars given to state and local governments?

Film Production For the period 1971–2001, the number y of fi lms produced in the world can be modeled by the function y 5 10x2 2 94x 1 3900 where x is the number of years since 1971. In what year were 4200 fi lms produced?

PROBLEM

METHOD

PRACTICE

LESSON

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In Exercises 1–5, the solution to a quadratic equation is given.Write an equation in standard form that has the solution.

Example: x 5 22 6 Ï

}

3 } 5

Solution: The solution to the quadratic equation ax2 1 bx 1 c 5 0 is given by

x 5 2b 6 Ï

}

b2 2 4ac }}

2a . Letting 2b 5 22 gives b 5 2, letting 2a 5 5 gives a 5

5 }

2 ,

and letting b2 2 4ac 5 3 gives c 5 b2 2 3

} 4a

. Substituting the values for a and b you get

c 5 1 } 10

. So the equation 5 }

2 x2 1 2x 1

1 } 10 5 0 has the desired solutions.

1. x 5 24 6 Ï

}

10 }

3

2. x 5 26 6 Ï

} 25 } 7

3. x 5 1 6 Ï

}

0 }

3

4. x 5 217 6 Ï

}

21 }

15

5. x 5 11 6 Ï

}

11 }

11


If the graph of a parabola has x-intercepts, then the axis of symmetry of the parabola can be found at the position that is the average of the two x-intercepts. Use this concept to fi nd the axis of symmetry for the parabola modeled by the equation.

6. y 5 3x2 1 5x 1 2

7. y 5 2x2 2 4x 1 1

8. y 5 6x2 1 x 2 1

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235Algebra 1


Identify the values of a, b, and c in the quadratic equation.

1. 2x2 1 x 2 10 5 0 2. 4x2 2 5x 1 2 5 0 3. x2 2 8x 1 11 5 0

4. 2x2 1 6x 2 3 5 0 5. 12 2 3x 2 x2 5 0 6. 3x2 2 4x 1 15 5 0

Find the discriminant of the quadratic equation.

7. x2 1 3x 1 6 5 0 8. x2 2 5x 1 12 5 0 9. x2 2 2x 2 10 5 0

10. 3x2 2 4x 1 1 5 0 11. 5x2 1 x 1 4 5 0 12. 2x2 1 8x 2 3 5 0

13. 24x2 2 6x 1 3 5 0 14. 10x2 2 3x 1 7 5 0 15. 2x2 2 9x 2 3 5 0

Tell whether the equation has two solutions, one solution, or no solution.

16. 3x2 1 x 1 1 5 0 17. 2x2 1 5x 1 7 5 0 18. x2 2 10x 1 8 5 0

19. 4x2 1 x 2 6 5 0 20. 2x2 2 5x 2 8 5 0 21. 26x2 2 2x 1 7 5 0

22. 10x2 1 12x 2 1 5 0 23. 8x2 2 x 1 15 5 0 24. 3x2 1 12x 1 12 5 0

Find the number of x-intercepts that the graph of the function has.

25. y 5 x2 2 5x 2 3 26. y 5 3x2 2 x 2 1 27. y 5 4x2 1 6x 1 1

28. y 5 2x2 2 7x 1 7 29. y 5 8x2 2 4x 1 1 30. y 5 x2 1 2x 1 1

31. Blueprints You want to build a shed in your backyard. 10 ft x

15 ft15 2 x

You have blueprints which show that the shed is 15 feet long and 10 feet wide. You want to change the dimensions as shown. The new area can be modeled by the function y 5 2x2 1 5x 1 150.

a. Write an equation that you can use to determine if there is a value of x that gives an area of 155 square feet.

b. Use the discriminant of your equation from part (a) to show that it is possible to fi nd a value of x for which the area is 155 square feet.

c. Find the value(s) of x for which the area is 155 square feet. Round your answer(s) to the nearest tenth.

32. House Painting You are painting a house. While standing on a ladder that is 15 feet above the ground, you ask your friend to toss you a paintbrush. The starting height of the paintbrush is 5.5 feet and its initial vertical velocity is 20 feet per second. Write an equation that you can use to determine whether or not the paint-brush reaches you. Then use the discriminant to determine whether the paintbrush reaches you.

LESSON


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1. x2 1 x 1 3 5 0 2. 2x2 2 4x 2 5 5 0 3. 22x2 1 10x 2 5 5 0

4. 3x2 2 9x 1 8 5 0 5. 10x2 2 8x 1 1 5 0 6. 24x2 1 9 5 0

7. 36x2 2 9x 5 0 8. 3x2 1 2 5 4x 9. 12 5 x2 2 6x

10. 1 }

6 x2 1 3 5 x 11. 28x2 2 9x 5

2 } 3 12. 8x2 1 12x 1 2 5 4x


13. y 5 x2 2 6x 2 3 14. y 5 5x2 2 x 2 1 15. y 5 6x2 2 6x 1 1

16. y 5 x2 1 x 1 6 17. y 5 24x2 1 x 1 1 18. y 5 4x2 1 5x 2 1

19. y 5 2x2 2 4x 1 2 20. y 5 10x2 2 5x 1 1 21. y 5 8x2 1 x 1 4

22. y 5 215x2 1 3x 1 5 23. y 5 1 } 2 x2 2 4x 1 8 24. y 5

2 } 3 x2 2 5x 1 2

Give a value of c for which the equation has (a) two solutions, (b) one solution, and (c) no solution.

25. x2 1 10x 1 c 5 0 26. x2 2 4x 1 c 5 0 27. 25x2 1 10x 1 c 5 0

28. 49x2 2 14x 1 c 5 0 29. 2x2 1 4x 1 c 5 0 30. 3x2 2 18x 1 c 5 0

31. Playhouse You want to build a playhouse for your sister in your 12 ft x

13 ft13 2 x

backyard. You have blueprints which show that the playhouse is 12 feet long and 13 feet wide. You want to change the dimensions as shown. The new area can be modeled by the function y 5 2x2 1 x 1 156.

a. Write an equation that you can use to determine if there is a value of x that gives an area of 150 square feet.

b. Use the discriminant of your equation from part (a) to show that it is possible to fi nd a value of x for which the area is 150 square feet.

c. Find the value(s) of x for which the area is 150 square feet.

32. Tennis You and your friend are walking around the exterior of a tennis court that has a 12-foot high fence around it. You pick up a ball and try to throw it from a height of 5 feet over the fence. You throw it with an initial vertical velocity of 20 feet per second. Did the ball make it over the fence?

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237Algebra 1


LESSON



1. x2 1 x 1 5 5 0 2. 100x2 2 36x 5 0 3. 5x2 1 4 5 6x

4. 14 5 x2 2 7x 5. 1 }

3 x2 1 6 5 x 6. 24x2 2 5x 5

3 } 4

7. 9x2 1 11x 1 1 5 5x 8. 6x2 1 10 5 3x2 2 3x 1 4 9. 4x2 1 4 5 12x 2 4x2


10. y 5 5x2 1 4x 2 1 11. y 5 3x2 2 15x 1 5 12. y 5 4x2 1 x 1 8

13. y 5 x2 2 4x 2 2 14. y 5 5x2 2 10x 1 5 15. y 5 26x2 1 5x 1 3

16. y 5 6x2 1 9x 1 1 17. y 5 1 } 5 x2 2 4x 2 3 18. y 5

3 } 4 x2 2 4x 1 3

Give a value of c for which the equation has (a) two solutions, (b) one solution, and (c) no solution.

19. x2 1 12x 1 c 5 0 20. x2 2 8x 1 c 5 0 21. 81x2 1 18x 1 c 5 0

22. 36x2 2 12x 1 c 5 0 23. 4x2 1 24x 1 c 5 0 24. 5x2 2 45x 1 c 5 0

Tell whether the vertex of the graph of the function lies above, below, or on the x-axis. Explain your reasoning.

25. y 5 x2 2 9x 1 20 26. y 5 4x2 2 24x 1 36 27. y 5 8x2 2 3x 1 5

28. Football You kick a football with an initial upward velocity of 42 feet per second from the ground.

a. Use the vertical motion model to write a function that models the height h (in feet) of the ball after t seconds.

b. Does the ball reach a height of 25 feet? If so, when?

29. Deck Box You want to build a deck box for the deck

48 in.x

18 in.18 2 x

off the back of your house. You have blueprints which show that the base of the deck box is 18 inches wide and 48 inches long. You want to change the dimensions as shown. The area can be modeled by the function y 5 2x2 2 30x 1 864.

a. Can you change the dimensions so that the area is 700 square inches?

b. Can you change the dimensions so that the area is 5 square feet? Explain how you got your answer.

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Use the value of the discriminant.

VocabularyIn the quadratic formula, the expression b2 2 4ac is called the discriminant of the associated equation ax2 1 bx 1 c 5 0.

GOAL

Use the discriminant

Equationax2 1 bx 1 c 5 0

Discriminantb2 2 4ac

Number ofsolutions

a. 9x2 1 30x 1 25 5 0 302 2 4(9)(25) 5 0 One solution

b. 7x2 2 4x 1 6 5 0 (24)2 2 4(7)(6) 5 2152 No solution

c. 4x2 2 8x 1 3 5 0 (28)2 2 4(4)(3) 5 16 Two solutions

EXAMPLE 1


LESSON

10.7

Find the number of solutions

Tell whether the equation 16x2 1 49 5 56x has two solutions, one solution, or no solution.

Solution


16x2 1 49 5 56x Write equation.

16x2 2 56x 1 49 5 0 Subtract 56x from each side.

STEP 2 Find the value of the discriminant.

b2 2 4ac 5 (256)2 2 4(16)(49) Substitute 16 for a, 256 for b, and 49 for c.

5 0 Simplify.

The discriminant is zero, so the equation has one solution.



1. 2x2 1 x 5 21

2. 4x2 1 5x 1 2 5 0

3. 25x2 1 4 5 20x

EXAMPLE 2

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239Algebra 1


Find the number of x-intercepts

Find the number of x-intercepts of the graph of y 5 x2 2 12x 1 36.

Solution

Find the number of solutions of the equation 0 5 x2 2 12x 1 36.

b2 2 4ac 5 (212)2 2 4(1)(36) Substitute 1 for a, 212 for b, and 36 for c.

5 0 Simplify.

The discriminant is zero, so the equation has one solution. This means that the graph of y 5 x2 2 12x 1 36 has one x-intercept.

CHECK You can use a graphing calculator to check your answer. Notice that the graph of y 5 x2 2 12x 1 36 intercepts the x-axis once.


Find the number of x-intercepts of the graph.

4. y 5 7x2 2 14x

5. y 5 x2 1 7x 1 13

6. y 5 4x2 2 12x 1 9

EXAMPLE 3


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In Exercises 1–5, fi nd the value(s) of k for which the equation has exactly one solution.

1. x2 1 kx 1 1 5 0

2. 4x2 1 2x 1 k 5 0

3. 5kx2 1 40x 1 6 5 0

4. k2x2 1 kx 1 k 5 0

5. kx2 1 3k2x 1 2k4 5 0

In Exercises 6–8, fi nd the value(s) of k for which the equation has no solution. Write your answer as an inequality.

6. 3x2 1 2x 1 k 5 0

7. kx2 1 21x 2 3 5 0

8. 12 } k x2 2 6x 1 k2 5 0

In Exercises 9 and 10, fi nd the values of k for which the equation has exactly two solutions. Write your answer as an inequality.

9. 7kx2 2 2x 1 3 5 0

10. k2x2 1 kx 1 2 5 0

11. Suppose a recreation equipment manufacturer determines that the profi t for the sale of x number of snowboards is given by the equation P(x) 5 2400x2 1 12,000x 2 80,000. How many snowboards would the manufacturer have to sell in order to earn a profi t? Write your answer as an inequality.

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241Algebra 1


TI-83 Plus

Example 1

From the home screen, enter the following to clear lists L1 and L2.

STAT 4 2nd [L1] , 2nd [L2] ENTER

STAT 1

Move cursor to list L1.

0 ENTER 1 ENTER 2 ENTER 3

ENTER 4 ENTER


15.8 ENTER 30.7 ENTER 46 ENTER

75.7 ENTER 104 ENTER

2nd [STAT PLOT] 1 ENTER

ENTER 2nd [L1] ENTER

2nd [L2] ENTER ENTER WINDOW

(�) 1 ENTER 9 ENTER 1 ENTER (�)

20 ENTER 160 ENTER 20 GRAPH

STAT 0 2nd [L1] , 2nd [L2]

ENTER Y= 17.5 ( 1.6 ) ^

X,T,�,n GRAPH

Example 2

From the home screen, enter the following to clear lists L1 and L2.

STAT 4 2nd [L1] , 2nd [L2] ENTER

STAT 1


0 ENTER 3 ENTER 6 ENTER 9

ENTER 12 ENTER 15 ENTER


500 ENTER 31000 ENTER 76000

ENTER 135500 ENTER 201500

ENTER 360000 ENTER

2nd [STAT PLOT] 1 ENTER

ENTER 2nd [L1] ENTER

2nd [L2] ENTER ENTER WINDOW

(�) 3 ENTER 18 ENTER 3 ENTER

(�) 50000 ENTER 400000 ENTER

50000 GRAPH STAT 5 2nd [L1] ,

2nd [L2] ENTER Y= 1440 X,T,�,n

x 2 � 1010 X,T,�,n � 8000 GRAPH


LESSON

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Match the function with the graph it represents.

1. Linear function 2. Exponential function 3. Quadratic function

A.

x

y

1

3

5

1 32121

2325

B.

x

y

1

3

5

1 32121

2325

C.

x

y

1

3

5

1 32121

23

Use the graph to tell whether the points represent a linear function, an exponential function, or a quadratic function.

4.

x

y

2

6

10

2 62226

5.

x

y

1

3

1 3 52121

6.

x

y

2

6

2 6222622

26

7.

x

y1

1 32121

23

25

23

8.

x

y

1

3

5

1 3 5 72121

9.

x

y

1

52121

23

Use a graph to tell whether the ordered pairs represent a linear function, an exponential function, or a quadratic function.

10. (24, 27), (22, 24), (0, 21), (2, 2), (4, 5) 11. (22, 8), (21, 4), (0, 2), (1, 1), 1 2, 1 }

2 2

x

y

2

6

2 6222622

26

x

y

2

6

10

1 3212322

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243Algebra 1


12. (23, 0), (22, 22), (21, 22), (0, 0), (1, 4) 13. (0, 25), (1, 1), (2, 7), (3, 13), (4, 19)

x

y

1

3

1212321

x

y

5

15

1 3 52125

14. (0, 1), (1, 2), (2, 4), (3, 8), (4, 16) 15. (1, 2), (2, 21), (3, 22), (4, 21), (5, 2)

x

y

4

12

20

1 3 52124

x

y

1

3

1 3 52121

23

Tell whether the table of values represents a linear function, an exponential function, or a quadratic function.

16. x 28 24 0 4 8

y 21 0 1 2 3

17.x 23 22 21 0 1

y 625 125 25 5 1

18. x 24 23 22 21 0

y 7 4 3 4 7

19. x 21 0 1 2 3

y 23 0 1 0 23

20. Baseball Salaries The graph shows a model for the salaries (in

00

5001000150020002500

1 3 52 4Years since 1999

Sal

ary

(th

ou

san

ds

of

do

llars

)

y

x

thousands of dollars) of baseball players for the period 199922003.

a. Is the model a linear function, a quadratic function, or an exponential function?

b. Is this model good for predicting the salaries of players after 2003? Explain your reasoning.

21. Consumer Spending The graph shows the amount of money spent

00

406080

100120

2 6 104 8 12Years since 1990

Am

ou

nt

spen

t(b

illio

ns

of

do

llars

) y

x

(in billions of dollars) in the United States on video and audio products, computer equipment, and musical instruments for the period 199022002. Tell whether the data should be modeled by a linear function, an exponential function, or a quadratic function.

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LESSON




A.

x

y

1

3

12121

23

23

B.

x

y

1

3

1 32121

C.

x

y

1

3

1212321


4. (22, 16), (21, 8), (0, 4), (1, 2), (2, 1) 5. (23, 4), (22, 0), (21, 22), (0, 22), (1, 0)

x

y

4

12

20

1 3212324

x

y

1

3

1212321

6. (24, 17), (22, 11), (0, 5), (2, 21), (4, 27) 7. (29, 21), (26, 22), (23, 23), (0, 24), (3, 25)

x

y

4

12

6222624

x

y1

3232921

25

8. 1 22, 1 }

9 2 , 1 21,

1 }

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

x

y

2

6

10

1 3212322

x

y

1

3

5

1 3 5

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245Algebra 1



10.x 0 1 2 3 4

y 1 5 25 125 625

11. x 22 21 0 1 2

y 210 27 24 21 2

12. x 21 0 1 2 3

y 4 1 0 1 4

13. x 210 25 0 5 10

y 4 3.5 3 2.5 2

14. x 22 21 0 1 2

y 32 8 2 1 }

2

1 }

8

15. x 24 23 22 21 0

y 23 0 1 0 23

16.x 22 21 0 1 2

y 1 3 5 7 9

17. x 23 22 21 0 1

y 27 9 3 1 1 }

3

18. Use the graph shown.

a. Which function does the graph represent, an exponential function

x

y

32

96

160

224

1 5(0, 1) (1, 4)(2, 16)

(3, 64)

(4, 256)

or a quadratic function? Explain your reasoning.

b. Make a table of values for the points on the graph. Then use differences or ratios to check your answer in part (a).

c. Write an equation for the function that the table of values from part (b) represents.

19. Pleasure Boats The graph shows total amount of sales

00

6,0008,000

10,00012,00014,00016,000

2 6 104 8 12Years since 1990

Pleasure Boats

Sal

es(m

illio

ns

of

do

llars

) y

x

(in millions of dollars) of pleasure boats in the United States for the period 1990–2002. Tell whether the data should be modeled by a linear function, an exponential function, or a quadratic function. Explain your reasoning.

20. Computer Value The value V of a computer between 1999 and 2003 is given in the table. Tell whether the data should be modeled by a linear function, an exponential function, or a quadratic function. Then write an equation for the function.

Years since 1999, t 0 1 2 3 4

Value, V (dollars) 800 725 650 575 500

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LESSON




A.

x

y

1

1 321

23

B.

x

y

1

3

5

1 3 5212321

C.

x

y

1

3

5

1 3212321


4. (25, 5), (23, 23), (21, 23), (0, 0), (1, 5) 5. (24, 222), (22, 212), (0, 22), (2, 8), (4, 18)

x

y

1212325

2

6

22

x

y

6

18

2 6222626

218

6. (0, 25), (2, 24.5), (4,24), (6, 23.5), (8, 23) 7. (22, 8), (21, 2), 1 0, 1 }

2 2 , 1 1,

1 }

8 2 , 1 2,

1 }

32 2

x

y1

2 6 102221

23

25

x

y

2

6

1 3212322

8. (27, 7), (26, 4), (25, 3), (24, 4), (23, 7) 9. (0, 1), (1, 4), (2, 16), (3, 64), (4, 256)

x

y

2

6

10

2222621022

x

y

64

192

320

1 3 521264

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247Algebra 1



10.x 0 1 2 3 4

y 2 2.1 2.2 2.3 2.4

11. x 1 2 3 4 5

y 26 23 22 23 26

12. x 24 23 22 21 0

y 1296 216 36 6 1

13. x 0 1 2 3 4

y 6 3 0 23 26

14. x 25 24 23 22 21

y 24 21 0 21 24

15. x 23 22 21 0 1

y 1024 128 16 2 1 }

4

16. x 23 22 21 0 1

y 15 11 7 3 21

17. x 2 3 4 5 6

y 2 21 22 21 2

18. Use the graph shown.

x

y

8

24

40

56

1 3

(0, 1)(1, 0.25)

2123

(23, 64)

(22, 16)

(21, 4)

a. Which function does the graph represent, an exponential function or a quadratic function? Explain your reasoning.

b. Make a table of values for the points on the graph. Then use differences or ratios to check your answer in part (a).

c. Write an equation for the function that the table of values from part (b) represents.

19. Printer Value The value V of a printer between 1999 and 2003 is given in the table. Tell whether the data should be modeled by a linear function, an exponential function, or a quadratic function. Then write an equation for the function.

Years since 1999, t 0 1 2 3 4

Value, V (dollars) 2000 1920 1840 1760 1680

20. Interest The balance B of an account is given in the table. Tell whether the data should be modeled by a linear function, an exponential function, or a quadratic function. Then write an equation for the function.

Time, t (years) 0 1 2 3 4

Balance, B (dollars) 1020.20 1040.60 1061.42 1082.64 1104.30

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Compare linear, exponential, and quadratic models.GOAL

Choose functions using sets of ordered pairs


a. (22, 216), ( 21, 215), (0, 212), (1, 27), (2, 0)

b. (22, 1), ( 21, 3), (0, 5), (1, 7), (2, 9)

c. 1 22, 1 }

25 2 , 1 21,

1 } 5 2 , (0, 1), (1, 5), (2, 25)

Solution

a.

2222

2

26

210

x

y b.

21 1

1

3

5

7

9

x

y c.

2123 1

3

9

15

21

27

x

y

Quadratic function Linear function Exponential function

EXAMPLE 1

Identify functions using differences or ratios

Use differences or ratios to tell whether the table of values represents a linear function, an exponential function, or a quadratic function.

Solution

x 21 0 1 2

y 1 3 9 27

Ratios: 3 }

1 5 3 3 3

The table represents an exponential function.

EXAMPLE 2


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249Algebra 1


Write an equation for a function

Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. Then write an equation for the function.

STEP 1 Determine which type of function the values in the table represent.

x 21 0 1 2 3

y 7 5 3 1 21

First differences: 2 2 2 2 2 2 2 2 The table of values represents a linear function because the fi rst differences

are equal.

STEP 2 Write an equation for the linear function. The equation has the form y 5 mx 1 b. When x 5 0, y 5 5, so b 5 5. Find m by substituting any two points into the slope formula.

m 5 5 2 7

} 0 2 (21)

5 22

} 1 5 22

21 121

3

1

5

7

x

y

The equation is y 5 22x 1 5.

CHECK Plot the ordered pairs from the table. Then graph y 5 22x 1 5 to see that the graph passes through the plotted points.


Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. Then write an equation for the function.

3. x 21 0 1 2

y 12 6 2 0

4. x 22 21 0 1 2

y 0.0625 0.125 0.25 0.5 1

EXAMPLE 3

Exercises for Examples 1 and 2 1. Tell whether the ordered pairs represent a linear function, a quadratic

function, or an exponential function: (21, 26), (0, 24), (1, 0), (2, 6).

2. Tell whether the table represents a linear function, a quadratic function, or an exponential function.

x 0 1 2 3

y 26 3 12 21


LESSON

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LESSONS

10.5–10.8

1. Multi-Step Problem Different currents (in amperes) are sent through an electric circuit. The powers (in volts) that are recorded from the electric current are shown in the table.

Current (amperes) Power (volts)

0.5 5

1 20

1.5 45

2 80

2.5 125

a. Tell whether the data can be modeled by a linear function, an exponential function, or a quadratic function.

b. Write an equation for the function.

2. Multi-Step Problem A lacrosse player throws a ball upward from his playing stick with an initial height of 6.5 feet above the ground at initial vertical velocity of 80 feet per second.

a. Write an equation for the height h (in feet) of the ball as a function of the time t (in seconds) after it is thrown.

b. Another player catches the ball when it is 4 feet above the ground. How long after the ball is thrown is the ball caught? Round your answer to the nearest second.

3. Multi-Step Problem From the edge of a ledge directly over a target, you throw a marker with an initial downward velocity of 230 feet per second from a height of 80 feet.

a. Write an equation for the height h (in feet) of the marker as a function of the time t (in seconds) after it is thrown.

b. How long will it take the marker to hit the target? Round your answer to the nearest tenth of a second.

4. Open-Ended Write a quadratic equation that has no solution. Use the discriminant to verify the quadratic equation has no solution.

5. Short Response For the period 1997–2002, the average monthly basic rate y (in dollars) for cable television can be modeled by y 5 0.15x2 1 0.93x 1 26.55 where x is the number of years since 1997.

a. Use the discriminant to determine the number of values of x that correspond to y 5 29.

b. Were there any years during the period 1997–2002 in which the average monthly basic rate for cable television reached $29? Explain.

6. Gridded Response The triangle below has an area of 50 square inches. What is the value of x? Round your answer to the nearest tenth.

(x 2 2) in.

(x 2 3) in.

(x 1 8) in.

7. Extended Response You want to place a walkway around a pool as shown.

x ft

x ft

x ft x ft

40 ft

28 ft

a. Write an equation for the area A (in square inches) of the walkway.

b. You have enough bricks to cover 450 square feet. What should the width of the walkway be? Round your answer to the nearest foot.

c. Explain why you could ignore one of the values of x in part (b).

Problem Solving Workshop:Mixed Problem SolvingFor use with pages 685–713

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251Algebra 1


In Exercises 1–3, use the following data.

(0, 3), (2, 7), (3, 9), (5, k)

1. Tell whether the data fi ts a linear model, quadratic model, or exponential model.

2. Find a value of k that makes the data fi t the model selected in Exercise 1.

3. Write the model for the value of k found in Exercise 2.


(1, 3), (3, 6.75), (5, 15.1875), (7, k)





(2, 10), (5, 73), (8, 190), (11, k)




10. The weight of a male African elephant increases during the fi rst year of life according to the model y 5 10,000 2 9650(k)x where y represents the weight (in pounds) of the elephant and x represents the number of months after birth.If a one-year-old male African elephant weights 2000 pounds, how much did the elephant weigh when it was 4 months old?

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Vertical Motion PuzzleThe vertical motion model h 5 216(t 1 3)(t 2 3) models the height h (in feet) of an object after t seconds. Use the coordinate plane to plot the points according to the directions below. Then use the resulting fi gure to determine the initial height (in feet) of the object.

Plot the following points and connect them in order.

1. Use the negative solution of x2 2 2x 2 3 5 0 as the x-coordinate of (x, 2).

2. Use the solution of x2 1 2x 5 21 as the x-coordinate of (x, 6).


3. Use the greatest positive solution of 2x2 2 11x 1 15 5 0 as the x-coordinate of (x, 2).

4. Use the discriminant of 5x2 1 4x 1 1 }

2 5 0 as the y-coordinate of (3, y).

5. Use the positive solution 10x2 2 10 5 0 as the x-coordinate of (x, 4).

6. Use the positive solution of 3x2 2 8x 2 3 5 0 as the x-coordinate of (x, 4).


7. Use the least positive solution of x2 2 15x 1 54 5 0 as the x-coordinate of (x, 2).

8. Use the solution of 22x2 1 24x 2 72 5 0 as the x-coordinate of (x, 6).

9. Use the discriminant of 23x2 1 8x 5 5 as the y-coordinate of (4, y).

10. Use the positive solution of 5x2 2 30x 5 0 as the x-coordinate of (x, 4).

21 1 3 5 7

1

3

5

7

x

y

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253Algebra 1


Family LetterFor use with Chapter 11

CHAPTER

11



11.1: Graph Square Root Functions

Graph square root functions.

• Microphone Sales

• Oceanography

• Long Jump

11.2: Simplify Radical Expressions

Focus on Operations

Simplify radical expressions.

Perform operations with cube roots.

• Astronomy

• Finance

• Horizon

11.3: Solve Radical Equations Solve radical equations. • Sailing

• Forests

• Biology

11.4: Apply the Pythagorean Theorem and Its Converse

Use the Pythagorean theorem and its converse.

• Construction

• Sails

• Screen Sizes

11.5: Apply the Distance and Midpoint Formulas

Use the distance and midpoint formulas.

• Sightseeing

• Subway

• Archaeology



1. Graphing square root functions

2. Using properties of radicals in expressions and equations

3. Working with radicals in geometry

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CHAPTER

11CHAPTER



Lesson Exercise

11.1 Graph the function y 5 2 Ï}

x 2 3 and identify its domain and range. Compare the graph with the graph of y 5 Ï}

x .

11.2 Simplify the expression.

(a) Ï}

220 (b) 3 Ï}

5x3 p Ï}

4x2 (c) (6 1 Ï}

7 )(3 2 Ï}

7 ) (d) Ï}

75xy2 }

Ï}

5x2y3

Focus onOperations

Simplify the expression. Assume variables are nonzero.

(a) 3 Ï}

4 • 3 Ï}

128 (b) 9 3 Ï}

6t 2 13 3 Ï}

6t

11.3 Solve the equation. Check for extraneous solutions.

(a) 3 Ï}

x 2 5 2 6 5 9 (b) Ï}

2x 1 8 5 x

11.4 A garage door has a height of x feet and a width of (x 1 2.5) feet. If the diagonal (hypotenuse) of the garage door is 12.5 feet, fi nd the actual height and width of the garage door.

11.5 Find the distance between the two points. Then fi nd the midpoint of the line segment connecting the two points.

(a) (25, 1), (7, 3) (b) (2, 28), (26, 22)


Directions Create a treasure map, where either the Pythagorean theorem or the distance formula must be used to get from one point (clue) to another. Have a parent, sibling, or guardian follow your directions to reach the treasure.

Answers11.1:

x

3

1

5

135

y Domain: x ≥ 3; Range: y ≥ 0; The graph of y 5 2 Ï}

x 2 3 is a vertical stretch (by a factor of 2) and a horizontal translation (of 3 units to the right) of the graph of y 5 Ï

} x .

11.2: (a) 2 Ï}

55 (b) 6x2 Ï}

5x (c)11 2 3 Ï}

7 (d) Ï}

15xy } xy Focus on Operations: (a) 8

(b) 24 3 Ï}

6t 11.3: (a) 30 (b) 4 11.4: 7.5 ft; 10 ft 11.5: (a) 2 Ï}

37 units; (1, 2)

(b) 10 units; (22, 25)

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255Algebra 1


Carta para la familiaUsar con el Capítulo 11

CAPÍTULO

11



11.1: Grafi car funciones de raíces cuadradas

Grafi car funciones de raíces cuadradas

• Ventas de micrófonos

• Oceanografía

• Salto largo

11.2: Simplifi car expresiones radicales


Simplifi car expresiones radicales

Hacer operaciones con raíces cúbicas

• Astronomía

• Finanzas

• Horizonte

11.3: Resolver ecuaciones radicales

Resolver ecuaciones radicales • Navegación

• Bosques

• Biología

11.4: Aplicar el teorema de Pitágoras y su recíproco

Usar el teorema de Pitágoras y su recíproco

• Construcción

• Velas

• Tamaños de pantalla

11.5: Aplicar las fórmulas de distancia y del punto medio

Usar las fórmulas de distancia y del punto medio

• Hacer turismo

• Metro

• Arqueología



1. Grafi car funciones de raíces cuadradas

2. Usar propiedades de radicales en expresiones y ecuaciones

3. Trabajar con radicales en geometría

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CHAPTER

11CAPÍTULO



Lección Ejercicio

11.1 Grafi ca la función y 5 2 Ï}

x 2 3 e identifi ca su dominio y rango. Compara la gráfi ca con la gráfi ca de y 5 Ï}

x .

11.2 Simplifi ca la expresión.

(a) Ï}

220 (b) 3 Ï}

5x3 p Ï}

4x2 (c) (6 1 Ï}

7 )(3 2 Ï}

7 ) (d) Ï}

75xy2 }

Ï}

5x2y3

Enfoque en las

operaciones

Simplifi ca la expresión. Asume que las variables no son cero.

(a) 3 Ï}

4 • 3 Ï}

128 (b) 9 3 Ï}

6t 2 13 3 Ï}

6t

11.3 Resuelve la ecuación. Busca soluciones extrañas.

(a) 3 Ï}

x 2 5 2 6 5 9 (b) Ï}

2x 1 8 5 x

11.4 Una puerta de garaje tiene una altura de x pies y un ancho de (x 1 2.5) pies. Si la diagonal (hipotenusa) de la puerta es 12.5 pies, halla la altura y el ancho verdaderos de la puerta.

11.5 Halla la distancia entre dos puntos. Luego halla el punto medio del segmento conectando los dos puntos.

(a) (25, 1), (7, 3) (b) (2, 28), (26, 22)


Instrucciones Haz un mapa de tesoro en que se puede usar o el teorema de Pitágoras o la fórmula de distancia para ir de un punto (pista) a otro. Pida a un padre, hermano o tutor que siga tus indicaciones para encontrar el tesoro.

Respuestas11.1:

x

3

1

5

135

y Dominio: x ≥ 3; Rango: y ≥ 0; La gráfi ca de y 5 2 Ï}

x 2 3 es una extensión vertical (por un factor de 2) y una traslación (de 3 unidades a la derecha) de la gráfi ca de y 5 Ï

} x .

11.2: (a) 2 Ï}

55 (b) 6x2 Ï}

5x (c)11 2 3 Ï}

7 (d) Ï}

15xy } xy Enfoque en las operaciones: (a) 8

(b) 24 3 Ï}

6t 11.3: (a) 30 (b) 4 11.4: 7.5 ft; 10 ft 11.5: (a) 2 Ï}

37 units; (1, 2)

(b) 10 units; (22, 25)

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257Algebra 1


TI-83 Plus

Y= 2nd [ Ï}

] 2 X,T,�,n � 3 )

WINDOW (�) 5 ENTER 5 ENTER 1

ENTER (�) 5 ENTER 5 ENTER 1

ENTER GRAPH



SHIFT ( [ Ï}

] 2 X, ,T� � 3 )

EXE SHIFT F3 (�) 5 EXE 5 EXE 1

EXE (�) 5 EXE 5 EXE 1 EXE EXIT

F6


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1. y 5 8 Ï}

x 2. y 5 28 Ï}

x 3. y 5 1 } 8 Ï

}

x

A.

x

y5

22225

215

225

6 10

B.

x

y

0.1

0.3

22220.1

6 10

C.

x

y

5

15

25

22225

6 10

Graph the function and identify its domain and range. Compare the graph with the graph of y 5 Ï

}

x.

4. y 5 6 Ï}

x 5. y 5 0.4 Ï}

x 6. y 5 22 Ï}

x

x

y

2

6

10

22222

6 10

x

y

222 6 10

0.2

0.6

1.0

20.2

x

y

1

12121

23

25

3 5


7. y 5 Ï}

x 1 5 2 2 8. y 5 Ï}

x 2 2 1 5 9. y 5 Ï}

x 2 5 1 2

10. y 5 Ï}

x 2 5 2 2 11. y 5 Ï}

x 1 2 2 5 12. y 5 Ï}

x 1 5 1 2

A.

x

y1

12121

23

25

3 5

B.

x

y

1

1212521

23

C.

x

y

1

3

5

121232521

D.

x

y

2

6

10

22222

6 10

E.

x

y

2

6

22222

26

10

F.

x

y

2

6

10

22222

6 10

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259Algebra 1



}

x.

13. y 5 Ï}

x 2 5 14. y 5 Ï}

x 1 3 15. y 5 Ï}

x 2 6

x

y

2

6

22222

26

26 6

x

y

1

3

12121

23

23 3

x

y

2

6

22222

26

26 6

16. y 5 Ï}

x 2 2 17. y 5 Ï}

x 1 3 18. y 5 Ï}

x 2 5

x

y

2

6

22222

26

26 6

x

y

1

3

12121

23

23 3

x

y

2

6

22222

26

6 10

19. Fire Hoses For a fi re hose with a nozzle that has a diameter of

00

100200300400500600700

5 15 2510 20 30 35Nozzle pressure (lb/in.2)

Flow

rat

e (g

al/m

in)

f

p

2 inches, the fl ow rate f (in gallons per minute) can be modeled by f 5 120 Ï

}

p where p is the nozzle pressure in pounds per square inch.

a. Graph the function and identify its domain and range.

b. If the fl ow rate is 720 gallons per minute, what is the nozzle pressure?

20. Horizon The distance d (in nautical miles) that a person can see

005

101520

100 200Eye level (feet)

Dis

tan

ce(n

auti

cal m

iles) d

h

to the horizon is given by the formula d 5 1.17 Ï}

h where h is the person’s eye level in feet.


b. A person can see 20 nautical miles to the horizon. What is the person’s eye level? Round your answer to the nearest nautical mile.

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LESSON



}

x.

1. y 5 7 Ï}

x 2. y 5 1 } 5 Ï

}

x 3. y 5 24 Ï}

x

x

y

2

6

10

14

22226 6

x

y

1

3

12121

23

23 3

x

y

2

22222

26

210

26 6

Describe how you would graph the function by using the graph of y 5 Ï}

x.

4. y 5 Ï}

x 2 8 5. y 5 Ï}

x 1 3 6. y 5 Ï}

x 1 7

7. y 5 Ï}

x 2 5 8. y 5 Ï}

x 1 3.5 9. y 5 Î} x 2 1 } 2


10. y 5 Ï}

x 1 4 2 3 11. y 5 Ï}

x 2 3 1 4 12. y 5 Ï}

x 2 4 1 3

13. y 5 Ï}

x 2 4 2 3 14. y 5 Ï}

x 1 3 2 4 15. y 5 Ï}

x 1 3 1 3

A.

x

y

1

3

5

12121

3 5

B.

x

y1

1212321

23

25

3

C.

x

y

1

3

5

12121

3 5

D.

x

y

1

3

5

1212321

3

E. x

y1

2123

23

25

F.

x

y1

21

23

25

1 3 5

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261Algebra 1



}

x.

16. y 5 Ï}

x 1 4 2 4 17. y 5 Ï}

x 1 5 1 1 18. y 5 Ï}

x 2 6 1 4

x

y

1

3

12121

23

23 3

x

y

1

12121

23

2325 3

x

y

2

6

22222

26

26 6 10

19. y 5 Ï}

x 2 5 2 7 20. y 5 Ï}

x 2 1 1 2 21. y 5 Ï}

x 1 5 2 4

x

y

2

6

22222

26

26 6 10

x

y

1

3

12121

23

23 3

x

y

1

3

12121

23

2325 3

22. Box Design You are designing a box with a square base that will hold

00

1

2

3

4

5

6

7

200 400Volume

(cubic inches)

Sid

e len

gth

(in

ch

es)

x

V

popcorn. The box must be 9 inches tall. The side length x (in inches)

of the box is given by the function x 5 1 } 3 Ï

}

V where V is the volume

(in cubic inches) of the box.


b. What is the volume of a box with a side length of 5 inches?

c. What is the volume of a box with a side length of 8 inches?

23. Steel Pipe The inside diameter d of a steel pipe (in inches)

00

3

6

9

12

15

20 40 60 80Weight (pounds)

Dia

mete

r (i

nch

es) d

w10 30 50 70 90

and the weight w of water in the pipe (in pounds) are related by the function d 5 1.71 Ï

}

w.


b. What does the water weigh in a pipe with an inside diameter of 17 inches? Round your answer to the nearest pound.

c. What does the water weigh in a pipe with an inside diameter of 3.5 inches? Round your answer to the nearest pound.

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LESSON



}

x.

1. y 5 2.5 Ï}

x 2. y 5 2 3 } 5 Ï

}

x 3. y 5 20.25 Ï}

x

x

y

1

3

12121

23

23 3

x

y

0.6

1.8

12120.6

21.8

23 3

x

y

0.25

0.75

12120.25

20.75

23 3

Describe how you would graph the function by using the graph of y 5 Ï}

x.

4. y 5 Ï}

x 1 2.5 5. y 5 Ï}

x 2 3 } 2 6. y 5 Ï

}

x 1 12

7. y 5 Î} x 2 1 } 4 8. y 5 Ï

} x 1 5.5 9. y 5 Ï

}

x 1 3 } 4


10. y 5 3 Ï}

x 1 2 2 1 11. y 5 2 Ï}

x 2 1 1 3 12. y 5 3 Ï}

x 2 1 1 2

13. y 5 Ï}

x 2 3 2 2 14. y 5 3 Ï}

x 1 1 2 2 15. y 5 Ï}

x 1 2 1 3

A.

x

y

1

1 3 7

B.

x

y

1

3

5

121 3 5

C.

x

y

1

3

5

121 3 5

D.

x

y

1

5

1212321

3

E.

x

y

1

3

5

121 3 5

F.

x

y

1

3

1 3 5

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263Algebra 1



}

x.

16. y 5 Ï}

x 1 6 2 4 17. y 5 2 Ï}

x 2 1 1 5 18. y 5 Ï}

x 2 3 2 3

x

y

2

6

22222

26

26 6

x

y

2

6

22222

26

26 6

x

y

1

3

12121

23

3 5

19. y 5 2 Ï}

x 1 6 1 2 20. y 5 Ï}

x 2 7 1 8 21. y 5 2 Ï}

x 2 4.5 1 2.5

x

y

2

6

22222

26

26 6

x

y

2

6

10

2 6 10 14

x

y

2

6

22222

26

6 10

22. Bridge The time t (in seconds) it takes an object dropped from a

00

0.51.01.52.02.5

5 15 2510 20Height (meters)

Tim

e (s

eco

nd

s)

t

h

height h (in meters) to reach the ground is given by the function

t 5 Ï

}

10 } 7 Ï

}

h .


b. You are on a bridge that passes over a river. It takes about 1.5 seconds for a stone dropped from the bridge to reach the river. About how high is the bridge?

23. Steel Pipe The radius of gyration of a steel pipe is a number

00

0.51.01.52.02.5

1 3 52 4 76Inside diameter (inches)

Rad

ius

of

gyra

tio

n(i

nch

es)

r

d

that describes a pipe’s resistance to buckling. The greater value of r, the more resistance to buckling. The radius of gyration r (in inches) of a steel pipe is given by the function

r 5 1 } 4 Ï}

D2 1 d2 where D is the outside diameter of the pipe

(in inches) and d is the inside diameter of the pipe (in inches). One standard outside pipe diameter is 4 inches. Write a function for r and d using D 5 4.


b. If you want a pipe with a 4-inch outside diameter and a radius of gyration of 1.3 inches, what must its inside diameter be? Round your answer to the nearest tenth.

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Graph square root functions.

VocabularyA radical expression is an expression that contains a radical, such as a square root, cube root, or other root.

A radical function involves a radical expression with the independent variable in the radicand.

If the radical is a square root, then the function is called a square root function.

The most basic square root function in the family of all square root

functions, called the parent square root function, is y 5 Ï}

x .

GOAL

Graph a function in the form y 5 a Ï}

x

Graph the function y 5 5 Ï}

x and identify its domain and range. Compare the graph with the graph of y = Ï

}

x .

Solution

STEP 1 Make a table. Because the square root of

10

14

6

2

x

y

y 5 5

2 6 10

x

y 5 x

a negative number is undefi ned, x must be non-negative. So the domain is x ≥ 0.

x 0 1 2 3

y 0 5 7.1 8.7


STEP 3 Draw a smooth curve through the points. From either the table or the graph, you can see the range of the function is y ≥ 0.

STEP 4 Compare the graph with the graph of y 5 Ï}

x . The graph of y 5 5 Ï}

x is a

vertical stretch (by a factor of 5) of the graph of y 5 Ï}

x .



}

x .

1. y 5 4 Ï}

x 2. y 5 26 Ï}

x

EXAMPLE 1


LESSON

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265Algebra 1


Graph a function in the form y 5 Ï}

x 1 k

Graph the function y 5 Ï}

x 22 and identify its domain and range. Compare the graph with the graph of y 5 Ï

}

x .

Solution

To graph the function, make a table, plot the

3

1

21x

y

y 5 2 2

5

x

y 5 xpoints, and draw a smooth curve through the points. The domain is x ≥ 0.

x 0 1 2 3 4

y 22 21 20.6 20.3 0

The range is y ≥ 22. The graph of y 5 Ï}

x 2 2 is a vertical translation (of 2 units down) of the graph of y 5 Ï

}

x .

EXAMPLE 2


LESSON

11.1

Graph a function in the form y 5 a Ï}

x 2 h 1 k

Graph the function y 5 3 Ï}

x 1 2 2 4.

Solution

STEP 1 Sketch the graph of y 5 3 Ï}

x .

x

y

6

10

22226 6 10

(0, 0)

(22, 24)

y 5 3 x

y 5 3 x 1 2 2 4

STEP 2 Shift the graph h units horizontally and k units vertically. Notice that

y 5 3 Ï}

x 1 2 2 4 5 3 Ï}

x 2 (22) 1 (24). So, h 5 22 and k 5 24. Shift the graph left 2 units and down 4 units.



}

x .

3. y 5 Ï}

x 1 1 4. y 5 Ï}

x 2 3

5. Identify the domain and range of the function in Example 3.

EXAMPLE 3

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In Exercises 1–5, graph the function and identify its domain and range. Compare the graph with the graph of y 5 Ï

}

x .

1. y 5 Ï}

6 2 x

2. y 5 2 Ï}

2x

3. y 5 2 Ï}

1 2 x

4. y 5 Ï}

1 }

2 x 1 2

5. y 5 Ï}

2 2 x 1 3

In Exercises 6–10, write a rule for a radical function that has the given properties.

6. The domain is all real numbers greater than or equal to 2. The range is all real numbers greater than or equal to 1.

7. The domain is all real numbers less than or equal to 4. The range is all real numbers greater than or equal to 0.

8. The domain is all real numbers greater than or equal to 0. The range is all real numbers less than or equal to 1.

9. The domain is all real numbers less than or equal to 5. The range is all real numbers less than or equal to 3.

10. The domain is all real numbers greater than or equal to 21. The range is all real numbers greater than or equal to 0.

LESSON


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267Algebra 1


Values of a and b

Value of Ï

}

a p Ï}

b Value of

Ï}

ab

a 5 4, b 5 9

a 5 9, b 5 16

a 5 25, b 5 4

a 5 16, b 5 36

Values of a and b

Value of Ï

}

a p Ï}

b Value of

Ï}

ab

a 5 2, b 5 3

a 5 10, b 5 5

a 5 7, b 5 11

a 5 13, b 5 6

Values of a and b

Value of

Ï}

a } Ï

}

b

Value of

Î}

a } b

a 5 4, b 5 16

a 5 9, b 5 25

a 5 36, b 5 4

a 5 4, b 5 49

Values of a and b

Value of

Ï}

a } Ï

}

b

Value of

Î}

a } b

a 5 1, b 5 2

a 5 3, b 5 8

a 5 12, b 5 7

a 5 6, b 5 11


LESSON

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Match the radical with the simplifi ed expression.

1. Ï}

150 2. Ï}

90 3. Ï}

60

A. 3 Ï}

10 B. 2 Ï}

15 C. 5 Ï}

6


4. Ï}

99 5. Ï}

28 6. Ï}

54

7. Ï}

50 8. Ï}

27a 9. Ï}

16x2

10. Ï}

100n3 11. Ï}

125p3 12. Ï}

3 p Ï}

15

Name the value of 1 that you would multiply the radical expression by to rationalize the denominator.

13. 1 }

Ï}

23 14.

3 }

Ï}

10 15.

1 }

Ï}

5x

Simplify the expression by rationalizing the denominator.

16. 1 }

Ï}

5 17.

1 }

Ï}

17 18.

7 }

Ï}

3


19. 3 Ï}

5 1 4 Ï}

5 20. 10 Ï}

2 2 3 Ï}

2 21. Ï}

7 2 4 Ï}

7

22. 4 Ï}

18 1 Ï}

18 23. 5 Ï}

8 2 4 Ï}

8 24. Ï}

12 1 3 Ï}

3

25. Ï}

2 (1 1 Ï}

2 ) 26. Ï}

3 ( Ï}

3 2 2) 27. Ï}

3 (1 1 Ï}

12 )

28. Electricity The voltage V (in volts) required for a circuit is given by V 5 Ï}

PR where P is the power (in watts) and R is the resistance (in ohms). Find the volts needed to light a 60-watt light bulb with a resistance of 110 ohms. Round your answer to the nearest tenth.

29. Drum Heads The radius r (in inches) of a circle with an area A (in square inches)

is given by the function r 5 Î}

A

} π .

a. The drum head on a conga drum has an area of 16π square inches. Find the diameter of the drum head.

b. The drum head on a bongo has an area of 9π square inches. Find the diameter of the drum head.

LESSON


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269Algebra 1



1. Ï}

200 2. Ï}

45 3. Ï}

112

4. Ï}

400d 5. Ï}

9y2 6. Ï}

25n3

7. Ï}

3 p Ï}

21 8. Ï}

20 p Ï}

15 9. Ï}

10x p Ï}

2x

10. Î}

16

} 81

11. Î}

5 }

49 12. Î}

x2

} 144


13. 4 }

Ï}

5 14. Î}

3 }

50 15. Î}

9 } 75

16. 2 }

Ï}

p 17.

1 }

Ï}

3y

18.

9 }

Ï}

2x


19. 10 Ï}

7 1 3 Ï}

7 20. 4 Ï}

5 2 7 Ï}

5 21. Ï}

7 (4 2 Ï}

7 )

22. Ï}

5 (8 Ï}

10 1 1) 23. (2 Ï}

3 1 5)2 24. (6 1 Ï}

3 )(6 2 Ï}

3 )

25. Water Flow You can measure the speed of water by using an

V

hL-shaped tube. The speed V of the water (in miles per hour) is

given by the function V 5 Î}

5 }

2 h where h is the height of the

column of water above the surface (in inches).

a. If you use the tube in a river and fi nd that h is 6 inches, what is the speed of the water? Round your answer to the nearest hundredth.

b. If you use the tube in a river and fi nd that h is 8.5 inches, what is the speed of the water? Round your answer to the nearest hundredth.

26. Walking Speed The maximum walking speed S (in feet per second) of an animal

is given by the function S 5 Ï}

gL where g is 32 feet per second squared and L is the length of the animal’s leg (in feet).

a. How fast can an animal whose legs are 9 inches long walk? Round your answer to the nearest hundredth.

b. How fast can an animal whose legs are 3 feet long walk? Round your answer to the nearest hundredth.

LESSON


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LESSON



1. Ï}

45s3 2. Ï}

196r4 3. Ï}

450c5

4. Ï}

124m4n10 5. 11 Ï}

x7y8 6. Ï}

a3b p Ï}

ab

7. Ï}

27xy p Ï}

5y3 8. Î} 121

} 16m2 9. Î}

5d2

} 125


10. Î}

5 }

8 11. Î}

7m5

} 11

12. Î}

125

} 4x3


13. Ï}

15 1 5 Ï}

3 2 2 Ï}

27 14. Ï}

7 (3 2 2 Ï}

7 ) 15. Ï}

2 (3 Ï}

14 2 Ï}

7 )

16. (3 Ï}

12 1 5)2 17. (8 Ï}

3 1 Ï}

2 )(1 2 Ï}

3 ) 18. Î} 250m3

} 2n

19. 5 }

Ï}

7 1

2 }

Ï}

14 20. 4 Ï

}

10 } Ï

}

30 2

2 }

Ï}

3 21.

4 }

Ï}

x 1

5 }

2 Ï}

x

22. Electricity Current, power, and resistance are related by the formula I 5 Î}

P

} R

where I is the current (in amps), P is the power (in watts), and R is the resistance (in ohms).

a. A light bulb with a 283-ohm resistor is using 0.42 amp of current. What is the wattage of the light bulb? Round your answer to the nearest whole watt.

b. A light bulb with a 145-ohm resistor is using 0.83 amp of current. What is the wattage of the light bulb? Round your answer to the nearest whole watt.

23. Medicine A doctor may need to know a person’s body surface area to prescribe the correct amount of medicine. A person’s body surface area A (in square meters) is given by the function

A 5 Î} hw

} 3131

where h is the height (in inches) and w is the weight (in pounds).

a. Find the body surface area of a person who is 5 feet 5 inches tall and weighs 110 pounds. Round your answer to the nearest tenth of a square meter.

b. Find the body surface area of a person who is 5 feet 10 inches tall and weighs 120 pounds. Round your answer to the nearest tenth of a square meter.

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271Algebra 1


Simplify radical expressions.

VocabularyA radical expression is in simplest form if the following conditions are true:

• No perfect square factors other than 1 are in the radicand.

• No fractions are in the radicand.

• No radicals appear in the denominator of a fraction.

The process of eliminating a radical from an expression’s denominator is called rationalizing the denominator.

GOAL

Use the product property of radicals


Solution

a. Ï}

28 5 Ï}

4 p 7 Factor using perfect square factor.

5 Ï}

4 p Ï}

7 Product property of radicals

5 2 Ï}

7 Simplify.

b. Ï}

50y3 5 Ï}}

25 p 2 p y2 p y Factor using perfect square factors.

5 Ï}

25 p Ï}

2 p Ï}

y2 p Ï}

y Product property of radicals

5 5y Ï}

2y Simplify.

EXAMPLE 1


LESSON

11.2

Multiply radicals


Solution

a. Ï}

2 p Ï}

18 5 Ï}

2 p 18 Product property of radicals

5 Ï}

36 Multiply.

5 6 Simplify.

b. 5 Ï}

2xy p Ï}

32y 5 5 Ï}

2xy p 32y Product property of radicals

5 5 Ï}

64xy2 Multiply.

55 Ï}

64 p Ï}

x p Ï}

y2 Product property of radicals

5 40y Ï}

x Simplify.

EXAMPLE 2

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Rationalize the denominator

3 } Ï

}

6x 5

3 }

Ï}

6x p Ï

}

6x }

Ï}

6x Multiply by

Ï}

6x }

Ï}

6x .

5 3 Ï

}

6x }

Ï}

36x2 Product property of radicals

5 3 Ï

}

6x }

6x 5

Ï}

6x }

2x Simplify.

EXAMPLE 4

Use the quotient property of radicals

a. Î}

17

} 25

5 Ï

}

17 }

Ï}

25 Quotient property of radicals

5 Ï

}

17 } 5 Simplify.

b. Î} 4 }

49y2 5 Ï

}

4 }

Ï}

49y2 Quotient property of radicals

5 2 } 7y Simplify.



1. Ï}

72 2. Ï}

3x2 3. Ï}

45y5 4. 3 Ï}

12x2

5. Ï}

5 p Ï}

10 6. Ï}

3x p Ï}

15xy 7. Ï}

5 }

81 8. Ï

}

2x2

} 9y2

EXAMPLE 3


LESSON

11.2

Add and subtract radicals 3 Ï

}

3 1 6 Ï}

27 5 3 Ï}

3 1 6 Ï}

9 p 3 Factor using perfect square factor.

5 3 Ï}

3 1 6 p Ï}

9 p Ï}

3 Product property of radicals

5 3 Ï}

3 1 18 Ï}

3 5 21 Ï}

3 Simplify.



9. 3 }

Ï}

2x 10. 6 Ï

} 7 1 8 Ï

}

10 2 3 Ï}

7 11. 3 Ï}

5 1 2 Ï}

500

EXAMPLE 5

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273Algebra 1


In Exercises 1–5, simplify the expression.

1. 1 }

Ï}

2 2

1 }

Ï}

8

2. (3 Ï}

6 2 Ï}

18 )(2 Ï}

6 1 2 Ï}

18 )

3. Ï}

5 (2 Ï}

10 2 3 Ï}

15 )

4. (4 Ï}

2x 2 Ï}

x ) ( Ï}

3x 1 3 Ï}

x ); x > 0

5. Ï}

y ( Ï}

2y 1 5 Ï}

4y ); y > 0


A student studying the falling velocity of a skydiver jumping out of an airplane at a height

of d feet above the ground decides to model the velocity by the equation v 5 Î} d 2 h

} c

where v is the velocity in feet per second, c is a constant measuring the coeffi cient of drag caused by the air resistance of the skydiver, and h is the height of the skydiver above the ground in feet.

6. Suppose a skydiver jumps from a height of 10,000 feet wearing a normal jumpsuit

with a coeffi cient of drag c 5 1 } 2 . What is the velocity of the skydiver, in miles per

hour, when the skydiver is 1000 feet above the ground? Round your answer to the nearest tenth.

7. Suppose a skydiver jumps from a height of 10,000 feet wearing a low drag

jumpsuit with a coeffi cient of drag c 5 1 } 4 . What is the velocity of the skydiver

when the skydiver is 1000 feet above the ground? Round your answer to the nearest tenth.

8. Suppose two skydivers, Ann and Bob, jump simultaneously from two different planes. Ann jumps from a height of 12,000 feet wearing a jumpsuit with a

coeffi cient of drag c 5 1 }

3 . Bob is wearing a jumpsuit with a coeffi cient of drag

c 5 3 } 4 . Ann and Bob both plan to open their parachutes at a height of 2000 feet.

From what height should Bob jump if he wants his velocity to be the same as Ann’s velocity when they open their parachutes? Round your answer to the nearest foot.

LESSON


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FOCUS ON



1. 3 Î}

27 ? 3 Î}

27 2. 3 Î}

3z }

281 3.

3 Î}

7 ? 3 Î}

49x

4. 3 Î}

2 ? 3 Î}

232 5. 3 Î}

5y

} 40

6. 6 }

3 Ï}

72

7. 4 }

3 Ï}

2 8.

12 }

3 Ï}

29 9. 7

3 Ï}

x 2 5 3 Ï}

x

10. 3 Ï}

128 2 4 3 Ï}

2 11. 3 3 Ï}

p 1 3 Ï}

27p 12. 3 Ï}

254 1 3 3 Ï}

2

13. 3 Ï}

2z 2 3 Ï

} z 14.

3 Ï}

3 1 4 1 3 Ï}

9x 2 15. 3 Ï}

22 1 2 1 3 Ï}

4 2

16. 1 3 Ï}

32 1 4 2 1 3 Ï}

2 2 1 2 17. 3 Ï}

25 1 4 2 3 Ï}

225 2 18. 1 3 Ï}

8 1 2 2 1 3 Ï}

28 2 2 2

22. Challenge Solve the equation 1 x }

3 Ï}

3 1

2 }

3 Ï}

23 2 3 Ï

}

281 5 3.

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275Algebra 1


Perform operations with cube roots.

Key ConceptsThe Product Property of Cube Roots states that the cube root of a product equals the product of the cube roots.

The Quotient Property of Cube Roots states that the cube root of a quotient equals the quotient of the cube roots of the numerator and denominator.

When rationalizing a denominator, multiply by a form of 1 that will make the radicand in the denominator a perfect cube.

You can use the distributive property to simplify sums and differences of cube roots when the expressions have the same radicand.

GOAL

Use properties of radicals

a. 3 Ï}

4 p 3 Ï}

54 5 3 Ï}

4 p 54 Product property of cube roots

5 3 Ï}

216 Multiply.

5 6 Simplify.

b. 3 Î}

a }

28 5

3 Ï}

a }

3 Ï}

28 Quotient property of cube roots

5 3 Ï}

a }

22 Simplify.



1. 3 Ï}

16 p 3 Ï}

32 2. 3 Î}

8x

} 64

3. 3 Ï}

4 p 3 Ï}

2y

EXAMPLE 1


FOCUS ON

11.2

Rationalize the denominator

4 }

3 Ï}

4 5

4 }

3 Ï}

4 p

3 Ï}

2 }

3 Ï}

2 Multiply by

3 Ï}

2 }

3 Ï}

2 .

5 4

3 Ï}

2 }

3 Ï}

8 Product property of cube roots

5 2 3 Ï}

2 Simplify.

EXAMPLE 2

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Multiply expressions involving cube roots

a. 3 Ï}

6 1 4 1 3 Ï}

36x 2 5 4 3 Ï}

6 1 3 Ï}

6 p 3 Ï}

36x Distributive property

5 4 3 Ï}

6 1 3 Ï}

216x Product property of cube roots

5 4 3 Ï}

6 1 6 3 Ï}

x Simplify.

b. 1 3 Ï}

9 2 3 2 1 3 Ï}

3 1 1 2

5 3 Ï}

9 1 3 Ï}

3 2 1 3 Ï}

9 (1) 1 (23) 3 Ï}

3 1 (23)(1) Multiply.

5

3 Ï}

27 1 3 Ï}

9 2 3 3 Ï}

3 2 3 Product property of cube roots

5 3 1 3 Ï}

9 2 3 3 Ï}

3 2 3 Simplify.

5 3 Ï}

9 2 3 3 Ï}

3 Combine like terms.



9. 3 Ï}

4 1 2 2 3 Ï}

16x 2 10. 1 3 Ï}

25 1 5 2 1 3 Ï}

5 2 1 2

EXAMPLE 4

Add and subtract cube roots

a. 3 3 Ï}

n 1 2 3 Ï}

n 5 (3 1 2) 3 Ï}

n Distributive property

5 5 3 Ï}

n Simplify.

b. 3 Ï}

81 2 2 3 Ï}

3 5 3 Ï}

3 p 27 2 2 3 Ï}

3 Factor using perfect cube factor.

5 3 3 Ï}

3 2 2 3 Ï}

3 Product property of cube roots

5 (3 2 2) 3 Ï}

3 Distributive property

5 3 Ï}

3 Simplify.



7. 4 3 Ï}

p 2 2 3 Ï}

p 8. 3 Ï}

135 1 3 3 Ï}

5

EXAMPLE 3


FOCUS ON

11.2



4. 2 }

3 Ï}

2 5.

1 }

3 Ï}

5 6.

3 }

3 Ï}

3

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277Algebra 1



1. Ï}

2x 1 5 5 3; 2 2. Ï}

3x 2 1 5 4; 25

3. Ï}

7x 1 3 5 10; 1 4. Ï}

2x 1 10 5 4; 23

5. Ï}

1 2 4x 5 5; 26 6. Ï}

6 1 3x 5 12; 22

Isolate the radical expression on one side of the equation. Do not solve the equation.

7. 7 Ï}

x 2 21 5 0 8. 22 Ï}

x 1 8 5 0

9. 3 Ï}

x 1 5 5 14 10. Ï}

x 1 5 2 1 5 8

11. Ï}

x 2 4 2 6 5 22 12. Ï}

2x 1 3 2 10 5 3

Solve the equation. Check for extraneous solutions.

13. Ï}

x 2 2 5 13 14. Ï}

x 1 6 5 14 15. 8 Ï}

x 2 24 5 0

16. 5 Ï}

x 2 15 5 0 17. Ï}

4x 1 3 5 15 18. Ï}

2x 2 7 5 5

19. Ï}

2x 2 1 5 7 20. Ï}

3x 1 7 5 4 21. 2 Ï}

x 1 5 5 12

Simplify each side of the equation.

22. ( Ï}

7x 1 3 )2 5 ( Ï

}

7x 2 1 )2 23. ( Ï}

5x 2 8 )2 5 ( Ï

}

1 2 6x )2

24. ( Ï}

9 2 2x )2 5 (5x)2 25. (2x)2 5 ( Ï

}

3x 1 1 )2

26. (x 1 1)2 5 ( Ï}

1 2 3x )2 27. ( Ï}

4x 2 3 )2 5 (x 2 2)2


28. Ï}

2x 1 5 5 Ï}

3x 1 4 29. Ï}

9x 2 3 5 Ï}

7x 1 9 30. x 5 Ï}

6 2 x

31. Free-Falling Object The velocity v of a free-falling object (in feet per second), the height h in which it falls (in feet), and the acceleration due to gravity, 32 feet per second squared, are related by the function v 5 Ï

}

64h .

a. Find the height from which a tennis ball was dropped if it hits the ground with a velocity of 32 feet per second.

b. How much higher than the ball in part (a) was a tennis ball dropped from if it hits the ground with a velocity of 40 feet per second?

32. Children’s Museum A new children’s museum opens. For the fi rst 12 weeks, the number of people N (in hundreds of people) that visit the museum can be

modeled by the function N 5 Ï}}

1000 1 300t where t is the number of weeks since the opening week.

a. After how many weeks did 4000 (or 40 hundred) people visit the museum?

b. After how many weeks did 5000 (or 50 hundred) people visit the museum?

LESSON


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1. 4 Ï}

2x 2 3 5 12; 2 2. 2 Ï}

9x 2 1 5 20; 7

3. Ï}

4x 1 8 5 Ï}

6 1 2x ; 21 4. Ï}

7x 2 2 5 Ï}

8 2 3x ; 21

5. x 5 Ï}

4x 2 3 ; 3 6. Ï}

4x 2 3 5 x 2 2; 7

Describe the steps you would use to solve the equation. Do not solve the equation.

7. Ï}

7x 1 3 2 5 5 2 8. 6 Ï}

4 2 x 2 3 5 1

9. Ï}

12x 2 7 5 Ï}

9x 1 3 10. 10 Ï}

6 2 x 5 2 Ï}

x 1 4

11. Ï}

5x 2 3 2 Ï}

10 2 4x 5 0 12. Ï}

9x 1 1 2 2 5 x


13. 8 Ï}

x 2 32 5 0 14. Ï}

5x 2 4 5 16 15. Ï}

x 1 3 1 8 5 15

16. Ï}

x 2 6 2 2 5 4 17. Ï}

x 1 9 2 5 5 2 18. Ï}

8 2 3x 1 5 5 6

19. Ï}

5x 1 4 2 12 5 26 20. 3 Ï}

x 1 5 2 3 5 6 21. 4 Ï}

2x 1 1 2 7 5 1

22. Ï}

x 5 Ï}

5x 2 1 23. Ï}

7x 2 6 5 Ï}

x 24. Ï}

6x 2 8 5 Ï}

4x 2 10

25. Ï}

7x 2 5 5 Ï}

3x 1 19 26. Î}

x 2 15 2 Î}

x 2 7 5 0 27. Ï}

10x 2 3 2 Ï}

8x 2 11 5 0

28. Ï}

5x 2 6 5 x 29. x 5 Ï}

2x 1 24 30. Ï}

2x 2 15 5 x

31. Market Research A marketing department determines that the price of a magazine subscription and the demand to subscribe are related by the function P 5 40 2 Ï

}}

0.0004x 1 1 where P is the price per subscription and x is the number of subscriptions sold.

a. If the subscription price is set at $25, how many subscriptions would be sold? Round your answer to the nearest whole subscription.

b. If the subscription price is set at $30, how many more subscriptions are sold in part (a) than when the price is $30. Round your answer to the nearest whole subscription.

32. Awning The area A of a portion of a circle bounded by two radii r and angle t

6 ft

53p

of a sector of a circle are related by the function

r 5 Î}

2A

} t .

The length of a side (radius) of the top view of the awning shown at the right

is 6 feet and the angle that is formed by the awning is 5π

} 3 . Find the area of the

awning. Round your answer to the nearest hundredth.

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279Algebra 1


Describe how you would solve the equation. Do not solve the equation.

1. 1 1 Ï}

x 1 6 5 13 2. 15 2 Ï}

2x 1 2 5 13 3. 4 2 2 Ï}

1 2 4x 5 26

4. 6 Ï}

5x 1 3 2 5 5 2 5. Ï}

10 2 6x 5 Î} 3 }

2 x 2 1 6. Ï

}

3 2 2x 2 Ï}

2 1 4x 5 0

7. 6 Ï}

5 2 2x 5 3 Ï}

5x 2 2 8. x 1 1 5 Ï}

3 2 2x 9. x 1 Ï}

1 2 3x 5 25


10. 3 Ï}

x 1 9 5 4 11. 7 Ï}

3x 2 4 1 7 5 35 12. 14 2 5 Ï}

8 2 3x 5 19

13. 3 Ï}

5 1 x 2 8 5 4 14. 10 1 4 Ï}

3 2 2x 5 14 15. 2 Ï}

5 2 2x 2 13 5 217

16. Ï}

4x 2 3 2 Ï}

6x 2 11 5 0 17. Î} 1 }

4 x 2 5 2 Î

}

x 2 9 5 0 18. Ï}

8 2 6x 5 3x

19. 2x 5 Ï}

11x 1 3 20. Ï}

3x 1 6 5 x 2 4 21. x 1 3 5 Ï}

2x 1 21

22. Ï}

x 1 3 5 Ï}

x 1 12 23. 4 2 Ï}

x 2 3 5 Ï}

x 1 5 24. Ï}

4x 1 3 1 Ï}

4x 5 3

25. Write a radical equation that has 22 and 3 as solutions.

26. Speed of Sound The speed of sound near Earth’s surface depends on the temperature. The speed v (in meters per second) is given by the function v 5 20 Ï

}

t 1 273 where t is the temperature (in degrees Celsius).

a. A friend is throwing a tennis ball against a wall 200 meters from you. You hear the sound of the ball hitting the wall 0.6 second after seeing the ball hit the wall. What is the temperature? Round your answer to the nearest tenth.

b. The temperature 2273°C is called absolute zero. What is the speed of sound at this temperature?

27. Pendulum The period T (in seconds) of a pendulum is the time it takes for the

L

pendulum to swing back and forth. The period is related to the length L (in inches)

of the pendulum by the model T 5 2πÎ}

L }

384 .

a. Find the length of a pendulum with a period of 2 seconds. Round your answer to the nearest tenth.

b. What is the length of a pendulum whose period is double the period of the pendulum in part (a)? Round your answer to the nearest tenth.

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Solve radical equations.

VocabularyAn equation that contains a radical expression with a variable in the radicand is a radical equation.

Squaring both sides of the equation a 5 b can result in a solution of a2 5 b2 that is not a solution of the original equation. Such a solution is called an extraneous solution.

GOAL

Solve a radical equation

Solve 16 Ï}

x 2 4 5 0.

Solution

16 Ï}

x 2 4 5 0 Write original equation.

16 Ï}

x 5 4 Add 4 to each side.

Ï}

x 5 4 }

16 Divide each side by 16.

Ï}

x 5 1 }

4 Simplify.

1 Ï}

x 2 2 5 1 1 } 4 2 2 Square each side.

x 5 1 }

16 Simplify.

The solution is 1 }

16 .

CHECK Check your solution by substituting it in the original equation.

16 Ï}

x 2 4 5 0 Write original equation.

16 Î}

1 }

16 2 4 0 0 Substitute

1 }

16 for x.

16 p 1 1 } 4 2 2 4 0 0 Simplify.

0 5 0 ✓ Solution checks.


1. Solve 5 Ï}

x 2 15 5 0.

EXAMPLE 1


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281Algebra 1


Solve an equation with radicals on both sides

Solve Ï}

x 1 3 5 Ï}

3x 2 5 .

Solution

Ï}

x 1 3 5 Ï}

3x 2 5 Write original equation.

1 Ï}

x 1 3 2 2 5 1 Ï}

3x 2 5 2 2 Square each side.

x 1 3 5 3x 2 5 Simplify.

22x 1 3 5 25 Subtract 3x from each side.

22x 5 28 Subtract 3 from each side.

x 5 4 Divide each side by 22.

The solution is 4. Check the solution.


Solve the equation.

2. 5 Ï}

x 2 3 2 12 5 18

3. Ï}

x 1 2 5 Ï}

4x 2 7

4. Ï}

5x 2 12 2 Ï}

2x 1 9 5 0

EXAMPLE 3

Solve a radical equation

Solve 3 Ï}

x 1 2 1 17 5 32.

Solution

3 Ï}

x 1 2 1 17 5 32 Write original equation.

3 Ï}

x 1 2 5 15 Subtract 17 from each side.

Ï}

x 1 2 5 5 Divide each side by 3.

1 Ï}

x 1 2 2 2 5 52 Square each side.

x 1 2 5 25 Simplify.

x 5 23 Subtract 2 from each side.

The solution is 23.

CHECK To check the solution using a graphing

X=23 Y=0

calculator, fi rst rewrite the equation so that on oneside is 0: 3 Ï

}

x 1 2 2 15 5 0. Then graph the relatedequation y 5 3 Ï

}

x 1 2 2 15. You can see that the graph crosses the x-axis at x 5 23.

EXAMPLE 2

Review for Mastery continuedFor use with pages 7532758

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1. Multi-Step Problem For the period 1999–2005, the annual revenue y (in millions of dollars) of a company can be modeled by y 5 170 1 38 Ï

}

x where x is the number of years since 1999.


b. In what year was the revenue about $255 million?

2. Multi-Step Problem The fi nal velocity v (in meters per second) of an object after traveling a distance of 200 meters with a constant acceleration of 0.5 meter per

second squared is given by v 5 Ï}

v02 1 200

where v0 is the initial velocity of the object.


b. What is the fi nal velocity of an object after 200 meters that has an initial velocity of 20 meters per second?

c. What is the initial velocity of an object that travels 200 meters and has a fi nal velocity of 35 meters per second?

3. Open-Ended The velocity v (in meters per second) of a car moving in a circular path that has radius r (in meters) is given by v 5 Ï

}

ar where a is the centripetal acceleration (in meters per second squared) of the car. A car is traveling at a constant velocity of 15 meters per second in a circular path of radius r where r ≥ 30. Choose two different values of r to show how the centripetal acceleration a of the car changes as the radius increases.

4. Gridded Response Many birds drop clams or other shellfi sh in order to break the shells and get the food inside. The time t (in seconds) it takes for a clam to fall a distance d (in feet) is given by

t 5 Ï

}

d }

4 . A bird drops a clam and it takes

1.75 seconds to hit the ground. What is the height of the bird, in feet?

5. Short Response The velocity v (in meters per second) of an object moving in a straight path can be modeled

by the equation v 5 Î}

2E

} m where E is the

kinetic energy (in joules) of the object and m is the mass (in kilograms) of the object.

a. A 50-kilogram boy is on a moped that is moving at 5 meters per second. What is the kinetic energy of the boy?

b. What happens to the kinetic energy of an object as its mass stays constant and its velocity increases? Explain.

6. Open-Ended Write a problem involving distance that can be solved by simplifying a radical expression. Find a solution of the expression. Explain what the solution means in the context of the problem.

7. Extended Response In chemistry, Graham’s Law of Effusion shows the relationship between the molecular mass of a gas and the rate at which it will effuse. Effusion is the process of gas molecules escaping through tiny holes in a container. To determine how many times greater the rate of a gas is to the rate of oxygen, use

the equation r 5 Î}

32

} M where r is how many

times greater the rate of effusion is for a gas compared to the rate of effusion for oxygen and M is the molecular mass (in grams) of the gas.

a. Helium has a molecular mass of 2 grams. How many times greater than the rate of effusion for oxygen is the rate of effusion for helium?

b. Nitrogen has a molecular mass of 28 grams. How many times greater than the rate of effusion for oxygen is the rate of effusion for nitrogen?

c. What happens to rate of effusion when using a gas that has a molecular mass greater than 28? Explain.

LESSONS

11.1–11.3 Problem Solving Workshop:Mixed Problem SolvingFor use with pages 7342758

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283Algebra 1


In Exercises 1–5, write a radical equation that has the given solutions.

1. The solutions are 1 and 2.

2. The solutions are 3, 6, and 22.



5. The solutions are 2 1 } 2 ,

3 }

2 , and

1 }

2 .

In Exercises 6–15, write a radical equation that has the given solution(s) and the given extraneous solution(s).

6. 1 is a solution; 23 is an extraneous solution.




10. 0 and 2 are solutions; 25 is an extraneous solution.


12. 1 is a solution; 23 and 25 are extraneous solutions.


14. 27 is a solution; 2 and 6 are extraneous solutions.


LESSON

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Name the legs and hypotenuse of the right triangle.

1.

x

y

z

2.m

p

n

3.

r

c

t

Let a and b represent the lengths of the legs of a right triangle, and let c represent the length of the hypotenuse. Find the unknown length.

4. c

a 5 2

b 5 4

5.

ca 5 1

b 5 3 6.

cb 5 3

a 5 5

7. a 5 6, b 5 4 8. a 5 3, b 5 7 9. a 5 5, b 5 5

10. a 5 9, c 5 12 11. a 5 8, b 5 6 12. b 5 2, c 5 10

Find the unknown lengths.

13. x

x24

14.

x

x23

15. 2x

x52

Tell whether the triangle with the given side lengths is a right triangle.

16. 3, 3, 9 17. 12, 16, 20 18. 6, 9, 12

19. Window A window in a house is in the shape of a square.

20 in.

20 in.

The side length of the window is 20 inches. What is the length of the diagonal from one corner of the window to the opposite corner? Round your answer to the nearest tenth.

20. Table Top Soccer The top of a soccer table is in the shape

42 in.

60 in.

of a rectangle. If the tabletop measures 60 inches by 42 inches, what is the length of the diagonal from one corner of the table to the opposite corner? Round your answer to the nearest tenth.

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285Algebra 1



1. a 5 1, b 5 5 2. b 5 4, c 5 9 3. a 5 6, b 5 6

4. b 5 7, c 5 12 5. a 5 2, b 5 8 6. a 5 6, b 5 30

7. a 5 4, b 5 15 8. b 5 7, c 5 11 9. a 5 10, b 5 20

10. a 5 30, b 5 40 11. a 5 15, c 5 25 12. a 5 11, b 5 22


13.

x 2 6

x

172

14.

3x 2 23x 1 2

2x

15.

x 1 64x 1 3

4x

16. A right triangle has one leg that is 3 inches longer than the other leg. The hypotenuse

is Ï}

65 inches. Find the lengths of the legs.


17. 4, 5, 6 18. 15, 20, 25 19. 9, 15, 20

20. Shuffl eboard The playing bed of a shuffl eboard table is in the shape of a 20 in.

154 in.

rectangle. If the playing bed measures 154 inches by 20 inches, what is the length of the diagonal from one corner of the playing bed to the opposite corner? Round your answer to the nearest inch.

21. Indirect Measurement You are trying to determine the distance

28 ft18 ft

B

A Cacross a pond. You put posts into the ground at A, B, and C so that angle B is a right angle. You measure and fi nd that the length of

} AB

is 18 feet and the length of }

CB is 28 feet. How wide is the pond from A to C? Round your answer to the nearest foot.

22. Badminton You are setting up a badminton net. To keep each pole

8 ft

4.5 ft4.5 ft

standing straight, you use two ropes and two stakes as shown. How long is each piece of rope? Round your answer to the nearest tenth.

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LESSON



1. a 5 9, b 5 12 2. b 5 25, c 5 30 3. a 5 4, b 5 1.5

4. b 5 2.5, c 5 7 5. a 5 4, b 5 1.8 6. a 5 2.6, b 5 3.5

7. a 5 14, b 5 8.8 8. b 5 1.4, c 5 2.5 9. a 5 0.2, b 5 0.6

10. a 5 10.5, b 5 6.4 11. a 5 14.1, c 5 20.5 12. a 5 0.3, b 5 0.7


13. A right triangle has one leg that is 4 inches shorter than the other leg. The hypotenuse is Ï

}


14. A right triangle has one leg that is 2 times as long as the other leg. The hypotenuse is Ï

}


15. A right triangle has one leg that is 3 } 5 of the length of the other leg.

The hypotenuse is 2 Ï}



16. 4.5, 6, 7.5 17. 15, 60, 61 18. 12, 71, 72

19. Guy Wire A tower that is being constructed will be 30 feet tall.

30 ft 39 ft

The correct length of the guy wire that will help tether the tower should be 39 feet long. If the correct length wire is used, how far away from the tower should the guy wire be attached to the ground? Round your answer to the nearest foot.

20. Shortest Route You are traveling from Valmont to Milesburg.

Milesburg

Valmont13.5 mi

25.75 mi

You can avoid the city traffi c by taking the L-shaped route shown. If you could travel straight through the city, how many miles could you save? Round your answer to the nearest mile.

21. Flag Each wilderness troop at a camping outing has created its own fl ag. Your troop’s fl ag is triangular with side lengths of 15 inches, 18 inches, and 23 inches. Is the fl ag a right triangle? Explain.

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287Algebra 1


Use the Pythagorean theorem and its converse.

VocabularyThe hypotenuse of a right triangle is the side opposite the right angle. It is the longest side of a right triangle.

The legs are the two sides that form the right angle.

The Pythagorean theorem states the relationship among the lengths of the sides of a right triangle.

The Pythagorean Theorem

Words If a triangle is a right triangle, then the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.

Algebra a2 1 b2 5 c2

Converse of the Pythagorean Theorem

If a triangle has side lengths a, b, and c such that a2 1 b2 5 c2, then the triangle is a right triangle.

GOAL

Use the Pythagorean theorem

Find the unknown length of the triangle shown.

b 5 8

a

c 5 12Solution

a2 1 b2 5 c2 Pythagorean theorem

a2 1 82 5 122 Substitute 8 for b and 12 for c.

a2 1 64 5 144 Simplify.

a2 5 80 Subtract 64 from each side.

a 5 Ï}

80 5 4 Ï}

5 Take positive square root of each side.

The side length a is 4 Ï}

5 .

Exercise for Example 1 1. The lengths of the legs of a right triangle are a 5 9 and b 5 12. Find c.

EXAMPLE 1


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Determine right triangles


a. 14, 48, 50 b. 8, 13, 14

Solution

a. 142 1 482 0 502 b. 82 1 132 0 142

196 1 2304 0 2500 64 1 169 0 196

2500 5 2500 ✓ 233 5 196 ✗

The triangle is a right triangle. The triangle is not a right triangle.

Exercises for Examples 2 and 3 2. A right triangle has one leg that is 6 inches shorter than the other leg. The hypotenuse is 5 Ï

}

2 inches. Find the unknown lengths.


3. 4, 7, 9 4. 10, 12, 26 5. 33, 180, 183

EXAMPLE 3

Use the Pythagorean theorem

A right triangle has one leg that is 3 inches shorter than the other leg. The hypotenuse is Ï

}

29 inches. Find the unknown lengths.

Solution x 2 3

x

29

Sketch a right triangle and label the sides with their lengths. Let x be the length of the longer leg.

a2 1 b2 5 c2 Pythagorean theorem

x2 1 (x 2 3)2 5 1 Ï}

29 2 2 Substitute.

x2 1 x2 2 6x 1 9 5 29 Simplify.

2x2 2 6x 2 20 5 0 Write in standard form.

2(x 2 5)(x 1 2) 5 0 Factor.



Because the length is non-negative, the solution x 5 22 does not make sense. The legs have lengths of 5 inches and 5 2 3 5 2 inches.

EXAMPLE 2


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289Algebra 1


In Exercises 1–5, fi nd the values of x so that the given set of values forms a Pythagorean triple.

1. (x, x 1 1, x 1 2)

2. 1 x 2 2 }

2 , x, x 1 1 2

3. 1 x } 2 1 1, x 2 2, x 2

4. (x, x 1 3, x 1 6)

5. 1 x 1 2, x 1 2

} 2 1 10, 2x 2

6. The circumference of a circle with radius 1 can be roughly approximated using the Pythagorean theorem in the following way. Within the circle of radius 1, draw a square whose corners just touch the circle.

x

y

0.25

0.2520.25

Four right triangles with legs of length 1 are formed within the diagram. Use the Pythagorean theorem to fi nd the hypotenuse of the triangles, then approximate the circumference of the circle by the sum of the hypotenuses. Round your answer to the nearest tenth.

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Match the pair of points with the expression that gives the distance between the points.

1. (26, 3), (24, 2) 2. (6, 23), (24, 2) 3. (6, 23), (4, 22)

A. Ï}}

(24 2 6)2 1 (2 1 3)2 B. Ï}}

(4 � 6)2 1 (22 � 3)2 C. Ï}}

(24 1 6)2 1 (2 2 3)2

Use the coordinate plane to estimate the distance between the two points. Then use the distance formula to fi nd the distance between the points.

4.

x

y

1

3

121

23 3

(23, 22)

(3, 3) 5.

x

y1

121

23

25

21 3 5

(3, 24)

(4, 1) 6.

x

y

1

3

1

23

23 3

(1, 23)

(23, 2)

Find the distance between the two points.

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

The distance d between two points is given. Find the value of b.

10. (0, b), (5, 12); d 5 13 11. (1, b), (4, 5); d 5 5 12. (2, 3), (b, 9); d 5 10

13. (1, 4), (10, b); d 5 15 14. (5, 2), (21, b); d 5 6 15. (b, 6), (3, 22); d 5 8

Find the midpoint of the line segment with the given endpoints.

16. (5, 3), (7, 11) 17. (23, 10), (9, 2) 18. (22, 24), (8, 4)

19. Bus Stop A student is taking the bus home. The student can get

x

y

0.5

1.5

2.5

1.50.5 2.5

Stop 2

HomeStop 1

off at one of two stops, as shown on the map. The distance between consecutive grid lines represents 0.5 mile.

a. Find the distance between stop 1 and home. Round your answer to the nearest hundredth.

b. Find the distance between stop 2 and home. Round your answer to the nearest hundredth.

c. Which distance is shorter? By how much?

20. Sales Use the midpoint formula to estimate the sales of a company in 2000, given the sales in 1995 and 2005. Assume that the sales followed a linear pattern.

Year 1995 2005

Sales (dollars) 740,000 980,000

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291Algebra 1



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

4. (0, 4), (8, 22) 5. (25, 3), (1, 2) 6. (1, 26), (22, 4)

7. (8, 27), (4, 23) 8. (210, 22), (6, 5) 9. (21, 28), (25, 22)


10. (b, 4), (2, 21); d 5 5 11. (23, 2), (7, b); d 5 10 12. (3, 2), (b, 29); d 5 11

13. (4, 1), (5, b); d 5 Ï}

17 14. (b, 2), (3, 21); d 5 Ï}

58 15. (24, b), (5, 22); d 5 Ï}

106


16. (2, 5), (4, 12) 17. (27, 2), (210, 14) 18. (29, 25), (7, 214)

19. (8, 28), (3, 5) 20. (20, 5), (30, 25) 21. (211, 7), (8, 23)

Use the distance formula and the converse of the Pythagorean theorem to determine whether the points are vertices of a right triangle.

22. (1, 1), (4, 4), (1, 4) 23. (6, 0), (6, 4), (2, 4) 24. (22, 1), (3, 5), (6, 22)

25. (6, 4), (21, 22), (24, 3) 26. (5, 3), (4, 22), (10, 2) 27. (2, 24), (2, 23), (6, 1)

28. Walking Trail A walking trail follows the path shown on the

x

y5

2121

3 5

FinishStart

Stop 1

Stop 2Stop 3

map. The distance between consecutive grid lines is 1 mile. Find the total distance of the trail from start to fi nish. Round your answer to the nearest mile.

29. Amusement Park An amusement park designer wants to place

x

y

500

1500

2500

3500

500 1500 2500 3500

Big coaster 2

Big coaster 1

a Ferris wheel midway between the two largest coasters. The distance between consecutive grid lines is 500 feet.

a. Determine the coordinates of where the Ferris wheel should be.

b. How far will the Ferris wheel be from each of the coasters? Round your answer to the nearest foot.

30. Reading You have 30 days left to read the books on your summer reading list. As of today, you have read 5 books. By the end of the 30 days, you have to have read 12 books. Assume that the books are all approximately the same length and you read at a relatively constant pace. After 15 days, how many books should you have read?

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LESSON



1. (1, 25), (6, 7) 2. (23, 23), (8, 22) 3. (14, 25), (23, 8)

4. (211, 24), (9, 22) 5. (4, 215), (22, 10) 6. (1.5, 6), (1.5, 22)

7. (4.1, 6), (5.1, 17) 8. 1 1 } 2 , 8 2 , 1 3 }

2 , 5 2 9. 1 2

1 } 3 ,

2 }

3 2 , 1 5 }

3 ,

1 }

3 2


10. (7, b), (21, 3); d 5 2 Ï}

17 11. (4, 22), (b, 9); d 5 5 Ï}

5 12. (b, 1), (22, 8); d 5 5 Ï}

2

13. (9, 25), (b, 6); d 5 Ï}

290 14. (28, b), (1, 23); d 5 3 Ï}

10 15. (10, 210), (b, 22); d 5 2 Ï}

65


16. (214, 3), (10, 24) 17. (211, 26), (16, 22) 18. (105, 2214), (97, 45)

19. (3.5, 8), (4, 10.5) 20. (7.25, 21.5), (2.25, 22) 21. (28.4, 3.5), (22.6, 4.5)

Use the distance formula and the converse of the Pythagorean theorem to determine whether the points are vertices of a right triangle.

22. (1, 24), (5, 6), (22, 3) 23. (22, 4), (5, 3), (0, 21) 24. (2, 1), (6, 23), (25, 1)

25. (22, 23), (4, 3), (3, 28) 26. (4, 22), (2, 3), (23, 1) 27. (7, 21), (26, 3), (29, 27)

28. Treasure Hunt You set up a treasure hunt with the items placed

x

y

200

1000

600200 1000

Basket

Backpack

Pen

Book

according to the map shown. The distance between consecutive grid lines is 200 feet.

a. Which two objects are closest together? What is the distance between these two objects? Round your answer to the nearest foot.

b. Which two objects are farthest apart? What is the distance between these two objects? Round your answer to the nearest foot.

29. Biking You are biking a straight-line distance between the two

x

y

1

3

5

7

9

11

1 3 5 7

Tipton

Larkintowns shown on the map. The distance between consecutive grid lines is 1 mile.

a. How far is your bike ride one way? Round your answer to the nearest mile.

b. You stop halfway between the two towns to eat a snack. What are the coordinates of your location?

c. On the way back, you stop one-quarter of the way from your destination to visit a friend. How far are you from your destination? Round your answer to the nearest mile. What are the coordinates of your location? Explain how you got your answers.

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293Algebra 1


Use the distance and midpoint formulas.

VocabularyThe Distance Formula

The distance d between any two points (x1, y1) and (x2, y2) is

d 5 Ï}}

(x2 2 x1)2 1 (y2 2 y1)2 .

The midpoint of a line segment is the point on the segment that is equidistant from the endpoints.

The Midpoint Formula

The midpoint M of the line segment with endpoints A(x1, y1) and

B(x2, y2) is 1 x1 1 x2 }

2 ,

y1 1 y2 }

2 2 .

GOAL

Find the distance between two points

Find the distance between (3, 22) and (22, 4).

Solution

Let (x1, y1) 5 (3, 22) and (x2, y2) 5 (22, 4).

d 5 Ï}}

(x2 2 x1)2 1 ( y2 2 y1)2 Distance formula

5 Ï}}}

(22 2 3)2 1 [4 2 (22)]2 Substitute.

5 Ï}}

(25)2 1 (6)2 5 Ï}

61 Simplify.

The distance between the points is Ï}

61 units.


Find the distance between the points.

1. (5, 2), (3, 8)

2. (22, 0), (24, 5)

3. (7, 21), (25, 3)

EXAMPLE 1


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Find a midpoint of a line segment

Find the midpoint of the line segment with endpoints (7, 21) and (5, 7).

Solution

Let (x1, y1) 5 (7, 21) and (x2, y2) 5 (5, 7).

1 x1 1 x2 }

2 ,

y1 1 y2 }

2 2 5 1 7 1 5

} 2 ,

21 1 7 }

2 2 Substitute.

5 (6, 3) Simplify.

The midpoint of the line segment is (6, 3).



5. (14, 3), (6, 9) 6. (211, 23), (2, 25)

EXAMPLE 3

Find a missing coordinate

The distance between (4, 1) and (a, 23) is Ï}

52 units. Find the value of a.

Solution

Use the distance formula with d 5 Ï}

52 . Let (x1, y1) 5 (4, 1) and (x2, y2) 5 (a, 23). Then solve for a.

d 5 Ï}}

(x2 2 x1)2 1 ( y2 2 y1)2 Distance formula

Ï}

52 5 Ï}}

(a 2 4)2 1 (23 2 1)2 Substitute.

Ï}

52 5 Ï}}

a2 2 8a 1 16 1 16 Multiply.

Ï}

52 5 Ï}}

a2 2 8a 1 32 Simplify.

52 5 a2 2 8a 1 32 Square each side.

0 5 a2 2 8a 2 20 Write in standard form.

0 5 (a 2 10)(a 1 2) Factor.

a 2 10 5 0 or a 1 2 5 0 Zero-product property

a 5 10 or a 5 22 Solve for a.

The value of a is 10 or 22.

Exercise for Example 2 4. The distance between (5, 7) and (23, b) is 17 units. Find the value of b.

EXAMPLE 2


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295Algebra 1


1. Multi-Step Problem Use the triangle below.

x

y3

121

23

23 3

A C

B

a. Find the length of each side of the triangle.

b. Find the midpoint of each side of the triangle.

c. Join the midpoints to form a new triangle. Find the length of each of its sides.

d. Compare the perimeters of the two triangles.

2. Multi-Step Problem A rescue helicopter and an ambulance are both traveling from the scene of an accident to the hospital. The distance between consecutive grid lines represents 1 mile.

x

y

1

3

5

7

12121

53 7 9

A(3, 0)

B(3, 5) C(7, 5)

Accident scene

Hospital

a. Find the distance that the ambulance traveled (solid route).

b. How much farther did the ambulance travel than the helicopter (dashed route)?

3. Gridded Response A lacrosse fi eld is a rectangle 60 yards by 110 yards. What is the length of the diagonal from one corner of the fi eld to the opposite corner? Round your answer to the nearest yard.

4. Open-Ended Andrew wants to build a frame for a rectangular garden. He wants the frame to have a diagonal that is 25 feet long and connects opposite corners of the frame. What is one possibility for the length and width of the frame?

5. Multi-Step Problem You and a friend go hiking. You hike 2 miles north and 3 miles east. Starting from the same point, your friend hikes 2 miles west and 1 mile south.

a. How far apart are you and your friend? (Hint: Draw a diagram on a grid.)

b. You and your friend want to meet for lunch. Where should you meet so that both of you hike the same minimum distance? How far do you have to hike?

6. Short Response You have just planted a new tree. To support the tree in bad weather, you attach guy wires from the trunk of the tree to stakes in the ground. You cut 25 feet of wire into four equal lengths to make the guy wires. You attach the four guy wires so they are evenly spaced around the tree. You put the stakes in the ground four feet from the base of the trunk. Approximately how far up the trunk should you attach the guy wires? Explain.

7. Extended Response Molly and Julie leave from the same point at the same time. Julie bicycles east at a rate that is 3 miles per hour faster than Molly, who bicycles north. After one hour they are 15 miles apart.

a. Let r represent Molly’s rate in miles per hour. Write an expression for the distance each girl has traveled in one hour.

b. Use the Pythagorean theorem to fi nd how fast each person is traveling.

c. They continue to bike at the same rate for another hour. How far apart are they after two hours? Explain how you found your answer.

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In Exercises 1–10, fi nd the values of a and b to fi t the given conditions.

1. (2, a) is the midpoint of 1 3, 2 1 } a 2 and (1, 3a).

2. (a, 3) is the midpoint of (1, 5) and (4, b).

3. (23, 2a) is the midpoint of 1 1, 1 }

a 2 and (b, 24a).

4. (a, b) is the midpoint of (21, 1) and (2a2, b2).

5. 1 1 } a ,

1 }

b 2 is the midpoint of (3a, 2b) and (2a, 2b).

6. The distance between (4a, a) and (3, 7) is Ï}

37 units.

7. The distance between (25a, 2) and (21, 26a) is Ï}

221 units.

8. The distance between 1 1 } 2a

, 3 2 and 1 22, 3 }

4a 2 is Ï

}

5013

} 400

units.

9. The distance between (6, 22) and (3, a) is Ï}

13 units.

10. The distance between (a, b) and (3a, 5b) is 2a units.


Park rangers in Yellowstone National Park receive word that there is a lost hiker somewhere in the Lamar valley. Two rangers are sent out on foot to search the trails nearest their ranger stations. One ranger heads directly south hiking at a speed of 4 miles per hour. The other ranger heads directly east hiking at a rate of 3 miles per hour. At these speeds the rangers should meet each other after 5 hours of hiking. Both rangers leave their stations at the same time, and plan to hike until their paths intersect. After hiking for three hours the fi rst ranger fi nds the lost hiker and stops hiking. The ranger decides to stay with the lost hiker and wait until the second ranger is within radio communication distance, which is 9 miles.

11. How far apart are the two ranger stations?

12. How far apart are the two rangers when the hiker is found?

13. How long must the fi rst ranger wait after fi nding the hiker until the second ranger is within radio communication distance? Round your answer to the nearest minute.

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297Algebra 1


Word Search

Use the clues at the bottom of the page to fi nd and circle the vocabulary words from Chapter 11 in the puzzle. Words can be found forward, backward, upward, downward, and diagonal.

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1. Eliminating a radical from the denominator 2. Side opposite the right angle of a triangle of an expression is called _______ the denominator.

3. The point on a line segment that is 4. A group of integers a, b, and c that represent equidistant from the endpoints the side lengths of a right triangle is called a Pythagorean _______.

5. y 5 Ï}

x is a _______ root function. 6. A function involving a radical expression with the independent variable in the radicand is called a _______ function.

7. The expressions 3 1 Ï}

5 and 3 2 Ï}

5 8. d 5 Ï}}

(x2 2 x1)2 1 ( y2 2 y1

)2 represents the _______ formula.

9. Two sides of a triangle that form a 10. a2 1 b2 5 c2 represents the _______ theorem. right angle

11. A solution that is not a solution of an 12. A statement that can be proved true original equation is called _______.

13. No perfect square factors in the radicand, no fractions in the radicand, no radicals appear in the denominator of a fraction

Chapter Review GameFor use after Chapter 11

CHAPTER

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299Algebra 1




12.1: Model Inverse Variation Write and graph inverse variation equations.

• Theater

• Bicycles

• Sports

12.2: Graph Rational Functions Graph rational functions. • Trip Expenses

• Team Sports

• Charity Events

12.3: Divide Polynomials

Focus on Operations

Divide polynomials.

Use synthetic division to divide polynomials.

• Printing Costs

• Movie Rentals

• Membership Fees

12.4: Simplify Rational Expressions

Simplify rational expressions. • Cell Phone Costs

• Television

• Car Radios

12.5: Multiply and Divide Rational Expressions

Focus on Operations

Multiply and divide rational expressions.

Simplify complex fractions.

• Advertising

• Vehicles

• Consumer Spending

12.6: Add and Subtract Rational Expressions

Add and subtract rational expressions.

• Boat Travel

• Canoeing

• Driving

12.7: Solve Rational Equations Solve rational equations. • Paint Mixing

• Ice Hockey

• Running Times



1. Graphing rational functions

2. Performing operations on rational expressions

3. Solving rational equations

CHAPTER


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CHAPTER

12 Family Letter continuedFor use with Chapter12


Lesson Exercise

12.1 A public pool plans to hire lifeguards for the summer season. The work time t (in hours per person) varies inversely with the number g of lifeguards hired. They estimate that they will need 20 lifeguards working 170 hours each to meet their needs. Find the total work time per lifeguard if the pool hires 25 lifeguards.

12.2 Graph y 5 4 } x 1 3 2 2.

12.3 Divide 6x2 2 x 2 12 by 3x 1 4.

Focus on Operations

Divide 2x3 2 x2 1 x 1 4 by x 1 1 using synthetic division.

12.4 Simplify the rational expression, 3x 1 4x

} x 2 4

, if possible. State the excluded values.

12.5 Find the quotient 3x2 2 9x

} x2 1 3x 2 18

4 x 1 7 }

x2 1 5x 2 6 .

Focus on Operations Simplify the complex fraction

4x

} 3 }

28x3 .

12.6 Find the sum of 4x 1 5

} x2 2 9

1 2x 2 3

} x2 2 9

.

12.7 Solve the equation 5x }

x 1 8 2 2 5

5 }

x 1 8 .


Directions Compare the cost of a season pass and additional expenses, such as parking and food, for a local amusement park or other summer attraction, to a per usage cost of the same attraction. Write an equation that gives the average cost C per use as a function of the number of times n you use the attraction. Graph the equation. How many times must you go for the season pass to save you money?

Answers12.1: 136 h 12.2:

22 26

2

22

26

x

y 12.3: 2x 2 3 Focus on Operations: 2x2 2 3x 1 4

12.4: 7x }

x 2 4 ; 4 12.5: 3x(x 2 1)

} x 1 7 Focus on Operations:2

1 }

6x2

12.6: 6x 1 2

} x2 2 9

12.7: x 5 7

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301Algebra 1




12.1: Modelar variación inversa Escribir y grafi car ecuaciones de variación inversa

• Teatro

• Bicicletas

• Deportes

12.2: Grafi car funciones racionales

Grafi car funciones racionales • Gastos de viaje

• Deportes en equipo

• Eventos de caridad

12.3: Dividir polinomios


Dividir polinomios

Usar división sintética para dividir polinomios

• Gastos de impresa

• Alquiler de películas

• Gastos de membresía

12.4: Simplifi car expresiones racionales

Simplifi car expresiones racionales


• Televisión

• Radios de carro

12.5: Multiplicar y dividir expresiones racionales


Multiplicar y dividir expresiones racionales

Simplifi car fracciones complejas

• Publicidad

• Vehículos

• Consumidores

12.6: Sumar y restar expresiones racionales

Sumar y restar expresiones racionales

• Viaje en bote

• Ir en canoa

• Manejar

12.7: Sumar y restar expresiones racionales

Resolver ecuaciones racionales • Mezclar pintura

• Hockey sobre hielo

• Tiempos de carrera



1. Grafi car funciones racionales

2. Hacer operaciones en expresiones racionales

3. Resolver ecuaciones racionales

CAPÍTULO


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CAPÍTULO



Lección Ejercicio

12.1 Una piscina comunitaria piensa emplear unos salvavidas para el verano. El tiempo de trabajo t (en horas por persona) varía inversamente con el número g de salvavidas que se emplean. Se calcula que se necesitarán 20 salvavidas trabajando 170 horas para satisfacer las necesidades. Halla el total del tiempo trabajado por salvavidas si se emplean 25 salvavidas.

12.2 Grafi ca y 5 4 } x 1 3 2 2.

12.3 Divide 6x2 2 x 2 12 por 3x 1 4.


Divide 2x3 2 x2 1 x 1 4 por x 1 1 usando división sintética.

12.4 Simplifi ca la expresión 3x 1 4x

} x 2 4

, si es posible. Nombra los valores excluidos.

12.5 Halla el cociente de 3x2 2 9x

} x2 1 3x 2 18

4 x 1 7 }

x2 1 5x 2 6 .

Enfoque en las operaciones Simplifi ca la fracción compleja

4x

} 3 }

28x3 .

12.6 Halla la suma de 4x 1 5

} x2 2 9

1 2x 2 3

} x2 2 9

.

12.7 Resuelve la ecuación 5x }

x 1 8 2 2 5

5 }

x 1 8 .


Instrucciones Compara el costo de una entrada de temporada y gastos adicionales, tales como estacionamiento y comida, para un parque de atracciones local u otra atracción, a un costo de uso por la misma atracción. Escribe una ecuación que indique el costo promedio C por uso como una función del número de veces n que usas la atracción. Grafi ca la ecuación. ¿Cuántas veces tendrías que entrar para que una entrada de temporada te ahorre dinero?

Respuestas12.1: 136 h 12.2:

22 26

2

22

26

x

y 12.3: 2x 2 3 Enfoque en las operaciones: 2x2 2 3x 1 4

12.4: 7x }

x 2 4 ; 4 12.5: 3x(x 2 1)

} x 1 7 Enfoque en las operaciones: 2

1 }

6x2

12.6: 6x 1 2

} x2 2 9

12.7: x 5 7

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303Algebra 1



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Tell whether the equation represents direct variation, inverse variation, or neither.

1. y 5 27x 2. xy 5 21 3. y 5 x 1 2

4. x 5 23

} y 5. xy 5 8 6. y }

x 5 9

7. x 5 11y 8. 2x 1 y 5 8 9. y 5 13x

Match the inverse variation equation with its graph.

10. xy 5 10 11. xy 5 210 12. xy 5 5

A.

22 2 6

2

22

6

x

y B.

2329 3

3

23

29

x

y C.

23 3 9

3

9

x

y

Graph the inverse variation equation. Then fi nd the domain and range of the function.

13. y 5 22

} x 14. y 5 8 } x 15. y 5

11 } x

2123 1 3

1

21

23

3

x

y

2226 2 6

2

22

26

6

x

y

2329 3 9

3

23

29

9

x

y

16. y 5 210

} x 17. y 5 29

} x 18. y 5 7 } x

2329 3 9

3

23

29

9

x

y

2329 3 9

3

23

29

9

x

y

2226 2 6

2

22

26

6

x

y

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305Algebra 1


Match the inverse variation equation with its description.

19. y varies inversely with x and the constant of variation is 4. A. y 5 24

} x

20. y varies inversely with x and the constant of variation is 1 }

4 . B. y 5

1 }

4x

21. y varies inversely with x and the constant of variation is 24. C. y 5 4 }

x

Given that y varies inversely with x, use the specifi ed values to write an inverse variation equation that relates x and y. Then fi nd the value of y when x 5 2.

22. x 5 1, y 5 3 23. x 5 4, y 5 2 24. x 5 3, y 5 6

25. x 5 22, y 5 8 26. x 5 7, y 5 22 27. x 5 5, y 5 21

Tell whether the table represents inverse variation. If so, write the inverse variation equation.

28. x 0 1 2 3 4

y 0 3 6 9 12

29. x 24 22 2 4 8

y 0.5 1 21 20.5 20.25

In Exercises 30 and 31, tell whether the variables in the situation described have direct variation, inverse variation, or neither.

30. Bike Ride You are riding your bike at an average speed of 14 miles per hour. The number of miles you ride d during t hours is given by d 5 14t.

31. Earning Money You want to fi nd out how many hours you need to work at your job to earn $500. The number of hours h you have to work at pay rate p is given by ph 5 500.

32. Volunteer Work Every spring, a volunteer group plants fl owers to beautify different areas of a city. The planting time t (in hours per person) varies inversely with the number p of people volunteering. The group estimates that 20 people working for 200 hours can get all of the fl ower beds planted.

a. Write an inverse variation equation that relates t and p.

b. Find the total amount of time it will take if 32 people volunteer to plant.

33. Walking You are walking to a bookstore that is 3 miles from

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 t

Time (hours)

Wal

kin

g s

pee

d (

mi/

ho

ur)

00

syour home. Write and graph an equation that relates your walking speed s (in miles per hour) and the time t (in hours) that it takes for you to get to the bookstore. Is the equation an inverse variation equation? Explain.

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LESSON



1. y 5 211x 2. xy 5 25 3. y 5 x 2 4

4. x 5 28

} y 5. xy 5 14 6. y }

x 5 13

7. 2x 1 y 5 8 8. 3y 5 9 } x 9. 4x 2 4y 5 0


10. xy 5 12 11. xy 5 26 12. xy 5 7

23 3 9

3

23

9

x

y

2226 2

2

22

26

x

y

22 2 6

2

22

6

x

y

13. y 5 28

} x 14. y 5 15

} x 15. y 5 14

} x

2226 2

2

22

26

6

x

y

24 4 12

4

12

x

y

4 12

4

12

x

y

16. y 5 29

} x 17. y 5 212

} x 18. y 5 5 } x

2329 3

3

23

29

x

y

2329 3

3

23

29

x

y

22 2 6

2

22

6

x

y

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307Algebra 1



19. x 5 7, y 5 2 20. x 5 3, y 5 9 21. x 5 23, y 5 1

22. x 5 11, y 5 21 23. x 5 212, y 5 212 24. x 5 218, y 5 24

25. x 5 10, y 5 5 26. x 5 7, y 5 24 27. x 5 6, y 5 6

28. x 5 23, y 5 12 29. x 5 25, y 5 40 30. x 5 25, y 5211


31. x 2 4 6 8 10

y 11 21 31 41 51

32. x 25 24 1 2 10

y 24 25 20 10 2

33. x 10 23 25 28 50

y 160 368 400 448 800

34. x 210 29 26 25 24

y 21.8 22 23 23.6 24.5

35. Catalog Orders A clothing company allows customers to place orders on the Internet or by phone. The orders must be entered into the computer inventory system. The amount of time t needed to enter 1000 orders varies inversely withthe number p of people working. The company estimates that 10 people can enter 1000 orders in 240 minutes.

a. Write an inverse variation equation that relates t and p.

b. Find the time needed to enter 1000 orders if 20 people are working.

c. Find the time needed to enter 1000 orders if 8 people are working.

36. Volume and Pressure The volume V of a gas at a constant temperature varies inversely with the pressure P. When the volume is 125 cubic inches, the pressure is 20 pounds per cubic inch.

a. Write the inverse variation equation that relates P and V.

b. Find the pressure of a gas with a volume of 250 cubic inches.

37. Running Every other day, weather permitting, you run 5 miles.

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 t

Time (hours)

Avera

ge s

peed

(m

i/h

ou

r)

00

sWrite and graph an equation that relates your average running speed s (in miles per hour) and the time t (in hours) that it takes for you to complete the run. Is the equation an inverse variation equation? Explain.

LESSON


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LESSON



1. y 5 20.5x 2. xy 5 20.25 3. 4y 5 x 2 8

4. x 5 27

} y 5. xy 5 22 6. y }

x 5 4.5

7. x 5 24 8. 8x 2 8y 5 0 9. 5xy 5 30


10. xy 5 0.75 11. xy 5 23 12. 2xy 5 14

21 1

1

21

x

y

2123 1 3

1

21

23

3

x

y

2226 2 6

2

22

26

6

x

y

13. y 5 211

} x 14. y 5 20

} x 15. y 5 24

} x

2329 3 9

3

23

29

9

x

y

25215 5 15

5

25

215

15

x

y

2123 1 3

1

21

23

3

x

y

16. y 5 1.5

} x 17. y 5 20.2

} x 18. y 5 13

} x

2123 1 3

1

21

23

3

x

y

20.421.2 0.4 1.2

0.4

20.4

21.2

1.2

x

y

24212 4 12

4

24

212

12

x

y

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309Algebra 1



19. x 5 17, y 5 23 20. x 5 212, y 5 212 21. x 5 26, y 5 7

22. x 5 9, y 5 4 23. x 5 10, y 5 23 24. x 5 7, y 5 7

25. x 5 23, y 5 50 26. x 5 26, y 5 220 27. x 5 4, y 5 211

28. x 5 219, y 5 6 29. x 5 7, y 5 15 30. x 5 214, y 5 25


31. x 232 220 216 210 25

y 20.5 20.8 21 21.6 23.2

32. x 2 4 20 25 40

y 25 22.5 20.5 20.4 20.25

33. Radio Waves The frequency f in hertz (vibrations per second) of a radio wave varies inversely with the wavelength w (in meters per vibration). When the frequency is 2.336 3 105 hertz, the wavelength is 1.28 meters.

a. Write the inverse variation equation that relates f and w.

b. What is the frequency when the wavelength is 2.92 meters?

34. Saving Money You plan to save the same amount of money each month so that you can afford a season pass to a local ski area. One season pass costs $400.

a. Let a represent the amount of money that you plan to save each month. Complete the table that gives the number m of months that you need to save money for different values of a. Describe how the number of months changes as the amount of money you save each month increases.

a 40 50 80 100 200 400

m

b. Use the values in the table to draw a graph of the situation.

123456789

a

Nu

mb

er o

f m

on

ths

00

m

100 200 300 400

Amount saved each month (dollars)

Does the graph suggest a situation that represents direct variation or inverse variation? Explain your choice.

c. Write the equation that relates a and m.

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Write and graph inverse variation equations.

Vocabulary

The variables x and y show inverse variation if y 5 a} x and a Þ 0.

The number a is the constant of variation, and y is said to vary inversely with x.

The graph of the inverse variation equation y 5 a} x (a Þ 0) is a

hyperbola. The two symmetrical parts of a hyperbola are called the branches of a hyperbola. The hyperbola also has two asymptotes, which are lines that a hyperbola approaches but does not intersect.

GOAL

Identify direct and inverse variation

Tell whether the equation represents direct variation, inverse variation,or neither.

a. xy 5 1 } 5 b. y 5 3x 2 1 c.

y }

3 5 x

Solution

a. xy 5 1 } 5 Write original equation.

y 5 1 } 5x Divide each side by x.

Because xy 5 1 } 5 can be written in the form y 5

a }

x , xy =

1 } 5

represents inverse variation.

b. Because y 5 3x 2 1 cannot be written in the form y 5 a }

x or

y 5 ax, y 5 3x 2 1 does not represent either direct variation or inverse variation.

c. y }

3 5 x Write original equation.

y 5 3x Multiply each side by 3.

Because y }

3 5 x can be written in the form y 5 ax,

y }

3 5 x represents

direct variation.



1. 8x 5 y 2 3 2. 2x 5 8y 3. xy 5 3 4. y} 2 5

x} 3

EXAMPLE 1


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311Algebra 1


Use an inverse variation equation

The variables x and y vary inversely, and y 5 22 when x 5 4.

a. Write an inverse variation equation that relates x and y.

b. Find the value of y when x 5 210.

Solution

a. Because y varies inversely with x, the equation has the form y 5 a} x .

Use the fact that x 5 4 and y 5 22 to fi nd the value of a.

y 5 a} x Write inverse variation equation.

22 5 a} 4 Substitute 4 for x and 22 for y in y 5

a }

x .

28 5 a Multiply each side by 4.

An equation that relates x and y is y 5 28

} x .

b. When x 5 210, y 5 28

} 210

5 4 } 5 .



5. y 5 12

} x 6. y 5

20 }

x 7. y 5

22 }

x 8. y 5

215 }

x

9. The variables x and y vary inversely. Write an inverse variation equation that relates x to y when x 5 2 and y 5 8. Then fi nd y when x 5 24.

EXAMPLE 3

Graph an inverse variation equation

Graph y 5 6 } x . Then fi nd the domain and range of the function.

STEP 1 Make a table by choosing several integer values of x and fi nding the values of y. Then plot the points. To see how the function behaves for values of x closer to 0 and farther from 0, make a second table for such values and plot the points.

x 26 23 21 0 1 3 6

y 21 22 26 undefi ned 6 2 1

x 212 210 20.6 20.5 0.5 0.6 10 12

y 20.5 20.6 210 212 12 10 0.6 0.5

x

y

4

12

4 12

STEP 2 Connect the points in Quadrant I by drawing a smooth curve through them. Repeat for the points in Quadrant III.

Both the domain and the range of the function are all real numbers except 0.

EXAMPLE 2


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The variables u and v vary inversely with a constant of variation a.

The variables x and y vary inversely with a constant of variation b.

The variables w and z vary inversely with a constant of variation c.

The variables u and x vary directly with a constant of variation d.

The variables x and z vary directly with a constant of variation k.

Determine an equation relating the given variables and tell whether the given variables vary directly or inversely.

1. x and v 2. v and y

3. u and w 4. u and z

5. v and z 6. y and u

7. w and x 8. v and w

9. w and y 10. y and z


The points (1, 2a) and (a 2 1, a2) are two of the points that lie on the graph of an inverse variation equation of the form y 5

c }

x .

11. Find the value of a.

12. Find the value of c.

13. Find the value of x when y is 6.

14. Find the value of y when x is 8.

15. Find the value of x when y is 1000.

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313Algebra 1



1. y 5 1 } 5x 2. y 5

1 } x 2 5 3. y 5

1 } x 1 5

A.

22 6 10

2

22

26

6

x

y B.

22210 2

2

22

26

6

x

y C.

1

1

x

y

Identify the domain and range of the function from its graph.

4.

23 3 9 15

3

23

29

9

x

y 5.

2

22

26

6

x

y 6.

22

22

26

6

x

y

Graph the function and identify its domain and range. Then compare the

graph with the graph of y 5 1 } x.

7. y 5 4 } x 8. y 5

1 } 3x 9. y 5

25 } x

2226 2 6

2

22

26

6

x

y

21 1

13

13

21

1

x

y

13

2

2226 2 6

2

22

26

6

x

y

10. y 5 1 } x 1 4 11. y 5

1 } x 2 2 12. y 5

1 } x 1 6

2226 2 6

2

22

6

10

x

y

2123 1 3

1

21

23

25

x

y

2226210

2

22

26

6

x

y

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Match the function with its asymptotes.

13. y 5 1 } x 1 3 2 2 14. y 5

1 }

x 2 2 1 3 15. y 5

1 } x 2 3 1 2

A. x 5 3, y 5 2 B. x 5 2, y 5 3 C. x 5 23, y 5 22

Determine the asymptotes of the graph of the function.

16. y 5 23

} x 2 8 17. y 5 211

} x 2 14 18. y 5 6 } x 2 6 1 5

19. y 5 24 } x 1 13 1 1 20. y 5

10 } x 1 10 2 2 21. y 5

8 } x 1 5 2 7

Graph the function.

22. y 5 4 } x 2 1 23. y 5

2 } x 1 2 24. y 5

1 } x 1 3 2 5

2226 2 6

2

22

26

6

x

y

2123 1 3

1

21

3

5

x

y

22 226210

2

22

26

210

x

y

25. Football Hall of Fame Your football team is planning a bus trip

0 10 20 30 40 50 60 70 p0

25

50

75

100

125

150

175C

Number of people

Co

st (

do

llars

/per

son

)to the Pro Football Hall of Fame. The cost for renting a bus is $500, and the cost will be divided equally among the people who are going on the trip. One admission costs $13.

a. Write an equation that gives the cost C (in dollars per person) of the trip as a function of the number p of people going on the trip.

b. Graph the equation.

26. Prom It’s prom season and a fl orist has orders for

0 2 4 6 8 10 12 14 p0

25

50

75

100

125

150

175f

Number of extra workers

Ave

rage

nu

mb

er o

f fl

ower

s p

er p

erso

n400 boutonnieres and corsages. Currently, 3 people are scheduled to put together the fl owers. The fl orist hopes to call in some extra workers to complete all of the fl owers. Write an equation that gives the average number f of boutonnieres and corsages made per person as a function of the number p of extra workers that can come in and help complete the work. Then graph the equation.

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315Algebra 1


LESSON


Identify the domain and range of the function from its graph.

1.

22 2 6

2

22

26

6

x

y 2.

22 2 6

2

22

26

6

x

y 3.

2329215 3

2

22

26

210

x

y

4. 2329215 3

22

26

210

x

y 5.

26 2

2

22x

y 6.

2 6

2

26

x

y



7. y 5 8 } x 8. y 5

1 } 6x 9. y 5

23 } 2x

22 2 6

2

22

6

x

y

1

1

x

y

2123 1

1

21

23

x

y

10. y 5 1 } x 2 7 11. y 5

1 } x 1 10 12. y 5

1 } x 2 4

2123 1 3

22

26

x

y

2123 1 3

2

x

y

6 10

1

21

23

3

x

y

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13. y 5 10 } x 2 6 1 4 14. y 5

28 } x 1 5 2 6 15. y 5

14 } x 2 3 2 8

16. y 5 12 } x 1 7 1 7 17. y 5

24 } x 2 8 1 12 18. y 5

9 } x 1 5 1 10

19. y 5 14 } x 2 14 1 1 20. y 5

212 } x 1 12 2 3 21. y 5

7 } x 2 5 2 14

Graph the function.

22. y 5 2 } x 1 5 23. y 5

1 } x 2 4 1 2 24. y 5

23 } x 1 6 2 1

2123 1 3

6

10

x

y

22 2 6 10

1

21

3

5

x

y

222

26

6

x

y

25. Baseball Hall of Fame Your baseball team is planning a bus trip

0 5 10 15 20 25 30 35 p0

20

40

60

80

100

120

140C

Number of people

Co

st (

do

llars

/per

son

)to the National Baseball Hall of Fame. The cost for renting a bus is $515, and the cost will be divided equally among the people who are going on the trip. One admission costs $14.50.

a. Write an equation that gives the cost C (in dollars per person) of the trip as a function of the number p of people going on the trip.


c. What would the cost per person be if 20 people go on the trip?

26. Fundraiser A pizza shop makes pizzas that organizations sell for

0 1 2 3 4 5 6 7 p0

20

40

60

80

100

120

140n


Ave

rag

e n

um

ber

of

piz

zas

fundraisers. One organization has placed an order for 450 pizzas. Currently, 4 people are scheduled to put together the pizzas. The owner of the shop hopes to call in some extra workers to complete all of the pizzas.

a. Write an equation that gives the average number n of pizzas made per person as a function of the number p of extra workers that can come in and help complete the work.


c. If 2 people come in to help out, what is the average number of pizzas made person?

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317Algebra 1


LESSON




1. y 5 21

} 8x 2. y 5 4 } 5x 3. y 5

25 } 3x

21 1

1

21

x

y

2123 1 3

1

21

23

3

x

y

2123 1 3

1

21

23

3

x

y

4. y 5 22

} 3x 5. y 5 7 } 2x 6. y 5

1 } x 2 9

2123 1 3

1

21

23

3

x

y

2123 1 3

1

21

23

3

x

y 2226 2 6

22

26

210

x

y

7. y 5 1 } x 1 5 8. y 5

1 } x 2 6 9. y 5

1 } x 1 8

2226 2 6

2

22

6

10

x

y

22 2 6 10

2

22

26

6

x

y

2226210

2

22

26

6

x

y


10. y 5 22 } x 1 13 2 10 11. y 5

4 } 4x 2 8 1 2 12. y 5

210 } 5x 1 5 2 3

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Graph the function.

13. y 5 4 } x 2 3 1 5 14. y 5

22 } x 1 2 2 1 15. y 5

5 } x 1 4 1 2

22 2 6 10

2

22

6

10

x

y

21 12325

1

21

23

x

y

2226210

2

22

6

x

y

16. y 5 22

} x 2 4 2 4 17. y 5 3 } x 1 6 2 2 18. y 5

24 } x 1 2 2 4

22 2 6 10

2

22

26

210

x

y

2226210

2

22

26

x

y

22 22622

2

26

210

x

y

19. Zoo Trip A grade school is taking a trip to the zoo. A parent

0 2 4 6 8 10 12 14 p0

5

10

15

20

25

30

35n

Number of extra parents

Ave

rage

nu

mb

er o

f b

ox lu

nch

es p

er p

erso

ngroup of 6 people is responsible for putting together 225 box lunches for the trip. The group hopes to recruit extra people for the task. Write an equation that gives the average number n of box lunches made per person as a function of the number p of parents that can come in and help complete the task. Then graph the equation. How many people need to come in so that the average number of box lunches made per person is 15 box lunches?

20. Video Games You rent games from a web site for $17.25 per

0 1 2 3 4 5 6 7 r0

1

2

3

4C

Number of additional rentals

Ave

rage

co

st p

er

ren

tal (

do

llars

)

month. You can rent any number of games per month, but you usually rent at least 4 games per month.

a. Write an equation that gives the average cost C per rental as a function of the number r of additional rentals beyond 4 rentals.

b. Graph the equation from part (a). Then use the graph to approximate the number of additional rentals needed per month so that the average cost is $2.25.

LESSON


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319Algebra 1


Graph rational functions.

VocabularyA rational function has a rule given by a fraction whose numerator and denominator are polynomials and whose denominator is not 0.

GOAL

Graph y 5 1 } x 1 k

Graph y 5 1 } x 2 2 and identify its domain and range. Compare the graph

with the graph of y 5 1 } x .

Graph the function using x y

22 22.5

21 23

20.5 24

0 undefi ned

0.5 0

1 21

2 21.5

x

y

1

3

121 3

y 51x

y 51x 22

a table of values.

The domain is all real numbers except 0. The range is all real numbers except 22.

The graph of y 5 1 } x 2 2 is a

vertical translation (of 2 units

down) of the graph of y 5 1 } x .

EXAMPLE 1

Graph y 5 1 } x 2 h

Graph y 5 1 } x 2 4

and identify its domain and range. Compare the graph

with the graph of y 5 1 } x .

Graph the function using x y

2 20.5

3 21

3.5 22

4 undefi ned

4.5 2

5 1

6 0.5

x

y

2

6

26

6

y 51x

y 51

x 2 4a table of values.

The domain is all real numbers except 4. The range is all real numbers except 0.

The graph of y 5 1 }

x 2 4 is a

horizontal translation (of 4 units

up) of the graph of y 5 1 } x .

EXAMPLE 2

Review for Mastery For use with pages 799–808

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Graph y 5 a} x 2 h

1 k.

Graph y 5 5 } x 1 3

2 2.

Solution

STEP 1 Identify the asymptotes of the graph.

x

y

2

6

26

26

y 5 5x 1 3 22

The vertical asymptote is x 5 23. The horizontal asymptote is y 5 22.

STEP 2 Plot several points on each side of the vertical asymptote.

STEP 3 Graph two branches that pass through the plotted points and approach the asymptotes.


4. Graph y 5 2 }

x 2 2 11.

EXAMPLE 3


Graph the function and identify its domain and range. Compare the

graph with the graph of y 5 1 } x .

1. y 5 8 }

x 2. y 5

1 }

x 1 5 3. y 5 1 }

x 1 10


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321Algebra 1


In Exercises 1–5, fi nd the asymptotes of the graph of the function, then graph the function.

1. f (x) 5 3 }

2 2 x 1 4

2. f (x) 5 5 }

2x 1 1 2 3

3. f (x) 5 21

} x 1 1

1 1

4. f (x) 5 22 }

3 2 4x 1 2

5. f (x) 5 6 }

1 }

2 x 1

2 } 3 2

1 } 4

In Exercises 6–10, fi nd a function whose graph satisfi es the given conditions.

6. f has the form f (x) 5 a }

bx 1 d 1 d; f has a vertical asymptote at x 5 3;

f has a horizontal asymptote at y 5 2; f (6) 5 3 }

2 .

7. f has the form f (x) 5 a }

ax 1 b 1 c; f has a vertical asymptote at x 5

1 } 7 ;

f has a horizontal asymptote at y 5 1; f (1) 5 13

} 6 .

8. f has the form f (x) 5 6 }

cx 1 b 1 c; f has a vertical asymptote at x 5 22;

f has a horizontal asymptote at y 5 21; f (1) 5 23.


ax 1 b ; f has a vertical asymptote at x 5 2

1 } 2 ;

f has a horizontal asymptote at y 5 0; f (0) 5 6 and f (1) 5 2.


ax 1 b 1 c; f has a vertical asymptote at x 5

3 }

2 ;

f has a horizontal asymptote at y 5 2; f (1) 5 22.

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TI-83 Plus

Example 1

Y= ( 2 X,T,�,n � 1 ) � ( 3 X,T,�,n x 2

� 4 X,T,�,n � 5 )

ENTER WINDOW (�) 10 ENTER

10 ENTER 2 ENTER (�) 1 ENTER 1ENTER .2 ENTER TRACE

Use the arrow keys to identify the asymptotes.

Example 2

Y= CLEAR ( 2 X,T,�,n x 2 � 1 )

� ( X,T,�,n x 2 � 9 ) ENTER

ZOOM 6 TRACE



Example 1

From the main menu, choose GRAPH.( 2 X, ,T� � 1 ) � ( 3 X, ,T�

x 2 � 4 X, ,T� � 5 ) EXE SHIFT F3 (�) 10 EXE 10 EXE 2 EXE (�) 1 EXE

1 EXE .2 EXE EXIT F6 SHIFT F1


Example 2

From the main menu, choose GRAPH.( 2 X, ,T� x 2 � 1 ) � ( X, ,T�

x 2 � 9 ) EXE SHIFT F3 F3 EXIT F6 SHIFT F1



LESSON

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323Algebra 1



1. 18x3

} 6x

2. 215x2 }

5x 3.

210x }

10x

Divide.

4. (9x3 2 6x2 1 18x) 4 3x 5. (14x3 1 21x2 2 28x) 4 7x

6. (16x4 2 16x3 2 24x2) 4 8x 7. (20x4 2 5x2 1 10x) 4 5x

8. (22x3 1 6x2 1 4x) 4 (22x) 9. (4x3 2 16x2 1 20x) 4 (24x)


10. (x2 1 3x 2 10) 4 (x 1 5) A. x 2 2

11. (x2 2 3x 2 10) 4 (x 1 5) B. x 1 5

12. (x2 1 3x 2 10) 4 (x 2 2) C. x 2 8 1 30 } x 1 5

Divide.

13. (x2 1 10x 1 24) 4 (x 1 6) 14. (x2 2 2x 2 15) 4 (x 1 3)

15. (x2 2 7x 1 6) 4 (x 2 1) 16. (x3y2 1 3x2y 1 2xy) 4 xy

17. Moped Rental While on vacation, you decide to rent a moped

0 1 2 3 4 5 6 7 h0

20

30

40

50C

Number of hours rented

Ave

rage

co

st p

er

ho

ur

(do

llars

)

to see the sights. A local rental store offers mopeds for $20 an hour plus a $5 gasoline fi ll-up fee.

a. Write an equation that gives the average cost C per hour as a function of the number h of hours you rent the moped.


18. Car Dealer The number of sports cars that a dealer sells per

1 2 3 4 5 6 7 8 t

Rat

io o

f sp

ort

s ca

rs

sold

to

to

tal c

ars

sold

00

R

Years since 1995

0.168

0.170

0.172

0.174

0.176

0.178year between 1995 and 2004 can be modeled by S 5 4t 1 21 where t is the number of years since 1995. The total number of cars sold by the dealer can be modeled by C 5 24t 1 120.

a. Use long division to fi nd a model for the ratio R of the number of sports cars sold to the total number of cars sold.

b. Graph the model.

Practice AFor use with pages 810–817

LESSON

12.3

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Divide.

1. (18x3 2 24x2 1 12x) 4 6x 2. (25x3 1 15x2 2 30x) 4 (25x)

3. (22x4 2 18x2 1 6x) 4 (22x) 4. (x2 1 6x 1 5) 4 (x 1 5)

5. (5x2 1 7x 2 6) 4 (x 1 2) 6. (4x2 1 x 2 5) 4 (x 2 1)

7. (6x2 1 22x 2 8) 4 (x 1 4) 8. (4x2 1 x 2 8) 4 (x 2 2)

9. (10x3y4 1 4x2y 2 2xy) 4 2xy 10. (24a5b 1 16a4b2 2 8a3b) 4 8a3b

Graph the function.

11. y 5 x 1 8

} x 12. y 5 3x 2 5

} x 13. y 5 x 1 5

} x 2 2

212 4 12

4

212

12

x

y

2226 2 6

2

22

6

x

y

6

2

6

x

y

14. Scootcar Rental A resort area offers rentals of scootcars

30

40

50

60

70

80

1 2 3 4 5 6 7 8 h

Ave

rag

e co

st p

er

ho

ur

(do

llars

)

00

C

Time (hours)

(a cross between a scooter and a small car) for $40 per hour plus a $4.50 gasoline fi ll-up fee.

a. Write an equation that gives the average cost C per hour as a function of the number h of hours the scootcar is rented.


15. Juice Bar Between 1995 and 2004, the number D of drinks

Rat

io o

f fr

uit

dri

nks

so

ldto

to

tal d

rin

ks s

old

0

0.3

0.6

0.9

1.2

1.8

1.5

1 2 3 4 5 6 7 8 t0

R

Years since 1995

(in thousands) sold at a juice bar can be modeled by D 5 4t 1 18 where t is the number of years since 1995. The number F of drinks (in thousands) made from fruit juice rather than vegetable juice can be modeled by F 5 2t 1 32.

a. Use long division to fi nd a model for the ratio R of the number of fruit drinks sold to the total number of drinks sold.

b. Graph the model.

LESSON


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325Algebra 1


LESSON


Divide.

1. (45x4 2 60x2 1 30x) 4 15x 2. (96x3 2 64x2 2 24x) 4 (28x)

3. (7x2 1 2x 2 5) 4 (x 2 2) 4. (9 2 3x 2 x2) 4 (1 2 x)

5. (22 2 4x 1 3x2) 4 (x 2 4) 6. (6x 1 x2 1 5) 4 (3 1 x)

7. (8x 1 x2 2 3) 4 (2 2 x) 8. (9x2 2 4) 4 (3x 1 1)

9. (15x8y5 2 3x6y4 2 2x2y2) 4 3x2y 10. (56a5b4 1 14a3b3 2 9a4b2) 4 7a3b2

Graph the function.

11. y 5 5 2 x

} x 1 7 12. y 5 3 1 6x

} x 2 2 13. y 5 8 2 5x

} x 1 4

2329215 3

3

9

23

29

x

y

25 5 15

5

25

15

x

y

25215 5

5

25

215

x

y

14. Car Rental A local car rental company offers an economy

4

8

12

16

20

24

28

1 2 3 4 5 6 7 8 m

Ave

rage

co

st p

er

mile

(d

olla

rs)

00

C

Number of miles

car rental for $24 per day plus $.06 per mile. You want to rent the car for three days.

a. Write an equation that gives the average cost C per mile as a function of the number m of miles you drive the rental.


15. Athletic Shoes Between 1999 and 2002, the sales S of

1 2 3 4 t

Rat

io o

f w

alki

ng

sh

oes

so

ld t

o t

ota

l sh

oes

so

ld

00

R

Years since 1999

0.230

0.235

0.240

0.245

0.250

0.255athletic and sport footwear (in millions of dollars) can be modeled by S 5 546t 1 12,552 where t is the number of years since 1999. The sales W of walking shoes (in millions of dollars) can be modeled by W 5 91t 1 3141.

a. Use long division to fi nd a model for the ratio R of walking shoe sales to all athletic shoe sales.

b. Graph the model.

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Divide polynomials.GOAL

Divide a polynomial by a monomial

Divide 15x3 2 10x2 2 20x by 25x.

Solution

METHOD 1: Write the division as a fraction.

(15x3 2 10x2 2 20x) 4 (25x) 5 15x3 2 10x2 220x

}} 25x Write as fraction.

5 15x3

} 25x 1

210x2 }

25x 1 220x

} 25x Divide each term by 25x.

5 23x2 1 2x 1 4 Simplify.

METHOD 2: Use long division.

Think: 15x3 4 (25x)

Think: 210x2 4 (25x)

Think: 220x 4 (25x)

23x2 1 2x 1 4

25x qww 15x3 2 10x2 2 20x

(15x3 2 10x2 2 20x) 4 (25x) 5 23x2 1 2x 1 4

CHECK 25x(23x2 1 2x 1 4) 0 15x3 2 10x2 2 20x

25x(23x2) 1 (25x)(2x) 1 (25x)(4) 0 15x3 2 10x2 2 20x

15x3 2 10x2 2 20x 5 15x3 2 10x2 2 20x ✓


Divide.

1. (14p3 2 35p2 1 42p) 4 7p

2. (12r3 1 8r2 2 22r) 4 2r

3. (15t3 1 6t2 2 18t) 4 (23t)

EXAMPLE 1


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327Algebra 1



LESSON

12.3

Insert missing terms

Divide 16y 2 2 7 by 3 1 4y.

Solution

4y 2 3

4y 1 3 qww 16y2 1 0y 2 7 Rewrite polynomials. Insert missing term.

16y2 1 12y Multiply 4y and 4y 1 3.

212y 2 7 Subtract 16y2 1 12y. Bring down 27.

212y 2 9 Multiply 23 and 4y 1 3.

2 Subtract 212y 2 9.

(16y2 2 7) 4 (3 1 4y) 5 4y 2 3 1 2 }

4y 1 3


Divide.

4. (8x2 2 22x 2 21) 4 (2x 2 7)

5. (24x2 2 19x 1 6) 4 (8x 2 1)

6. (4x2 2 25) 4 (25 1 2x)

7. (16x2 2 46) 4 (4x 1 7)

EXAMPLE 3

Divide a polyomial by a binomial

Divide 6x2 2 13x 1 2 by 2x 2 5.

Solution

3x 1 1

2x 2 5 qww 6x2 2 13x 1 2

6x2 2 15x Multiply 3x and 2x 2 5.

2x 1 2 Subtract 6x2 2 15x. Bring down 2.

2x 2 5 Multiply 1 and 2x 2 5.

7 Subtract 2x 2 5.

(6x2 2 13x 1 2) 4 (2x 2 5) 5 3x 1 1 1 7 }

2x 2 5

EXAMPLE 2

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In Exercises 1–5, divide.

1. (x3 1 3x2 2 4x 2 12) 4 (x2 2 4)

2. (x4 1 2x3 2 10x2 2 23x 2 6) 4 (x2 2 3x 2 1)

3. (x2 1 1) 4 (x2 2 1)

4. (x3 1 3x2 1 3x 1 1) 4 (x 1 1)

5. (5x4 2 3x2 1 6) 4 (x2 1 3x 1 1)

In Exercises 6–10, fi nd the polynomial p(x) that satisfi es the given equation.

6. p(x) 4 (6x 1 1) 5 3x2 1 5

7. p(x) 4 (x2 1 3x 2 5) 5 x2 1 6x 1 1

8. p(x) 4 (2x2 1 1) 5 3x 1 1 1 5 }

2x2 11

9. p(x) 4 (x3 1 x 1 1) 5 x2 1 5 1 2x 2 1

} x3 1 x 1 1

10. p(x) 4 (x4 1 1) 5 1 1 x3 1 x2 1 x 1 1

}} x4 1 1

In Exercises 11–15, fi nd the polynomial q(x) that satisfi es the given equation.

11. (x2 1 8x 1 15) 4 q(x) 5 x 1 5

12. (x3 2 2x2 2 8x 2 3) 4 q(x) 5 x2 1 2x 1 1

13. (x4 2 5x3 1 4x2 2 5x 1 3) 4 q(x) 5 x2 1 1

14. (2x7 1 5x5 2 x4 1 2x3 1 5x 2 1) 4 q(x) 5 x4 1 1

15. (x5 1 3x2 2 1) 4 q(x) 5 x3 1 5x 1 3 1 25x2 1 14

} x2 2 5

LESSON


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329Algebra 1


Divide using synthetic division.

1. (x2 1 2x 2 3) 4 (x 2 1) 2. (x2 1 5x 1 4) 4 (x 1 4)

3. (x3 2 4x2 1 4x 2 2) 4 (x 2 2) 4. (2x4 2 x2 1 2x 2 4) 4 (x 1 1)

5. (x3 2 6x2 1 4x 1 5) 4 (x 2 5) 6. (2x4 1 6x3 2 x2 2 5x 2 6) 4 (x 1 3)

7. (x3 1 6x2 1 6x 1 4) 4 (x 1 5) 8. (x3 2 3x2 – 3x 1 1) 4 (x 2 3)

9. (x4 2 3x2 2 3) 4 (x 2 1) 10. (x3 2 3x2 1 2x 2 24) 4 (x 2 4)

11. 1 x3 2 1 }

2 x2 1 x 2

3 }

2 2 4 1 x 2

1 }

2 2 12. 1 x3 2

1 }

3 x2 1 2x 2

2 }

3 2 4 1 x 2

1 }

3 2

13. Application Can you use synthetic division to divide x4 2 2x2 1 1 by x2 2 1? Explain why or why not.

14. Challenge What value of a makes the remainder of (5x3 1 52x2 1 15x 2 a) 4 (x 1 10) equal to zero?

FOCUS ON


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Use synthetic division to divide polynomials.

VocabularySynthetic division is a convenient method to use when dividing a polynomial by a binomial of the form x 2 k where k is a constant. Synthetic division is derived from polynomial long division but uses only the value of k and the coeffi cients of the dividend.

GOAL

Use synthetic division

Divide 2x3 1 6x2 2 8x 2 12 by x 2 2 using synthetic division.

Solution

STEP 1 Write the value of k from the divisor and coeffi cients of the dividend in order of descending exponents.

STEP 2 Bring down the leading coeffi cient. Multiply the leading coeffi cient by the k-value. Write the product under the second coeffi cient. Add.

STEP 3 Multiply the previous sum by the k-value, and write the product under the next coeffi cient. Add. Repeat for all of the remaining coeffi cients.

STEP 4 Identify the quotient and remainder. The bottom row gives the coeffi cients of the quotient and the remainder.

(2x3 1 6x2 2 8x 2 12) 4 (x 2 2) 5 2x2 1 10x 1 12 1 12 }

x 2 2



1. (x3 2 3x2 1 2x 2 3) 4 (x 2 1) 2. (2x4 2 4x2 2 6x 2 7) 4 (x 2 2)

EXAMPLE 1


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331Algebra 1



FOCUS ON

12.3

Use synthetic division

Divide x3 1 6x2 1 6x 2 9 by x 1 3 using synthetic division.

Solution

STEP 1 Write the value of k from the divisor and the coeffi cients of the dividend in order of descending exponents.

STEP 2 Bring down the leading coeffi cient. Multiply the leading coeffi cient by the k-value. Write the product under the second coeffi cient. Add.

STEP 3 Multiply the previous sum by the k-value, and write the product under the next coeffi cient. Add. Repeat for all of the remaining coeffi cients.

STEP 4 Identify the quotient and remainder from the bottom row. The quotient is x2 1 3x 2 3, and the remainder is 0.

(x3 1 6x2 1 6x 2 9) 4 (x 1 3) 5 x2 1 3x 2 3



3. (x3 2 2x2 1 4x 1 2) 4 (x 1 1) 4. (x3 1 4x2 1 6x 1 4) 4 (x 1 2)

EXAMPLE 2

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Find the excluded values, if any, of the expression.

1. 8x

} 24

2. 15

} 4x

3. 10 }

x 2 6

4. 24

} x 1 3

5. 1 }

2x 2 2 6.

5 }

8x 2 16

7. 8 }

3x 1 6 8.

5 }

2x 2 1 9.

21 }

3x 1 2

Determine whether the expression is in simplest form.

10. x 2 1

} 3x 2 3

11. x 1 1 } x2 2 1

12. x 1 10

} x2 2 4

13. x 1 3

} x2 2 4x

14. x 1 5 } x2 1 5x

15. x }

x2 2 4x 1 4

Simplify the rational expression, if possible. Find the excluded values.

16. 14

} 21x

17. 42 } 12x

18. 2x 1 4

} x 1 2

19. x 1 5

} x 2 5

20. x 2 6 } x2 2 36

21. 10x }

x2 2 100

22. Deck You have drawn up preliminary plans for a rectangular

2x

x

deck that will be attached to the back of your house. You have decided that the length of the deck should be twice the width as shown.

a. Write a rational expression for the ratio of the perimeter to the area of the deck.

b. Simplify your expression from part (a).

23. School Enrollment The total enrollment (in thousands) of students in public schools from kindergarten through college from 1996 to 1999 can be modeled by E 5 465t 1 56,780 where t is the number of years since 1996. The total enrollment (in thousands) of students in public schools from kindergarten through grade 8 can be modeled by K 5 245t 1 32,800.

a. Write a model for the ratio R of the number of enrollments in kindergarten through grade 8 to the total number of enrollments.

b. Simplify your model from part (a).

LESSON


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333Algebra 1



1. 14

} 3x

2. 28 } x 2 5

3. 5x }

x 1 10

4. 2x }

4x 2 8 5. 3x }

7x 1 21 6.

x 1 1 }

3x 1 7

7. x 1 6 }

x2 2 2x 1 1 8. 8 }}

x2 1 4x 2 12 9.

7x }

x2 2 25


10. 236x2

} 18x

11. 6x 2 24 } x 2 4

12. 4x 2 12

} 3 2 x

13. x 1 11

} x2 2 121

14. x 1 3 }} x2 1 10x 1 21

15. x 2 4 }}

x2 1 11x 1 24

Write and simplify a rational expression for the ratio of the perimeter to the area of the given fi gure.

16. Square 17. Rectangle 18. Triangle

8x

8x

2x

x 1 5

2x2x 1 1

2x 1 2

2x 1 1

19. Zoo Exhibit The directors of a zoo have drawn up

4x 1 3

4x 2 2

preliminary plans for a rectangular exhibit. They have decided on dimensions that are related as shown.

a. Write a rational expression for the ratio of the perimeter to the area of the exhibit.


20. Materials Used The material consumed M (in thousands of pounds) by a plastic injection molding machine per year between 1995 and 2004 can be modeled by

M 5 8t2 1 66t 1 70

}} (3 2 0.2t 1 0.1t2)(t 1 7)

where t is the number of years since 1995. Simplify the model and approximate the number of pounds consumed in 2000.

LESSON


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LESSON



1. 2x }}

3x2 1 11x 2 4 2. 12 }

8x2 2 3x 2 5 3.

5x2 }}

x2 2 14x 1 49


4. x 2 7 }

x2 2 6x 2 7 5. 28x3

} 12x2 2 20x

6. 9x2 2 36x

} 12x 2 24x2

7. 15x4 }

15x2 1 20x 8. 2x 2 4 }

x2 1 8x 2 20 9.

4x2 2 12x }}

2x2 2 5x 2 3

10. x2 1 4x 2 60

}} 2x2 1 23x 1 30

11. x 2 4 }} x3 2 8x2 1 16x

12. x2 1 7x 1 10

} 2x3 2 8x

13. The expression a }}

15x2 1 13x 1 2 simplifi es to

5x 1 1 }

3x 1 2 . What is the value of a?

Explain how you got your answer.

14. Find two polynomials whose ratio simplifi es to 3x 2 1

} 5x 1 1

and whose sum is

8x2 1 24x. Describe your steps.

15. Gazebo You have drawn up a preliminary plan for a gazebo that

xx 2 1

2x 1 4

x 1 3

xyou want to build in your backyard. Your plan for the base is to use two identical trapezoids as shown at the right.

a. Write a rational expression for the ratio of the perimeter to the area of the fl oor of the gazebo.


16. Advertisement Flyers The number A (in hundreds of thousands) of advertising fl yers sent out by a department store between 1995 and 2004 can be modeled by

A 5 6t2 1 102t 1 312

}}} (18 2 0.5t 1 0.01t2)(t 1 13)


a. Simplify the model.

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 t

Nu

mb

er o

f fl

yers

(hu

nd

red

s o

f th

ou

san

ds)

00

A

Years since 1995

b. Use the model to approximate how many fl yers were sent out in 2001.

c. Graph the model. Describe how the number of fl yers sent out changed over time.

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335Algebra 1


Simplify rational expressions.

VocabularyA rational expression is an expression that can be written as a ratio of two polynomials.

A rational expression is undefi ned when the denominator is 0. A number that makes a rational expression undefi ned is called an excluded value.

A rational expression is in simplest form if the numerator and denominator have no factors in common other than 1.

GOAL

Find excluded values


a. 8 }

22x b.

x} 3x 2 9

c. x 1 2 }

x2 1 2x 2 15 d.

12 }

5x2 1 2x 1 7

Solution

a. The expression 8 }

22x is undefi ned when 22x 5 0, or x 5 0. The

excluded value is 0.

b. The expression x

} 3x 2 9

is undefi ned when 3x 2 9 5 0, or x 5 3.

The excluded value is 3.

c. The expression x 1 2 }

x2 1 2x 2 15 is undefi ned when x2 1 2x 2 15 5 0,

or (x 2 3)(x 1 5) 5 0.

The solutions of the equation are 3 and 25. The excluded values are 3 and 25.

d. The expression 12 }

5x2 1 2x 1 7 is undefi ned when 5x2 1 2x 1 7 5 0.

The discriminant is b2 2 4ac 5 22 2 (4)(5)(7) < 0. So, the qua-dratic equation has no real roots. There are no excluded values.



1. 9x }

5x 2 15 2.

x 21 }

x2 2 16

3. 7 }

2x2 2 5x 1 6 4.

x 1 6 }

x2 1 4x 2 12

EXAMPLE 1


LESSON

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Simplify expressions by dividing out monomials

Simplify the rational expression, if possible. State the excluded values.

a. 2m }

8m(m 2 1) 5

2 p m }}

2 p 4 p m p (m 2 1) Divide out common factor.

5 1 }

4(m 2 1) Simplify.

The excluded values are 0 and 1.

b. The expression 11 }

y 1 6 is already in simplest form. The excluded

value is 26.

c. 7q2 2 14q

} 14q3 5

7 p q p (q 2 2) }}

7 p 2 p q p q2 Divide out common factors.

5 q 2 2

} 2q2 Simplify.

The excluded value is 0.

EXAMPLE 2


LESSON

12.4

Simplify an expression by dividing out binomials

Simplify x2 1 4x 2 21 }}

x2 2 5x 1 6 . State the excluded values.

Solution

x2 1 4x 2 21

} x2 2 5x 1 6

5 (x 1 7)(x 2 3)

}} (x 2 2)(x 2 3)

Factor numerator and denominator.

5 (x 1 7)(x 2 3)

}} (x 2 2)(x 2 3)

Divide out common factors.

5 x 1 7

} x 2 2

Simplify.

The excluded values are 2 and 3.


Simplify the expression. State the excluded value(s).

5. 3x3

} 15x5 6.

28x }

7x 2 21

7. 3x2 }

2x 1 6 8.

5x 1 10 }

3x2 1 6x

9. x2 1 13x 1 42

}} x2 2 2x 2 63

10. 4x2 1 20x 1 25 }} 4x2 2 25

EXAMPLE 3

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337Algebra 1


1. Multi-Step Problem The number N (in thousands) of people attending private colleges in the United States during the period 1995–2001 can be modeled by

N 5 3176 2 124x

} 1 2 0.056x

where x is the number of years since 1995.

a. Rewrite the model so that it has only whole number coeffi cients. Then simplify the model.

b. Approximate the number of people attending private colleges in 2000.

2. Multi-Step Problem The amount S (in millions of dollars) of federal budget outlays for Social Security and the amount O (in millions of dollars) of federal budget outlays in the United States during the period 1994–2001 can be modeled by

S 5 15x 1 320 and O 5 55x 1 1455


a. Write and simplify a rational model for the percent p (in decimal form) of federal budget outlays that were for Social Security as a function of x.

b. Make a table for the percent of federal budget outlays that were for Social Security for the years 1994–2001.

c. Was the percent increasing or decreasing from 1994–2001?

3. Gridded Response You pay $75 for an annual membership to an aerobics club and pay $2 per aerobics class. How much less (in dollars) will the average cost per class be if you go to 30 aerobics classes than if you go to 10 aerobics classes?

4. Open-Ended Write an equation whose graph is a hyperbola that has a vertical asymptote of x 5 2 and a horizontal asymptote of y 5 21.

5. Short Response The table shows the relationship between the density D (in kilograms per cubic meter) and the volume V (in liters) of a substance in a rectangular prism container.

Density D (kg/m3) Volume V (L)

1.4 10

2 7

3.5 4

14 1

a. Explain why the density and the volume are inversely related. Then write an equation that relates the density and the volume.

b. Suppose that only the height of the container can be changed. Describe how the density changes as the height increases.

6. Gridded Response A rectangular garden has an area of 6x2 1 7x 2 20 and a width of 2x 1 5. What is the length of the garden when x 5 5?

7. Extended Response The number N (in millions) of new trucks sold in the United States during the period 1993–2002 can be modeled by

N 5 5.684 1 0.674x

}} 1 1 0.028x


a. Rewrite the model so that it has only whole number coeffi cients. Then simplify the model and approximate the number of new trucks sold in 2001.

b. Graph the model. Describe how the number of new trucks sold changed during the period.

c. Can you use the model to conclude that the revenue of new trucks sold increased over time? Explain.

Problem Solving Workshop:Mixed Problem SolvingFor use with pages 791–828

LESSONS

12.1–12.4

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1. Find the value of a so that 6x2 2 ax 2 5

}} 9x2 2 3ax 1 10

5 2x 1 1

} 3x 2 2

.

2. Find the value of b so that 26x2 1 (b 1 10)x 1 20

}} 18x2 2 bx 2 5

5 2x 1 4

} 3x 2 1

.

3. Find the value of c so that x3 2 5x2 1 cx 2 5

}} x3 2 5x2 2 cx 1 5

5 x2 1 1

} x2 2 1

.

4. Find the value of d so that x4 1 x3 1 dx2 1 x 1 1

}}} x4 2 5x3 1 (d 1 1)x2 2 5x 1 2

5 x2 1 x 1 1

} x2 2 5x 1 2

.

5. Find the value of e so that x3 1 2x2 2 x 2 e

}} x3 2 3x 1 e

5 x 1 1

} x 2 1

.

6. Find the expressions for p(x) and q(x) so that p(x)

} q(x)

5 2x 2 1

} 4x 2 1

and

p(x) 2 q(x) 5 22x2 1 5x.


} q(x)

5 2x 1 5

} x 2 3

and

p(x) 1 q(x) 5 3x2 1 5x 1 2.


} q(x)

5 x 1 1

} x 1 5 and

p(x) 2 q(x) 5 28x 1 4.


} q(x)

5 x2 2 1

} 2x2 2 1

and

p(x) 1 q(x) 5 3x4 1 x2 2 2.


} q(x)

5 2x 1 3

} x 2 1

and

p(x) 2 q(x) 5 3x3 1 14x2 1 9x 1 4.

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339Algebra 1



1. 4x2

} 10

p 5 }

22x 2. 4x2

} 10

4 5 }

22x 3. 2x

} 5 p 10 }

4x2

A. 1 }

x B. 24x3

} 25

C. 2x

Find the product.

4. 14x2

} 3 p 9 }

2x 5. 7 }

9x4 p 3x2

} 2 6.

6x2 } 5 p 10

} 12x3

7. x 1 3

} 4x

p 2x2 }

4x 1 12 8. 3x 2 6 }

5x2 p 10x4

} x 2 2

9. x 1 5

} 6x3 p 15x

} 2x 1 10

10. x 1 3

} x2 2 2x

p x 2 2 }

x2 1 4x 1 3 11. 5x 1 5 }

x 1 3 p x

2 1 5x 1 6 }

x 1 1 12.

x 1 2 }

x 2 3 p x

2 2 4x 1 3 }

x2 1 6x 1 8

Find the quotient.

13. 4x2

} 5 4 8x

} 15 14. 11 } 6x

4 22

} 9x2 15.

x 1 4 } 5x 4

x 1 4 }

9x2

16. 2x 1 2

} 3x2 4

x 1 1 } 4 17. 8x 2 16 }

5x2 4

4x 2 8 } 10x 18.

x 1 1 }

14x 4

x2 1 3x 1 2 }

27x2

19. Model Cars You want to create a display box that will hold

4x

5x

5 in.

3 in.

your model cars. You want each section of the box to be 5 inches by 3 inches and you want the box’s dimensions to be related as shown. Write and simplify an expression that you can use to determine the number of sections you can have in the display box.

20. Total Cost The cost C (in dollars) of producing a product from 1995 to 2004 can be modeled by

C 5 10 1 3t

} 80 2 t

where t is the number of years since 1995. The number N (in hundreds of thousands) of units made each year from 1995 to 2004 can be modeled by

N 5 160 2 2t

} 11 2 t


a. Write a model that gives the total production cost T.

b. Approximate the total production cost in 2000.

LESSON


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Find the product.

1. 4x2

} 15

p 5 }

8x5 2. 24 } 7x2

p 14x6 }

40 3.

21 }

2x 1 12 p 4x 1 24

} 15

4. 5x 1 10

} 2x 2 6

p x 2 3 }

10x 1 20 5. x 2 3 }

2x 1 8 p x 1 4

}} x2 1 2x 2 15

6. x2 1 4x 2 12

}} x2 1 7x 1 10

p x 1 5 }

2x 2 4

7. 6x }

4x2 2 1 p 2x2 1 7x 1 3

}} 18

8. x4 }

x4 1 5x3 p (x 1 5) 9.

3x 2 6 }

x2 2 x 2 2 p (x2 1 6x 1 5)

Find the quotient.

10. 24

} 5x3 4

6 }

25x2 11. 11x4 }

18 4

22 }

9x2

12. 7x 1 21

} 30

4 21x 1 63

} 20 13. 4x 2 24

} 3x 1 15

4 12x 2 72

} x 1 5

14. x 1 2 } 3x 2 3

4 x2 1 11x 1 18

}} x 2 1 15. x2 1 4x

} 4x

4 x2 1 x 2 12

} x 2 3

16. 2x 1 10

} x2 2 25

4 4x2

} 2x2 2 10x

17. 2x 2 14 }} x2 2 4x 2 21

4 (x 1 3)

18. Wall Art You want to create a rectangular picture from

4x

6x

3 in.

2 in.

2-inch by 3-inch tiles. You want the picture’s dimensions to be related as shown.

a. Write and simplify an expression that you can use to determine the number of 2-inch by 3-inch tiles that will be needed for the picture.

b. If x 5 5, how many tiles will you need?

19. Profi t The total profi t P (in millions of dollars) earned by a company from 1995 to 2004 can be modeled by

P 5 3500 1 500t

} 98 2 t

where t is the number of years since 1995. The number N (in hundreds of thousands) of units sold can be modeled by

N 5 (t 1 7)(3000 2 20t)

}} 490 2 5t

where t is the number of years since 1995. Write a model that gives the profi t earned per unit per year. Then approximate the profi t per unit in 2002.

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341Algebra 1


LESSON


Find the product.

1. 8x }

2x2 1 x 2 3 p 4x2 1 2x 2 6

}} 16

2. x2 2 x 2 2 }

18x3 p 14x2

} x2 1 x 2 6

3. 2x 2 3

} 5x2 1 10x

p 10x2 1 20x } 4x 1 12

4. x2 1 8x 1 15

} x2 1 7x 1 10

p x2 2 2x 2 8

} 3x2 1 9x

5. x6 }

9x3 1 63x p (x2 1 7) 6.

4x 2 12 }

x2 1 5x 2 24 p (2x2 1 11x 2 40)

Find the quotient.

7. x2 2 2x 2 48

}} 4x2 1 24

4 x 2 8

} 8x 1 24 8. x2 2 5x 2 36 } 5x2 1 16x

4 x2 2 8x 2 9

} x 1 1

9. 2x2 2 9x 2 5

} 5 2 x 4 2x2 1 7x 1 3

}} x 1 3 10. x2 1 4x

} 5x3 1 20x2 4

x2 2 16 } 10x 2 40

11. 4x4 2 20x2 }

x 1 7 4

16x2 2 112 }

x2 2 49 12.

3x2 2 10x 2 8 }}

5x2 2 20x 4

6x2 1 x 2 2 }

30x2 2 120x

13. x2 1 2x 2 35

}} x2 2 3x 2 10

4 3x2 1 21x

} 9x 1 18 14. x3 2 x2 1 4x 2 4 }}

10x3 4

x2 1 7x 2 8 }

5x2 1 40x

Let a be a polynomial in the given equation. Find a.

15. a } x 1 5 p 2x2 1 11x 1 5

}} x 1 6

5 2x2 2 11x 2 6 16. 4x2 1 7x 2 15 }} 2x 1 1

4 x 1 3

} a 5 4x2 2 33x 1 35

17. Snow Tires The average amount C (in dollars) of money spent per snow tire and the number N of snow tires bought by an auto body shop from 2000 to 2004, can be modeled by

C 5 t 1 80

} 1 2 0.05t and N 5 500(t 1 20)

} t 1 80

where t is the number of years since 2000. Write a model that gives the total amount A spent by the shop each year on snow tires. Then approximate the amount spent in 2003.

18. Drive-in Movies The average monthly revenue R (in dollars) from admissions at a drive-in theater and the average price p (in dollars) per car from 1988 to 2000 can be modeled by

R 5 13,124 1 3122t

}} 26 2 t and p 5 294 1 7t

} 130 2 5t


a. Write a model that gives the average number x of cars admitted per month to the theater.

b. Graph the model on a graphing calculator and describe how the number of cars admitted changed over time.

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Multiply and divide rational expressions.GOAL

Multiply rational expressions involving polynomials

Find the product x2 1 x 2 6 }}

10x2 2 20x p 5x2 1 15x }}

x2 2 2x 2 15 .

x2 1 x 2 6 }

10x2 2 20x p 5x2 1 15x

} x2 2 2x 2 15

5 (x2 1 x 2 6)(5x2 1 15x)

}}} (10x2 2 20x)(x2 2 2x 2 15)

Multiply numerators and denominators.

5 (x 2 2)(x 1 3)5x(x 1 3)

}}} 2 p 5x(x 2 2)(x 2 5)(x 1 3)

Factor and divide out common factors.

5 x 1 3

} 2(x 2 5)

Simplify.

EXAMPLE 1

Multiply a rational expression by a polynomial

Find the product 4x2 }}

2x3 1 10x2 2 48x p (x 1 8).

4x2 }}

2x3 1 10x2 2 48x p (x 1 8)

5 4x2 }}

2x3 1 10x2 2 48x p x 1 8

} 1 Rewrite polynomial as a fraction.

5 4x2(x 1 8)

}} 2x3 1 10x2 2 48x


5 2x(2x)(x 1 8)

}} 2x(x 1 8)(x 2 3)


5 2x }

x 2 3 Simplify.


Find the product.

1. x2 2 1 }

2x2 2 3x 1 1 p 4x 2 2

} 3x 1 18

2. 9x }}

3x2 1 9x 2 30 p (x 1 5)

EXAMPLE 2


LESSON

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343Algebra 1


Divide a rational expression by a polynomial

Find the quotient 5x2 2 10x } 4x2 1 12

4 (x 2 2).

5x2 2 10x

} 4x2 1 12

4 (x 2 2)

5 5x2 2 10x

} 4x2 1 12

4 x 2 2

} 1 Rewrite polynomial as a fraction.

5 5x2 2 10x

} 4x2 1 12

p 1 }

x 2 2 Multiply by multiplicative inverse.

5 5x2 2 10x

}} (4x2 1 12)(x 2 2)


5 5x(x 2 2)

}} 4(x2 1 3)(x 2 2)


5 5x }

4(x2 1 3) Simplify.


Find the quotient.

3. x2 1 3x 2 10

} 3x2 2 3x

4 x2 2 8x 1 12

} x 2 1

4. 2x4 2 6x3 2 56x2

}} x3 2 5x2 4 (x 2 7)

EXAMPLE 4

Divide rational expressions involving polynomials

Find the quotient 8x2 1 24x } x2 2 5x

4 x2 1 7x 112 }}

x2 27x 110 .

8x2 1 24x

} x2 2 5x

4 x2 1 7x 1 12

} x2 2 7x 1 10

5 8x2 1 24x

} x2 2 5x

p x2 2 7x 1 10

} x2 1 7x 1 12

Multiply by multiplicative inverse.

5 (8x2 1 24x)(x2 2 7x 1 10)

}} (x2 2 5x)(x2 1 7x 1 12)


5 8x(x 1 3)(x 2 2)(x 2 5)

}} x(x 2 5)(x 1 4)(x 1 3)


5 8(x 2 2)

} x 1 4

Simplify.

EXAMPLE 3


LESSON

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Find the missing polynomial p(x) in the equation.

1. (x 1 1)(2x 1 3)

}} (3x 2 1)

p (x 2 1)(3x 2 1)p(x)

}} (x2 2 1)

5 8x2 1 10x 2 3

2. (3x 2 5)(24x 1 3)

}} p(x)

4 (3x 2 5)(24x 1 3)

}} (x2 1 1)(x 1 2)

5 x2 1 1

3. (27x 1 1)(22x 1 3)

}} (214x2 2 47x 1 7)

p (4x 2 5)(2x 1 7)p(x)

}} (28x2 1 22x 2 15)

5 22x2 1 3x 2 1

4. (4x 1 5)(22x 1 3)

}} (3x2 2 1)

4 (22x 1 3)

} (3x2 2 1)p(x)

5 24x3 2 5x2 1 12x 1 15

5. (x2 1 1)(x 1 1)

}} (26x 1 7)

p (x 2 1)(26x 1 7)p(x)

}} (x2 2 1)

5 16x3 2 3x2 1 16x 2 3

6. (5x 2 4)(26x 1 1)

}} (4x 2 1)

4 (4x 1 1)(26x 1 1)p(x)

}} (16x2 2 1)

5 5x 2 4

7. (x 1 1)(x 1 2)(x 1 3)

}} (x 2 1)(x 2 2)(x 2 3)

p (x2 2 3x 1 2)p(x)

}} (x2 1 3x 1 2)

5 x2 2 9

8. (28x 1 5)(23x 1 2)

}} (7x 1 1)

4 (23x 1 2)(2x 1 1)

}} (14x2 1 9x 1 1)p(x)

5 216x2 2 14x 1 15

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345Algebra 1


Find the product.

1. 5x4

} 3 }

10x4 2. 2x

}

8x2

} 3x3

3. 9x5

} 24

}

23x

4. 2x2

} 2x

} 4 5.

22x8

} 3 }

8x5 6.

3x4 }

2x3

} 4x

7. 4x2

} 5 }

22x 8.

5 }

215x2

} x3

9. x2 2 9

} x 2 3

} x 1 3

10. x4 2 16

}

x2 1 4

} x2 – 4

11.

x2 1 4x

} 3x 2 9

}

x2 2 16 12.

x3 1 3x2

} x2 2 3x

}

x2 2 9

13. 4x2 2 x

} x2 2 9

}

4x3 2 4x

} x 1 3

14.

3x 1 6

} 2x 1 6

}}

3x2 1 12x 1 12

}} x 1 3

15.

2x3 1 10x2

} 2x 1 2

}

x3 1 4x2 2 5x

}} x 2 7

16. Are the complex fractions 1 }

2 }

3 and

1 }

2 }

3 equivalent? Explain your answer.

17. Challenge Are the complex fractions a2 2 b2

} a 2 b

}

a + b and

a2 2 b2

} a + b

}

a 2 b equivalent? Explain your

answer.

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Simplify complex fractions

VocabularyA complex fraction is a fraction that contains a fraction in its numerator, denominator, or both. To simplify a complex fraction, divide its numerator by its denominator.

Key ConceptSimplifying a Complex Fraction

Let a, b, c and d be polynomials where b Þ 0, c Þ 0, and d Þ 0.

Algebra a }

b }

c }

d 5

a }

b 4

c }

d 5

a }

b p d } c

Example x }

2 }

x }

3 5

x }

2 4

x }

3 5

x }

2 p 3 } x 5

3x }

2x 5

3 }

2

GOAL

Simplify a complex fraction

Simplify the complex fraction.

a. 9x

} 4 }

23x3 5 9x

} 4 4 (23x3) Write the fraction as quotient.

5 9x

} 4 •

1 }

23x3 Multiply by multiplicative inverse.

5 9x }

212x3 Multiply numerators and denominators.

5 2 3 }

4x2 Simplify.

b. x2 2 4

} x 1 2

} x 2 2

5 (x2 2 4) 4

x 1 2 }

x 2 2 Write fraction as quotient.

5 (x2 2 4) 3 x 2 2

} x 1 2


5 (x2 2 4) • (x 2 2)

}} x 1 2


5 (x 1 2)(x 2 2)(x 2 2)

}} x 1 2

Factor and divide out common factor.

5 (x 2 2)2 Simplify.

EXAMPLE 1


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347Algebra 1


Simplify a complex fraction

Simplify 3x2 2 6x }} x2 2 4x 1 4

}}

x3 2 9x } x 1 3

.

3x2 2 6x } x2 2 4x 1 4

}

x3 2 9x

} x 1 3

5

3x2 2 6x }

x2 2 4x 1 4 ÷

x3 2 9x }

x 1 3 Write fraction as quotient.

5 3x2 2 6x

} x2 2 4x 1 4

• x 1 3

} x3 2 9x


5 (3x2 2 6x)(x 1 3)

}} (x2 2 4x 1 4)(x3 2 9x)


5 3x(x 2 2)(x 1 3)

}}} (x 2 2)(x 2 2)x(x 1 3)(x 2 3)


5 3 }}

(x 2 2)(x 2 3) Simplify.



5. x2 1 4x

} 3x2 2 75

}

x2 2 16

} x 1 5 6.

4x 1 24

}} 2x2 2 24x 1 72

}}

x2 1 12x 1 36

}} x 2 6

EXAMPLE 2


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1. 2 }

9 }

4 2.

3 }

6 } 5 3.

2x2

} 5 }

4x 4.

3x3 }

6x2

} 7

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1. 1 }

4x 1

2 } 4x 2.

4 } 5x 1

6 } 5x 3.

8 }

3x2 2 7 }

3x2

4. 20

} 7x3 2

6 }

7x3 5. x 2 3

} 2x

1 7 } 2x 6.

x 2 10 }

9x 2

17 } 9x

7. 2x 1 1

} 5x 1 6 } 5x 8.

x 1 4 }

2x2 2 x }

2x2 9. x 1 6

} x 2 1

1 x 2 2

} x 2 1

Find the LCD of the rational expressions.

10. 2 } 5x ,

4 }

10x 11.

1 }

12x ,

x 1 1 }

4x3

12. 3 }

x 1 1 ,

1 }

x 13.

5 }

x 2 4 ,

3 }

x

14. 6x }

x 1 2 ,

5 }

x 1 4 15.

9 }

x 2 3 ,

8x } x 1 7


16. 8x

} 3 1

1 } 5x 17. 7x }

2 2

4 } 8x

18. 5 }

4x 1

7 } 9x 19.

2 }

3x2 2 8 } 5x

20. 4 }

x 1

3 } x 1 4 21.

4 }

x 2 2 1

5 } x 1 7

22. Cabin Cruiser A cabin cruiser travels 48 miles upstream (against the current) and 48 miles downstream (with the current). The speed of the current is 4 miles per hour.

a. Write an expression for the time it takes the cruiser going upstream and write an expression for the time it takes the cruiser going downstream.

b. Use your answers from part (a) to write an equation that gives the total travel time t (in hours) as a function of the boat’s average speed r (in miles per hour) in still water.

c. Find the total travel time if the cabin cruiser’s average speed in still water is 12 miles per hour.

23. Driving You drive 40 miles to visit a friend. On the drive back home, your average speed decreases by 4 miles per hour. Write an equation that gives the total driving time t (in hours) as a function of your average speed r (in miles per hour) when driving to visit your friend. Then fi nd the total driving time if you drive to your friend’s house at an average speed of 52 miles per hour. Round your answer to the nearest tenth.

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349Algebra 1



1. 8 } x 1 5 1

x } x 1 5 2. 10x }

x 2 4 2

6x }

x 2 4

3. x 1 3

} x 2 9

1 5x } x 2 9 4.

x 2 5 }

x 1 2 2

x 1 6 } x 1 2

5. 3x 2 4 } x2 2 9

1 7x 2 3

} x2 2 9

6. 2x 1 4

} 3x2 2

x 2 1 }

3x2

Find the LCD of the rational expressions.

7. 6 }

5x3 , 7 }

15x 8.

10 }

x ,

9x } x 1 7

9. 3x 1 1

} x 2 4

, x 2 4

} x 1 6

10. x 1 5

} 2x 2 4

, 4x }

x 2 2

11. 1 } x2 2 5x

, 8 }

x2 2 3x 2 10 12.

3 }

x2 1 5x 1 4 ,

4x }

x2 1 2x 1 1


13. 11

} 2x

1 4 } 7x 14.

8 }

3x3 2 5 } 12x 15.

8x } x 2 5 2

3x } x 1 2

16. x }

6x 2 5 1

1 } 5x 2 3 17.

4 }

x2 2 7x 2

3 } x 18.

5} x2 1

x 1 3 } x 2 1

19. x 1 3

} x 2 1

1 x 1 2

} x 1 1 20. 2x }

x2 2 3x 1

x 1 4 } x 2 3 21.

1 }

x2 1 5x 1 4 2

1 }

x2 2 16

22. Paddle Boat You paddle boat 8 miles upstream (against the current) and 8 miles downstream (with the current). The speed of the current is 1 mile per hour.

a. Write an equation that gives the total travel time t (in hours) as a function of your average speed r (in miles per hour) in still water.

b. Find your total travel time if your average speed in still water is 3 miles per hour.

c. How much faster is your total travel time if you increased your average speed in still water to 3.5 miles per hour? Round your answer to the nearest tenth.

23. Bike Ride You bike 50 miles from home. On your way back home, your average speed increases by 3 miles per hour.

a. Write an equation that gives the total biking time t (in hours) as a function of your average speed r (in miles per hour) when you are biking away from home

b. Find the total biking time if you bike away from your home at an average speed of 15 miles per hour. Round your answer to the nearest tenth.

c. How much longer is your total biking time if you bike away from your home at an average speed of 12 miles per hour?

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LESSON



1. x 2 9

} x 1 3

1 2x 1 3

} x 1 3 2. 2x 2 4

} x 2 5 2 x 1 4

} x 2 5

3. 3x }

2x 2 5 2

6x 2 2 } 2x 2 5 4.

10x } x 2 5 1

x 1 4 } x 1 2

5. x 1 9

} x 1 10

2 3x } x 2 1 6.

6x 2 5 }

2x 2 3 2

4x 1 3 } x 1 5

7. 3x 2 5

} x 2 2

2 x 2 1

} 3x2 8.

x 1 6 }

5x2 1 x 2 4

} x 1 2

9. x 2 5

} 8x

2 2x } x 1 6 10.

4x }

x2 2 1 2

x 1 1} x2 1 8x 1 7

11. x 2 2 }

x2 2 6x 1 9 2

x 1 1 }

x2 1 2x 2 15 12.

x 1 6 }}

x2 2 4x 2 12 1

x 2 1 }

x2 1 3x 1 2

Use the order of operations to write the expression as a single rational expression.

13. 4 1 x }

x 1 2 2 2 5 1 x 2 5

} x 1 1

2 14. 6 1 4x }

x 2 3 1

7 }

x2 1 5x 2 24 2

15. x 2 2 }}

x2 1 10x 1 24 1

4x } x 1 1 p 5

} x 1 6

16. x 1 3 } x 2 7

2 2x2 1 3x 1 1

}} x 2 3

4 x2 1 3x 1 2

} x2 2 9

17. Suppose that a 5 4b 2 b2 and b 5 c 2 5

} 3c 1 4 . Write a in terms of c.

18. Inline Skating You inline skate 10 miles from the beginning of a trail. On your way back, your average speed decreases by 2.75 miles per hour.

a. Write an equation that gives the total skating time t (in hours) as a function of your average speed r (in miles per hour) when you are skating away from the beginning of the trail.

b. Find the total skating time if you skate away from the beginning of the trail at an average speed of 10 miles per hour. Round your answer to the nearest tenth.

c. How much faster is your total skating time if you skate away from the beginning of the trail at an average speed of 10.75 miles per hour?

19. Advertisement Delivery You and your friend plan to spend 45 minutes delivering pizza shop advertisements to houses in the shop’s delivery area. You can deliver all of the advertisements on your own in two and a half hours.

a. Write an equation that gives the fraction y of advertisements that your friend can deliver alone as a function of the time t (in minutes).

b. Suppose that your friend can deliver the advertisements alone in two hours and fi fteen minutes. Can you deliver all of the advertisements if you and your friend work together for 45 minutes? Explain.

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351Algebra 1


Find the LCD of rational expressions

Find the LCD of the rational expression.

a. 3x }

x2 2 5x 1 6 ,

x 1 2 }

x2 2 7x 1 10 b.

1 }

2x 2 1 ,

7 }

4x 2 5

Solution

a. Find the least common multiple of b. Find the least common multiple of x2 2 5x 1 6 and x2 2 7x 1 10. 2x 2 1 and 4x 2 5.

x2 2 5x 1 6 5 (x 2 2) p (x 2 3) Because 2x 2 1 and 4x 2 5 cannot be

x2 2 7x + 10 5 (x 2 2) p (x 2 5) factored, they don’t have any factors in

The LCD of x2 2 5x 1 6 and common. The LCD is their product,

x2 2 7x 1 10 is (x 2 2)(x 2 3)(x 2 5). (2x 2 1)(4x 2 5).

EXAMPLE 2

Add and subtract rational expressions.

VocabularyThe least common denominator (LCD) of two or more rational expressions is the product of the factors of the denominators of the rational expressions with each common factor used only once.

GOAL

Add and subtract with the same denominator


a. 2 } 5x 1

8 } 5x 5

10 } 5x Add numerators.

5 5 p 2

} 5x Factor and divide out common factors.

5 2 }

x Simplify.

b. 11r

} r 2 7 2 3r 2 5

} r 2 7 5 11r 2 (3r 2 5)

}} r 2 7 Subtract numerators.

5 8r 1 5

} r 2 7 Simplify.



1. x 1 3

} 7x 1 x 2 2

} 7x 2. 5x 1 7

} 3x 2 4

2 2x 2 9

} 3x 2 4

EXAMPLE 1


LESSON

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Subtract expressions with different denominators

Find the difference 12 } x 1 2

2 4x } x 2 3

.

12 }

x 1 2 2

4x } x 2 3 5

12(x 2 3) }}

(x 1 2)(x 2 3) 2

4x(x 1 2) }}

(x 2 3)(x 1 2) Rewrite fraction using LCD,

(x + 2)(x 2 3).

5 12(x 2 3) 2 4x(x 1 2)

}} (x 1 2)(x 2 3)

Subtract fractions.

5 24x2 1 4x 2 36

}} (x 1 2)(x 2 3)

Simplify numerator.



6. 7 }

18r2 1 12

} 9r3

7. x

} x2 2 2x 2 15

1 3 }

x2 2 9

8. t 1 1

} t 2 7 2 t 2 2

} t 1 3

EXAMPLE 4


Find the LCD of the rational expression.

3. 3 }

10x2 , x 1 7

} 15x5 4.

9 }

3x 2 1 , 2x }

x 1 6 5. 8x }

(x 1 5)2 ,

4x 1 1 }

x2 1 8x 1 15


LESSON

12.6

Add expressions with different denominators

Find the sum 11 } 12x2

1 15 } 16x5 .

11

} 12x2 1

15 }

16x5 5 11 p 4x3

} 12x2 p 4x3 1

15 p 3 }

16x5 p 3 Rewrite fraction using LCD,

48x5.

5 44x3

} 48x5 1

45 }

48x5 Simplify numerator and denominator.

5 44x3 1 45

} 48x5 Add fractions.

EXAMPLE 3

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353Algebra 1


1. What If? Suppose in the example that on the drive back home, Beth’s average speed decreases by 15 miles per hour because of construction. Write an equation that gives the total driving time t (in hours) as a function of heraverage speed r (in miles per hour) when driving to the city. Find the total travel time if her average driving speed to the city is 45 miles per hour.

2. Boat Travel A boat travels 25 kilome-ters against the current and 25 kilome-ters with the current. The speed of the current is 5 kilometers per hour. Write an equation that gives the total travel time t (in hours) as a function of the boat’s average speed r (in kilometers per hour) in still water. The boat’s speed in still water is 15 kilometers per hour. Find the total travel time.

Driving Beth drives 135 miles to another city. On the drive back home, her average speed decreases by 9 miles per hour. Write an equation that gives the total driving time t (in hours) as a function of her average speed r (in miles per hour) when driving to the city. Then fi nd the total driving time if she drives to the city at an average speed of 45 miles per hour.


What do you know?

The distance that Beth drives and the decrease of her average speed on the way back.

What do you want to fi nd out?

The total driving time.

STEP 2 Make a Plan Use what you know to write and solve an equation.

STEP 3 Solve the Problem An equation to represent the situation is t 5 135

} r 1

135 }

r 2 9

where 135

} r is the time to drive to the other city and 135

} r 2 9

is the time to drive

back home. Find the sum of the expressions.

t 5 135(r 2 9)

} r(r 2 9)

1 135r

} r(r 2 9)

Rewrite fractions using the LCD, r(r 2 9).

5 270r 2 1215

} r(r 2 9)

Add fractions and simplify.

Calculate the value of t when r 5 45.

t 5 270(45) 2 1215

}} 45(45 2 9)

5 10,935

} 1620 5 6.75 hours

The total travel time is 6.75 hours.

STEP 4 Look Back The time of the trip to the city is 135

} 45

5 3 hours and the time of

the trip back home is 135

} 45 2 9

5 3.75 hours. The total time of the trip is 6.75 hours.

PROBLEM

PRACTICE

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In Exercises 1–5, write w in terms of x.

1. w 5 3u 1 3u2 and u 5 x 2 5

} 3x 1 1

2. w 5 2u 2 1

} u2 1 5

and u 5 x2 1 1

3. w 5 u 1 1

} u 1 2

and u 5 x 1 1

} x 2 1

4. w 5 u2 1 1

} u2 2 1

and u 5 x 2 3

} x 1 2

5. w 5 u 2 2u2, u 5 3 1 5v, and v 5 2x 1 1

} x 2 1


Billy and Mark are painting a fence together. Working alone, it would take Billy 60 hours to paint the fence. Working alone, it would take Mark x hours to paint the fence.

6. Let y represent the fraction of the fence that is painted after t hours by Billy and Mark working together. Write y as a function of t and x.

7. If Mark working alone can paint the fence in 45 hours, then how long would it take Billy and Mark working together to paint the fence?

8. If working together Billy and Mark can paint the fence in 30 hours, then how long would it take Mark to paint the fence alone?

9. If working together Billy and Mark can paint the fence in 20 hours, then how long would it take Mark to paint the fence alone?

10. Suppose Tom, who paints as fast as Mark, also helps paint. If working together, Billy, Mark, and Tom can paint the fence in 20 hours, then how long would it take Mark to paint the fence alone?

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355Algebra 1


Identify the excluded values for the rational expressions in the equation.

1. 5x }

x 2 6 5 0 2.

x 1 4 }

x 1 10 5

1 } x 1 4 3.

x 1 2 }

x2 2 9 5

1 } x 2 3

Solve the equation. Check your solution.

4. 4 }

x 5

x } 9 5.

x }

2 5

32 } x

6. 5 }

x 5

4 } x 2 3 7.

10 }

x 1 4 5

12 } x

8. 1 } x 1 5 5

2 } x 2 6 9.

5 }

x 1 2 5

x } 3

Find the LCD of the rational expressions in the equation.

10. 7 }

x 1 4 1

1 } x 5 8 11.

4 }

x 2 3 1 3 5

1 } x 12. 7 2

3 } x 2 5 5

1 }

x 1 2


13. 1 }

3 1

4 } x 5

1 } x 14.

1 } 5 2

6 } 5x 5

1 } x

15. 1 }

x 2 4 1 2 5

2x } x 2 4 16.

2x } x 2 5 1 1 5

5 } x 2 5

17. x }

x 1 6 2 4 5

21 } x 1 6 18. 3 1

x } x 2 2 5

3 } x 2 2

19. Rain It has rained 3 of the last 8 days. How many consecutive days does it have to rain in order for the percent of the number of rainy days to be raised to 75%?

20. Field Goal Average A fi eld goal kicker has made 25 out of 40 attempted fi eld goals so far this season. How many consecutive fi eld goals must he make to increase his average to at least 0.680?

21. Paint Mixing You have a 4-pint mixture of paint that is made up of equal amounts of blue paint and red paint. To create a certain shade of purple, you need a paint mixture that is 60% blue.

a. Let p represent the number of pints of blue paint that you have to add. Write an expression for the number of pints of blue paint that will be in the new mixture. Write an expression for the total number of pints of blue and red paint that will be in the new mixture.

b. Use your expressions from part (a) to write an equation that represents a paint mixture that is 60% blue.

c. How many pints of blue paint do you need to add?

d. How many total pints of paint are there in the new mixture?

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1. x }

27 5

3 } x 2.

3 }

x 5

2 } x 1 4 3.

4 } x 2 7 5

2 } x

4. 10 }

x 1 2 5

7 } x 2 4 5.

25 }

x 1 4 5

x } x 1 4 6.

8 }

x 1 8 5

x } x 1 2

7. 21

} x 1 2

5 x } x 1 2 8.

2 }

3x 5

x 1 3 } 2x 2 5 9.

6x }

x 1 2 5

22 } x 1 2

Find the LCD of the rational expressions in the equation.

10. 7x }

x 2 3 1 4 5

x 1 1 } x 2 3 11.

3 }

2x 2 2 1 4 5

7x } x 2 1 12.

7 }

x 2 2 1 1 5

4 } x 2 3


13. 3x }

x 1 4 2 2 5

212 } x 1 4 14.

3 }

x 1 2 1 5 5

4 } x 1 2 15.

2x }

x 2 1 1 2 5

10 } x 1 2

16. x 2 1

} x 1 5 1 6 5 22

} x 1 2 17. 4x } x 2 5 1 1 5

9 } x 2 1 18.

x }

x 2 4 2

5x } x 2 2 5

218 } x 2 2

19. Stain Mixing You are staining a coffee table you just made. After testing some sample pieces of wood, you decide that you want a mix of a yellow stain and a red stain. You estimate that you want a mix that contains 75% of the yellow stain. You only have 1 pint that is made up of equal parts of the stain. How many pints of the yellow stain do you have to add to the current mixture?

20. Wallpaper Working together, an expert wallpaper hanger and an assistant can hang the wallpaper in a room in 3 hours. The assistant can hang the wallpaper in one and one-half times the time it takes the expert wallpaper hanger to hang the wallpaper alone. Let x represent the time (in hours) that the assistant can hang the wallpaper alone.

a. Copy and complete the table.

PersonFraction of room

papered each hourTime

(hours)Fraction of

room papered

Assistant 1 }

x 3 ?

Expert ? 3 ?

b. Explain why the sum of the expressions in the last column must be 1.

c. Write a rational equation that you can use to fi nd the amount of time it takes the assistant to wallpaper the room alone. Then solve the equation.

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357Algebra 1


LESSON



1. 14 }

2 2 x 5

2 } x 2.

x 1 2 }

x 1 1 2 x 5

26 } x 1 1 3.

2 }

x 2 4 1 2 5

6 } x 2 4

4. 10 }

x 1 2 5

4 } x 2 1 5.

3x 1 2 }

3x 2 5 5

x } x 2 1 6.

8 }

x 2 4 5

2 } x 2 2

7. 1 }

x 1

1 } x 1 1 5

5 } 4 8.

x } x 2 5 1 1 5

4x 2 3 } x 2 4 9.

7 }

x 2 2 2

4 } x 1 2 5

3 }

x2 2 4

10. 2x 1 3

} x 1 2

1 3x 5 22

} x 1 2 11. 2 }

x 1 3 2

6 } 2x 1 6 5

x } 2 12.

5 }

2x 2 2 5

3 } x 2 1 1

x 1 3 }

8

13. 1 2 x 2 1

} (x 1 1)2 5

1 } x 1 1 14. 2x 1

3x 2 4 } x 2 2 5

2 } x 2 2 15.

x 1 2 }

x 2 4 2

2x } x 2 1 5

18 }

x2 2 5x 1 4

16. Let a and b be real numbers. The solutions of the equation ax 1 b 5 24 } x 1 3 2 1 are

29 and 9. What are the values of a and b?

17. Paint Mixing You have a 6-pint mixture of paint that is made up of equal amounts of red paint and yellow paint. To create a certain shade of orange, you need a paint mixture that is 30% red. How many pints of yellow paint do you need to add to the mixture?

18. Investing Mrs. Jackson invested a total of $4000 in two accounts earning simple interest at annual rates of r% and (r 1 1)%. After 1 year, she earned $50 in interest on the fi rst account, and $180 in interest on the second account. How much did Mrs. Jackson invest in each account?

19. Roofi ng Working together, an expert roofer and an assistant can complete the roof on a certain building in 24 hours. The expert roofer can roof the building alone in about three fi fths of the time it takes the assistant to roof the building alone. Let x represent the time (in hours) that the expert can roof the building alone.

a. Copy and complete the table.

PersonFraction of roof

completed each hourTime

(hours)Fraction of

roof completed

Expert 1 }

x24 ?

Assistant ? 24 ?

b. Explain why the sum of the expressions in the last column must be 1.

c. Write a rational equation that you can use to fi nd the time that the expert can roof the building alone. Then solve the equation.

d. How long does it take the assistant to roof the building alone?

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Solve rational equations.

VocabularyA rational equation is an equation that contains rational expressions.

GOAL

Use the cross products property

Solve 6 } x 1 5

5 x } 6 . Check your solution.

Solution

6 } x 1 5

5 x} 6 Write original equation.

36 5 x2 1 5x Cross products property

0 5 x2 1 5x 2 36 Subtract 36 from each side.

0 5 (x 1 9)(x 2 4) Factor polynomial.




CHECK If x 5 29: If x 5 4:

6 }

29 1 5 0

29 }

6

6 }

4 1 5 0

4 }

6

21.5 5 21.5 ✓ 2 }

3 5

2 } 3 ✓

EXAMPLE 1

Multiply by the LCD

Solve 3x } x 1 5

2 5 } 2 5 5 }

x 1 5 .

3x } x 1 5 2

5 } 2 5

5 } x 1 5 Write original equation.

3x } x 1 5 p 2(x 1 5) 2

5 } 2 p 2(x 1 5) 5

5 } x 1 5 p 2(x 1 5) Multiply by LCD.

3x p 2(x 1 5)

} x 1 5 2 5 p 2(x 1 5)

} 2 5 5 p 2(x 1 5)

} x 1 5 Multiply. Divide out common factors.

6x 2 5x 2 25 5 10 Simplify.

x 2 25 5 10 Combine like terms.

x 5 35 Add 25 to each side.

The solution is 35.

EXAMPLE 2


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359Algebra 1



LESSON

12.7

Factor to fi nd the LCD

Solve 4 } x 2 4

1 2 5 16 }} x2 1 x 2 20

. Check your solution.

Solution

Write each denominator in factored form. The LCD is (x 2 4)(x 1 5).

4 }

x 2 4 1 2 5

16 }

x2 1 x 2 20

4 }

x 2 4 p (x 2 4)(x 1 5) 1 2 p (x 2 4)(x 1 5) 5

16 }}

(x 2 4)(x 1 5) p (x 2 4)(x 1 5)

4(x 2 4)(x 1 5)

}} x 2 4

1 2(x 2 4)(x 1 5) 5 16(x 2 4)(x 1 5)

}} (x 2 4)(x 1 5)

4(x 1 5) 1 2(x2 1 x 2 20) 5 16

2x2 1 6x 2 20 5 16

2x2 1 6x 2 36 5 0

2(x2 1 3x 2 18) 5 0

2(x 2 3)(x 1 6) 5 0

x 2 3 5 0 or x 1 6 5 0

x 5 3 or x 5 26


CHECK If x 5 3: If x 5 26:

4 }

3 2 4 1 2 0

16 }

32 1 3 2 20

4 }

26 2 4 1 2 0

16 }}

(26)2 2 6 2 20

22 5 22 ✓ 1.6 5 1.6 ✓



1. 212

} x 5

x 2 14 }

4

2. 6 }

x 2 3 5

x} 18

3. x }

x 1 10 1

1 } 5 5

27 }

x 1 10

4. x

} x 1 4

2 4 }

x 2 2 5

11 }

x2 1 2x 2 8

EXAMPLE 3

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LESSONS


1. Multi-Step Problem You hike 6 miles on steep trails and 8 miles on fl at trails. Your average speed on steep trails is 2 miles per hour slower than your average speed on fl at trails.

a. Write an equation that gives the total time t (in hours) of the hike as a function of your average speed x (in miles per hour) on fl at trails.

b. Your average speed on fl at trails is 4 miles per hour. Find the total time of the hike.

2. Short Response Baseball player Roberto Clemente’s career number B of times at bat and career number H of hits during the period 1955–1972 can be modeled by

B 5 355 1 555x

} 1 1 0.001x

and H 5 67 1 168x

} 1 2 0.003x


a. A baseball player’s batting average is the number of hits divided by the number of times at bat. Write a model that gives Roberto Clemente’s career batting average A as a function of x.

b. The table shows Clemente’s actual career number of times at bat and actual career number of hits for three different years. For which year does the model give the best approximation of A? Explain your choice.

Year 1955 1964 1972

Career times at bat 474 5321 9454

Career hits 121 1633 3000

3. Open-Ended Describe a real-world situation that can be modeled by a rational equation and can be solved using the cross products property. Explain what the solution means in this situation.

4. Multi-Step Problem The number M (in thousands) of males and the number F (in thousands) of females participating in high school athletic programs during the period 1996–2003 can be modeled by

M 5 3634 1 332x

} 1 1 0.07x

and F 5 2369 1 355x

} 1 1 0.1x


a. Write a model that gives the total number S of high school students who participated in high school athletic programs as a function of x.

b. Approximate the total number of high school students who participated in high school athletic programs in 2001.

5. Gridded Response After 25 times at bat, a major league baseball player has a batting average of 0.160. How many consecutive hits must the player get to raise his batting average to 0.300?

6. Extended Response The amount A (in millions of dollars) of passenger fares by all commuter rails and the number P (in millions) of passengers who traveled by commuter rails in the United States during the period 1997–2002 can be modeled by

A 5 1175.8 1 213.3x

}} 1 1 0.107x

and

P 5 356.2 1 28.1x 2 3.2x2


a. Write a model that gives the average cost C (in dollars) per passenger as a function of x.

b. Approximate the average cost in 2000.

c. Graph the equation in part (a) on a graphing calculator. Describe how the average cost changed during the period. Can you use the graph to describe how the amount of passenger fares changed during the period? Explain your answer.

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361Algebra 1


In Exercises 1–5, let a and b be real numbers. Find the values of a and b that satisfy the equation.

1. The solutions to the equation ax 1 b 5 216

} x 1 6

1 7 are x 5 22 and x 5 2.

2. The solutions to the equation 1 }

3 x 2 2 5

215 }

ax 1 b 1 1 are x 5 26 and x 5 6.

3. The solutions to the equation 2x 1 3 5 2a }

2x 2 7 2 b are x 5 2

5 } 2 and x 5

5 }

2 .

4. The solutions to the equation ax 1 b 5 240

} x 2 1

2 44 are x 5 2 3 } 7 and x 5

3 } 7 .

5. The solutions to the equation 32x 1 1 5 2a }

2x 2 3 2 b are x 5 2

5 } 8 and x 5

5 }

8 .


The octane rating of a gasoline, which is a measure of the gasoline’s tendency to cause “engine knock” is regulated by many states in the United States. Typically a refi nery will manufacture gasoline in two octane ratings, 87 and 93, and then mix these two octane levels to make a variety of grades. For example, the mixing of one gallon of 87 octane gasoline with one gallon of 93 octane gasoline, results in two gallons of 90 octane gasoline. Suppose a refi nery has 100,000 barrels of 87 octane gasoline available and 50,000 barrels of 93 octane gasoline available.

6. If an order comes in for 60,000 barrels of 91 octane gasoline, can the refi nery fi ll this order? If so, how many barrels of each octane must they mix?



9. If an order comes in for 20,000 barrels of 88 octane gasoline and 40,000 barrels of 91 octane gasoline, can the refi nery fi ll this order? If so, how many barrels of each octane must they mix?

LESSON


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Mathematics Terminology

Solve the following exercises. Find the answer at the right of the page. Place the letter associated with the correct answer on the line with the exercise number to answer the following question.

What is the correct term for the division bar symbol in the expression a }

b ?

Exercises Answers

1. What is the horizontal asymptote of y 5 4 } x 2 5 1 2? (U) x 5 9 (S) x 5 2

2. Divide: (12x2 1 7x 2 10) 4 (3x 2 2) (T) 4x 2 5 (N) x 1 3

} 2x 1 1

3. Simplify: x2 2 9 }

2x2 2 5x 2 3 (V) y 5 2 (I) 4x 1 5

4. Multiply: 2x2 1 14x

} x2 2 3x 1 2

p x2 1 3x 2 4

} 4x 1 28

(E) x 5 5 (C) x(x 1 4)

} 2(x 2 2)

5. Divide: 3x2 1 12x

}} 3x2 1 14x 2 5

4 x2 1 10x 1 24

}} 3x 2 1

(M) x 5 0 or x 5 2

6. Subtract: x 2 1 }}

x2 1 10x 1 21 2

x 1 5 }

x2 1 3x (K)

3x(x 1 4)2(x 1 6) }}

(x 1 5)(3x 2 1)2

7. Solve: x 1 1

} x 2 4

5 x 2 3

} x 2 6

(U) 3x }}

(x 1 5)(x 1 6)

8. Solve: 1 }

x 2 1 1

3 } x 1 1 5 2 (L)

213x 2 35 }}

x(x 1 3)(x 1 7)

1 2 3 4 5 6 7 8

CHAPTER


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Algebra 1Chapter 7 Resource Book A1

Answers

Lesson 7.1Practice Level A

1. yes 2. no 3. yes 4. no 5. no 6. yes

7. B 8. A 9. C 10. F 11. E 12. D

13. (2, 21) 14. (3, 4) 15. (21, 21)

16. (4, 2) 17. (2, 23)

x

y

1

3

5

12121

3 5

x

y

1

12121

23

3 5

18. (23, 0) 19. (3, 3)

x

y

3

32923

29

9

x

y

3

9

329 23

29

9

20. (21, 4) 21. (3, 22)

x

y

1

123 2121

x

y

2

6

2 626 2222

26

22. between 1995 and 1996; about 1175 thousand people

23.

14 1600

2

4

6

8

10

12

14

16

2 4 6 8 10 12Bottles of apple juice

Bo

ttle

s o

f o

ran

ge ju

ice

x 1 y 5 15

1.5x 1 2y 5 26

y

x

8 bottles of apple juice and 7 bottles of orange juice

Practice Level B

1. no 2. yes 3. yes 4. no 5. no 6. yes

7. (3, 25) 8. (21, 4) 9. (22, 2) 10. (4, 22)

11. (25, 3) 12. (0, 4)

13. (1, 3) 14. (24, 4)

x

y

1

3

12123 3

x

y

2

6

226

26

15. (2, 25) 16. (23, 0)

x

y1

12325 21

23

25

21

x

y

3

15

329215

17. (25, 3) 18. (22, 24)

x

y35

5215

x

y

1

123 21

25

21

19. (23, 6) 20. (4, 25)

x

y

9

929 23

29

x

y

6 102222

26

21. (2, 7)

x

y

3

9

15

21

929 23

22.

3500

5

10

15

20

25

30

35

5 10 15 20 25 30Blooming annuals

No

n-b

loo

min

g a

nn

uals

3.2x 1 1.5y 5 49.6

x 1 y 5 24

x

y

8 blooming annuals and 16 non-blooming annuals

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LAHA111FLCRB_BMv2_C07_A01-A09.indd A1LAHA111FLCRB_BMv2_C07_A01-A09.indd A1 9/24/09 10:01:59 PM9/24/09 10:01:59 PM

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Algebra 1Chapter 7 Resource BookA2

23. a. x 1 y 5 27 and 0.25x 1 y 5 12

b.

28 3200

4

8

12

16

20

24

28

32

4 8 12 16 20 24Outs made by infielders

Ou

ts m

ad

e b

y o

utf

ield

ers

x 1 y 5 27

0.25x 1 y 5 12

x

y

c. infi elders: 20 outs; outfi elders: 7 outs

Practice Level C

1. no 2. yes 3. no 4. yes 5. no 6. yes

7. (6, 1) 8. (28, 4)

x

y

1

3

121

3 5 7

x

y

4

12

421224

9. (5, 25) 10. (23, 5)

x

y

121

23

25

3 5

x

y

6

22221022

11. (2, 2) 12. (21, 7)

x

y

1

3

1 3 5

23

21

x

y

15

21

329 23

13. (5, 6) 14. (29, 2)

x

y

1

3

5

1 3 521

x

y

2

6

10

2222621022

15. (4, 4)

x

y

2

6

6 102222

16. Sample answer: m 5 1, b 5 24

17.

x

y

3

5

7

12121

23

(1, 5)

(22, 2)

(4, 21)

18. 5.5%: $20,000; 6.5%: $25,000

19.

t700

5

10

15

20

25

30

35

1 2 3 4 5 6

Nu

mb

er

of

um

bre

llas

y 5 22t 1 25

y 5 2t 1 15

Years since 2000

y

mid-2002

20. 20% off;

x20000

50

100

150

200

50 100 150Amount of purchase

(dollars)

Am

ou

nt

yo

u p

ay

(do

llars

)

y 5 x 2 25

y 5 x 2 0.2x(125, 100)

y

For purchases greater than $125, 20% off is the better deal.

Review for Mastery

1. (21, 1) 2. (2, 4) 3. (24, 22) 4. 60 mi

Problem Solving Workshop: Worked Out Example

1. 10 square feet 2. 115 student tickets, 98 general admission tickets 3. 124 student tickets, 117 general admission tickets 4. 4

Challenge Practice

1. yes 2. yes 3. no 4. Yes, if a2 1 b2 Þ 0.

Lesson 7.1, continuedA

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Lesson 7.1, continued

5. Bayside: y 5 500t 1 100,000; Coal Flats: y 522000t 1 105,000

6. 1992

t

y

0 2 4 6 8 100

85,000

90,000

95,000

100,000

105,000

110,000

Years since 1990

Nu

mb

er

of

ho

useh

old

s

7. Hockey: y 5 1200t 1 20,000; Soccer: y 5 2000t; Baseball: y 5 21000t 1 90,000

8. 1975 9. 1980

0 10 20 30 40 500

20,000

40,000

60,000

80,000

100,000

120,000

Years since 1950

Sp

ecta

tors

t

y

0 10 20 30 40 500

20,000

40,000

60,000

80,000

100,000

120,000

Years since 1950

Sp

ecta

tors

t

y


1. y 5 7 2 9x 2. y 5 3x 2 10 3. x 5 4y 1 1

4. x 5 3 2 2y 5. y 5 x 2 4 6. x 5 6y 1 14

7. Equation 1. 8. Equation 2.



13. (1, 0) 14. (2, 23) 15. (21, 2) 16. (3, 22)

17. (4, 1) 18. (2, 2) 19. (6, 5) 20. (1, 5)

21. (23, 21) 22. (21, 4) 23. (3, 3)

24. (5, 22) 25. brother: 6 hr; sister: 5 hr

26. a. x 5 2y b. 3x 1 4.5y 5 252 c. popcorn: 48 boxes; nuts: 24 cans

Practice Level B

1. y 5 22x 1 3 2. y 5 3 }

4 x 2 3

3. x 5 2 }

3 y 1

4 }

3 4. Equation 1.


7. (2, 1) 8. (23, 4) 9. (4, 21) 10. (25, 5)

11. (3, 22) 12. (24, 22) 13. (6, 23) 14. (7, 4)

15. (3, 8) 16. (1, 1) 17. (4, 24) 18. (1, 2) 19. 4 pairs of sticks and 2 pairs of brushes

20. a. x 1 y 5 12; 225x 1 200y 5 2600 b. households mowed: 8; households shoveled: 4 21. length of hole: 16 cm; length of sheet: 17 cm

Practice Level C


3. Equation 2. 4. (6, 10) 5. (28, 24)

6. (3, 7) 7. (29, 5) 8. (2, 10) 9. (212, 8)

10. (27, 6) 11. (8, 1) 12. (23, 23)

13. 1 1, 3 }

4 2 14. 1 2

3 }

2 ,

1 }

2 2 15. 1 2

2 }

3 ,

1 }

3 2

16. a 5 25, b 5 22 17. cleanups: 250 hr; painting: 150 hr 18. x 5 16, y 5 4 19. yes; The linear system x 1 y 5 8 and x 1 0.5y 5 6.4 where x is the amount of soil and y is the amount of the half and half mix has a solution of x 5 4.8 and y 5 3.2. So 3.2 buckets are needed and there are 4 buckets.

Review for Mastery

1. (2, 24) 2. (23, 6) 3. (6, 2) 4. (3, 8)

5. (27, 6) 6. (4, 2)

Challenge Practice

1. (2, 3) 2. 1 3 } 2 , 2

15 } 16 2 3. 1 2 Î}

23

} 6 , 2 Î}

5 }

6 2 ,

1 2 Î}

23

} 6 , Î}

5 }

6 2 , 1 Î}

23

} 6 , 2 Î}

5 }

6 2 , 1 Î}

23

} 6 , Î}

5 }

6 2

4. (214, 2 Ï}

10 ), (214, Ï}

10 )


1. 3x 2 y 5 23 and 8x 1 y 5 11 2. 8x 2 y 5 1 and 8x 1 3y 5 7 3. 7x 2 4y 5 8 and 7x 1 4y 5 9

4. 7x 2 y 5 13 and 214x 1 y 5 23

5. x 2 3y 5 14 and x 1 10y 5 23

6. 8x 2 4y 5 21 and 214x 1 4y 5 23

7. Add the equations. 8. Arrange the terms.

9. Subtract the equations. 10. Arrange the terms. 11. Add the equations. 12. Arrange the terms.

13. (1, 1) 14. (215, 6) 15. 1 22, 4 }

3 2

16. (6, 25) 17. (3, 2) 18. (24, 1) 19. (2, 1)

20. (23, 4) 21. (21, 5) 22. (6, 0) 23. (8, 5)

24. 1 2 19

} 3 , 2 1 } 2 2 25. Your speed with no wind:

5.5 mi/h; Wind speed: 2.5 mi/h 26. Car wash: $6; One gallon of regular gasoline: $2.10

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Practice Level B

1. 8x 2 y 5 19 and 3x 1 y 5 7

2. 4x 2 y 5 211 and 4x 1 6y 5 23

3. 9x 2 2y 5 5 and 11x 1 2y 5 8 4. Arrange the terms. 5. Arrange the terms. 6. Arrange the terms. 7. Add the equations. 8. Arrange the terms. 9. Subtract the equations. 10. (3, 5)

11. (22, 4) 12. (7, 23) 13. (26, 2)

14. (10, 5) 15. (29, 25) 16. (3, 11)

17. (10, 9) 18. (15, 8) 19. (21, 21)

20. (24, 3) 21. 1 8, 37

} 3 2 22. Speed of barge in

still water: 5.9 mi/h; Speed of current: 2.1 mi/h

23. a. Flat fee: $15; Hourly fee: $12 b. $147

Practice Level C

1. (24, 5) 2. (8, 6) 3. (210, 3) 4. (26, 25)

5. (9, 14) 6. (21, 7) 7. (18, 18) 8. (26, 24)

9. (15, 20) 10. (3, 5) 11. (28, 24)

12. (11, 12) 13. (23, 8) 14. (9, 16)

15. (28, 27) 16. 1 5, 12

} b 2 17. (1, 2, 1);

Answers will vary. 18. a. 5x 1 30y 5 207.5 and 5x 1 50y 5 212.5; Let x be the cost of one day of rental and let y be the cost per mile over 150 miles. Because a person is only charged for miles over 150, subtract the number of miles traveled from 150 to get the number of miles a person is charged for. b. Daily rental fee: $40; Per mile fee: $.25 19. $24.72; Use the table to set up a linear system to fi nd the cost of one stamp and one package of cards. Then use this information to fi nd the total cost of 3 stamps and 3 packages of cards.

Review for Mastery

1. (4, 2) 2. (23, 4) 3. (21, 22) 4. (6, 5)

5. (2, 26) 6. (3, 3)

Problem Solving Workshop: Using Alternative Methods

1. speed of Calvin in still air: 7.95 miles per hour, speed of wind: 0.45 miles per hour 2. speed ofCalvin in still air: 7.2 miles per hour, speed of wind: 1.2 miles per hour 3. speed of boat in still water: 25 miles per hour, speed of current: 5 miles per hour

Challenge Practice

1. 1 2 4 } 11 ,

1 }

2 2 2. 1 2

4 } 15 , 2 2

3. ( Ï}

7 , 3 Ï}

23 ), (2 Ï}

7 , 3 Ï}

23 ) 4. 1 1 } a ,

1 }

2b 2

5. 1 37 }

6a ,

23 }

6b 2 6. 1 2

13b } 3a , 2

5b } 3 2

Lesson 7.4

Practice Level A

1. C 2. B 3. A 4. Sample answer: Multiply the fi rst equation by 7. 5. Sample answer: Multiply the fi rst equation by 2. 6. Sample answer: Multiply the second equation by 6.

7. Sample answer: Multiply the fi rst equation by 22. 8. Sample answer: Multiply the second equation by 3. 9. Sample answer: Multiply the

second equation by 1 }

2 . 10. (1, 2) 11. (23, 4)

12. (5, 5) 13. (6, 23) 14. (22, 22)

15. (8, 10) 16. (25, 7) 17. (8, 21) 18. (4, 4)

19. (10, 12) 20. (22, 24) 21. (1, 5)

22. a. Adult: $9; Youth: $5 b. $43

23. a. y 5 30 1 45x and y 5 45 1 40x b. x 5 3, y 5 165 c. 3 h

Practice Level B

1. Sample answer: Multiply the fi rst equation by 2. 2. Sample answer: Multiply the second equation by 23. 3. Sample answer: Multiply the fi rst equation by 23. 4. Sample answer: Multiply the second equation by 22. 5. Sample answer: Multiply the fi rst equation by 25.

6. Sample answer: Multiply the fi rst equation by 2. 7. (4, 21) 8. (3, 6) 9. (22, 25)

10. (26, 7) 11. (9, 5) 12. (2, 22) 13. (10, 8)

14. (21, 12) 15. (5, 4) 16. (25, 23)

17. (15, 24) 18. (8, 8) 19. a. 2x 1 4y 5 28 and 4x 1 5y 5 45.5 b. Adult: $7; Youth: $3.50 c. $31.50 20. a. 3x 1 2y 5 557 and 5x 1 4y 5 974 b. Hotel: $140/night; Tickets: $68.50/pair 21. x 1 y 5 15 and 180x 1 155y 5 2400; $180/day: 3 workers; $155/day: 12 workers

Practice Level C

1. (4, 8) 2. (23, 21) 3. (5, 29)

4. (210, 10) 5. (22, 25) 6. (6, 7)

7. (0, 3) 8. (8, 14) 9. (6, 4) 10. (1, 9)


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11. (2, 7) 12. (23, 23) 13. (2, 21) 14. (4, 3)

15. (25, 22) 16. (6, 1) 17. 1 2 1 }

2 ,

1 }

4 2

18. (24, 4) 19. a 5 2, b 5 1

20. a. 2x 1 4y 5 166 and 4x 1 5y 5 263 b. Adult: $37; Youth: $23 c. $189

21. Thai: 5 people; Szechwan: 3 people

22. To school: 2.72 mi/h; Home: 2.04 mi/h

Review for Mastery

1. (3, 25) 2. (6, 24) 3. (7, 2) 4. (2, 3)

5. (9, 21) 6. (5, 6)

Problem Solving Workshop: Mixed Problem Solving

1. a. x 5 student tickets, y 5 general admission tickets; x 1 y 5 556, 5x 1 8y 5 3797 b. 217 student tickets, 339 general admission tickets.

2. a. 6 miles per hour into the wind, 10 miles per hour with the wind b. x 5 speed of bicyclist, y 5 speed of wind; x 2 y 5 6, x 1 y 5 10 c. The bicyclist’s speed in still air is 8 miles per hour. The speed of the wind is 2 miles per hour.

3. a. x 5 amount in the 3% annual interest ac-count, y 5 amount in the 4% annual interest account; x 1 y 5 30,000, 0.03x 1 0.04y 5 1020 b. $18,000 at 3%, $12,000 at 4% 4. 11

5. Answer will vary. 6. By solving the linear system, 1 pound of chicken costs $2.25 and 1 pound of fi sh costs $3.75. So, 2 pounds of chicken and 2 pounds of fi sh costs $12.

7. Answer will vary. Sample answer: m 5 2, b 5 25 8. 5 9. a. x 5 amount of 20% acid solution, y 5 amount of 70% solution; x 1 y 5 900, 0.2x 1 0.7y 5 360 b. 540 milliliters of 20% acid solution, 360 milliliters of 70% acid solution c. No; The chemist needs 450 milliliters of both acid solutions.

Challenge Practice

1. x 1 2y 5 5

3x 1 3y 5 8

2. 1 1 } 3 ,

7 }

3 2 ; it takes Terry

1 }

3 hour to mow a small

lawn and 7 }

3 hours to mow a large lawn.

3. 13 small lawns

4. x 1 20y 5 43

x 1 30y 5 63

5. 3 mi 6. 2 mi


1. 3 2. 22 3. 3 4. A; infi nitely many solutions 5. C; one solution 6. B; no solution

7. no solution 8. one solution

x

y

1

3

12121

3

x

y

1

3

1212321

23

3

9. infi nitely many solutions

x

y

1

3

1212321

23

3

10. one solution 11. one solution

x

y

1

3

123 3

x

y

1

3

212321

23

3

12. no solution

x

y

3

21

23

3

13. no solution 14. no solution 15. (0, 0)

16. (21, 21) 17. no solution 18. infi nitely many solutions 19. one solution 20. one solution 21. infi nitely many solutions 22. one solution 23. no solution 24. one solution

25. one solution 26. one solution 27. one solution 28. Yes; the system 15x 1 8y 5 263.25 and 20x 1 13y 5 358 can be used to model the situation, and this system has one solution.


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29. a. 45x 1 10y 5 425 and 225x 1 50y 5 2125 b. infi nitely many solutions c. No, because one equation in the system is a multiple of the other, so specifi c values for neither x nor y can be found.

Practice Level B

1. C; infi nitely many solutions 2. A; no solution

3. B; one solution


x

y

3

123 3

x

y

1

3

12121

23

5

6. one solution 7. one solution

x

y

1

1212321

3

x

y

1

3

3

23

521

8. no solution 9. infi nitely many solutions

x

y

1

3

121 3 5

x

y1

1212321

23

3

10. (8, 0) 11. infi nitely many solutions

12. (20, 30) 13. (21, 21) 14. (3, 4) 15. no solution 16. no solution 17. no solution

18. one solution 19. one solution 20. one solution 21. one solution 22. infi nitely many solutions 23. no solution 24. infi nitely many solutions 25. Yes; the system 2x 1 12y 5 1859.3 and 2x 1 22y 5 3158.8 can be used to model the situation, and this system has one solution, (about $153, about $130).

26. a. 30x 1 20y 5 910 and 45x 1 30y 5 1365 b. infi nitely many solutions c. No, because one equation in the system is a multiple of the other, no specifi c values for x or y can be found.

Practice Level C

1. C; infi nitely many solutions 2. B; one solution 3. A; no solution 4. one solution 5. one solution

x

y

2

6

2222

26

6 10

x

y

1

3

12121

6. infi nitely many solutions

x

y

1

3

1212321

23

3


x

y

2

22222

210

x

y

1

3

12321

23

3

9. no solution

x

y

3

123

23

3

10. (212, 28) 11. infi nitely many solutions

12. 1 6, 5 }

4 2 13. no solution 14. 1 1,

1 }

2 2 15. 1 4 }

3 ,

8 }

3 2

16. one solution 17. no solution 18. one solution 19. infi nitely many solutions 20. one solution 21. no solution 22. one solution 23. one solution 24. infi nitely many solutions 25. a. 28x 1 44y 5 964.4 and 21x 1 33y 5 723.30 b. infi nitely many solutions c. No, because one equation in the system is a multiple of the other, no specifi c values for x or y can be found. 26. y 5 10x and y 5 8(x 2 10) b. x 5 240, y 5 2400 c. No, because x and y both represent quantities that are never negative.


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Review for Mastery

1. infi nitely many solutions 2. no solution

3. infi nitely many solutions 4. one solution

5. one solution

Challenge Practice

1. a 5 1 } 2 2. No value of a gives infi nitely many

solutions. 3. a Þ 1 }

2 4. The number of solutions

depends only on the value of c. 5. When c 5 4 there are an infi nite number of solutions. When c Þ 4 there are no solutions.

6. x 5 c1b2 2 c2b1

} a1b2 2 a2b1

; y 5 a1c2 2 a2c1

} a1b2 2 a2b1

7. a1b2 Þ a2b1 8. a1b2 5 a2b1 and c2 Þ b2

} b1

c1

9. a1b2 5 a2b1 and c2 5 b2

} b1

c1


1. yes 2. no 3. yes 4. no 5. yes 6. yes

7. D 8. B 9. A 10. C 11. F 12. E

13.

x

y

1

3

123 3

14.

x

y

1

3

1212321

3

15.

x

y

1

3

1212321

23

3

16.

x

y

1

3

1212321

23

3

17.

x

y

1

3

12123

23

3

18.

x

y

1

1212321

23

3

19. a. x 1 y ≤ 10 and 15x 1 18y ≤ 90 b.

x00

1

2

3

4

5

6

7

8

9

1 2 3 4 5 6 7 8 9Boxes of 5-ounce cups

Bo

xes o

f 8-o

un

ce c

up

s

y c. Answers will vary.

20. a. 6 h

b.

x00

1

2

3

4

5

6

1 2 3 4 5 6Hours spent on science

Ho

urs

sp

en

t o

n h

isto

ry

x 1 y < 6

x 1 y > 4

y

Practice Level B

1. yes 2. yes 3. no 4. B 5. A 6. C

7.

x

y

323

3

1

23

21

8.

x

y

3123 21

1

23

21

9.

x

y

3123 21

3

23

10.

x

y

3121

3

1

21

11.

x

y

212325

3

1

21

12.

x

y

3123 21

1

23

21

13. y ≥ 24 and y < 1 14. x ≥ 24 and y < 23 15. y ≥ x 1 1 and x ≤ 0 16. y ≤ 4 2 x and y > 2 17. y ≤ x and y < 1 2 x

18. x ≥ 0, y ≥ 0, and y ≤ x 1 2


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19. a. x 1 y ≥ 5 and 1.9x 1 5.2y ≤ 20 b.

x00

1

2

3

4

5

6

1 2 3 4 5 6Packages of hot dogs

Packag

es o

f h

am

bu

rgers

x 1 y > 5

1.9x 1 5.2y < 20


20. a. 5.5 h

b.

x00

1

2

3

4

5

6

1 2 3 4 5 6Hours cleaning

Ho

urs

weed

ing x 1 y < 5.5

x 1 y > 4


Practice Level C

1. no 2. no 3. yes 4. C 5. A 6. B

7.

x

y

12123

1

3

21

8.

x

y

3123 21

3

1

23

9.

x

y

3123 21

3

21

10.

x

y

3121

3

1

21

11.

x

y

1224

12

20

4

12.

x

y

312123

3

1

23

13. x ≤ 21 and y > 5 14. y ≤ 2x and y < 21

15. y ≤ 4 2 x and y ≥ 2x 16. y ≥ 1 2 x and x ≥ 0

17. x ≥ 0, y ≥ 0, and y ≤ x 1 2

18. x < 3, x > 21, and y < 2x 2 1

19. a. x 1 y ≤ 525 and 8x 1 5y ≥ 3000 b.

00

Adult tickets

Stu

den

t ti

ckets

x

200

400

600

100

300

500

200 400 600

y

c. Yes. If there are twice as many student tickets sold, then 175 adult tickets are sold and 350 student tickets are sold, which is a solution of the system.

20. a. x 1 y 5 15 and y ≥ 2x, where x is the number of hours you run and y is the number of hours you swim.

b.

1400

2

4

6

8

10

12

14

2 4 6 8 10 12

Sw

imm

ing

Running

y

x

c. The solution of the system is the portion of the graph x 1 y 5 15 for which 0 ≤ x ≤ 5. This means that if you run for no more than 5 hours, you can spend the remaining time swimming.

Review for Mastery

1.

x

y

3

212321

31

2.

x

y

3

1

21252721

1

3.

x

y

3

1

5

7

2123 31

4. y > 3x 1 1; y ≤ x 1 2 5. y < 23; 2x 1 3y > 6


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1. a. y 5 32x, y 5 28(x 2 0.25) b. (21.75, 256) c. The solution of the linear system does not make sense because you do not consider negative reading times.

2. a. x 1 y ≥ 260, y ≥ x, 6x 1 4y ≤ 1600

b.

x

y

00

40

80

120

160

200

240

280

320

360

80 160Salmon orders

Ch

icken

ord

ers

c. Yes, 120 orders of salmon and 160 orders of chicken can be ordered. 3. Answers will vary.

4. 24 5. No; Solving the linear system produces infi nitely many solutions, so you need more information.

6. a. x 1 y ≤ 25, and 9x 1 6y ≥ 120

x0

0

5

10

15

20

25

30

5 10 15 20 25 30Babysitting

Gro

cery

Sto

re

x 1 y < 25

9x 1 6y > 120

y

b. No, you will earn $111. c. You can work between 2 and 13 hours at the grocery store. 7. Yes, just pick any value that is not equal to 22. 8. a. 6x 1 8y 5 94, and 12x 1 16y 5 188 b. No, solving the linear system produces infi nitely many solutions. c. A large brick costs $9 and a small brick costs $5.

Challenge Practice

1.

x

y

312123

3

5

1

2.

x

y

312123

3

1

23

21

3. 0 ≤ 2x 1 y ≤ 160 0 ≤ x 1 y ≤ 88 0 ≤ x 1 2y ≤ 140 0 ≤ x ≤ 80 0 ≤ y ≤ 70

4.

x0 20 40 60 800

10

20

30

40

50

60

70

Pounds of Country Blend

Po

un

ds o

f P

rem

ium

Mix

y

5. 36 bags of Country Blend and 52 bags of Pre ium Mix 6. $544

Chapter Review GameRow 1: 8, 1, 6; Row 2: 3, 5, 7; Row 3: 4, 9, 2Each row, column, and diagonal has a sum of 15.


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A10Algebra 1Chapter 8 Resource Book

Answers


1. power of a product property 2. product of powers property 3. power of a power property

4. (z3)5 5 z3p5 5 z15 5. (5x)4 5 54x4 5 625x4

6. 33 p 31 5 3311 5 34

7. (24y2)3 5 (24)3( y2)3 5 264y6

8. (x2y4)3 5 (x2)3( y4)3 5 x6y12

9. x2(x3y)2 5 x2(x3)2y2 5 x2x6y2 5 x8y2

10. 87 11. 56 12. 79 13. 220 14. 621

15. 418 16. 132 p 182 17. 215 p 255 18. 76 p 1546

19. x4 20. y8 21. z13 22. m28 23. b18

24. p15 25. 27n3 26. 32x5 27. x6y6

28. Wisconsin: 106; Nebraska: 105; New Jersey: 106; Oregon: 105 29. 103 mi2

30. 106 metric tons

Practice Level B

1. 512 2. (24)10 3. (210)7 4. 87 5. 210

6. 310 7. 921 8. 158 9. (24)45 10. 134 p 194

11. 486 p 276 12. 1355 p 85 13. x7 14. y8

15. a18 16. z25 17. b14 18. (b 1 1)6

19. 81x4 20. 281x4 21. 32a5b5 22. 64x18y6

23. 81m31 24. 36p12 25. 6 26. 3 27. 6

28. 106 newspapers 29. a. 104 tons b. 1012 tons c. 1018 tons 30. a. x4 square units b. 625 square units c. 10,000 square units

Practice Level C

1. (29)14 2. 109 3. (27)8 4. 456 5. 1127

6. (26)18 7. 205 p 315 8. 1258 p 88

9. (216)6 p 266 10. x12 11. (c 1 5)18

12. 264c21 13. 264c21 14. 625x32y20

15. 2100,000a35b5 16. 250p13 17. 640m34

18. 22304x21 19. 768n17 20. 3z20

21. 32,000c13 22. 4 23. 5 24. 3 25. Answers will vary. 26. 105 computers; First fi nd the number of computers in use in Bahrain by fi nding 103 p 101. Then fi nd the number of computers in use in Australia by fi nding (103 p 101) p 101.

27. 104 metric tons; Solve the equation 10? p 102 5 106. 28. a. x6 cubic units b. 15,625 cubic units c. 1,000,000 cubic units

Review for Mastery

1. 814 2. 64 3. y11 4. (210)8 5. 1330

6. (28)21 7. f 16 8. (w 1 8)18 9. 56 p 186

10. 21331p3 11. 9x4y10 12. 8m23

Challenge Practice

1. a (x 1 9)/3 2. a12yb9y 3. xy1/2 4. x8y12

5. (x 1 2)5a 2 4 6. a3 cubic feet 7. 1 }

4 8.

2 }

3

9. (a 1 1)3 cubic feet 10. 210610 11. 220310


1. quotient of powers 2. power of a quotient

3. quotient of powers 4. 38

} 35 5 3825 5 33

5. 1 3 } 4 2

4 5

34 }

44 6. 86 }

84 p 82 5 86

} 86 5 80 7. 44 8. 93

9. 35 10. (25)1 11. (27)4 12. 15

} 45 13.

57 }

37

14. 29

} 79 15. 43 16. y6 17. z1 18. m4 19. x

3 }

y3

20. a13 }

b13 21.

19 }

z9 22. a. 103 b. 10 c. 102 d. 102

Practice Level B

1. 66 2. 141 3. (25)5 4. 124 5. 87

6. 35

} 45 7.

(21)6

} 56 8. 37 9. 48 10. y6 11. z9

12. a8

} b8 13. 2

216 }

z3 14. a12

} 16b20 15.

243x20 }

y30

16. m12

} 125n27 17. 81x28

} 16y 48

18. 32m25 }

243n45 19. 103

20. a. 104 b. 103 21. 3087π

} 2 in.3

Practice Level C

1. 155 2. 64 3. 2 87

} 97 4. 87 5. 510 6. 2105

7. 2 a7

} b7 8.

81x24 }

y36 9. m42

} 64n60 10.

64a6 }

125b9

11. 49x6

} 64y14 12.

27x11 }

200y6 13. 8x5

} y15 14.

20x14 }

27y13

15. 2 8x17

} 9y35 16. x 5 8, y 5 3; Use the properties

of exponents to write two equations in x and y. Then solve the system of equations.

17. 102 18. 1012 19. π

} 6 ft3;

9π }

16 ft3;

9π }

2 ft3

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LAHA111FLCRB_BMv2_C08_A10-A19.indd A10LAHA111FLCRB_BMv2_C08_A10-A19.indd A10 11/6/09 11:40:09 AM11/6/09 11:40:09 AM

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A11Algebra 1



Review for Mastery

1. 129 2. (28)4 3. 135 4. w5 5. b7

} c7 6.

81 }

w4

7. 27s15

} t12 8.

9m2 }

n3 9. 10

Challenge Practice

1. a 5 1 2. b 5 3, or b 5 1 }

3 3. x 5 1, y 5 4

4. 4 5. 0 6. $65,155.79 7. $63,814.08

8. $265,329.77


1. C 2. A 3. B 4. 1 }

125 5.

1 }

64 6.

1 }

32 7.

1 }

81

8. 2 1 }

9 9. 1 10. 1 11. 1 12. 36 13.

4 }

3

14. 125

} 8 15. undefi ned 16.

1 }

x5 17. 1 }

m9 18. 6 }

y3

19. 8 }

a10 20. 1 }

81b4 21. x3

} y2 22.

y3

} x4 23.

1 }

ab2

24. 2y

} x3

25. 3 }

4 in. 26.

1 }

4 in.;

1 }

2 in.

27. π

} 16

in.2; 3π

} 16

in.2

Practice Level B

1. 1 }

243 2.

1 }

1000 3.

1 }

64 4. 1 5. 1 6. 1 7.

64 }

25

8. 343

} 64

9. undefi ned 10. 1 }

100,000 11.

1 }

64

12. 625 13. 1 }

x7 14. 6 }

y4 15. 1 }

32b5 16. 1 } 81m4

17. a2 }

b4 18.

3 }

x2y5 19. x12

} 64y6 20. 1 21.

d 5 }

c3

22. x2y4 23. 1 } 4x6y5

24. x3y7

} 3 25. a.

1 }

20 in.

b. 2 }

25 in. 26.

4π }

375 cm3 27. a. 1026 m

b. 10215 m c. 1022 m

Practice Level C

1. 1 }

243 2. 6561 3.

1 }

625 4. 100,000 5. 125

6. 1 }

64 7. 25 8.

1 }

4 9. 210 10.

x6 }

16y8 11. x4y8

} 9

12. y10

} 6x4 13.

16 }

x10 14. 128d8 15. 2 y11

} 16x9

16. 81x2y5 17. 5x11

} y2 18.

2y12

} 3x2 19. false; a 5 2;

225

} 226 5 2 Þ

1 }

2 20. true 21. false; a 5 1, b 5 1;

1 }

1 1 1 5

1 }

2 Þ 2 5 1 1 1 22. 106 23. a. 2

b. 7π

} 2 cm3;

7π }

8 cm3 c. 4 d. overestimated;

A knitting needle narrows at one end.

Review for Mastery

1. 1 2. 1 }

625 3. 36 4.

8 }

125 5. 1 6. 81 7.

1 }

16

8. 100,000 9. 625x8z4

} y12 10.

n6p2

} 3m4 11.

s4t }

48r11


1. a. 3375

} 64

cubic inches

b. power of a quotient property

2. a. Blood (cubic millimeters)

Number of white corpuscles

10 104

100 105

1000 106

10,000 107

100,000 108

b. 103 p 105 5 108 3. No, the mass of a sweet corn seed is 0.1 gram.

4. a. Answers will vary. b. Answers will vary.

5. 2800 6. a. 1022 in. b. 1 cubic inch c. Assuming the same thickness, the amount of oil needed to cover a container of water with a surface area of 10x square inches is 10x 2 2 cubic inches.

Challenge Practice

1. Always true 2. Never true 3. Never true

4. Always true 5. Sometimes true; true when a 5 1 and b 5 1, false when a 5 2 and b 5 2.

6. True if a > 1. 7. 1 8. 1 9. 21 10. 9 }

256

Focus On 8.3Practice

1. 25 2. 1 }

13 3. 32 4.

1 }

27 5.

1 }

14 6. 343

7. 5 8. 1 } 7 9. 9 10.

1 }

256 11. 24 12. 16

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13. 625 14. 1 }

216 15. 6 16. 24 17. 9 18. 16

19. b3 5 a, defi nition of cube root; (ak)3 5 a, substitute ak for b; a3k 5 a1, product of powers

property; 3k 5 1, set exponents equal; k 5 1 }

3 ,

solve for k; a1/3 • a1 5 a4/3, substitute value of k into equation; a1/3 ? a3/3 5 a4/3, fi nd LCD for exponents; a4/3 5 a4/3, sum of powers property

20. 1 }

8

Review for Mastery

1. 1 }

12 2. 27 3.

1 }

512 4. 5 5. 1 }

6 6. 81 7. 16

8. 1 } 625

9. 64 10. 3 11. 32 12. 2


1. C 2. A 3. B 4. 6.4 3 100 5. 8.52 3 101

6. 2.5 3 1021 7. 1.04 3 1021 8. 5.4 3 102

9. 9.1245 3 103 10. 9.5 3 1023 11. 6.3 3 105

12. 3 3 1022 13. 2.396 3 104 14. 4.57 3 1022

15. 4.5 3 1025 16. 52,000 17. 910,000,000

18. 625,000 19. 605 20. 8,125,000

21. 11,130,000,000 22. 0.0047

23. 0.000000016 24. 0.00000445

25. 0.000924 26. 0.0071123 27. 0.000020123

28. 4.5 3 103; 15,625; 21,000; 3 3 104

29. 7.8 3 1026; 0.0006; 0.0012; 2.15 3 102

30. 0.0125; 1.3 3 1022; 6.15 3 1021; 1.765

31. Oxygen: 9.75 3 101 lb; Chlorine: 0.3 lb; Cobalt: 2.4 3 1024 lb; Magnesium: 0.06 lb; Sodium: 1.65 3 1021 lb; Hydrogen: 15 lb

32. about 28.5%

Practice Level B

1. 1.04 3 101 2. 6.751 3 103 3. 5.4 3 1021

4. 1.03 3 1024 5. 4.1562 3 105

6. 8.104 3 1022 7. 3.412 3 106

8. 5.255 3 102 9. 1.0425 3 102

10. 4.56 3 1025 11. 2.07 3 1027

12. 2.3551 3 104 13. 158,000 14. 321,000,000

15. 4,502,100,000 16. 810,450 17. 17,220,000

18. 101.2 19. 0.000812 20. 0.0000004014

21. 0.0081025 22. 0.00000000312056

23. 0.01211 24. 0.0000700135

25. 9.287 3 103; 1.3759 3 104; 14,205; 3.0214 3 104

26. 1.04 3 1023; 2.5 3 1023; 0.0985; 0.16

27. 8.79 3 102; 1.0085 3 103; 1023; 1146

28. 1.2 3 1025; 0.001023; 1.045 3 1023; 0.01036 29. 3 3 107 30. 5 3 1024

31. 3.2 3 10224 32. 5.4 3 105 pixels

33. about 13.57 people/km2 34. a. Titania, Oberon, Ariel, Umbriel, Miranda b. about 53

Practice Level C

1. 1.5 3 1023 2. 3.04 3 104 3. 4.6 3 1026

4. 9.120006 3 106 5. 2.45 3 101

6. 1.256 3 1021 7. 7.05 3 102

8. 1.00456 3 105 9. 5.01 3 1027

10. 132,500 11. 705,123,000

12. 0.0000000815 13. 0.09044 14. 5100

15. 31,112,000,000 16. 0.000081101

17. 0.00000077 18. 62,500,000 19. 758.4; 7.208 3 103; 7.914 3 103; 72,164 20. 0.000526; 1.305 3 1023; 2.018 3 1023; 0.00205

21. 3.016 3 1024; 0.000316; 3.28 3 1024; 0.003028 22. 1.254 3 1022 23. 5 3 1024

24. 2.43 3 10243 25. about 82.48 people/km2

26. about 6.68 3 10224 g 27. a. Dione, Tethys, Mimas, Phoebe, Calypso b. 275,000 c. Mimas: 8.25 3 1019 lb; Calypso: 8.8 3 1015 lb; Tethys: 1.38 3 1021 lb; Dione: 2.42 3 1021 lb; Phoebe: 8.8 3 1017 lb

Review for Mastery

1. 7.9 3 1026 2. 1.356 3 106 3. 1012

4. 0.000037 5. 2.8 3 105; 361,000; 2.1 3 106

6. 4.0 3 1029 7. 2.093 3 103 8. 8.41 3 1012

Challenge Practice

1. 1.44 3 102 2. 5 3 1010 3. 3.5 3 106

4. 4.24 3 100 5. 1 3 106 6. 5% 7. 2%

8. about 2.4% 9. 1 10. 20.00000005


1. y 5 3x 2. y 5 5x 3. C 4. B 5. A

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A13Algebra 1


6.

x

y

1

3

5

1212321

3

domain: all real numbers; range: all positive real numbers

7.

x

y

2

6

10

1212322

3


8.

x

y

1

3

5

1212321

3


9.

x

y

1

3

5

1212321

3


10.

x

y

3

5

1212321

3


11.

x

y

3

5

1212321

3


12.

x

y

3

5

1212321

3


13.

x

y

3

5

1212321

3


14.

x

y

3

5

1212321

3


15.

x

y

1

3

12321

23

3

16.

x

y

1

3

1212321

23

3

refl ection in x-axis vertical stretch

17.

x

1

3

1212321

23

3

y vertical shrink

18. initial amount: 3; growth rate: 0.05; growth factor: 1.05 19. initial amount: 2; growth rate: 0.25; growth factor: 1.25

20. initial amount: 0.1; growth rate: 0.75; growth factor: 1.75 21. a. $206 b. $212.18 c. $231.85 22. Freshmen: 2; Sophomore: 2.5; Junior: 3.125; Senior: about 3.906

Practice Level B

1. y 5 11x 2. y 5 0.25(2)x

3.

x

y

2

6

10

1212322

3


4.

x

3

1212321

23

3

y domain: all real numbers; range: all positive real numbers


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5.

x

3

1212321

23

3


6.

x

1

3

5

1212321

3


7.

x

1

12321

23

25

3

y domain: all real numbers; range: all negative real numbers

8.

x

1

1

23

25

3


9.

x

6

2

10

123 2122

3


10.

x

1

3

5

1212321

3


11.

x

1

12121

23

25

3


12.

x

y

2

6

10

1212322

3

13.

x

1

12321

23

25

3

y

vertical stretch refl ection in x-axis

14.

x

1

3

5

1212321

3

y 15.

x

3

1212323

29

215

3

y

vertical shrink vertical stretch and refl ection in x-axis

16.

x

1

1212321

23

25

3

y 17.

x

2

1212322

26

210

3

y

vertical shrink and vertical stretch and refl ection in x-axis refl ection in x-axis

18. a. $512.50 b. $565.70 c. $819.31 19. y 5 8000(1.07)t 20. a. y 5 10,000(1.08)t b. $19,990.05

Practice Level C

1. y 5 24x 2. y 5 5 p 2x 3.

x

y

3

9

15

12123

23 3


4.

x

y

3

12121

23

23 3


5.

x

y

1

3

5

12121

23 3



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A15Algebra 1


6.

x

y

3

12121

23

23 3


7. x

y

12121

23

25

27

23 3 domain: all real numbers;

range: all negative real numbers

8.

x

y

1

3

1

23

23 3

domain: all real numbers; range: all negative real numbers

9.

x

y

3

9

15

12123

23 3


10.

x

y

2

6

10

12122

23 3


11.

x

y2

122

26

210

23 3


12.

x

y

2

6

10

12122

23 3

13.

x

y

1

121

23

25

23 3


14.

x

y

1

3

12121

23

23 3

15.

x

y3

123

29

215

23 3

vertical shrink vertical stretch andrefl ection in x-axis

16.

x

y

1

3

12121

23

23 3

17.

x

y

1

3

121

23

23 3

vertical shrink and vertical shrink andrefl ection in x-axis refl ection in x-axis

18. Subtract the amount deposited from the balance. a. $10.31 b. $54.48 c. $270.16

19. a. y 5 65,000(1.025)t b. about 71,748 people

20. a. 100% b. y 5 10(2)t c. 160 students

Review for Mastery 1. y 5 9 p 3x

2.

2123 1 3

1

5

7

x

y 5 4(3)x


3. 23 1 3

25

27

y 5 25(6)x

x

y

Because the y-values for y 5 25 p 6x are 25 times the corresponding y-values for y 5 6x, the graph of y 5 25 p 6x is a vertical stretch and refl ection in the x-axis of the graph of y 5 6x.

Problem Solving Workshop: Worked Out Example 1. $389.78 2. The value raised to the x power should have been 1 1 0.36; and the fi nal calculation of 0.10 is also incorrect. The spending per person per year on the Internet in 2007 is $389.78.


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3. y 5 179,323,175(1.011)x; 309,880,465

4. 7.59 feet 5. 16.41 feet

Challenge Practice

1. y 5 3x 2. y 5 3 p 2x 3. y 5 1 } 2 p 5x

4. y 5 1 1 } 9 2 3x or y 5 3x 2 2 5. y = 1 3 }

2 2 2x or

y 5 3 p 2x 2 1

6. f (x) 5 3 p 28x and g(x) 5 3 p 212x, so g(1) > f (1)

7. f (x) 5 1 }

2 p 16x and g(x) 5 1280 p 16x,

so g(1) > f (1)

8. f (x) 5 25 p 52x and g(x) 5 52x, so f (1) > g(1)

9. f (x) 5 6 p 42x and g(x) 5 1 }

2 p 43x, so

f (1) > g(1) 10. f (x) 5 1000 p (1.5)10x and g(x) 5 2000 p (1.5)3x, so f (1) > g(1)


1. yes; y 5 1 1 } 10

2 x 2. no 3. C 4. A 5. B

6.

x

y

1

5

12121

23 3


7.

x

y

1

3

5

12121

23 3


8.

x

y

1

3

5

12121

23 3


9.

x

y

1

3

5

12121

23 3


10.

x

y

3

5

12121

23 3


11.

x

y

1

3

5

12121

23 3


12.

x

y

1

5

12121

23 3

13.

x

y

1

3

2121

23 3


14.

x

y

1

3

12121

23

23 3

vertical shrink

15. exponential decay 16. exponential growth

17. exponential decay 18. exponential decay

19. exponential growth 20. exponential growth

21. a. $10,200 b. $7369.50 c. $5324.46

22. a. y 5 4000(0.98)t b. 3689 employees

Practice Level B

1. yes; y 5 1 1 } 5 2 x 2. no

3.

x

6

10

1212322

3


4.

x

3

5

2222621

6



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A17Algebra 1


5.

x12123 3

y2


6.

x12123 3

y

6

2

10

22


7.

x

1

5

7

12123 3


8.

x

0.5

3212320.5


9.

x

20

1212324

3

y 10.

x

2

12123 3

y


11.

x

3

5

1212321

3

y vertical shrink

12. never; The graphs are refl ections in the x-axis.

13. always; At x 5 0, the graphs will intersect at (0, 1).

14. sometimes; Sample answer: If a 5 1, then the graphs are identical. If a 5 2, then the graphs are not identical.

15. exponential decay; y 5 3(0.75)x


17. exponential growth; y 5 4(2)x

18. a. $2400 b. $1536 c. $983.04

19. a. y 5 7(0.979)t b. about 5.4%

20. a. y 5 18,000(0.945)t b. about 13,565 people

Practice Level C

1. yes; y 5 1 9 } 10

2 x 2. no

3.

x

y

9

15

12123

23 3


4.

x

y

1

3

12121

23

23 3


5.

x

y

1

3

2121

23 3


6.

x

y

12126

23 3


7.

x

y

2

12122

23 3


8.

x

y0.1

2120.1

23 3



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9.

x

y

5

25

12125

23 3

10.

x

y

1

2121

23 3


11.

x

y

1

3

121

23

23 3

vertical shrink and refl ection in x-axis

12. always; The graphs are refl ections in the x-axis.

13. always; The graphs are refl ections in the y-axis, so they have the same range.

14. never; The function is an exponential decay function.


16. exponential growth; y 5 3(1.25)x

17. exponential decay; y 5 6(0.8)x 18. a. $2700 b. $6729.48 c. $9438.90 19. a. y 5 8(0.982)t b. about 7.6 h 20. no; At the beginning of the second 5-year period, there was more money being lost than there was being gained at the beginning of the fi rst 5-year period.

Review for Mastery

1. yes; y 5 9 p 3x

2.

2123 1 3

3

5

7

x

y 5 (0.7)x


3.

2123 1 3

1

3

x

y 5 4 18( )x

y

exponential decay; y 5 4 p 1 1 } 8 2

x


1. a. 7.1492 3 104, 2.4 3 103 b. The surface area of Jupiter is about 6.423 3 1010 km2. The surface area of Callisto is about 7.238 3 107 km2. c. about 8.874 3 102, The surface area of Jupiter is about 887.4 times larger than the surface area of Callisto.

2. a. y 5 20(0.5)x, where x is the number of 45-day periods b. 1.25 ounces 3. a. exponential growth b. y 5 91(1.59)x c. 365.79 million

4. Yes; After two years of depreciation, the value of the boat is $5057.50. The family is getting more for the boat than it is worth. 5. 0.14 6. Answers will vary.

7. a. y 5 20,000(0.94)x b. 6%; The decay rate for the car is 0.06, or 6%.

8. a. y 5 200(1.04)x

b.

7 8 900

50

100

150

200

250

1 2 3 4 5 6Time (years)

Valu

e (

do

llars

)

x

y

c. No; After 3 years there is $224.97 in the account.

Challenge Practice

1. f (x) 5 3 p 1 1 } 2 2

x 2. f (x) 5 2 p 1 1 }

3 2

x

3. f (x) 5 4 p 1 3 } 5 2

x 4. f (x) 5

5 } 2 p 1 2 }

5 2

x

5. f (x) 5 7 } 3 p 1 3 }

7 2

x 6. f (x) 5 3 p 1 1 }

9 2

5x and

g(x) 5 4 p 1 1 } 9 2

3x, so g(1) > f (1)

7. f (x) 5 8 p 1 1 } 16

2 x and g(x) 5

5 } 256 p 1 1 }

16 2

x, so

f (1) > g(1) 8. f (x) 5 1 } 5 p 1 1 }

5 2

x and g(x) 5 1 1 }

5 2

4x,

so f (1) > g(1) 9. f (x) 5 6 p 1 9 } 16

2 x and

g(x) 5 1 } 2 p 1 9 }

16 2

x, so f (1) > g(1)


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A19Algebra 1



1. arithmetic;

2. geometric;

3. arithmetic;

4. an 5 64 1 1 } 4 2

n 2 1;

D: 1, 2, 3, ...R: 64, 16, 4, 1, ...

5. an 5 (26)n 2 1; D: 1, 2, 3, ...R: 1, –6, 36, 2216, ...

6. an 5 3(2)n 2 1; D: 1, 2, 3, ...R: 3, 6, 12, 24, ...

7. an 5 1 1 } 4 2

n 2 1;

D: 1, 2, 3, ...

R: 1, 1 }

4 ,

1 }

16 ,

1 }

64 , ...

8. an 5 21 ? 1 2 1 } 2 2

n 2 1;

D: 1, 2, 3, ...

R: 21, 1 }

2 , 2

1 } 4 ,

1 }

8 , ...

9. an 5 281 1 1 } 3 2

n 2 1;

D: 1, 2, 3, ...R: 281, 227, 29, 23, ...

10. an 5 6n 2 1;

Review for Mastery

1. geometric; 2500 2. arithmetic; 3.5

3. geometric; 2

4. 5.

6.

7. an 5 (22)n 2 1; a10 5 2512

8. an 5 1 1 } 3 2

n 2 1; a10 5

1 }

19683

9. an 5 10(2)n 2 1; a10 5 5120

Chapter Review Game

1. x8 2. 3.10091 3 107 3. 1 4. x6

} y3 5. 0.891

6. 2324x10 7. 0.0000987 8. 2x7y

9. 3.0 3 1025 10. y14

} 16x6 11. 2

500 }

x4y6 12. 0.055

13. 1.495 3 1011

RENE DESCARTES


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Answers


1. 8n6; degree: 6; leading coeffi cient: 8

2. 29z 1 1; degree: 1; leading coeffi cient: 29

3. 2x5 1 4; degree: 5; leading coeffi cient: 2

4. 2x2 1 18x 1 2; degree: 2; leading coeffi cient: 21 5. 3y3 1 4y2 1 8; degree: 3; leading coeffi cient: 3 6. 220m3 1 m 1 5; degree: 3; leading coeffi cient: 220 7. 23a7 1 10a4 2 8; degree: 7; leading coeffi cient: 23

8. 6z4 1 z3 2 5z2 1 4z; degree: 4; leading coeffi cient: 6 9. h7 2 6h4 1 8h3; degree: 7; leading coeffi cient: 1 10. polynomial; degree: 2; monomial 11. not a polynomial; variable exponent 12. not a polynomial; negative exponent 13. polynomial; degree: 2; binomial 14. polynomial; degree: 2; trinomial 15. polynomial: degree: 3; binomial 16. 7x 1 9 17. 7m2 2 7 18. 9y2 1 5y 2 4 19. 2x2 1 3 20. 7a2 1 2a 2 6 21. 2m2 2 8m 1 3 22. 4x 1 423. 4x 1 9 24. B 5 0.014t2 1 0.13t 1 12 25. Area: 4x2 2 12πx 1 6π

Practice Level B 1. 4n5; degree: 5; leading coeffi cient: 4

2. 22x2 1 4x 1 3; degree: 2; leading coeffi cient: 22 3. 4y4 1 6y3 2 2y2 2 5; degree: 4; leading coeffi cient: 4 4. not a polynomial; variable exponent 5. polynomial; degree: 3; trinomial 6. not a polynomial; negative exponent 7. 5z2 1 3z 2 7 8. 5c2 2 3c 1 6

9. 3x2 1 6 10. 6b2 2 8b 1 1

11. 24m2 1 2m 2 3 12. 22m2 1 9m 2 1

13. 10x 1 2 14. 9x 2 1

15. Area: 17

} 4 x2 1 8x 2 32

16. P 5 1 }

6 t2 1 2t 1 200

Practice Level C 1. polynomial; degree: 0; monomial 2. not a polynomial; negative exponent 3. polynomial; degree: 2; trinomial 4. 3m3 1 4m2 2 m 1 2 5. 25y2 2 2y 1 9 6. c3 1 c2 2 9c 1 5 7. 24z2 1 4z 1 14 8. 14x4 2 3x3 2 7x2 2 3 9. 2x4 2 2x3 1 6x2 2 5x 10. f (x) 1 g(x) 5 6x3 2 3x2 1 2x 2 6; f (x) 2 g(x) 5 26x3 2 7x2 1 2x 1 4

11. 24a3b2 1 15a2b2 2 10a2b 1 5

12. 3m2n 2 11mn2 2 8n 1 2m

13. a. T 5 4.93t4 2 56.78t3 1 177.65t2 2 126.42t 1 1367.51 b. In 1997, 1367.51 thousand metric tons were produced and in 2003, 1129.19 thousand metric tons were produced. So more peat and perlite were produced in 1997.

14. a. N 5 187,443 1 13,857t; M 5 151,629 1 5457t b. 1997: $35,814; 2003: $86,214; Northeast: $83,142; Midwest: $32,742

Review for Mastery

1. 22x2 1 9; degree: 2; coeffi cient: 22

2. 3y3 1 2y 1 16; degree: 3; coeffi cient: 3

3. 23z5 1 6z3 1 7z2; degree: 5; coeffi cient: 23

4. 9a2 1 4a 1 4 5. 13b2 2 2b 1 5

6. 22c3 1 5c2 2 5c 1 4 7. 13d2 2 23d 1 11


1. 22,055,300 people 2. $1,115,940

Challenge Practice

1. x 1 x 1 4 5 2x 14 5 2(x 1 2); Because the number of quarters and dimes is a multiple of 2, it is even. 2. x 1 2x 1 1 5 3x 11; If x is even, then 3x is even and 3x 1 1 is odd. If x is odd, then 3x is odd and 3x 1 1 is even. So, whether the total number of coins is even or odd can’t be determined. 3. x 1 3x 1 5 5 4x 1 5; Whether x is even or odd, 4x is even, so 4x 1 5 is odd.

4. x 1 4 1 3x 1 5 5 4x 1 9; Whether x is even or odd, 4x is even, so 4x 1 9 is odd.

5. x 1 4 1 2x 1 1 1 3x 1 5 5 6x 1 10 5 2(x 1 5); Because the number of dimes, nickels, and pennies is a multiple of 2, it is even.

6. 0 7. 1 8. x 9. 3 10. 81

} 4 11. 25 12. 19


1. 3x3 2 2x2 1 x 2. 6y4 1 2y3 2 8y

3. 23m3 2 12m2 1 3m 4. 4d4 2 3d3 1 d2

5. 2w5 2 3w4 6. 2a4 2 3a3 1 a2

7. x2 2 3x 2 4 8. y2 1 8y 1 12

9. a2 2 8a 1 15 10. 2m2 1 7m 1 3

11. 3z2 2 11z 2 20 12. 3d2 1 17d 2 6

13. y2 1 5y 2 24 14. n2 1 11n 1 30

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A21Algebra 1



15. 3x2 1 13x 2 10 16. 8a2 2 2a 2 1

17. w3 1 3w2 1 3w 1 1

18. m3 2 4m2 1 7m 2 6 19. 8y2 2 23y 2 3

20. 15b2 1 7b 2 2 21. 6d2 2 14d 1 4

22. 6x2 1 8x 1 2 23. 6x2 1 22x 2 8

24. 2s2 1 s 2 15 25. 40c2 2 46c 2 14

26. 16p2 2 46p 1 15 27. 14t2 1 26t 2 4

28. a. V 5 288x2 1 1152x 1 1152 b. 41,472 in.3

29. a. A: 76,226; P: 0.6; A p P indicates the number of acres (in thousands) that are parks. b. A p P 5 20.1688t3 2 59.0818t2 1 812.634t 1 45,735.6

Practice Level B

1. 6x4 2 3x3 2 x2 2. 220a7 1 15a4 2 5a3

3. 28d5 1 20d4 2 24d3 1 8d2

4. 6x2 2 13x 2 5 5. 2y2 2 7y 2 15

6. 24a2 2 18a 1 3 7. 5b2 2 42b 1 16

8. 16m2 1 38m 1 21 9. 23p3 1 6p2 2 p 1 2

10. 22z2 1 13z 2 21 11. 26d2 1 23d 2 10

12. n3 1 5n2 1 9n 1 5 13. w3 1 5w2 2 23w 2 3

14. 2s3 1 11s2 1 13s 2 5

15. 5x3y 2 20x2y2 1 5xy3 16. 4a2 1 a 2 1

17. 23x2 1 8x 1 10 18. 2m2 1 5m 2 41

19. 3x2 1 15x 20. x2 1 6x 1 8

21. a. A 5 4x2 1 22x 1 30 b. 72 ft2

22. a. S: 66,939; P: 0.4; S p P indicates the number of students (in thousands) that were between 7 and 13 in 1995. b. A p P 5 0.000163t7 2 0.01166225t6 1 0.218856t5 2 1.510115t4 1 0.46605t3 1 38.8676t2 1 181.107t 1 26,775.6 c. about 26,775,600 students

Practice Level C

1. 216y7 1 40y5 2 24y3 2. 3b3 1 7b2 2 5b 1 3

3. 218w2 1 33w 2 12

4. 36m5 2 9m3 1 4m2 2 1

5. 2x3 1 11x2 1 13x 2 6

6. 24n4 2 32n3 1 37n2 1 4n 2 5

7. 6p6 2 12p4 2 10p2 1 20

8. 248r5 1 8r3 1 12r2 2 2 9. 10z4 2 39z2 2 27

10. x3y 1 2xy2 11. 26x2y 2 15xy

12. x2y3 1 xy4 13. 5x2 1 xy 2 6y2

14. 2xy3 1 3x2y2 1 210x 1 140y

15. 5x3y 2 20x2y2 1 5xy3 16. 33n2 1 36n 1 3

17. w6 1 13w5 1 3w4 2 10w3 1 5w2

18. 1 }

2 x2 1

7 } 2 x 1 6 19. 22x2 2 2x 1 96

20. a. A 5 2330.6934t5 1 14,967.1039t4 2 149,699.734t3 1 178,230.4684t2 1 18,574.268t 1 106,563,461.4 b. $106,563,461,400

21. a. E: 14,439.09; P: 0.126; E p P indicates the amount of money spent (in millions of dollars) on exercise equipment.

b. E p P 5 0.0001112t8 2 0.0002186t7 2 0.06424t6 1 0.983634t5 2 6.7188068t4 1 22.667885t3 2 120.819698t2 1 568.42959t 1 1819.32534 c. $1,819,325,340

Review for Mastery 1. 21x4 2 6x3 1 9x2

2. 12x8 2 8x7 2 32x6 1 36x5

3. 3m3 1 17m2 1 6m 2 4 4. 6n2 1 29n 1 28

5. 2p3 1 13p2 2 p 1 42

6. 12q3 2 28q2 1 7q 1 12 7. 15t2 2 13t 2 72

8. 72s2 2 119s 1 49 9. 2y21 15y 2 27

Challenge Practice 1. x7 1 3x5 1 2x3 2. 2y7 1 3y5 2 y4 1 3y2

3. 2x7 1 4x3y3 1 2x4y 1 4y4

4. 2x12 1 11x10 1 12x8

5. x5 1 2x4 1 3x3 1 6x2 1 2x 1 4 6. 0

7. 4x 8. 4x2 9. 2x3 2 x2 2 6x 1 1

10. 2x3 1 8x2 1 5

11. V 5 9x(50x 1 150) (8x 1 16)

12. V 5 3600x3 1 18,000x2 1 21,600x

13. 168 trailers


1. 2ab 2. 2mn 3. 2x 4. 10x 5. y2 6. 9

7. C 8. A 9. B 10. x2 1 8x 1 16

11. m2 2 16m 1 64 12. a2 1 20a 1 100

13. p2 2 24p 1 144 14. 4y2 1 4y 1 1

15. 9y2 2 6y 1 1 16. 100r2 2 20r 1 1

17. 16n2 1 16n 1 4 18. 9c2 2 12c 1 4

19. z2 2 25 20. b2 2 4 21. n2 2 64

22. a2 2 100 23. 4x2 2 1 24. 25m2 2 1

25. 16d2 2 1 26. 9p2 2 4 27. 4r2 2 9

28. Find the product (10 2 3)(10 1 3).

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29. Find the product (30 2 6)(30 1 6).

30. Find the product (60 1 9)(60 2 9).

31. T 5 9t2 2 4 32. a. 0.25B2 1 0.5Bb 1 0.25b2

b. 25%

Practice Level B

1. x2 2 18x 1 81 2. m2 1 22m 1 121

3. 25s2 1 20s 1 4 4. 9m2 1 42m 1 49

5. 16p2 2 40p 1 25 6. 49a2 2 84a 1 36

7. 100z2 2 60z 1 9 8. 4x2 1 4xy 1 y2

9. 9y2 2 6xy 1 x2 10. a2 2 81

11. z2 2 400 12. 25r2 2 1 13. 36m2 2 100

14. 49p2 2 4 15. 81c2 2 1 16. 16x2 2 9

17. 2w2 1 16 18. 24y2 1 25 19. Find the product (20 2 5)(20 1 5). 20. Find the product (50 2 7)(50 1 7). 21. Find the product (20 2 2)2.

22. 16x2 2 0.25 23. 16x2 1 4x 1 0.25

24. 16x2 2 4x 1 0.25

25. a. S

S

s

s

SS Ss

sS ss

b. 0.25S2 1 0.5Ss 1 0.25s2 c. 25%

26. a. 75%; Three of the four squares in the area model represent at least one foul shot being made. b. The chance of making a foul shot is 50% and the chance of not making a foul shot is 50%. So the polynomial (0.5C 1 0.5I)2 5 0.25C2 1 0.5CI 1 0.25I2 represents this situation where C represents a foul shot made and I represents a foul shot missed.

Practice Level C

1. 64x2 2 80x 1 25 2. 16p2 1 32p 1 16

3. 100 m2 2 220m 1 121 4. 121s2 2 220s 1 100

5. 400b2 2 600b 1 225 6. m2 1 8mn 1 16n2

7. r2 2 16rs 1 64s2 8. 100a2 1 60ab 1 9b2

9. 4x2 2 16xy 1 16y2 10. 64p2 2 9

11. 121t2 2 16 12. 49n2 2 25 13. 81z2 2 144

14. 2w2 1 225 15. 225p2 1 36

16. 29m2 1 400 17. 100a2 2 25b2

18. 16x2 2 9y2 19. Find the product (40 2 4)(40 1 4). 20. Find the product (20 1 3)2.

21. Find the product (50 2 1)2. 22. 81x2 2 0.25

23. 324x2 24. 1 25. (x 2 12)(x 1 12)

26. (a 2 b)3 5 (a 2 b)2(a 2 b) 5

(a2 2 2ab 1 b2)(a 2 b) 5 a3 2 3a2b 1 3ab2 2 b3

27. a. 75%; Three of the four squares in the area model represent at least one goal being made. b. The chance of making a goal is 50% and the chance of not making a goal is 50%. So the polynomial (0.5C 1 0.5I)2 5 0.25C2 1 0.5CI 1 0.25I2 represents this situation where C represents a goal made and I represents a goal missed. 28. The expression 8(122) represents the original volume. If the side lengths are changed as described, the expression 8(12 2 x)(12 1 x) 5 8(122) 2 8x2 represents the new volume. Because x is positive, subtracting 8x2 will always decrease the original volume.

Review for Mastery

1. y2 1 18y 1 81 2. 9z2 1 42z 1 49

3. 4w2 2 12w 1 9 4. 100r2 2 60rs 1 9s2

5. g2 2 121 6. 49f 2 2 1 7. 4h2 2 81

8. 36k2 2 64 9. Square of a binomial pattern; (50 1 5)2 10. Sum and difference pattern; (40 2 9)(40 1 9)

Challenge Practice

1. 8x2 1 18 2. 2x4 1 2x2 1 5

3. 2a2x2 1 2b2y2 4. 2a2x4 1 2b2y4 5. 34x 1 50

6. (a 2 b 1 c)2

5 a(a 2 b 1 c) 2 b(a 2 b 1 c) 1 c(a 2 b 1 c)5 a2 2 ab 1 ac 2 ab 1 b2 2 bc 1 ac 2 bc 1 c2

5 a2 1 b2 1 c2 2 2ab 1 2ac 2 2bc

7. 9x2 1 4y2 1 25z2 2 12xy 1 30xz 2 20yz

8. a2x2 1 b2y2 1 c2z2 2 2abxy 1 2acxz 2 2bcyz

9. 8x3 1 24x2 1 16x 10. Because 8x3 1 24x2 1 16x 5 2(4x3 1 12x2 1 8x), the expression represents an even number.

11. 8x3 1 36x2 1 46x 1 15

12. Because 8x3 1 36x2 1 46x 1 15 5 2(4x3 1 18x2 1 23x) 1 15, the expression represents the sum of an even number and an odd number, which gives an odd number.


1. B 2. A 3. C 4. 26, 22 5. 23, 5 6. 7, 10

7. 21, 8 8. 29, 9 9. 215, 212 10. 250, 25


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A23Algebra 1


11. 23, 1 12. 2, 3 13. 2(2m 2 1) 14. 5(x 2 2)

15. 3(2y 1 5) 16. 8(x 1 y) 17. 7(a 2 b)

18. 2(a 1 5b) 19. 9(m 2 2n) 20. 3(5p 2 q)

21. 4(3x 1 y) 22. 2c(c 1 2) 23. m2(9m 1 1)

24. 2w(w 1 2) 25. C 26. B 27. A 28. 28, 0

29. 0, 7 30. 21, 0 31. 0, 1 32. 0, 2

33. 22, 0 34. 9 sec 35. about 0.33 sec

Practice Level B

1. 214, 3 2. 25, 12 3. 224, 215 4. 8, 9

5. 28, 1 }

2 6. 2

3 } 4 , 6 7. 25, 4 8. 22, 3

9. 2 }

3 , 8 10. 2

1 } 2 ,

1 }

2 11. 23,

1 }

2 12. 2

5 } 4 ,

5 }

4

13. 10(x 2 y) 14. 4(2x2 1 5y) 15. 6(3a2 2 b) 16. 4x(x 2 1) 17. r(r 1 2s) 18. 2m(m 1 3n)

19. 5q(p2 1 2) 20. a3(9a2 1 1) 21. 2w2(3w 2 7)

22. 0, 10 23. 214, 0 24. 0, 1 25. 21, 0

26. 0, 3 27. 22, 0 28. 0, 5 }

2 29. 2

5 } 4 , 0

30. 0, 5 }

2 31. 0,

1 }

2 32. 2

1 } 2 , 0 33. 2

3 } 8 , 0

34. 1.5 sec; Yes. From the equation, you can see that the factor t 2 1.5 will be zero when t 5 1.5.

35. a. h 5 216t2 1 14t b. 7 }

8 sec

36. a. w(w 1 3) 5 w(7 2 w) b. 2 ft c. 20 ft2

Practice Level C

1. 23, 2 } 5 2. 2

3 } 2 ,

5 }

2 3. 24, 6 4. 28, 2

5. 2 3 } 2 , 9 6. 2

2 } 3 ,

8 } 5 7.

2 }

9 ,

3 } 7 8. 2

1 } 5 , 4 9.

9 }

8 ,

5 }

2

10. 3(3x2 2 7y) 11. 4m(m2 1 6) 12. 5pq(2p 2 q) 13. 3y(2x3 1 3y) 14. 5ab(7ab 2 1) 15. 4mn(3m 2 2n)

16. w(w3 2 2w2 1 1) 17. 3p(2p3 1 5p 1 2)

18. 4r2(2r3 2 5r2 2 3) 19. 0, 3 }

4 20. 2

2 } 3 , 0

21. 0, 4 }

3 22. 2

6 } 5 , 0 23. 2

1 } 2 , 0 24. 0,

5 }

6

25. 2 3 } 50 , 0 26. 0,

3 }

10 27. 2

13 } 17 , 0

28. 0, 1 }

4 29. 0,

4 }

9 30. 0,

3 } 5

31. a. 0, 0.21875; These are the times at which the fi sh leaves and enters the water. b. Sample answer: Any value of t ≥ 0 because time should be positive. 32. a. Locate the zeros and fi nd the horizontal distance between them.

b. Check student’s work; 40 ft c. Check student’s work; (0, 60)

Review for Mastery

1. 7, 9 2. 22, 23 3. 0, 216 4. 0, 2 5. 0, 3

6. 0.625 sec


1. a. 4x2 1 24x 1 35 b. 99 square inches

2. a. E 5 4.3791t2 1 235.3518t 1 2944.308 b. $3,955,780,800 3. a–c. Answers will vary.

4. 2 5. 0, 1.125; The kangaroo jumped off the ground at 0 seconds and landed back on the ground at 1.125 seconds. 6. Brian was in the air longer during his fi rst jump since he had a larger initial velocity, which means that he landed on the ground in the fi rst jump later than he landed on the second jump.

7. a. S 5 144.3t3 2 841.1t2 1 520.5t 1 6559.3 b. $6,559,300,000, $4,418,900,000 c. a loss of $535,100,000 per year; Find the difference in the sales fi gures from part (b) and divide by the number of years.

Challenge Practice

1. (x 2 1)(x 2 2)(x 2 3); x3 2 6x2 1 11x 2 6

2. (x 1 1)x(x 2 1); x3 2 x

3. x p x(x 2 1)(x 2 1); x4 2 2x3 1 x2

4. x 1 x 2 1 }

2 2 (x 2 2); x3 2

5 }

2 x2 1 x

5. (x 1 1) 1 x 1 2 }

3 2 (x 1 3); x3 1

14 } 3 x2 1

17 }

3 x 1 2

6. (x 1 10)(2x 1 15) 5 1650 7. 20 feet wide by 40 feet long 8. x 5 0, or y 5 x, or y 5 2x

9. x 5 0, or y 5 x, or y 5 2x

10. y 5 x, or y 5 2x


1. B 2. C 3. A 4. (x 1 1)(x 1 5)

5. (a 1 7)(a 1 3) 6. (w 1 5)(w 1 3)

7. (p 2 5)(p 1 2) 8. (c 2 1)(c 1 11)

9. (y 1 7)(y 2 2) 10. (n 2 1)(n 2 3)

11. (b 2 3)(b 2 2) 12. (r 2 7)(r 2 5)

13. (z 1 3)(z 1 4) 14. (s 2 6)(s 1 3)

15. (d 2 8)(d 1 3) 16. 24, 21 17. 25, 22


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18. 27, 22 19. 1, 11 20. 22, 3 21. 5, 7

22. 21, 5 23. 25, 3 24. 27, 1 25. C 26. A

27. B 28. 24, 3 29. 22, 5 30. 21, 6

31. 27, 3 32. 29, 4 33. 21, 4 34. 30 ft

35. a. x(x 1 1) 5 6 b. 2 ft, 3 ft c. 3 ft2

Practice Level B

1. (x 1 7)(x 1 1) 2. (b 2 5)(b 2 2)

3. (w 2 13)(w 1 1) 4. (p 1 5)2

5. (m 2 6)(m 2 4) 6. (y 2 8)(y 1 3)

7. (a 1 9)(a 1 4) 8. (n 2 6)(n 1 8)

9. (z 2 10)(z 2 4) 10. 29, 28 11. 23, 12

12. 6, 7 13. 22, 7 14. 28, 23 15. 3, 9

16. 210, 5 17. 212, 24 18. 25, 6 19. 24, 9

20. 210, 2 21. 3, 8 22. 27, 24 23. 212, 1

24. 26, 3 25. 212, 25 26. 24, 8

27. 25, 23 28. 27, 1 29. 22, 5 30. 29

31. a. x2 1 150x 1 5000 b. 20 ft

32. a. x2 2 7x 1 12 b. 144 in.

Practice Level C

1. (x 2 8)(x 1 7) 2. (m 1 6)(m 1 8)

3. (y 2 9)(y 2 6) 4. (p 1 10)(p 1 2)

5. (w 2 9)(w 2 5) 6. (x 1 6)(x 2 4) 7. 24, 15

8. 211 9. 12 10. 225, 20 11. 212, 11

12. 29, 28 13. 26, 10 14. 26, 12 15. 8

16. 215 17. 210, 15 18. 3, 10 19. 220, 30

20. 214, 22 21. 28, 25 22. 23, 7

23. 26, 4 24. 3, 8 25. 29, 24 26. 22, 12

27. 7, 8 28. 25, 4 29. 214, 23 30. 2, 9

31. a. x2 1 600x 1 80,000 b. 25 ft c. $234,375

32. a. 50 ft by 40 ft b. 110 ft by 90 ft

Review for Mastery

1. (x 1 8)(x 1 2) 2. ( y 1 5)( y 1 1)

3. (z 2 3)(z 2 4) 4. (x 1 1)(x 2 11)

5. ( y 2 7)( y 1 9) 6. (z 1 4)(z 2 9) 7. 6, 5

Challenge Practice

1. (y1/3 1 4)(y1/3 1 2)

2. (y 2 2)(y 1 2)(y2 1 3) 3. 1 1 } y 2 9 2 1 1 }

y 1 1 2

4. 1 5 Ï}

y 1 12 2 1 5 Ï}

y 1 4 2 5. 1 4 Ï}

y 1 11 2 1 4 Ï}

y 1 1 2

6. x 5 22, 2 7. x 5 22, 2, 23, 3

8. x 52 1 }

3 ,

1 } 4 9. x 5 9

10. x 5 22 Ï}

3 , 2 Ï}

3 , 22, 2


1. B 2. A 3. C 4. 2(x 2 3)(x 1 5)

5. 2(m 2 1)(m 2 2) 6. 2(p 1 2)(p 2 7)

7. (2w 1 1)(w 1 3) 8. (3y 1 2)(y 1 1)

9. (2b 2 1)(b 1 1) 10. 3(n 2 1)(n 1 1)

11. (5a 2 2)(a 1 3) 12. (2z 2 1)(z 1 5)

13. (7d 2 1)(d 2 2) 14. 2(r 2 5)(r 2 1)

15. (6s 2 1)(s 2 2) 16. 25, 3 }

2 17. 24, 2

1 } 3

18. 21, 1 }

2 19. 23,

1 }

2 20. 24,

1 }

3 21. 22, 2

5 } 3

22. 2 3 } 2 , 2

1 } 2 23. 2

2 } 3 ,

1 }

3 24. 2

2 } 5 , 2 25. 27,

2 }

3

26. 2 1 } 2 , 2

1 } 4 27. 2

3 } 2 ,

1 }

3 28. 25, 1 29. 2

2 } 3 , 5

30. 2 1 } 2 , 5 31. 2, 3 32.

1 }

4 , 2 33. 26,

3 }

2

34. 24, 1 }

2 35. 2

3 } 2 , 1 36. 22, 2

1 } 4 37. 1 sec

38. a. 4x2 2 39x 1 90 b. 18 in. by 72 in.

Practice Level B

1. 2(x 2 4)(x 1 7) 2. 2(p 2 2)(p 2 6)

3. 2(m 1 8)(m 1 5) 4. (2y 1 1)(y 1 7)

5. (3a 2 1)(a 2 4) 6. (5d 1 2)(d 2 4)

7. (3c 1 2)(2c 1 1) 8. 2(5n 2 3)(n 2 2)

9. (2w 1 3)(6w 2 5) 10. 2(b 1 4)(2b 2 3)

11. 2(r 1 5)(3r 1 2) 12. 22(s 2 2)(2s 1 1)

13. 24, 5 14. 28, 22 15. 6, 7 16. 1 }

2 , 5

17. 2 5 } 2 , 2 18. 2

5 } 8 , 2

1 } 2 19. 26, 2

1 } 3

20. 1 }

4 ,

2 }

3 21. 2

2 } 3 ,

4 } 5 22. 24, 2

1 } 2 23. 22,

5 }

3

24. 2 1 } 2 ,

5 }

4 25. 23, 9 26. 28,

1 }

2 27. 26,

4 }

3

28. 21, 2 29. 5 }

3 , 4 30. 27,

3 }

8 31. 26, 2

5 } 4

32. 210, 3 }

2 33. 21,

1 }

2 34. $90 35. 3 sec

36. a. 4x2 1 24x 1 32 b. 8 in. by 16 in.

Practice Level C

1. 2(x 2 9)(x 1 20) 2. 2(2m 2 3)(m 2 8)

3. 2(3p 1 4)(p 2 10) 4. (2r 1 5)(4r 1 3)

5. 2(b 1 3)(7b 2 2) 6. 2(y 2 3)(5y 2 3)


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A25Algebra 1


7. 2 5 } 4 ,

3 }

8 8. 2

3 } 2 , 2

1 } 2 9. 2

2 } 5 ,

2 }

3 10. 2

5 } 2 , 2

1 } 3

11. 2 2 } 9 ,

5 } 7 12.

3 }

10 ,

3 }

4 13. 2

3 } 8 ,

1 }

2 14. 2

4 } 5 ,

2 }

3

15. 1 }

3 , 5 16.

5 }

8 ,

5 }

2 17.

7 }

10 ,

3 }

2 18. 22,

11 }

10

19. 2 1 } 2 , 2

1 } 3 20. 2

3 } 2 ,

5 }

6 21. 21,

1 }

3 22. 2

7 } 2 , 5

23. 21, 2 } 5 24. 2

3 } 2 ,

1 }

4 25.

2 } 5 , 2 26.

3 }

4 ,

4 }

3 27.

5 }

6

28. 3.5 sec 29. 2 sec 30. a. h 5 216t2 1 8t b. 0.25 sec c. It takes the frog 0.25 second to reach a height of 12 inches and it reaches the ground at 0.5 second, so it can’t go any higher because it will take another 0.25 second to reach the ground. d. h 5 216t2 1 8t 1 4 e. No, because the frog is higher when it jumps, it will take the frog longer to reach the ground.

Review for Mastery

1. (7a 2 1)(a 2 7) 2. (2b 2 5)(2b 1 1)

3. (6c 2 7)(c 1 2) 4. 2(3r 1 4)(r 1 1)

5. 2(3s 1 4)(s 2 4) 6. 2(4t 2 1)(2t 2 1)


1. 2.75 seconds 2. The linear term should be positive in the vertical motion equation. The diver enters the water after 2.75 seconds. 3. 1.5 seconds 4. 1 second 5. 2 seconds

Challenge Practice

1. (2y1/3 1 1)(2y1/3 1 5) 2. (4y2 1 1)(2y2 2 3)

3. 1 3 } y 1 1 2 1 3 }

y 2 5 2 4. 1 7 3 Ï

}

y 1 1 2 1 5 3 Ï}

y 1 1 2

5. 1 2 Ï}

y 2 3 2 1 2 Ï}

y 1 1 2 6. x 5 3 Î}

2 2 } 3 , 3 Î}

1 }

2

7. x 5 2 Î}

5 }

3 , Î}

5 }

3 8. x 52

1 } 5 , 2

5 }

3 9. x 5

49 }

9

10. x 5 2 2 Ï

}

5 } 5 ,

2 Ï}

5 } 5


1. B 2. A 3. C 4. (x 2 1)(x 1 1)

5. (b 2 9)(b 1 9) 6. (m 2 10)(m 1 10)

7. (p 2 15)(p 1 15) 8. (2y 2 1)(2y 1 1)

9. (4n 2 5)(4n 1 5) 10. (3w 2 10)(3w 1 10)

11. (8z 2 6)(8z 1 6) 12. (7d 2 5)(7d 1 5)

13. (2r 2 11)(2r 1 11) 14. (3s 2 12)(3s 1 12)

15. (c 2 25)(c 1 25) 16. (x 1 3)2 17. (b 1 5)2

18. (w 2 6)2 19. (m 2 4)2 20. (r 2 10)2

21. (z 1 8)2 22. (s 1 11)2 23. (x 2 8)2

24. (2c 1 1)2 25. (4d 1 1)2 26. (3y 2 1)2

27. (3p 2 2)2 28. 23, 3 29. 27 30. 5

31. 2 1 } 5 ,

1 } 5 32. 1 33. 210 34. 2

3 } 2 ,

3 }

2

35. 2 4 } 3 ,

4 }

3 36. 22 37. a. π(x 2 y)(x 1 y)

b. 55π cm2 38. a. 8.36 ft b. 3 }

4 sec

39. a. 6 ft b. about 0.79 sec

Practice Level B

1. (x 2 6)(x 1 6) 2. (5p 2 12)(5p 1 12)

3. 4(b 2 5)(b 1 5) 4. 9(2m 2 3)(2m 1 3)

5. 22(x 2 4)(x 1 4) 6. 24(r 2 5s)(r 1 5s)

7. (y 1 12)2 8. (3c 1 4)2 9. (5w 2 2)2

10. (4n 2 7)2 11. 22(3a 1 1)2 12. 5(2z 2 7)2

13. 27 14. 2 5 } 2 ,

5 }

2 15.

1 }

8 16. 23, 3 17. 25

18. 4 19. 25, 5 20. 10 21. 1 }

2 22. 2

5 } 3

23. 2 3 } 5 24. 2

3 } 8 ,

3 }

8 25. 8 26. 3 27. 1 sec

28. a. 0; 3.75; 5; 3.75; 0 b. Any other values between 0 and 20 because the ladder is on the ground at x 5 0 and meets the ground again at x 5 20. c.

00

1

2

3

4

5

5 10 15 20Distance from left end (feet)

Heig

ht

(feet)

x

y

d. 10 ft

Practice Level C

1. (5x 2 9)(5x 1 9) 2. 25(3p 2 2)(3p 1 2)

3. (11w 2 25)(11w 1 25) 4. 4(3m 2 4)(3m 1 4)

5. 1 }

16 (3r 2 1)(3r 1 1) 6. (9x 2 7y)(9x 1 7y)

7. 23(y 1 8)2 8. 4(n 2 5)2 9. 3(2z 1 1)2

10. 6(2a 2 5b)2 11. 22(3s 1 4t)2

12. 1 } 5 (5z 1 1)2 or 5 1 z 1

1 } 5 2

2 13. 2

8 } 5 ,

8 } 5 14. 29

15. 2 7 } 2 ,

7 }

2

16. 10 17. 11

} 6 18.

4 }

15 19. 2

3 } 5 20. 4 }

7

21. 22, 2 22. 8 }

3 23. 2

1 } 5 24.

3 }

4 25. 90


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26. 140 27. 36 28. 16 29. 9 30. 28

31. Only once, because the squirrel reaches the ground in 1 second and it takes the squirrel 0.5 second (half this time) to reach the height of 4 feet. 32. a. 0; 4.5; 6; 4.5; 0 b. Any other values between 0 and 80 because the bridge is on the ground at x 5 0 and meets the ground again at x 5 80. c.

00

1

2

3

4

5

6

20 40 60 8010 30 50 70Distance from left end (feet)

Heig

ht

(feet)

x

y

d. 40 ft

Review for Mastery

1. (m 1 11)(m 2 11) 2. (3n 2 8)(3n 1 8)

3. 3( y 1 7z)( y 2 7z) 4. 1 m 2 1 } 4 2 2

5. (4r 1 5s)2 6. 9(2x 2 1)2 7. 5 8. 2 9 } 4 ,

9 }

4

Challenge Practice

1. (x 2 3y)2 2. (2x 2 5y)2 3. (5xy 1 4)2

4. 4(x 2 10)2 5. (5x 1 13)2

6. 5 2 Ï}

5 , 5 1 Ï}

5 7. 1 } 2 3 Î}

7 }

2 1

7 }

2 8. 2

13 } 5

9. 2 4 } 7 10. 2

5 } 2


1. C 2. A 3. B 4. (x 1 1)(x 1 4)

5. (b 2 1)(b 1 3) 6. (2m 1 1)(m 1 1)

7. (5r 2 1)(r 1 2) 8. (w 1 3)(w 1 6)

9. (y 2 6)(y 1 4) 10. (n 2 7)(n 2 3)

11. (3z 1 8)(z 2 4) 12. (2p 2 3)(p 1 5)

13. (x 1 3)(x 1 1) 14. (x 1 2)(x 2 1)

15. (x 2 1)(x 1 8) 16. (x2 1 2)(x 2 5)

17. (x2 2 6)(x 2 4) 18. (x2 1 5)(x 1 3)

19. (x2 1 7)(x 2 1) 20. (x2 2 3)(x 1 3)

21. (x2 2 1)(x 1 3) 22. not completely factored

23. completely factored 24. not completely factored 25. x3(x 2 1)(x 1 1)

26. a2(2a 2 5)(2a 1 5) 27. 5y4(y 2 5)(y 1 5)

28. 25, 21, 5 29. 24, 21, 4 30. 22, 1, 2

31. 23, 1, 3 32. 22, 0, 2 33. 28, 0, 8

34. 4(x 1 1)(x 1 2) 35. a. 8πr2 2 72π 5 0

b. 3 in. 36. 2 sec

Practice Level B

1. (4x 2 3)(x 1 5) 2. 22(a 2 6)(a 2 3)

3. (w2 2 5)(w 1 8) 4. (2b2 1 3)(b 1 6)

5. (y 2 1)(x 1 15) 6. 3(x 2 2)(y 1 4)

7. (x2 1 5)(x 1 1) 8. (y2 1 1)(y 2 14)

9. (m2 1 2)(m 2 6) 10. (p2 1 4)(p 1 9)

11. (t2 2 2)(t 1 12) 12. (3n2 1 1)(n 2 1)

13. 7x2(x 1 4) 14. 4m(m 2 2)(m 1 2)

15. 22p(8p2 1 1) 16. 6r2(8r 2 5)

17. 15y(1 2 4y) 18. 6x(3y 2 4x)

19. 5(m2 1 4m 1 8) 20. 6(x 1 5)(x 2 4)

21. 4z(z 2 2)(z 1 1) 22. 9(x3 1 4x2 1 4) 23. (x2 1 5)(x 1 1) 24. (d2 1 5)(d 1 4)

25. 24, 22 26. 25, 5 27. 7 }

2 28. 2

1 } 2 , 21

29. 4 }

3 30. 2

5 } 3 31. 2(2x 1 3)(x 1 1)

32. a. 8πr2 2 32π 5 0 b. 2 cm

33. a. h 5 216t2 1 12t 1 4 b. 5.04 ft c. 6 ft

d. 1 sec

Practice Level C

1. 13a(1 2 2a) 2. 15x(2y 2 3x)

3. 22(m 1 1)(m 1 7) 4. 7(2p 2 3)(p 2 1)

5. r(r 1 5)2 6. 5b2(b 1 4)2 7. 4n3(n 1 6)(n 2 5)

8. 7c(c 2 2)2 9. 25(2t 2 5)(t 1 3)

10. (x 2 y)(x 1 9) 11. (x2 2 8)(x 1 5)

12. (3x 2 8y)(3x 1 8y) 13. 3x3y(x 2 9)(x 1 9)

14. 8rs4(r 2 3)(r 1 3) 15. 25x2y(x 2 4)

16. 23, 21, 0 17. 22, 2 18. 2 5 } 3 , 0,

1 }

2

19. 2 15

} 4 , 15

} 4 20. 2

1 } 7 , 0, 5 21. 26, 0

22. 22, 0, 3 } 5 23.

5 }

6 ,

9 }

2 24. 2

5 } 6 ,

5 }

6

25. 2 9 } 4 , 0,

9 }

4 26. 25, 7 27. 14

28. 2 3 } 5 , 0 29. 23, 9 30. 2

10 } 9 , 0,

10 }

9

31. a2 2 2ab 1 b2 5 a2 2 ab 2 ab 1 b2 5 a(a 2 b) 2 b(a 2 b) 5 (a 2 b)(a 2 b) 5 (a 2 b)2

32. 3(2x 2 1)(3x 1 2) 33. a. 2πr2 2 1 }

2 π 5 0

b. 1 }

2 ft 34. about 11 sec


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A27Algebra 1


Review for Mastery

1. (11x 1 3)(x 2 8) 2. (9x2 2 7)(x 1 1)

3. (5x2 2 3y)(2x 2 7) 4. 0, 27, 3 5. 0, 26

6. 0, 1, 3


1. a.

h 1 12 in.

h in.

h 2 3 in.

b. h3 1 9h2 2 36h c. length: 18 inches; width: 3 inches; height: 6 inches

2. Answers will vary. 3. a. 6x2 2 8x 2 30 b. length: 12 inches; width: 4 inches; height: 9 inches 4. 14 feet; The zeros of the function, 0 and 14, are where the underpass touches the ground. The difference between the zeros is the width of the underpass at its base.

5. a. h 5 216t2 1 60t 1 4 b. 1.25 seconds and 2.5 seconds c. Yes; The ball reaches a height of 54 feet on the way up and on the way down.

6. 0.75 second 7. a. 4x3 2 44x2 1 117x b. 77 cubic inches; 90 cubic inches; 63 cubic inches; 20 cubic inches; 2 inches c. No; You cannot cut two squares with a side length of 5 inches from a side of a piece of cardboard that is 9 inches.

Challenge Practice

1. (y 1 3)(2y 1 9)(4y 1 17)

2. (y 2 3)(y 1 1)(y2 2 2y 1 5) 3. (3x 2 5)(3x 1 1) 4. (7x 1 5)(3x 1 2)

5. 2y(y 2 2)(y 1 2)(y2 1 4) 6. x 5 28, 21

7. x 5 23, 0, 3 8. x 5 2 7 }

4 , 2

3 } 2 9. x 5

5 }

2 , 6

10. x 5 1 }

3 11. 150 mi/h

12. t 5 1 min and t 5 3 min

Chapter Review GameAcross 2. Perfect 4. Roots 7. Polynomial

9. Leading 11. Monomial 13. Degree

14. Projectile

Down 1. Grouping 3. Trinomial

5. FOIL pattern 6. Factoring 8. Binomial

10. Prime 12. Vertical

1

4

7

8

6

11

14

13

12

109

5

32

P E R F E C T

R O O

U

P

IM A L

N

G

R

G

T S

I

N

ONYLO

I

L E A D N

P

AIO

M

I

A

L

N

I

B

NO

R

I

N

G

T

C

A

F

M M L

T

T

E

R O J E C TP

N

F

P

M

I

A

L V

E

RGED E E

M

I

R

P

T

I L E

C

A

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Answers

Lesson 10.1

Practice Level A

1. x 22 21 0 1 2

y 20 5 0 5 20

2. x 22 21 0 1 2

y 216 24 0 24 216

3. x 22 21 0 1 2

y 10 7 6 7 10

4. x 22 21 0 1 2

y 24 27 28 27 24

5. C 6. B 7. A

8.

x

y

3

5

12121

23 3

domain: all reals; range: y ≥ 0; vertical stretch by a factor of 5

9.

x

y

1

21

23

23 3

domain: all reals; range: y ≤ 0; vertical shrink by

a factor of 1 }

3 and

refl ection in x-axis

10. x

y

12123 3 domain: all reals;

range: y ≤ 0; vertical stretch by a factor of 6 and refl ection in x-axis

11. (0, 8); x 5 0 12. (0, 24); x 5 0

13. (0, 21.5); x 5 0 14. A 15. C 16. B

17.

x

y1

12121

23

23 3

domain: all reals; range: y ≥ 25; vertical shift 5 units down

18.

x

y

2

6

10

12122

23 3

domain: all reals; range: y ≥ 7; vertical shift 7 units up

19.

x

y

1

3

23 3

domain: all reals; range: y ≥ 23; vertical stretch by a factor of 2 and vertical shift 3 units down

20. 5 units up 21. vertically stretching; 10

22. a. 210 ≤ x ≤ 10 b. 0 ≤ y ≤ 8

23. a. 26 ≤ x ≤ 6 b. 0 ≤ y ≤ 2

Practice Level B

1. x 22 21 0 1 2

y 36 9 0 9 36

2. x 22 21 0 1 2

y 220 25 0 25 220

3. x 24 22 0 2 4

y 41 11 1 11 41

4. x 216 28 0 8 16

y 234 210 22 210 234

5. x 22 21 0 1 2

y 213 21 3 21 213

6. x 22 21 0 1 2

y 19 1 25 1 19

7. F 8. A 9. D 10. B 11. C 12. E

13. shift the graph 8 units down 14. shift the graph 4 units up and refl ect over x-axis

15. stretch vertically by a factor of 2 and shift 3 units up 16. stretch vertically by a factor of 5, refl ect in x-axis, and shift 1 unit up

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A29Algebra 1



17. shrink vertically by a factor of 1 }

2 and shift

2 units down 18. shrink vertically by a factor

of 3 }

4 , refl ect over x-axis, and shift 5 units up

19.

x

y

3

9329 2323

domain: all reals; range: y ≥ 9; vertical shift 9 units up

20.

x

y

1

3

32321

23

domain: all reals; range: y ≤ 0; vertical shrink by a factor

of 1 } 5 and refl ection in x-axis

21.

x

y

1

3

1 32123

23

domain: all reals; range: y ≤ 0; vertical stretch by a factor

of 3 }

2 and refl ection in x-axis

22.

x

y1

1 3212321

25

domain: all reals; range: y ≥ 23.5; vertical shift 3.5 units down

23.

x

y

3

9

9329 23

29

domain: all reals; range: y ≥ 29; vertical stretch by a factor of 2 and shift 9 units down

24.

x

y3

1 32123

domain: all reals; range: y ≤ 2; vertical stretch by a factor of 5, refl ection in x-axis, and vertical shift 2 units up

25. a. 218 ≤ x ≤ 18 b. 0 ≤ y ≤ 20

26. a.

00

20

40

60

80

100

1 2Time (seconds)

Heig

ht

(feet)

y

t

b. 0 ≤ t ≤ 2.5; 0 ≤ y ≤ 100

c. 84 ft

d. about 1.8 sec

e. 2.5 sec

Practice Level C

1. x 22 21 0 1 2

y 36 6 24 6 36

2. x 22 21 0 1 2

y 23 1.5 3 1.5 23

3.

x

y

1

3

12121

23

23 3

domain: all reals; range: y ≥ 2; vertical shrink by a

factor of 1 }

6 and vertical

shift 2 units up

4.

x

y2

12122

23 3

domain: all reals; range: y ≤ 23; vertical stretch by a factor of 4, refl ection in x-axis, and shift 3 units down

5.

x

y

2

6

12123 3

domain: all reals;

range: y ≥ 2 7 } 2 ;

vertical stretch by a factor

of 9 and vertical shift 7 } 2

units down

6.

x

y

1

3

5

12121

23 3

domain: all reals;

range: y ≥ 1 } 5 ;

vertical shrink by a factor

of 3 } 5 and vertical shift

1 } 5

unit up

7.

x

y

1

3

12121

domain: all reals; range: y ≤ 4; vertical shrink by a factor

of 1 }

2 , refl ection in x-axis,

and vertical shift 4 units up

8.

x

y

6

18

30

12126

23 3

domain: all reals;

range: y ≥ 3 } 4 ;


of 6 and vertical shift 3 }

4

unit up

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9.

x

y

3

9

15

12123

23 3

domain: all reals;

range: y ≥ 2 2 } 3 ;


of 4 and vertical shift 2 }

3

unit down

10.

x

y2

121

26

210

23 3

domain: all reals;

range: y ≤ 2 1 } 2 ;

vertical stretch by a factor of 2, refl ection in x-axis, and

vertical shift 1 }

2 unit down

11.

x

y

5

12125

215

23 3

domain: all reals; range: y ≤ 15; vertical stretch by a factor of 5, refl ection in x-axis, and vertical shift 15 units up

12. shift the graph of f 8 units down 13. shift the graph of f 5 units down 14. shift the graph of f 4 units down 15. shift the graph of f 16 units up

16. stretch the graph of f vertically by a factor of 3

17. shrink the graph of f vertically by a factor of 1 }

2

18.

x

y

2

10

12123 3

y 5 x2 1 6

19.

x

y

21

23

23 3

y 5 2x2 1 1

20.

x

y

2

6

12122

26

23 3

y 5 x2 2 4

21. a.

00

20,000

40,000

60,000

80,000

100,000

0.5 1.0 1.5 2.0Diameter (inches)

Weig

ht

(po

un

ds) w

d

b. about 1.5 in.

22. a.

00

4

8

12

16

20


Dis

tan

ce (

feet)

y

t

b.

00

4

8

12

16

20


Heig

ht

(feet)

y

t

c. The second graph is a transformation of the fi rst graph. The fi rst graph has been refl ected in the x-axis and shifted 20 units up to obtain the second graph. For the fi rst graph, fi nd the value of t when y 5 8. For the second graph, fi nd the value of t when y 5 12.

Review for Mastery

1.

x

3

1

12123 3

y

Both graphs have the same vertex, (0, 0), and the same axis of symmetry, x 5 0. However, the graph of y 5 28x2 is narrower than the graph of y 5 x2 and it opens down. This is because the graph of y 5 28x2 is a vertical stretch (by a factor of 8) of the graph of y 5 x2 and a refl ection in the x-axis of the graph of y 5 x2.

2.

x

5

3

1

12123 3

y Both graphs have the same vertex, (0, 0), and the same axis ofsymmetry, x 5 0. Both graphs open upward.However, the graph of

y 5 1 } 7 x2 is wider than the graph of y 5 x2. This is

because the graph of y 5 1 } 7 x2 is a vertical shrink

1 by a factor of 1 } 7 2 of the graph of y 5 x2.

3.

x

3

1

21

23

23 3

y

Both graphs have the same vertex, (0, 0), and the same axis of symmetry, x 5 0. However, the graph

of y 5 2 1 } 3 x2 is wider than the graph of y 5x2

and it opens down. This is because the graph of

y 5 2 1 } 3 x2 is a vertical shrink 1 by a factor of

1 } 3 2 of

the graph of y 5 x2 and a refl ection in the x-axis of the graph of y 5 x2.


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A31Algebra 1



4.

x

3

1

12123 3

y

21

Both graphs have the same axis of symmetry, x 5 0, and both open up. However, the graph ofy 5 x2 2 3 has a lower vertex than the graph of y 5 x2. This is because the graph of y 5 x2 2 3 is a vertical translation (3 units down) of the graph of y 5 x2.

5.

x

5

3

1

12123 3

y

Both graphs open up, and have the same axisof symmetry, x 5 0. However, the graph of

y 5 1 }

4 x2 1 2 is wider than the graph of y 5 x2,

and has a higher vertex. This is because the

graph of y 5 1 }

4 x2 1 2 is a vertical shrink

1 by a factor of 1 } 4 2 and a vertical translation (2 units

up) of the graph of y 5 x2.

6.

x

3

1

12123 3

y

Both graphs have the same axis of symmetry,

x 5 0. However, the graph of y 5 2 1 }

2 x2 2 1 is

wider than the graph of y 5 x2, opens down and has a lower vertex. This is because the graph of

y 5 2 1 }

2 x2 2 1 is a vertical shrink 1 by a factor of

1 } 2 2 ,

a refl ection in the x-axis, and a vertical translation of the graph of y 5 x2.

Challenge Practice

1. y 5 3x2 1 4 2. y 5 22x2 1 1

3. y 5 4x2 2 10

4. y 5 2x2 1 5 5. y 5 2 1 } 2 x2 1 2

6. 1 kilogram 7. about 5.2 3 1027 kilograms

8. 1 3 1010 meters per second

9. about 1 3 1029 kilograms 10. 3.125 kilograms


1. a 5 7, b 5 2, c 5 11 2. a 5 3, b 5 25, c 5 1

3. a 5 4, b 5 2, c 5 22 4. a 5 23, b 5 9,

c 5 4 5. a 5 1 }

2 , b 5 21, c 5 25 6. a 5 21,

b 5 7, c 5 26 7. upward; x 5 0 8. downward; x 5 0 9. upward; x 5 23 10. upward; x 5 2

11. upward; x 5 21 12. downward; x 5 4

13. upward; x 5 2 3 } 2 14. downward; x 5

7 }

2

15. upward; x 5 21 16. (0, 5) 17. (0, 3)

18. (25, 222) 19. (2, 2) 20. (21, 22)

21. (2, 5) 22. 1 1 } 2 ,

9 }

2 2 23. 1 2 1 }

2 ,

11 }

4 2 24. 1 1 }

2 ,

3 }

4 2

25. x 1 2 3 4 5

y 3 0 21 0 3

26. x 4 5 6 7 8

y 27 30 31 30 27

27. x 23 22 21 0 1

y 23 2 25 2 23

28. x 23 22 21 0 1

y 25 1 3 1 25

29. C 30. B 31. A

32.

x

y2

12122

210

23 3

(0, 26)

x 5 0

33.

x

y

2

6

10

12122

23 3

(0, 7)

x 5 0

34.

x

y

1

3

5

7

12325 3

(21, 4)

x 5 21

35. x

y

2 6 10

(4, 215)

x 5 4

2222

26

210

214

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36. x

y

12121

23

27

23 3

x 5 14

( )14

238, 2

37.

x

y

1

3

7

2123

(22, 7)

x 5 22

38. minimum value; 27 39. maximum value; 9 40. minimum value; 22

41. 10 ft 42. 3.5 ft

Practice Level B

1. a 5 6, b 5 3, c 5 5 2. a 5 3 }

2 , b 5 21, c 5 8

3. a 5 7, b 5 23, c 5 21 4. a 5 22, b 5 9,

c 5 0 5. a 5 3 }

4 , b 5 0, c 5 210 6. a 5 28,

b 5 3, c 5 27 7. upward; x 5 0; (0, 25)

8. downward; x 5 0; (0, 9) 9. downward; x 5 3 }

2 ;

1 3 } 2 ,

23 }

2 2 10. upward; x 5 2; (2, 211)

11. upward; x 5 21; (21, 25) 12. downward;

x 5 7 }

4 ; 1 7 }

4 , 2

119 } 8 2 13. upward; x 5 25; 1 25, 2

33 } 2 2

14. downward; x 5 0; (0, 224) 15. downward;

x 5 3 } 2 ; 1 3 }

2 , 2

5 } 4 2 16. upward; x 5

1 }

3 ; 1 1 }

3 ,

8 }

3 2

17. downward; x 5 7 }

4 ; 1 7 }

4 ,

57 }

8 2

18. upward; x 5 2 1 } 3 ; 1 2

1 } 3 , 2

16 } 3 2

19. vertex: (5, 222)

x 3 4 5 6 7

y 218 221 222 221 218

20. vertex: (3, 7) x 1 2 3 4 5

y 3 6 7 6 3

21. vertex: 1 1, 13

} 2 2 x 21 0 1 2 3

y 17 } 2 7

13 }

2 7

17 }

2

22. vertex: (3, 0) x 1 2 3 4 5

y 4 }

3

1 }

3 0

1 }

3

4 }

3

23.

x

y5

5 152521525

(0, 210)

x 5 024.

x

y

1

5

1 3212321

(0, 3)x 5 0

25.

x

y

1

3

1 32123

x 5

( )12

12

32,

26.

x

y

1

3

1 32123

( )152

152,

x 5 215

27.x

y

1 3212321

23

x 5 14

( )14

3182,

28.

x

y2

2 6 102222

26

210

x 5 4

(4, 211)

29.

x

y

10230

x 5 28

(28, 35)

10

30.

x

yx 5 26

(26, 210)

2

22210

210

31.

x

y

1

2521

23

( )43

103,2

x 5 243

32. minimum; 240 33. maximum; 3

34. minimum; 7}2

35. 12 ft 36. 24 in.

Practice Level C

1. downward; x 51}2 ; 11

}2 ,

23}4 2 2. upward; x 5

2}5 ;

12}5 ,

3}5 2 3. upward; x 5

1}8 ; 11

}8 ,

23}8 2 4. downward;

x 51}2 ; 11

}2 ,

9}4 2 5. upward; x 5 0; (0, 29)

6. downward; x 51}5 ; 11

}5 , 2

14}5 2 7. upward;

x 5 8; (8, 28) 8. downward; x 5 0; (0, 7)

9. downward; x 5 1; (1, 11) 10. upward; x 53}2 ;

13}2 , 21 2 11. upward; x 5 21; (21, 28)

12. downward; x 52}3 ; 12

}3 , 2

22}3 2


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A33Algebra 1


13. vertex: (4, 1) x 2 3 4 5 6

y 2 5 }

4 1

5 }

4 2

14. vertex: (2, 9) x 0 1 2 3 4

y 21 13

} 2 9

13 }

2 21

15.

x

y5

52525

215

215 15

(0, 215)

x 5 0 16.

x

2

6

22226 6

(0, 8)

x 5 0

17.

x

y

3

12123 3

x 5

( )12

12

, 4 18.

x

y

5

215 15

(0, 20)

x 5 0

19.

x

y

1

3

5

2123 321

(1, 21)

x 5 1

20.

x

y

4

12

20

42424

212 12

(3, 23)x 5 3

21.

x

y

6

18

30

2626

218 18

x 5 3

( )5323,

22.

x

y

2

6

10

22226 6 10

x 5 4

(4, 6)

23.

x

y

2

6

10

14

22226 6

x 5 1

(1, 7)

24.

x

y 6

2 22 26

218

230

26 6 10

( )92

1274, 2

x 5 92

25.

x

y30

525215 15

(210, 2179)

x 5 210

26.

x

y

502150 150

50

(225, 255)

x 5 225

27. minimum; 236 28. maximum; 101

29. minimum; 217 30. a. lamp A: 25 mm; lamp B: 20 mm b. 5 mm 31. 6 ft; Find the maximum of the top part of the window and subtract 1.5 from the result.

Review for Mastery

1. x 5 22: (22, 211) 2. x 5 6: (6, 231)

3. minimum value; 219

4.

6

2

x

y

22 6

(2, 3)

x 5 2


1. about 215 feet 2. 28 feet 3. 8 feet 4. about 2.54 feet

Challenge Practice

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

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

5. y 5 23x2 1 6x 1 9 6. f (x) 5 x2 2 3x 1 2

7. f (x) 5 22x2 1 3x 1 5

8. f (x) 5 2x2 1 5x 2 7

9. f (x) 5 26x2 1 5x 2 1

10. f (x) 5 3x2 2 19x 1 6


1. domain: all real numbers; range: y ≥ 29


3. domain: all real numbers; range: y ≤ 4


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8. domain: all real numbers;

range: y ≥ 2 25

} 2


10. a. 23 and 1; b. 21; c. y 5 2(x + 3)(x 2 1)

11. y 5 2 1 } 25 (x 2 0)(x 2 50)

Review for Mastery



3. x

y

O 1 (2, 0)

(0, 12)

2( 2, 0)

x 0

domain: all real numbers; range: y ≥ 212



1. x2 1 3x 1 12 5 0 2. x2 2 8x 2 14 5 0

3. x2 2 9x 1 1 5 0 4. x2 1 10x 2 6 5 0

5. x2 1 3x 2 14 5 0 6. 1 }

2 x2 1 3x 1 7 5 0

7. not a solution 8. solution 9. solution

10. solution 11. not a solution 12. not a solution 13. no solution 14. 22, 2 15. 23, 21

16. 24, 4 17. no solution 18. 24, 2

19.

x

y

9

15

12123 3

20.

x

y

1

3

5

12123 3

no solution 21, 2 1 } 2

21.

x

y

6

222

22.

x

y3

3

215

29 9

26, 22 23, 5


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A35Algebra 1


23.

x

y3

121

29

215

23 3

24.

x

y

1

3

2121

no solution 23, 1

25.

x

y5

215 15

26.

x

y

3

23

29

29 9

25, 5 23, 3

27.

x

y

1

3

121

23

23 3

28.

x

y4

2

212

220

22, 0 22, 6

29.

x

y

10

30

50

222210

26

30. x

y

222212

236

260

6

28, 5 0, 10

31. a.

002468

1012

4 8 12 162 6 10 14Width (inches)

Hei

gh

t (i

nch

es)

y

x

b. 0 ≤ x ≤ 16; 0 ≤ y ≤ 12

c. 16 in.

d. 12 in.

Practice Level B

1. not a solution 2. not a solution

3. not a solution 4. solution 5. not a solution

6. solution 7. 24 8. 26, 6 9. 28, 3

10. 26, 25 11. 25, 5 12. no solution

13.

x

y

22222

14.

x

y

1

3

323

23

26, 0 21, 1

15.

x

y

2

6

10

62222

16.

x

y5

5 152525

215

225

2, 5 0, 10

17.

x

y

2

6

10

2 6222622

18.

x

y3

9232923

29

215

3 3, 6

19.

x

y2

2 6222622

210

20.

x

y

30

6 1826218

no zeros 26

21. x

y

6 1826218212

22.

x

y10

2 622210

230

212, 0 27, 7

23.

x

y3

32321

23

24. x

y

2 6222622

21, 1 24, 0

25. a.

00

1020304050

0.5 1.0 1.5 2.0Time (seconds)

Hei

gh

t (f

eet)

h

t

h 5 216t2 1 10t 1 50 b. about 2.1 sec

26. a.

00

0.20.40.60.81.01.21.4

0.1 0.2 0.3 0.4Time (seconds)

Hei

gh

t (f

eet)

h

t

h 5 216t2 1 3t 1 1.3


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b. about 0.39 sec

c. about 0.34 sec

Practice Level C

1.

x

y

1

12121

23 3

2.

x

y

2

6

22222

26 6

22, 2 24, 1

3.

x

y5

2225

215

210

4.

x

y

4

20

2 6

27 22, 8

5.

x

y25

52525

215 15

6.

x

y

3

9

15

2123

2325

25 25, 23

7.

x

y

1

3

12121

23

23 3

8.

x

y

4

2

212

26

no solution 22, 6

9.

x

y

5

25

215 15

10.

x

y

4

1

212

3

25, 5 21, 5

11.

x

y

6

18

30

12126

23

12.

x

y

2

123 3

24, 2 21, 2

13.

x

y

1

3

12123

14.

x

y1

2123 3

20.2, 2.7 21.3, 1.3

15.

x

y

1

12123 3

20.7, 1.4

16. 4.5 in. 17. 1.9 ft 18. 9.9 cm

19. a.

00

0.5

1.0

1.5

2.0

2.5


Heig

ht

(feet)

h

t

h 5 216t2 1 5t 1 2.5; First, write 30 inches in feet and then use the vertical motion model.

b. 1 ft

c. about 0.6 sec

20. a.

00

10

20

30

40

1 2 3Time (seconds)

Heig

ht

(feet)

h

t

h 5 216t2 1 50t 1 6

b. about 3.2 sec

c. about 3.1 sec; Determine t when y 5 5.

Review for Mastery

1. 23, 5 2. 22 3. about 25.2, about 20.8

4. 22, 2 5. 27, 2

Challenge Practice

1.

2123 1 3

10

14

x

y

y 5 3x2 1 1y 5 2x2 1 5

2.

22 2 6 10

2

22

26

6

10

x

yy 5 x2 2 11

2

y 5 2 x2 1 812

(22, 13), (2, 13) 1 23, 7 }

2 2 , 1 3,

7 }

2 2


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A37Algebra 1


3.

21 3 51

1

3

5

7

x

y

y 5 x2 1 72

y 5 2x2 2 12

4.

x

y

5

7

9

12123 3 5

y 5 x2 1 x 1 3

y 5 2x2 1 4x 1 3

1 22, 15

} 2 2 , 1 2,

15 }

2 2 (23, 9), (0, 3)

5.

x

y

1

3

12121

23

3 5

y 5 2x2 1 3x 1 1

y 5 22x2 2 3x 1 1

1 2 3 } 2 , 1 2 , (0, 1)

6. The baseball hits the fence. 7. The baseball hits the ground before reaching the fence. 8. The baseball goes over the fence.


1. 7 2. 15 3. 10 4. x2 5 2 5. x2 5 3

6. x2 5 4 7. 26, 6 8. 23, 3 9. 22, 2

10. 23, 3 11. 23, 3 12. 0, 4 13. 2.24

14. 3.16 15. 3.46 16. 22.83, 2.83

17. 21.73, 1.73 18. 21.41, 1.41 19. 5 m

20. 11 in. 21. about 9.59 cm

22. about 5.96 knots 23. 6 in., 7 in., 10 in.

Practice Level B

1. 22, 2 2. 24, 4 3. 26, 6 4. 27, 7

5. 25, 5 6. 29, 9 7. 25, 5 8. 23, 3

9. 21, 1 10. 22.83, 2.83 11. 21.73, 1.73

12. no solution 13. 22.24, 2.24 14. 0

15. 22.45, 2.45 16. 24.12, 4.12 17. 22.5, 2.5

18. no solution 19. 0.76, 5.24 20. 25.16, 1.16

21. 1.55, 6.45 22. 13 m 23. about 6.16 in.

24. about 13.42 cm 25. about 3 in.

26. 5 ft, 8 ft, 10 ft

Practice Level C

1. 23, 3 2. 27, 7 3. 24, 4 4. 22, 2

5. 26, 6 6. 28, 8 7. 22.24, 2.24

8. 23.32, 3.32 9. no solution 10. 24.12, 4.12

11. 22.45, 2.45 12. 6.27, 9.73 13. 211.45, 26.55 14. 0.26, 7.74 15. 210.46, 23.54

16. 22.32, 10.32 17. 23.16, 3.16

18. 21, 1 19. 23, 3 20. 226, 34

21. 21.03, 1.03 22. 27, 1 23. 28, 12

24. 26, 16 25. 1, 13 26. 22, 4 27. 28, 0

28. about 12 cm 29. about 64 ft/sec

Review for Mastery

1. 23, 3 2. 22, 2 3. no solution 4. 2 11

} 6 , 11

} 6

5. 0 6. 2 15

} 2 , 15

} 2 7. 21.73, 1.73

8. 22.24, 2.24 9. 21.12, 1.12

10. 21.83, 3.83 11. 21, 27 12. 2.17, 7.83


1. a. 1994 b. $3,582,000 2. a. 4 b. 48 square inches 3. Yes; The vertex, which is a maximum, of the parabola occurs at around 1 year after 1998, or 1999. 4. Answers will vary.

5. 45 6. a. h 5 216t2 1 25t 1 6 b. h 5 216t2 1 30t 1 5.5 c. The second throw is in the air longer. Find the x-intercept of the graph of each equation. The second equation has a larger x-intercept. 7. 4

8. a. R 5 25n2 160n 1 800 b. 980 c. The T-shirts should be sold for $14 each. The maximum occurs at an x-coordinate of 6, which means that there should be six $1 increases on the price of a T-shirt. Since the price was $8, you need to add $6 to this.

Challenge Practice

1. x 5 212, x 5 6 2. x 5 2 9 } 2 , x 5 2

1 } 2

3. x 5 24 4. x 5 2 1 } 2 , x 5

1 }

6

5. x 5 2 13

} 7 , x 5 2 3 } 7 6. 20 min 7. 12 min

8. 32 min


1. B 2. C 3. A 4. (x 1 1)2 5. (x 2 7)2

6. (x 1 9)2 7. (x 2 2)2 8. (x 1 11)2


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9. (x 2 12)2 10. 25; (x 2 5)2 11. 16; (x 2 4)2

12. 9; (x 2 3)2 13. 121; (x 1 11)2

14. 36; (x 2 6)2 15. 100; (x 1 10)2

16. 225; (x 2 15)2 17. 169; (x 1 13)2

18. 400; (x 1 20)2 19. 9 }

4 ; 1 x 1

3 } 2 2 2

20. 121

} 4 ; 1 x 1

11 } 2 2

2 21.

49 }

4 ; 1 x 2

7 } 2 2

2

22. 26.32, 0.32 23. 210.10, 0.10

24. 20.65, 4.65 25. a. 64 5 216t2 1 64t 1 32

b. about 0.59 sec, about 3.41 sec

c. 32 5 216t2 1 64t 1 32; 0 sec; 4 sec

26. a. 4 ft b. 152 ft2; Subtract the interior area, 28 square feet, from the total area, 12(15) 5 180 square feet.

Practice Level B

1. 36; (x 1 6)2 2. 625; (x 1 25)2

3. 169; (x 2 13)2 4. 81; (x 2 9)2

5. 169

} 4 ; 1 x 1

13 } 2 2 2 6.

81 }

4 ; 1 x 2

9 } 2 2 2

7. 121

} 4 ; 1 x 2

11 } 2 2 2 8.

1 }

16 ; 1 x 1

1 } 4 2 2

9. 9 }

25 ; 1 x 2

3 } 5 2 2 10. 26.16, 0.16

11. 26.12, 2.12 12. 21.32, 11.32

13. 29.10, 1.10 14. 21.83, 3.83

15. 21.55, 13.55 16. 23.56, 0.56

17. 25.54, 0.54 18. 20.62, 1.62 19. 6

20. 5 21. about 272 mi by about 383 mi

22. about 4.05 sec 23. a. l 1 2w 5 60; lw 5 400 b. 20 ft by 20 ft, 40 ft by 10 ft

Practice Level C

1. 3.24; (x 1 1.8)2 2. 1 }

64 ; 1 x 2

1 } 8 2

2

3. 1 }

9 ; 1 x 1

1 } 3 2

2 4. 20.5, 3.5 5. 210.65, 20.35

6. 22.67, 3 7. 20.82, 9.82 8. 0.21, 4.79

9. 26.89, 20.11 10. 21.27, 6.27

11. 217.66, 20.34 12. 0.76, 13.24

13. 28.89, 20.11 14. 21.05, 6.05

15. 22.08, 1.08 16. about 4.71 ft 17. 6

18. 216, 215 19. about 39 mi/h

20. a. l 1 2w 5 100; lw 5 1000 b. about 27.6 ft by 36.2 ft, about 72.4 ft by 13.8 ft

Review for Mastery

1. 81

} 4 ; 1 x 2

9 }

2 2 2 2.

121 }

4 ; 1 x 1

11 }

2 2 2

3. 64; (x 2 8)2 4. 20.80, 8.80

5. 20.26, 211.74 6. 13.71, 0.29

Challenge Practice

1. 14 and 16 2. 11 and 13 3. 23 and 24

4. 14 and 15 5. 17

6. x 5 2b 2 Ï

}

b2 128 }}

2 , x 5

2b 1 Ï}

b2 128 }}

2

7. x 5 5 2 Ï

}

37 2 4c }}

2 , x 5

5 1 Ï}

37 2 4c }}

2

8. x 5 2b 2 Ï

}

b2 2 4c }}

2 , x 5

2b 1 Ï}

b2 2 4c }}

2

9. x 5 2b 2 Ï

}

b2 2 4ac }}

2a , x 5

2b 1 Ï}

b2 2 4ac }}

2a

10. 17 feet wide by 35 feet long 11. 2.08 sec


1. 2.

3. 4.

5. 6.

7. y 5 2(x 2 3)2 2 16


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A39Algebra 1


8. y 5 24(x + 1)2 1 20;

x

y

1

( 1, 20)

x 14

O

9. y 5 1 }

2 (x 2 2)2 2 3;

10. y 5 3 }

2 1 x 2

1 }

2 2 2 2 1

11. y 5 2 3 } 64 (x 2 16)2 1 12

Review for Mastery

1. 2.

3. y 5 22(x + 2)2 1 1;

4. y 5 2(x + 1)2 2 3;

x

y

1

1

( 1, 3)

x 1


1. a 5 5, b 5 7, c 5 1 2. a 5 2, b 5 26, c 5 11 3. a 5 21, b 5 17, c 5 223 4. a 5 10, b 5 28, c 5 213 5. a 5 23, b 5 1, c 5 22

6. a 5 5, b 5 218, c 5 23 7. B 8. C 9. A

10. 27.36, 1.36 11. 21.61, 5.61 12. 21, 3 } 5

13. 27.74, 20.26 14. 29.90, 20.10

15. no solution 16. 22, 1 }

3 17. no solution

18. 0.42, 3.58 19. no solution 20. 1 }

3 , 2

21. no solution

22. a. 300 5 1.55x2 2 5.1x 1 197; 2000 b. 237 5 1.55x2 2 5.1x 1 197; 1997

23. a. 73 5 20.31x2 1 3.8x 1 61.6; 2000, 2002

b.

Years since 1995

Mil

lio

ns o

f acre

s

0

58

62

66

74

70

2 41 3 5 6 7 8

X=5.3191489 Y=73.0418

Years since 1995

Mil

lio

ns o

f acre

s

0

58

62

66

74

70

2 41 3 5 6 7 8

X=7.0212766 Y=72.9984

Practice Level B

1. 213.10, 6.10 2. 22.15, 2.48 3. 21.82, 2.07

4. 23.73, 20.27 5. 23, 4 6. 22.61, 1.28

7. 24.61, 21.39 8. 213.44, 7.44

9. 22.11, 2.36 10. 21.45, 1.25 11. 2 3 } 2 ,

1 }

3

12. no solution 13. Sample answer: Use fi nding square roots because the equation can be written in the form x2 5 d. 14. Sample answer: Use fi nding square roots because the equation can be written in the form x2 5 d. 15. Sample answer: Use factoring because the equation is easily factored.

16. Sample answer: Use factoring because the equation is easily factored.

17. Sample answer: Use the quadratic formula because the equation cannot be factored easily.


19. 22.24, 2.24 20. 8 21. 24.70, 1.70

22. 7 23. 27.80, 1.80 24. no solution

25. a. 500 5 1.36x2 1 27.8x 1 304; 1995 b. 575 5 1.36x2 1 27.8x 1 304; 1997

26. a. 80 5 20.27x2 1 3.3x 1 77; 1998

b.

Years since 1997

Nu

mb

er

of

eg

gs

(billio

ns)

0

65

75

85

70

80

9095

2 41 3 5 6 7

X=1.0425532 Y=80.146958

Practice Level C

1. 2 1 } 3 , 2

1 } 5 2.

1 }

2 , 1 3. 21.10, 0.10

4. 21.90, 7.90 5. 20.27, 2.77 6. 23.30, 0.30

7. 20.30, 3.30 8. 4.35, 9.65 9. 20.87, 3.67


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10. no solution 11. 20.34, 0.80

12. 21.78, 0.28 13. Sample answer: Use factoring because the equation is easily factored.



16. 22.45, 2.45 17. no solution 18. 23.30, 0.30

19. no solution 20. 27.36, 1.36 21. no solution

22. 2 3 } 2 ,

3 }

2 23. 23 24. 217, 21

25. a. 1992

b. 1999

c.

Years since 1990

Bil

lio

ns o

f d

oll

ars

0

0

20

10

30

40

4 82 6 10 12

X=2.6808511 Y=20.149205

26. a. 1994

b.

Years since 1990

Billio

ns o

f d

ollars

0

2

68

4

101214

4 82 6 10 12

X=4.0851064 Y=7.0589632

Review for Mastery

1. 21.07, 13.07 2. 2.72, 20.52 3. 20.93, 0.60

4. 2005 5. factor or complete the square

6. quadratic formula 7. complete the square


1. 2001 2. The steps taken are to fi nd the zero of the function and not when 50 million cassettes were shipped. There were 50 million cassettes shipped in 2001. 3. 1.1 seconds 4. 2000

Challenge Practice

1. 3 }

2 x2 1 4x 1 1 5 0 2.

7 }

2 x2 1 6x 1

41 } 14 5 0

3. 3 }

2 x2 2 x 1

1 } 6 5 0 4.

15 }

2 x2 117x 1

134 } 15 5 0

5. 11

} 2 x2 2 11x 1 5 5 0 6. x 5 2

5 } 6

7. x 5 1 8. x 5 2 1 } 12


1. a 5 2, b 5 1, c 5 210 2. a 5 4, b 5 25, c 5 2 3. a 5 1, b 5 28, c 5 11 4. a 5 21, b 5 6, c 5 23 5. a 5 21, b 5 23, c 5 12

6. a 5 3, b 5 24, c 5 15 7. 215 8. 223

9. 44 10. 4 11. 279 12. 52 13. 84

14. 2271 15. 105 16. no solution

17. two solutions 18. two solutions



23. no solution 24. one solution 25. two

26. two 27. two 28. none 29. none

30. one 31. a. 155 5 2x2 1 5x 1 150 b. discriminant: 5 > 0 c. about 1.4 ft; about 3.6 ft

32. 15 5 216t2 1 20t 1 5.5; no

Practice Level B

1. no solution 2. two solutions 3. two solutions

4. no solution 5. two solutions 6. two solutions

7. two solutions 8. no solution 9. two solutions

10. no solution 11. two solutions

12. one solution 13. two 14. two 15. two

16. none 17. two 18. two 19. one 20. none

21. none 22. two 23. one 24. two

25. Answers will vary. 26. Answers will vary.



31. a. 150 5 2x2 1 x 1 156 b. discriminant: 25 > 0 c. 3 ft 32. no

Practice Level C

1. no solution 2. two solutions 3. no solution

4. two solutions 5. no solution

6. two solutions 7. one solution 8. no solution

9. two solutions 10. two 11. two 12. none

13. two 14. one 15. two 16. two 17. two

18. two 19. Answers will vary.



24. Answers will vary. 25. below; the graph opens upward and the discriminant is positive


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A41Algebra 1


26. on the x-axis; the graph opens upward and the discriminant is 0 27. above; the graph opens upward and the discriminant is negative

28. a. h 5 216t2 1 42t b. yes; at about 0.9 sec; at about 1.7 sec 29. a. yes b. yes; First write 5 square feet as 720 square inches, substitute 720 for y in the equation and solve.

Review for Mastery

1. two solutions 2. no solution 3. one solution

4. 2 5. 0 6. 1

Challenge Practice

1. k 5 22, k 5 2 2. k 5 1 }

4 3. k 5

40 }

3 , k 5 0

4. k 5 0, k 5 1 }

4 5. k 5 0, k 5

9 }

8 6. k >

1 }

3

7. k < 2 147

} 4 8. k >

3 }

4 9. k <

1 }

21

10. No solution 11. 10 < x < 20


1. C 2. A 3. B 4. quadratic

5. quadratic 6. linear 7. linear

8. exponential 9. exponential

10.

x

y

2

6

2 6222622

26

11.

x

y

2

6

10

1 3212322

linear exponential

12.

x

y

1

3

1212321

13.

x

y

5

15

1 3 52125

quadratic linear

14.

x

y

4

12

20

1 3 52124

15.

x

y

1

3

1 3 52121

23

exponential quadratic

16. linear 17. exponential

18. quadratic 19. quadratic 20. a. quadratic b. no; The salaries should not continue to fall; at some point they would rise. 21. linear

Practice Level B

1. B 2. C 3. A

4.

x

y

4

12

20

1 3212324

5.

x

y

1

3

1212321


6.

x

y

4

12

6222624

7.

x

y1

3232921

25

linear linear

8.

x

y

2

6

10

1 3212322

9.

x

y

1

3

5

1 3 5


10. exponential 11. linear 12. quadratic

13. linear 14. exponential 15. quadratic

16. linear 17. exponential

18. a. exponential; The graph rises quickly.

b. x 0 1 2 3 4

y 1 4 16 64 256

c. y 5 4x 19. Answers will vary.

20. linear; V 5 275t 1 800

Practice Level C

1. C 2. A 3. B

4.

x

y

2

6

121232522

5.

x

y

6

18

2 6222626

218

quadratic linear


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6.

x

y1

2 6 102221

23

25

7.

x

y

2

6

1 3212322

linear exponential

8.

x

y

2

6

10

2222621022

9.

x

y

64

192

320

1 3 521264

quadratic exponential

10. linear 11. quadratic 12. exponential

13. linear 14. quadratic 15. exponential

16. linear 17. quadratic

18. a. exponential; The graph falls quickly.

b. x 23 22 21 0 1

y 64 16 4 1 0.25

c. y 5 (0.25)x 19. linear; V 5 280t 1 2000

20. exponential; B 5 1020.20(1.02)t

Review for Mastery

1. quadratic function 2. linear function

3. quadratic function: y 5 x2 2 5x 1 6

4. exponential function: y 5 (0.25)(2)x


1. a. quadratic function b. y 5 20x2, where y is the power and x is the current

2. a. h 5 216t2 1 80t 1 6.5 b. 5.0 seconds

3. a. h 5 216t2 2 30t 1 80 b. 1.5 seconds

4. Answers will vary. 5. a. The discriminant is positive, so there are two x-values that correspond to y 5 29. b. The average monthly basic rate for cable television reached $29 in 1999. The other value can be disregarded since it is negative.

6. 8.9 7. a. A 5 4x2 1 136x b. 3 feet c. You can ignore the negative value because a negative width does not make sense.

Challenge Practice

1. linear model 2. 13 3. y 5 2x 1 3

4. exponential model 5. 34.171875

6. y 5 2(1.5)x 7. quadratic model

8. 361 9. y 5 3x2 2 2 10. about 935 pounds

Chapter Review Game 1. (21, 2) 2. (21, 6) 3. (3, 2) 4. (3, 6)

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

10. (6, 4)

21 1 3 5 7

1

3

5

7

x

y

144 feet


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Answers

Lesson 11.1

Practice Level A

1. C 2. A 3. B

4.

x

y

2

6

10

22222

6 10

domain: x ≥ 0; range: y ≥ 0; vertical stretch by a factor of 6

5.

x

y

222 6 10

0.2

0.6

1.0

20.2

domain: x ≥ 0; range: y ≥ 0; vertical shrink by a factor of 0.4

6.

x

y

1

12121

23

25

3 5

domain: x ≥ 0; range: y ≤ 0; vertical stretch by a factor of 2 and refl ection in x-axis

7. B 8. F 9. D 10. E 11. A 12. C

13.

x

y

2

6

22222

26

26 6

domain: x ≥ 0; range: y ≥ 25; vertical translation 5 units down

14.

x

y

1

3

12121

23

23 3

domain: x ≥ 0; range: y ≥ 3; vertical translation 3 units up

15.

x

y

2

6

22222

26

26 6

domain: x ≥ 0; range: y ≥ 26; vertical translation 6 units down

16.

x

y

2

6

22222

26

26 6

domain: x ≥ 2; range: y ≥ 0; horizontal translation 2 units right

17.

x

y

1

3

12121

23

23 3

domain: x ≥ 23; range: y ≥ 0; horizontal translation 3 units left

18.

x

y

2

6

22222

26

6 10

domain: x ≥ 5; range: y ≥ 0; horizontal translation 5 units right

19. a.

00

100200300400500600700

5 15 2510 20 30 35Nozzle pressure (lb/in.2)

Flow

rat

e (g

al/m

in)

f

p

domain: p ≥ 0; range: f ≥ 0

b. 36 lb/in.2

20. a.

005

101520

100 200Eye level (feet)

Dis

tan

ce(n

auti

cal m

iles) d

h

domain: h ≥ 0; range: d ≥ 0

b. about 292 nautical miles

Practice Level B

1.

x

y

2

6

10

14

22226 6

domain: x ≥ 0; range: y ≥ 0; vertical stretch by a factor of 7

2.

x

y

1

3

12121

23

23 3

domain: x ≥ 0; range: y ≥ 0; vertical shrink by a

factor of 1 } 5

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3.

x

y

2

22222

26

210

26 6

domain: x ≥ 0; range: y ≤ 0; vertical stretch by a factor of 4 and refl ection in x-axis

4. translate graph of y 5 Ï}

x horizontally 8 units right 5. translate graph of y 5 Ï

}

x vertically 3 units up 6. translate graph of y 5 Ï

}

x horizontally 7 units left 7. translate graph of y 5 Ï

}

x vertically 5 units down 8. translate graph of y 5 Ï

}

x vertically 3.5 units up 9. translate

graph of y 5 Ï}

x horizontally 1 }

2 unit right 10. E

11. C 12. A 13. F 14. B 15. D

16.

x

y

1

3

12121

23

23 3

domain: x ≥ 24; range: y ≥ 24; vertical translation 4 units down and horizontal translation 4 units left

17.

x

y

1

12121

23

2325 3

domain: x ≥ 25; range: y > 1; vertical translation 1 unit up and horizontal translation 5 units left

18.

x

y

2

6

22222

26

26 6 10

domain: x ≥ 6; range: y ≥ 4; vertical translation 4 units up and horizontal translation 6 units right

19.

x

y

2

6

22222

26

26 6 10

domain: x ≥ 5; range: y ≥ 27; vertical translation 7 units down and horizontal translation 5 units right

20.

x

y

1

3

12121

23

23 3

domain: x ≥ 1; range: y ≥ 2; vertical translation 2 units up and horizontal translation 1 unit right

21.

x

y

1

3

12121

23

2325 3


22. a.

00

1

2

3

4

5

6

7

200 400Volume

(cubic inches)

Sid

e l

en

gth

(in

ch

es)

x

V

domain: V ≥ 0; range: x ≥ 0

b. 225 in.3 c. 576 in.3

23. a.

00

3

6

9

12

15

20 40 60 80Weight (pounds)

Dia

mete

r (i

nch

es) d

w10 30 50 70 90

domain: w ≥ 0; range: d ≥ 0

b. about 99 lb c. about 4 lb

Practice Level C

1.

x

y

1

3

12121

23

23 3

domain: x ≥ 0; range: y ≥ 0; vertical stretch by a factor of 2.5

2.

x

y

0.6

1.8

12120.6

21.8

23 3

domain: x ≥ 0; range: y ≤ 0; vertical shrink by a

factor of 3 } 5 and refl ection

in x-axis

3.

x

y

0.25

0.75

12120.25

20.75

23 3

domain: x ≥ 0; range: y ≤ 0; vertical shrink by a factor of 0.25 and refl ection in x-axis


x horizontally 2.5 units left 5. translate graph of y 5 Ï

}

x

vertically 3 }

2 units down 6. translate graph of


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y 5 Ï}

x vertically 12 units up 7. translate graph

of y 5 Ï}

x horizontally 1 }

4 unit right 8. translate

graph of y 5 Ï}

x horizontally 5.5 units left


x vertically 3 }

4 unit up

10. D 11. C 12. E 13. A 14. F 15. B

16.

x

y

2

6

22

26

26


17.

x

y

2

6

22222

26

26 6

domain: x ≥ 1; range: y ≤ 5; vertical translation 5 units up, horizontal translation 1 unit right, and refl ection in x-axis

18.

x

y

1

3

12121

23

3 5

domain: x ≥ 3; range: y ≥ 23; vertical translation 3 units down and horizontal translation 3 units right

19.

x

y

2

6

2222

26

26

domain: x ≥ 26; range: y ≤ 2; vertical translation 2 units up, horizontal translation 6 units left, and refl ection in x-axis

20.

x

y

2

6

10

2 6 10 14

domain: x ≥ 7; range: y ≥ 8; vertical translation 8 units up and horizontal translation 7 units right

21.

x

y

2

6

22222

26

6 10

domain: x ≥ 4.5; range: y ≤ 2.5; vertical translation 2.5 units up, horizontal translation 4.5 units right; and refl ection in x-axis

22. a.

00

0.5

1.0

1.5

2.0

2.5

5 15 2510 20Height (meters)

Tim

e (

seco

nd

s)

t

h

domain: h > 0; range: t > 0

b. 11.025 m

23. a.

00

0.5

1.0

1.5

2.0

2.5

1 3 52 4 76Inside diameter (inches)

Rad

ius o

f g

yra

tio

n

(in

ch

es)

r

d

domain: d > 0; range: r > 1

b. about 3.3 in.

Review for Mastery

1.

10

14

6

2

x

y

y 5 4

2 6 10

x

y 5 x

domain: x ≥ 0; range y ≥ 0; The graph is a vertical stretch (by a factor of 4) of the graph of y 5 Ï

}

x .

2. 2

22

26

210

x

y

y 5 26

531

x

y 5 x domain: x ≥ 0; range y ≤ 0; The graph is a vertical stretch (by a factor of 6) and a refl ection in the x-axis of the graph of y 5 Ï

}

x .

3.

5

7

3

1

x

y

y 5 1 1

1 3 5

x

y 5 x

domain: x ≥ 0; range y ≥ 1; The graph is a vertical translation (of 1 unit up) of the graph of y 5 Ï

}

x .

4. 3

1

21

23

x

y

y 5 2 3

31

x

y 5 x

domain: x ≥ 0; range y ≥ 23; The graph is a vertical translation (of 3 units down) of the graph of y 5 Ï

}

x .

5. domain: x ≥ 22; range: y ≥ 24


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Challenge Practice

1.

x

y

1

3

121 3 5

The domain is all real numbers less than or equal to 6. The range is all real numbers greater than or equal to 0. The graph of y 5 Ï

}

6 2 x is a refl ection in the y-axis and a horizontal translation of 6 units right of the graph of y 5 Ï

}

x .

2. x

y

12121

23

3 5 7

The domain is all real numbers greater than or equal to 0. The range is all real numbers less than or equal to 0. The graph of y 5 2 Ï

}

2 x is a refl ection in the x-axis and a vertical stretch (by a factor of 2) of the graph of y 5 Ï

}

x .

3. x

y

21232527

23

The domain is all real numbers less than or equal to 1. The range is all real numbers less than or equal to 0. The graph of y 5 2 Ï

}

1 2 x is a refl ection in the x-axis, a refl ection in the y-axis, and a horizontal translation of 1 unit right of the graph of y 5 Ï

}

x .

4.

x

y

1

3

12123 3

The domain is all real numbers greater than or equal to 24. The range is all real numbers greater

than or equal to 0. The graph of y 5 Î} 1 }

2 x 1 2 is a

vertical shrink 1 by a factor of 1 }

2 2 and a horizontal

translation of 4 units left of the graph of y 5 Ï}

x .

5.

x

y

1

3

5

1212325

The domain is all real numbers less than or equal to 2. The range is all real numbers greater than or equal to 3. The graph of y 5 Ï

}

2x 1 3 is a refl ection in the y-axis, a horizontal translation of 2 units right, and a vertical translation of 3 units up of the graph of y 5 Ï

}

x .

6. y 5 Ï}

x 2 2 1 1 7. y 5 Ï}

4 2 x

8. y 5 2 Ï}

x 1 1 9. y 5 2 Ï}5 2 x 1 3

10. y 5 Ï}

x 1 1


1. C 2. A 3. B 4. 3 Ï}

11 5. 2 Ï}7 6. 3 Ï

}

6

7. 5 Ï}

2 8. 3 Ï}

3a 9. 4 x 10. 10 n Ï}

n

11. 5p Ï}5p 12. 3 Ï

}5 13.

Ï}

23 } Ï

}

23 14.

Ï}

10 } Ï

}

10

15. Ï

} 5x

} Ï

} 5x

16. Ï

} 5

}5 17.

Ï}

17 }17

18.7 Ï

}

3 }

319. 7 Ï

}5

20. 7 Ï}

2 21. 23 Ï}7 22. 15 Ï

}

2 23. 2 Ï}

2

24. 5 Ï}

3 25. 2 1 Ï}

2 26. 3 2 2 Ï}

3

27. 6 1 Ï}

3 28. about 81.2 volts 29. a. 8 in.b. 6 in.

Practice Level B

1. 10 Ï}

2 2. 3 Ï}5 3. 4 Ï

}7 4. 20 Ï

}

d 5. 3y

6. 5nÏ}

n 7. 3 Ï}7 8. 10 Ï

}

3 9. 2 x Ï}5

10.4}9

11. Ï

} 5

}7 12.

x}12

13.4 Ï

} 5

}5 14.

Ï}

6 }10

15. Ï

}

3 }5 16.

2 Ï}

p }p 17.

Ï}

3y }3y

18.9 Ï

}

2x }

2 x

19. 13 Ï}7 20. 23 Ï

}5 21. 27 1 4 Ï

}7

22. 40 Ï}

2 1 Ï}5 23. 37 1 20 Ï

}

3 24. 33

25. a. about 3.87 mi/h b. about 4.61 mi/h

26. a. about 4.90 ft/sec b. about 9.80 ft/sec

Practice Level C

1. 3sÏ}5s 2. 14r2 3. 15c2Ï

}

2c

4. 2m2n5Ï}

31 5. 11x3y4Ï}

x 6. a2b


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7. 3y2 Ï}

15x 8. 11

} 4 m

9. d } 5 10.

Ï}

10 }

4

11. m2 Ï

}

77m }

11 12.

5 Ï}

5x }

2x2 13. Ï}

15 2 Ï}

3

14. 214 1 3 Ï}

7 15. 6 Ï}

7 2 Ï}

14

16. 133 1 60 Ï}

3 17. 8 Ï}

3 1 Ï}

2 2 Ï}

6 2 24

18. 5m Ï

}

5mn } n 19.

5 Ï}

7 1 Ï}

14 } 7 20.

2 Ï}

3 }

3

21. 13 Ï}

x}

2 x 22. a. about 50 watts

b. about 100 watts 23. a. about 1.5 m2 b. about 1.6 m2

Review for Mastery

1. 6 Ï}

2 2. x Ï}

3 3. 3y2 Ï}

5y 4. 6x Ï}

3 5. 5 Ï}

2

6. 3x Ï}

5y 7. Ï

}

5 }

9 8.

x Ï}

2 }

3y 9.

3 Ï}

2x }

2x

10. 3 Ï}

7 1 8 Ï}

10 11. 23 Ï}

5

Challenge Practice

1. Ï

}

2 }

4 2. 24 Ï

}

3 3. 10 Ï}

2 2 15 Ï}

3

4. 4x Ï}

6 1 12x Ï}

2 2 x Ï}

3 2 3x 5. y(10 1 Ï}

2 ) 6. 91.5 mi/h 7. 129.4 mi/h 8. 24,500 ft

Focus On 11.2 Practice

1. 9 2. 2 3 Ï

} z } 3 3. 7

3 Ï}

x 4. 24 5. 3 Ï

} y }

2 6.

3 Ï}

3

7. 2 3 Ï}

4 8. 24 3 Ï}

3 9. 2 3 Ï}

x 10. 0 11. 6 3 Ï}

p

12. 0 13. 22 3 Ï

} z 14. 4

3 Ï}

3 1 3 3 Ï}

x

15. 22 3 Ï}

2 2 2 16. 4 3 Ï}

2 2 2 3 Ï}

4

17. 24 3 Ï}

5 2 5 18. 216 19. x 5 1

Review for Mastery

1. 8 2. 3 Ï}

x }

2 3. 2 3 Ï

} y 4.

3 Ï}

4 5. 3 Ï}

25 } 5 6.

3 Ï}

9

7. 2 3 Ï}

p 8. 6 3 Ï}

5 9. 2 3 Ï}

4 2 4 3 Ï}

x

10. 5 3 Ï}

5 2 3 Ï}

25


1. solution 2. not a solution 3. not a solution

4. not a solution 5. solution 6. not a solution

7. Ï}

x 5 3 8. Ï}

x 5 4 9. Ï}

x 5 3

10. Ï}

x 1 5 5 9 11. Ï}

x 2 4 5 4

12. Ï}

2x 1 3 5 13 13. 225 14. 64 15. 9

16. 9 17. 36 18. 72 19. 25 20. 3 21. 31

22. 7x 1 3 5 7x 2 1 23. 5x 2 8 5 1 2 6x

24. 9 2 2x 5 25x2 25. 4x2 5 3x 1 1

26. x2 1 2x 1 1 5 1 2 3x

27. 4x 2 3 5 x2 2 4x 1 4 28. 1 29. 6 30. 2

31. a. 16 ft b. 9 ft 32. a. 2 wk b. 5 wk

Practice Level B

1. not a solution 2. not a solution 3. solution

4. not a solution 5. solution 6. solution

7. Add 5 to each side, then square each side, subtract 3 from each side, and divide each side by 7. 8. Add 3 to each side, divide each side by 6, square each side and solve the linear equation for x. 9. Square each side and solve the resulting linear equation for x. 10. Divide each side by 2, square each side, and solve the result-ing linear equation for x. 11. Add the second radical expression to each side, square each side, and solve the resulting linear equation for x.

12. Add 2 to each side, square each side, and then solve the resulting quadratic equation for x.

13. 16 14. 80 15. 46 16. 42 17. 40

18. 7 }

3 19.

32 } 5 20. 4 21.

3 }

2 22.

1 }

4 23. 1

24. no solution 25. 6 26. no solution

27. no solution 28. 2, 3 29. 6 30. no solution

31. a. about 560,000 subscriptions b. 312,500 subscriptions 32. about 94.25 ft2

Practice Level C

1. Subtract 1 from each side, square each side, and then solve the resulting linear equation for x.

2. Subtract 15 from each side, square each side, and then solve the resulting equation for x.


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3. Subtract 4 from each side, divide each side by 22, square each side, and then solve the resulting equation for x. 4. Add 5 to each side, divide each side by 6, square each side, and then solve the resulting linear equation for x. 5. Square each side and solve the resulting linear equation for x.

6. Add the second radical expression to each side, square each side, and solve the resulting linear equation for x. 7. Divide each side by 3, square each side, and then solve the resulting linear equation for x. 8. Square each side and solve the resulting quadratic equation for x. 9. Subtract x from each side, square each side, and solve the resulting quadratic equation for x.

10. no solution 11. 20

} 3 12. no solution 13. 11

14. 1 15. no solution 16. 4 17. no solution

18. 2 }

3 19. 3 20. 10 21. 2 22.

1 }

4 23. 4 24.

1 }

4

25. Answers will vary. 26. a. about 4.8°C b. 0 m/sec 27. a. about 38.9 in. b. about 155.6 in.

Review for Mastery

1. 9 2. 39 3. 3 4. 7


1. a.

0

170

0

180

190

200

210

220

230

240

250

260

2 41 3 5Years since 1999

Reven

ue (

millio

ns o

f d

ollars

)

y

x

b. 2004

2. a.

0

5

0

10

15

20

25

30

10 205 15 25Initial velocity

Fin

al velo

cit

y

v

v0

b. about 24.49 meters per second

c. about 32 meters per second

3. Answers will vary. 4. 49 5. a. 625 joules b. The kinetic energy increases. Since the velocity increases, that means the right-hand side of the equation must increase. Since the mass stays constant, that means the kinetic energy must increase. 6. Answers will vary. 7. a. 4 times

b. about 1.07 times c. When the mass increases, r decreases. That means that the rate of effusion for the gas as compared to oxygen will get lower and eventually when the mass is larger than 32, the rate of effusion for the gas will be less than 1 time greater than the rate of effusion for oxygen.

Challenge Practice

1. Ï}

x2 2 3x 1 2 5 0 2. Ï}}

x3 2 7x2 1 36 5 0

3. Ï}

x3 2 x 5 0 4. Ï}}

x3 2 2x2 1 x 5 0

5. Ï}}

8 x3 2 12 x2 2 2 x 1 3 5 0

6. x 5 Ï}

3 2 2x 7. x 5 2 Ï}

3 2 2x

8. x 5 2 Ï}

20 1 x 9. x 5 Ï}

20 1 x

10. x 5 Ï}

10x 2 x3

} 3 11. x 5 2 Ï

}

10x 2 x3

} 3

12. x 5 Ï}}

15 2 x3 2 7x

} 7

13. x 5 2 Ï}}

15 2 x3 2 7x

} 7

14. x 5 2 Ï}}

x3 2 44x 1 84

15. x 5 Ï}}

x3 2 44x 1 84


1. legs: x, y; hypotenuse: z 2. legs: m, n; hypotenuse: p 3. legs: c, t; hypotenuse: r

4. 2 Ï}

5 5. Ï}

10 6. Ï}

34 7. 2 Ï}

13 8. Ï}

58

9. 5 Ï}

2 10. 3 Ï}

7 11. 10 12. 4 Ï}

6 13. 4

14. 3 15. 2, 4 16. not a right triangle

17. right triangle 18. not a right triangle

19. about 28.3 in. 20. about 73.2 in.

Practice Level B

1. Ï}

26 2. Ï}

65 3. 6 Ï}

2 4. Ï}

95 5. 2 Ï}

17

6. 6 Ï}

26 7. Ï}

241 8. 6 Ï}

2 9. 10 Ï}

5 10. 50

11. 20 12. 11 Ï}

5 13. 2, 8 14. 12, 16, 20

15. 9, 12, 15 or 15, 36, 39 16. 4 in., 7 in.

17. not a right triangle 18. right triangle

19. not a right triangle 20. about 155 in.

21. about 33 ft 22. about 9.2 ft

Practice Level C

1. 15 2. 5 Ï}

11 3. Ï}

18.25 4. Ï}

42.75


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5. Ï}

19.24 6. Î}

19.01 7. Ï}

273.44 8. Ï}

4.29

9. Ï}

0.4 10. Ï}

151.21 11. Ï}

221.44

12. Ï}

0.58 13. 5 in., 9 in. 14. 4 in., 8 in.

15. 6 in., 10 in. 16. right triangle

17. not a right triangle 18. not a right triangle

19. about 25 ft 20. about 10 mi

21. No, because 152 1 182 5 549 Þ 529 5 232

Review for Mastery

1. 15 2. 7 inches 3. no 4. no 5. yes

Challenge Practice

1. 3 2. 12 3. 10 4. 9 5. 10 6. 5.7


1. C 2. A 3. B 4. Ï}

61 5. Ï}

26 6. Ï}

41

7. Ï}

13 8. 2 Ï}

10 9. Ï}

17 10. 0, 24 11. 1, 9

12. 26, 10 13. 28, 16 14. 2 15. 3 16. (6, 7)

17. (3, 6) 18. (3, 0) 19. a. about 2.06 mi b. about 2.24 mi c. the distance between stop 1 and home; 0.18 mi 20. $860,000

Practice Level B

1. Ï}

5 2. Ï}

10 3. 5 Ï}

2 4. 10 5. Ï}

37

6. Ï}

109 7. 4 Ï}

2 8. Ï}

305 9. 2 Ï}

13 10. 2

11. 2 12. 3 13. 23, 5 14. 24, 10 15. 27, 3

16. 1 3, 17

} 2 2 17. 1 2

17 } 2 , 8 2 18. 1 21, 2

19 }

2 2

19. 1 11 }

2 , 2

3 } 2 2 20. (25, 0) 21. 1 2

3 } 2 , 2 2

22. right triangle 23. right triangle

24. not a right triangle 25. not a right triangle


28. 15 mi 29. a. (1750, 2000) b. 1953 ft

30. 8.5 books

Practice Level C

1. 13 2. Ï}

122 3. Ï}

458 4. 2 Ï}

101 5. Ï}

661

6. 8 7. Ï}

122 8. Ï}

10 9. Ï

}

37 }

3 10. 1, 5

11. 2, 6 12. 23, 21 13. 24, 22 14. 26, 0

15. 24, 24 16. 1 22, 2 1 } 2 2 17. 1 5 }

2 , 8 2

18. 1 101, 2 169

} 2 2 19. (3.75, 9.25)

20. (4.75, 21.75) 21. (25.5, 4)


24. not a right triangle 25. right triangle


28. a. book and basket; about 447 ft b. book and backpack; about 894 ft 29. a. about 12 mi b. (4, 7) c. about 3 mi;

1 5 } 2 ,

9 }

2 2 ; Find the midpoint between (1, 2) and

(4, 7) and then fi nd the distance between these points.

Review for Mastery

1. 2 Ï}

10 2. Ï}

29 3. 4 Ï}

10 4. 28, 22

5. (10, 6) 6. (24.5, 24)


1. a. Ï}

17 , 4 Ï}

2 , 5 b. 1 2 3 } 2 , 0 2 , (1, 0), 1 1 }

2 , 22 2

c. 5 }

2 , 2 Ï

}

2 , Ï

}

17 }

2 d. The perimeter of the original

triangle is twice the perimeter of the triangle using the midpoints. 2. a. 9 miles b. about 2.6 miles 3. 125 4. Answers will vary.

5. a. Ï}

34 ø 5.83 miles

b. You should meet 1 }

2 mile north and

1 }

2 mile east

of your original starting point. You have to

hike Ï

}

34 }

2 miles, or about 2.92 miles.

6. You should attach the guy wires about 4.8 feet up the tree. Each guy wire is 6.25 feet long. The guy wire is the hypotenuse of the triangle and one of the legs is the distance from the trunk to the stake, which is 4 feet.

7. a. Molly: r miles, Julie: r 1 3 miles b. Molly: 9 miles per hour, Julie: 12 miles per hour c. They are 30 miles apart after 2 hours. After two hours Molly has biked 18 miles and Julie has biked 24 miles. These are two sides of a right triangle. The hypotenuse is how far apart they are.

Challenge Practice

1. a 5 61 2. a 5 5 }

2 , b 5 1

3. a 5 6 Ï

}

2 } 2 , b 5 27 4. a 5 21, b 5 1

5. a 5 61, b 5 6 Ï}

2 6. a 5 1, 21

} 17


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7. a 5 22, 108

} 61

8. a 5 5, 65

} 187

9. a 5 0, a 5 24 10. b 5 0, a > 0 11. 25 mi

12. 10 mi 13. 38 min

Chapter Review Game 1. rationalizing 2. hypotenuse 3. midpoint

4. triple 5. square 6. radical 7. conjugates

8. distance 9. legs 10. Pythagorean

11. extraneous 12. theorem 13. simplest form

F

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A

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I

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P

O

T

Z

E

I

Q

T

S

D

T

S

D

W

A

E

X

D

R

S

P

P

I

E

I

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M

I

D

P

O

I

N

T

U

L

S

O

F

S

F

C

D

R

I

S

T

A

C

N

E

U

N

S

T

A

M

A

P

L

E

F

S

T

E

U

O

A

H

A

H

J

H

L

Q

U

Q

N

C

T

F

E

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N

F

C

G

K

B

U

R

P

Y

O

I

N

I

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C

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A51Algebra 1


Answers

Lesson 12.1

Practice Level A

1. direct variation 2. inverse variation

3. neither 4. inverse variation

5. inverse variation 6. direct variation

7. direct variation 8. neither

9. direct variation 10. C 11. B 12. A

13. Domain and range:

2123 1

1

21

23

x

y all real numbers except 0


22 2 6

2

22

6

x

y

all real numbers except 0


23 3 9

3

23

9

x

y


16. Domain and range

2329 3

3

23

29

x

y

are all real numbers except 0.


2329 3

3

23

29

x

y



22 2 6

2

22

6

x

y


19. C 20. B 21. A 22. y 5 3 } x ;

3 }

2

23. y 5 8 } x ; 4 24. y 5

18 } x ; 9 25. y 5

216 } x ; 28

26. y 5 214

} x ; 27 27. y 5 25

} x ; 25

} 2 28. no

29. yes; y 5 22

} x 30. direct variation

31. inverse variation

32. a. t 5 4000

} p b. 125 h

33.

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 t

Time (hours)

Wal

kin

g s

pee

d (

mi/

ho

ur)

00

s

s 5 t3

yes; Answers will vary.

Practice Level B


3. neither 4. inverse variation 5. inverse variation 6. direct variation 7. neither



23 3 9

3

23

9

x

y



2226 2

2

22

26

x

y



22 2 6

2

22

6

x

y



2226 2

2

22

26

6

x

y


AN

SW

ER

S


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24 4 12

4

12

x

y



4 12

4

12

x

y



2329 3

3

23

29

x

y



2329 3

3

23

29

x

y



22 2 6

2

22

6

x

y


19. y 5 14

} x ; 7 20. y 5 27

} x ; 27

} 2 21. y 5

23 } x ; 2

3 } 2

22. y 5 211

} x ; 2 11

} 2 23. y 5 144

} x ; 72

24. y 5 72

} x ; 36 25. y 5 50

} x ; 25

26. y 5 228

} x ; 214 27. y 5 36

} x ; 18

28. y 5 236

} x ; 218 29. y 5 2200

} x ; 2100

30. y 5 55

} x ; 55

} 2 31. no 32. yes; y 5

20 } x

33. no 34. yes; y 5 18

} x

35. a. t 5 2400

} p b. 120 minutes c. 300 minutes

36. a. V 5 2500

} P b. 10 lb/in.3

37.

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 t

Time (hours)

Avera

ge s

peed

(m

i/h

ou

r)

00

s

s 5 t5

yes; Answers will vary.

Practice Level C


3. neither 4. inverse variation


7. neither 8. direct variation

9. inverse variation


21 1

1

21

x

y



2123 1 3

1

21

23

3

x

y



22 2 6

2

22

6

x

y



2329 3

3

23

29

x

y



5 15

5

15

x

y



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A53Algebra 1



2123 1 3

1

21

23

3

x

y



21 1 3

1

21

3

x

y



20.421.2 0.420.4

21.2

x

y



4 12

4

12

x

y


19. y 5 251

} x ; 2 51

} 2 20. y 5 144

} x ; 72

21. y 5 242

} x ; 221 22. y 5 36

} x ; 18

23. y 5 230

} x ; 215 24. y 5 49

} x ; 49

} 2

25. y 5 2150

} x ; 275 26. y 5 120

} x ; 60

27. y 5 244

} x ; 222 28. y 5 2114

} x ; 257

29. y 5 105

} x ; 105

} 2 30. y 5

70 } x ; 35

31. yes; y 5 16

} x 32. yes; y 5 210

} x

33. a. f 5 299,008

} w b. 1.024 3 105 hertz

34. a. a 40 50 80 100 200 400

m 10 8 5 4 2 1

As the amount of money you save each month increases, the number of months you need to save decreases.

b.

1

2

3

4

5

6

7

8

9

a

Amount saved each month (dollars)

Nu

mb

er

of

mo

nth

s

00

m

100 200 300 400

inverse variation; Answers will vary.

c. m 5 400

} a

Review for Mastery

1. neither 2. direct 3. inverse 4. direct


x

y

2

6

22222

26 6



x

y

2

6

22222

26

26 6



x

y

1

3

12121

23

23



x

y

2

6

2 62222

26

26


9. y 5 16

} x ; y 5 24

Challenge Practice

1. x 5 a }

dv ; inverse variation

2. v 5 a }

bd y; direct variation

3. u 5 dkc

} w

; inverse variation


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4. u 5 dkz; direct variation

5. v 5 a }

dkz ; inverse variation

6. y 5 bd

} u ; inverse variation

7. w 5 ck

} x ; inverse variation

8. v 5 a }

dkc w; direct variation

9. w 5 ck

} b y; direct variation

10. y 5 b }

kz ; inverse variation

11. a 5 21 12. c 5 22 13. x 5 2 1 } 3

14. y 5 2 1 } 4 15. x 5 2

1 } 500


1. C 2. A 3. B 4. domain: all reals except 6; range: all reals except 1 5. domain: all reals except 22; range: all reals except 21 6. domain: all reals except 1; range: all reals except 21

7.

22 2 6

2

6

x

y domain: all reals except 0; range: all reals except 0; vertical stretch

8.

21 1

13

13

1

x

y

13

2

domain: all reals except 0; range: all reals except 0; vertical shrink

9.

2226 2

2

22

26

x

y domain: all reals except 0; range: all reals except 0; vertical stretch and refl ection in x-axis

10.

2226 2 622

6

10

x

y domain: all reals except 0; range: all reals except 4; vertical translation 4 units up

11.

2123 1 3

1

21x

y domain: all reals except 0; range: all reals except 22; vertical translation 2 units down

12.

22210

2

22

26

6

x

y domain: all reals except 26; range: all reals except 0; horizontal translation 6 units to the left

13. C 14. B 15. A 16. x 5 8, y 5 0

17. x 5 0, y 5 214 18. x 5 6, y 5 5

19. x 5 213, y 5 1 20. x 5 210, y 5 22

21. x 5 25, y 5 27

22.

222

22

6

x

y 23.

23 1 3

1

3

5

x

y

24.

226210

2

22

26

210

x

y

25. a. C 5 500

} p

1 13

b.

0 10 20 30 40 50 60 70 p0

25

50

75

100

125

150

175C

Number of people

Co

st

(do

llars

/pers

on

)

26.

0 2 4 6 8 10 12 14 p0

25

50

75

100

125

150

175f


Avera

ge n

um

ber

of

flo

wers

per

pers

on

f 5 400

3 1 p


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A55Algebra 1


Practice Level B

1. domain: all reals except 3; range: all reals except 1 2. domain: all reals except 4; range: all reals except 3 3. domain: all reals except 26; range: all reals except 24 4. domain: all reals except 26; range: all reals except 28

5. domain: all reals except 23; range: all reals except 3 6. domain: all reals except 3; range: all reals except 22

7.

22 2 6

2

22

6

x


8.

1

1

x

y domain: all reals except 0; range: all reals except 0; vertical shrink

9.

2123 1

1

21

23

x


10. 2123 1 3

22

26

x


11.

2123 1 3

2

x


12.

6 10

1

21

23

3

x

y domain: all reals except 4; range: all reals except 0; horizontal translation 4 units right

13. x 5 6, y 5 4 14. x 5 25, y 5 26

15. x 5 3, y 5 28 16. x 5 27, y 5 7

17. x 5 8, y 5 12 18. x 5 25, y 5 10

19. x 5 14, y 5 1 20. x 5 212, y 5 23

21. x 5 5, y 5 214

22.

2123 1 3

6

10

x

y 23.

22 2 6 10

1

21

3

5

x

y

24.

222

26

6

x

y

25. a. C 5 515

} p 1 14.5

b.

0 5 10 15 20 25 30 35 p0

20

40

60

80

100

120

140C

Number of people

Co

st (

do

llars

/per

son

)

c. $40.25

26. a. n 5 450

} 4 1 p

b.

0 1 2 3 4 5 6 7 p0

20

40

60

80

100

120

140n


Ave

rage

nu

mb

er o

f p

izza

s c. 75 pizzas

Practice Level C

1.

21

21

x

y domain: all reals except 0; range: all reals except 0; vertical shrink and refl ection in x-axis

2.

1 3

1

3

x

y domain: all reals except 0; range: all reals except 0; vertical shrink


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3.

2123 1

1

21

23

x


4.

212321

23

x

y domain: all reals except 0; range: all reals except 0; vertical shrink and refl ection in x-axis

5.

2123 1 3

1

21

23

3

x


6. 2226 2 6

22

26

x


7.

2226 2 6

2

22

6

10

x


8.

22 2 10

2

22

26

6

x

y domain: all reals except 6; range: all reals except 0; horizontal translation 6 units to the right

9.

2226

2

22

26

6

x

y domain: all reals except 28; range: all reals except 0; horizontal translation 8 units to the left

10. x 5 213, y 5 210 11. x 5 2, y 5 2

12. x 5 21, y 5 23

13.

22 6 10

2

22

6

10

x

y 14.

21 123

1

x

y

15.

2221022

6

x

y 16.

22 2 6 10

2

22

26

210

x

y

17.

210

2

26

x

y 18.

22622

2

26

x

y

19.

0 2 4 6 8 10 12 14 p0

5

10

15

20

25

30

35n

Number of extra parents

Avera

ge n

um

ber

of

bo

x lu

nch

es p

er

pers

on

n 5 225

6 1 p

9 people

20. a. C 5 17.25

} 4 1 r

b.

0 1 2 3 4 5 6 7 r0

1

2

3

4C

Number of additional

rentals

Avera

ge c

ost

per

ren

tal (d

ollars

) about 4 rentals

Review for Mastery

1.

x

y

1

3

121 3

y 51x

y 58x

The graph of y 5 8 } x is a

vertical stretch of the

graph of y 5 1 } x . Domain:

all real numbers except 0; Range: all real numbers

except 0.

2.

x

y

2

6

226 6

y 51x

26

y 51x 1 5

The graph of y 5 1 } x 1 5 is

a vertical translation (of 5 units up) of the graph of

y 5 1 } x . Domain: all real

numbers except 0; Range: all real numbers except 5.


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A57Algebra 1


3.

x

y

2

6

226

26

y 51x

y 51

x 1 10

The graph of y 5 1 } x 1 10

is a horizontal translation(of 10 units to the left) of

the graph of y 5 1 } x .

Domain: all real numbers

except 10; Range: all real numbers except 0.

4.

x

y

2

6

26

2226 22 2 6

Challenge Practice

1. vertical asymptote: x 5 2;

horizontal asymptote: y 5 4

x

y

2

6

622

2. vertical asymptote: x 5 2 1 } 2 ;


x

y

22226 6

2

6

3. vertical asymptote: x 5 21;


x

y

2 22 26 6

2

4. vertical asymptote: x 5 3 } 4 ;


x

y

22226 6

6

22

5. vertical asymptote: x 5 2 4 } 3 ;

horizontal asymptote: y 5 2 1 } 4

x

y

22226

2

6

6. f (x) 5 23 }

2x 2 6 1 2 7. f (x) 5

7 }

7x 2 1 1 1

8. f (x) 5 6 }

2x 2 2 2 1 9. f (x) 5

6 }

2x 1 1

10. f (x) 5 24 }

22x 1 3 1 2


1. 3x2 2. 23x 3. 21 4. 3x2 2 2x 1 6

5. 2x2 1 3x 2 4 6. 2x3 2 2x2 2 3x

7. 4x3 2 x 1 2 8. x2 2 3x 2 2

9. 2x2 1 4x 2 5 10. A 11. C 12. B

13. x 1 4 14. x 2 5

15. x 2 6 16. x2y 1 3x 1 2

17. a. C 5 20h 1 5

} h

b.

0 1 2 3 4 5 6 7 h0

20

30

40

50C

Number of hours rented

Avera

ge c

ost

per

ho

ur

(do

llars

)


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18. a. R 5 1 } 6 1

1 }

24(t 1 5)

b.

1 2 3 4 5 6 7 8 t

Rati

o o

f sp

ort

s c

ars

so

ld t

o t

ota

l cars

so

ld

00

R

Years since 1995

0.168

0.170

0.172

0.174

0.176

0.178

Practice Level B

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

3. 211x3 1 9x 2 3 4. x 1 1 5. 5x 2 3

6. 4x 1 5 7. 6x 2 2 8. 4x 1 9 1 10 } x 2 2

9. 5x2y3 1 2x 2 1 10. 3a2 1 2ab 2 1

11.

212 4 12

4

212

12

x

y 12.

2226 2 6

2

22

6

x

y

13.

6

2

6

x

y

14. a. C 5 40h 1 4.5

} h

b.

30

40

50

60

70

80

1 2 3 4 5 6 7 8 h

Avera

ge c

ost

per

ho

ur

(do

llars

)

00

C

Time (hours)

15. a. R 5 1 } 2 1

23 }

2(2t 1 9)

b.

Rati

o o

f fr

uit

dri

nks

so

ld t

o t

ota

l d

rin

ks s

old

0

0.3

0.6

0.9

1.2

1.8

1.5

1 2 3 4 5 6 7 8 t0

R

Years since 1995

Practice Level C

1. 3x3 2 4x 1 2 2. 212x2 1 8x 1 3

3. 7x 1 16 1 27 } x 2 2 4. x 1 4 2

5 } x 2 1

5. 3x 1 8 1 54 } x 2 4 6. x 1 3 2

4 } x 1 3

7. 2x 2 10 1 17 }

2x 1 2 8. 3x 2 1 2 3 } 3x 1 1

9. 5x6y4 2 x4y3 2 2 } 3 y 10. 8a2b2 1 2b 2

9 } 7 a

11.

293

9

23

29

x

y 12.

25 5 1525

15

x

y

13.

25215

215

x

y

14. a. C 5 24 1 0.06m

} m

b.

4

8

12

16

20

24

28

1 2 3 4 5 6 7 8 m

Avera

ge c

ost

per

mile (

do

llars

)

00

C

Number of miles

15. a. R 5 1 } 6 1

1049 }} 546t 1 12,552

b.

1 2 3 4 t

Rati

o o

f w

alk

ing

sh

oes

so

ld t

o t

ota

l sh

oes s

old

00

R

Years since 1999

0.230

0.235

0.240

0.245

0.250

0.255

Review for Mastery

1. 2p2 2 5p 1 6 2. 6r2 1 4r 2 11

3. 25t2 2 2t 1 6 4. 4x 1 3

5. 3x 2 2 1 4 }

8x 2 1 6. 2x 1 5

7. 4x 2 7 1 3 }

4x 2 7


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A59Algebra 1


Challenge Practice

1. x 1 3 2. x2 1 5x 1 6 3. 1 1 2 }

x2 2 1

4. x2 1 2x 1 1

5. 5x2 2 15x 1 37 2 96x 1 31

} x2 1 3x 1 1

6. 18x3 1 3x2 1 30x 1 5

7. x4 1 9x3 1 14x2 2 27x 2 5

8. 6x3 1 2x2 1 3x 1 6

9. x5 1 6x3 1 x2 1 7x 1 4

10. x4 1 x3 1 x2 1 x 1 2 11. x 1 3

12. x 2 4 1 2x 1 1

} x2 1 2x 1 1

13. x2 2 5x 1 3

14. 2x3 1 5x 2 1 15. x2 2 5


1. x 1 3 2. x 1 1 3. x2 2 2x 2 2 }

x 2 2

4. 2x3 2 2x2 1 x 1 1 2 5 }

x 1 1 5. x2 2 x 2 1

6. 2x2 2 x 2 2 7. x2 1 x 1 1 2 1 } x 1 5

8. x2 2 3 2 8 }

x 2 3 9. x3 1 x2 2 2x 2 2 2

5 }

x 2 1

10. x2 1 x 1 6 11. x2 1 1 2 1 }

x 2 1 }

2 12. x2 1 2

13. No. To use synthetic division, the divisor must be of the form x 2 k.

14. 50

Review for Mastery

1. x2 2 2x 2 3 }

x 2 1

2. 2x3 1 4x2 1 4x 1 2 2 3 }

x 2 2

3. x2 2 3x 1 7 2 5 }

x 1 1

4. x2 1 2x 1 2


1. none 2. x 5 0 3. x 5 6 4. x 5 23

5. x 5 1 6. x 5 2 7. x 5 22 8. x 5 1 }

2

9. x 5 2 2 } 3 10. not in simplest form

11. not in simplest form 12. in simpest form

13. in simplest form 14. not in simplest form

15. in simplest form 16. 2 }

3x ; x 5 0

17. 7 }

2x ; x 5 0 18. 2; x 5 22

19. in simplest form; x 5 5

20. 1 }

x 1 6 ; x 5 26, 6

21. in simplest form; x 5 210, 10

22. a. 2(2x) 1 2x

} 2x(x)

b. 3 }

x

23. a. R 5 245t 1 32,800

}} 465t 1 56,780

b. R 5 49t 1 6560

} 93t 1 11,356

Practice Level B

1. x 5 0 2. x 5 5 3. x 5 210 4. x 5 2

5. x 5 23 6. x 5 2 7 } 3 7. x 5 1 8. x 5 26, 2

9. x 5 25, 5 10. 22x; x 5 0 11. 6; x 5 4

12. 24; x 5 3 13. 1 }

x 2 11 ; x 5 211, 11

14. 1 } x 1 7 ; x 5 23, 27

15. in simplest form; x 5 23, 28 16. 1 }

2x

17. 3x 1 5

} x(x 1 5)

18. 3x 1 2

} x(x 1 1)

19. a. 2(4x 1 3) 1 2(4x 2 2)

}} (4x 1 3)(4x 2 2)

b. 8x 1 1 }}

(4x 1 3)(2x 2 1)

20. 2(4t 1 5)

}} 0.1t2 2 0.2t 1 3

; about 11 thousand pounds

Practice Level C

1. x 5 24, 1 }

3 2. x 5 2

5 } 8 , 1 3. x 5 7

4. 1 }

x 1 1 , x 5 21, 7 5.

22x2 }

3x 2 5 , x 5 0,

5 }

3

6. 3(x 2 4)

} 4(1 2 2x)

; x 5 0, 1 }

2 7.

3x3 }

3x 1 4 ; x 5 2

4 } 3 , 0

8. 2 }

x 1 10 ; x 5 210, 2 9.

4x }

2x 1 1 ; x 5 2

1 } 2 , 3

10. x 2 6

} 2x 1 3

; x 5 210, 2 3 } 2 11.

1 }

x(x 2 4) ; x 5 0, 4

12. x 1 5

} 2x(x 2 2)

; x 5 22, 0, 2

13. (5x 1 1)2; Answers will vary.

14. Answers will vary.


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15. a. 2[x 1 x 1 (x 1 3)]

}}} 2 F 1 }

2 (x 1 3 1 2x 1 4)(x 2 1) G

b. 6(x 1 1)

}} (3x 1 7)(x 2 1)

16. a. 6(t 1 4) }}

0.01t2 2 0.5t 1 18

b. about 4 hundred thousand

c.

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 t

Nu

mb

er

of

flyers

(hu

nd

red

s o

f th

ou

san

ds)

00

A

Years since 1995

The number of fl yers increased as time went by.

Review for Mastery

1. 3 2. 24, 4 3. no excluded values 4. 26, 2

5. 1 }

5x2 , x Þ 0 6. 4x }

x 2 3 x Þ 3

7. simplest form, x Þ 23 8. 5 }

3x , x Þ 0, x Þ 22

9. x 1 6

} x 2 9

, x Þ 27, x Þ 9

10. 2x 1 5

} 2x 2 5

, x Þ 22.5, x Þ 2.5


1. a. N 5 397,000 2 15,500x

}} 125 2 7x

b. 3,550,000 people

2. a. p 5 3x 1 64

} 11x 1 291

b. x 0 1 2 3

p 0.2199 0.2219 0.2236 0.2253

x 4 5 6 7

p 0.2269 0.2283 0.2297 0.2310

c. increasing 3. 5 4. Answers will vary.

5. a. The density and the volume are inversely related because as the density increases, the

volume decreases; V 5 14

} D b. Because the height

is increasing, the volume is increasing which means that the density in decreasing. 6. 11

7. a. N 5 2842 1 337x

} 500 1 14x

; about 9 million

b.

00

4

5

6

7

8

9

1 2 3 4 5 6 7 8 x

N

Years since 1993

Nu

mb

er

of

new

tru

cks s

old

(m

illi

on

s) The number of

trucks sold from 1993 to 2002 increased.

c. You cannot use the model to conclude that the revenue of the new trucks sold had increased because the prices may have decreased which led to more trucks being sold.

Challenge Practice

1. a 5 7 2. b 5 9 3. c 5 1

4. d 5 2 5. e 5 2

6. p(x) 5 2x2 2 6x 1 5 } 2 , q(x) 5 4x2 2 11x 1

5 } 2

7. p(x) 5 2x2 1 7x 1 5, q(x) 5 x2 2 2x 2 3

8. p(x) 5 2x2 1 x 2 1, q(x) 5 2x2 1 9x 2 5

9. p(x) 5 x4 2 1, q(x) 5 2x4 1 x2 2 1

10. p(x) 5 6x3 1 13x2 1 8x 1 3,q(x) 5 3x3 2 x2 2 x 2 1


1. C 2. B 3. A 4. 21x 5. 7 }

6x2 6. 1 }

x

7. x }

8 8. 6x2 9.

5 }

4x2 10. 1 }

x(x 1 1)

11. 5(x 1 2) 12. x 2 1

} x 1 4

13. 3x

} 2 14.

3x }

4

15. 9x

} 5 16. 8 }

3x2 17. 4 }

x 18.

2x }

2(x 1 2) 19.

4x2 }

3

20. a. T 5 2(3t 1 10)

} 11 2 t

b. about 8 hundred thousand dollars

Practice Level B

1. 1 }

6x3 2. 6x4

} 5 3. 14

} 5 4. 1 }

4 5.

1 }

2(x 1 5)

6. x 1 6

} 2(x 1 2)

7. x(x 1 3)

} 3(2x 2 1)

8. x 9. 3(x 1 5)

10. 20

} x 11.

x6 }

4 12.

2 }

9 13.

1 }

9 14.

1 }

3(x 1 9)

15. 1 }

4 16.

1 }

x 17.

2 }

(x 1 3)2

18. a. 4x2 b. 100 tiles


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A61Algebra 1


19. 1250

} 150 2 t

; about $8.74 per unit

Practice Level C

1. x 2. 7(x 1 1)

} 9x(x 1 3)

3. 2 1 } 2 4.

x 2 4 }

3x 5.

x5 }

9

6. 4(2x 2 5) 7. 2(x 1 3)(x 1 6)

}} x2 1 6

8. x 1 4

} x(5x 1 16)

9. 21 10. 2 }

x(x 1 4) 11.

x2(x2 2 5)(x 2 7) }}

4(x2 2 7)

12. 6(x 2 4)

} 2x 2 1

13. 3 }

x 14.

x2 1 4 }

2x2

15. (x 1 6)(x 2 6) 16. (2x 1 1)(x 2 7)

17. A 5 500(t 1 20)

} 1 2 0.05t ; about $13,529

18. a. x 5 5(3122t 1 13,124)

}} 7(t 1 42)

b.

200

400

600

2 4 6 8 10

Avera

ge n

um

ber

of

cars

ad

mit

ted

per

mo

nth

0

0Years since 1988

12

800 Answers will vary.

Review for Mastery

1. 2(x 1 1)

} 3(x 1 6)

2. 3x }

x 2 2 3.

x 1 5 }

3x(x 2 6) 4.

2(x 1 4) } x 2 5

Challenge Practice

1. p(x) 5 4x 2 1 2. p(x) 5 x 1 2

3. p(x) 5 22x2 1 3x 2 1 4. p(x) 5 2x2 1 3

5. p(x) 5 16x 2 3 6. p(x) 5 1

7. p(x) 5 x2 2 6x 1 9 8. p(x) 5 2x 1 3


1. 1 }

6 2.

3x2 }

4 3.

3x4 }

4 4. 22x 5.

2x3 }

12 6. 6x2

7. 2x

} 25 8.

x }

23 9. (x 1 3)2 10. (x2 2 4)2

11. x }}

3(x 2 3)(x 2 4) 12.

x }

(x 2 3)2

13. 4x 2 1

}} 4(x 2 3)(x 2 1)(x 1 1)

14. 1 }

2(x 1 2)

15. x(x 2 7)

}} (x 1 1)(x 2 1)

16. No, when the fraction is in the denominator,

the value of the expression is 1 ? 3 }

2 5

3 }

2 . When

the fraction is in the numerator, the value of the

expression is 1 }

2 ?

1 }

3 5

1 }

6 .

17. Yes. a2 2 b2

} a 2 b

}

a 1 b 5

a2 2 b2 }

a 2 b ?

1 }

a 1 b 5

a2 2 b2 }

a2 2 b2 and

a2 2 b2

} a 1 b

}

a 2 b 5

a2 2 b2 }

a 1 b ?

1 }

a 2 b 5

a2 2 b2 }

a2 2 b2 , so the

complex fractions are equivalent.

Review for Mastery

1. 1 }

18 2.

5 }

2 3.

x }

10 4.

7x }

2 5.

x }}

3(x 2 5)(x 2 4)

6. 2 }}

(x 1 6)(x 2 6)


1. 3 }

4x 2. 2 }

x 3. 1 }

3x2 4.

2 }

x3 5. x 1 4

} 2x

6. x 2 27

} 9x

7. 2x 1 7

} 5x 8. 2 } x2

9. 2x 1 4 } x 2 1

10. 10x

11. 12x3 12. x(x 1 1) 13. x(x 2 4)

14. (x 1 2)(x 1 4) 15. (x 2 3)(x 1 7)

16. 40x2 1 3

} 15x

17. 7x2 2 1

} 2x

18. 73 } 36x

19. 2(5 2 12x)

} 15x2

20. 7x 1 16

} x(x 1 4)

21. 9(x 1 2)

}} (x 2 2)(x 1 7)

22. a. 48 }

r 2 4 ;

48 } r 1 4 b. t 5

96r }}

(r 2 4)(r 1 4) c. 9 h

23. t 5 80r 2 160

} r(r 2 4)

; about 1.6 h

Practice Level B

1. x 1 8

} x 1 5 2. 4x } x 2 4

3. 3(2x 1 1)

} x 2 9

4. 211

} x 1 2

5. 10x 2 7

} x2 2 9

6. x 1 5

} 3x2 7. 15x3 8. x(x 1 7)

9. (x 2 4)(x 1 6) 10. 2(x 2 2)

11. x(x 2 5)(x 1 2) 12. (x 1 1)2(x 1 4)

13. 85

} 14x

14. 32 2 5x2 }

12x3 15.

x(5x 1 31) }}

(x 2 5)(x 1 2)

16. 5x2 1 3x 2 5 }} (6x 2 5)(5x 2 3)

17. 25 2 3x } x(x 2 7)

18. x3 1 3x2 1 5x 2 5 }}

x2(x 2 1) 19. 2x2 1 5x 1 1 }}

(x 2 1)(x 1 1)


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20. x 1 6

} x 2 3

21. 25 }}

(x 1 1)(x 1 4)(x 2 4)

22. a. t 5 8 } r 2 1 1

8 } r 1 1 b. 6 h c. about 1.0 h

23. a. t 5 50

} r 1 50 } r 1 3 b. about 6.1 h

c. about 1.4 h

Practice Level C

1. 3(x 2 2)

} x 1 3

2. x 2 8

} x 2 5 3. 23x 1 2

} 2x 2 5

4. 11x2 1 19x 2 20

}} (x 2 5)(x 1 2)

5. 22x2 2 22x 2 9

}} (x 1 10)(x 2 1)

6. 22x2 1 31x 2 16

}} (2x 2 3)(x 1 5)

7. 9x3 2 16x2 1 3x 2 2 }} 3x2(x 2 2)

8. 5x3 2 19x2 1 8x 1 12

}} 5x2(x 1 2)

9. 215x2 1 x 2 30

}} 8x(x 1 6)

10. 3x2 1 28x 1 1

}} (x 2 1)(x 1 1)(x 1 7)

11. 5x 2 7 }}

(x 2 3)2(x 1 5)

12. 2(x2 1 6)

}} (x 2 6)(x 1 2)(x 1 1)

13. 2x2 1 19x 1 50 }} (x 1 2)(x 1 1)

14. 6(4x2 1 32x 1 7)

}} (x 1 8)(x 2 3)

15. 21x2 1 79x 2 2

}} (x 1 1)(x 1 4)(x 1 6)

16. 22x3 1 8x2 1 51x 1 27

}} (x 2 7)(x 1 2)

17. a 5 11c2 2 34c 2 105

}} (3c 1 4)2

18. a. t 5 10

} r 1 10 }

r 2 2.75 b. about 2.4 h

c. about 0.2 h 19. a. y 5 1 } t

b. no; Answers will vary.

Review for Mastery

1. 2x 1 1

} 7x 2. 3x 1 16

} 3x 2 4

3. 30x5

4. (3x 2 1)(x 1 6) 5. (x 1 5)2(x 1 3)

6. 7r 1 24

} 18r3 7.

x2 2 15 }}

(x 2 3)(x 1 3)(x 2 5)

8. 13t 2 11 }} (t 2 7)(t 1 3)


1. t 5 270r 2 2025

} r(r 2 15)

; 7.5 hours

2. t 5 50r }}

(r 1 5)(r 2 15) ; 3.75 hours

Challenge Practice

1. w 5 12x2 2 72x 1 60

}} 9x2 1 6x 1 1

2. w 5 2x2 1 1

} x4 1 2x2 1 6

3. w 5 2x }

3x 2 1 4. w 5

2x2 2 2x 1 13 }}

210x 1 5

5. w 5 2325x2 2 115x 2 10

}} x2 2 2x 1 1

6. y(t) 5 1 }

60 t 1

1 } x t

7. Approximately 25 hours and 43 minutes

8. 60 hours 9. 30 hours 10. 60 hours


1. x 5 6 2. x 5 210, 24 3. x 5 23, 3

4. 26, 6 5. 28, 8 6. 15 7. 224 8. 216

9. 25, 3 10. x(x 1 4) 11. x(x 2 3)

12. (x 1 2)(x 2 5) 13. 29 14. 11

15. no solution 16. 10 } 3 17. 2

23 } 3 18.

9 }

4

19. 12 days 20. 7 fi eld goals 21. a. 2 1 p; 4 1 p

b. 2 1 p

} 4 1 p

5 0.6 c. 1 pt d. 5 pt

Practice Level B

1. 29, 9 2. 212 3. 27 4. 18 5. 25

6. 24, 4 7. 21 8. no solution 9. 2 1 } 3

10. x 2 3 11. 2(x 2 1) 12. (x 2 2)(x 2 3)

13. no solution 14. 2 9 } 5 15. no solution

16. 24, 2 17

} 7 17. no solution 18. 3, 6

19. 1 pt 20. a.

Person

Fractionof roompapered

each hour

Time(hours)

Fraction of room papered

Assistant 1 }

x3

3 }

x

Expert 3 }

2x 3

9 }

2x

b. Answers will vary. c. 3 }

x 1

9 } 2x 5 1; 7.5 h

Practice Level C

1. 1 }

4 2. 62 Ï

}

2 3. 6 4. 3 5. 1 }

2 6.

4 }

3


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right

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ight

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ed.

A63Algebra 1


7. 3 6 Ï

}

89 }

10 8.

5 6 Ï}

35 }

2 9. 2

19 } 3 10. 2

5 } 3 , 21

11. 22, 21 12. 21 13. no solution 14. 2 3 } 2

15. 5 16. a 5 1 }

3 ; b 5 22 17. 4 pt 18. $1000

in the r% account and $3000 in the (r 1 1)% account

19. a.

Person

Fraction of roof

completedeach hour

Time(hours)

Fraction of roof

completed

Expert 1 }

x24

24 }

x

Assistant 3 } 5x 24

72 } 5x

b. Answers will vary. c. 24

} x

1 72

} 5x 5 1; 38.4 h

d. 64 h

Review for Mastery

1. 8, 6 2. 29, 12 3. 27.5 4. 23, 9


1. a. t 5 6 }

x 2 2 1

8 }

x b. 5 hours

2. a. A 5 (67 1 168x)(1 1 0.001x)

}} (1 2 0.003x)(355 1 555x)

b. The best

approximation of the model for the years shown is 1964. In 1964, Clemente had a career batting average of 0.307 and the estimate of the model is 0.306. 3. Answers will vary.

4. a. S 5 6003 1 1216.23x 1 58.05x2

}}} 1 1 0.17x 1 0.007x2

b. about 6,684,000 students 5. 5

6. a. C 5 1175.8 1 213.3x

}}} (1 1 0.107x)(356.2 1 28.1x 2 3.2x2)

b. $3.34 c.

00.00

3.00

3.10

3.20

3.30

3.40

3.50

1 2 3 4 x

C

Years since 1997

Avera

ge c

ost

(do

llars

per

passen

ger)

The average cost decreased from 1997 to 1998 and then increased from 1998 to 2002. You cannot use the graph to describe how the amount of passenger fares changed during the period because this graph just shows the average cost. You do not know what happened to the number of passengers during this period.

Challenge Practice

1. a 5 1 }

2 , b 5 4 2. a 5 1, b 5 9

3. a 5 24, b 5 4 4. a 5 49, b 5 5

5. a 5 119, b 5 47 6. Yes, 20,000 barrels of 87 octane and 40,000 barrels of 93 octane.

7. Yes, 60,000 barrels of 87 octane and 30,000 barrels of 93 octane. 8. The order cannot be fi lled. 9. Yes, 16,667 barrels of 87 octane and 3333 barrels of 93 octane for the 88 octane order, and 13,333 barrels of 87 octane and 26,667 barrels of 93 octane for the 91 octane order.

Chapter Review Game 1. y 5 2 2. 4x 1 5 3.

x 1 3 }

2x 1 1 4.

x(x 1 4) }

2(x 2 2)

5. 3x }}

(x 1 5)(x 1 6) 6.

213x 2 35 }}

x(x 1 3)(x 1 7) 7. x 5 9

8. x 5 0 or x 5 2

VINCULUM


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Holt McDougal Florida Larson Algebra 1