alg 1 ch 08 toc - mr. muscarello - home

Copyright © Big Ideas Learning, LLC Algebra 1 All rights reserved. Resources by Chapter

285

Chapter 8 Family and Community Involvement (English) ......................................... 286

Family and Community Involvement (Spanish) ......................................... 287

Section 8.1 ................................................................................................... 288

Section 8.2 ................................................................................................... 293

Section 8.3 ................................................................................................... 298

Section 8.4 ................................................................................................... 303

Section 8.5 ................................................................................................... 308

Section 8.6 ................................................................................................... 313

Cumulative Review ..................................................................................... 318

Algebra 1 Copyright © Big Ideas Learning, LLC Resources by Chapter All rights reserved. 286

Chapter

8 Graphing Quadratic Functions

Name _________________________________________________________ Date _________

Dear Family,

In this chapter, your student will learn how to graph a quadratic function. The graph of a quadratic function is called a parabola. The graph of a parabola is in the shape of a U. Your student will learn how to change the parent form of the quadratic function by altering the width and shifting the direction of the opening of the parabola.

Try this: You will need an open area, a softball, and three people. Have two people stand 10 to 15 feet away from each other. The third person should stand back, but be about in the middle of the other two (this person will be observing the throw). Give one person the ball and have him gently, throwing underhand, toss the ball to the person standing across from him.

The third person should observe the path of the ball. Talk over these observations using the questions below.

• What did you notice about the path of the ball?

• About how high did the ball travel off the ground?

• What happened to the ball once it reached its maximum height?

• What shape did the path of the ball make?

Now stand about 5 feet from each other and toss the ball as high as you can to the person standing across from you. Make the same observations as before.

Perform the task one last time, but this time stand at least 20 feet away from each other and toss the ball. Make the same observations.

• What changed with each throw?

Using the Internet, research the formula scientists/mathematicians use to plot parabolas on coordinate planes. This formula is a quadratic function.

Brainstorm together as a family other uses of parabolas in the real world. Can you think of any careers that would utilize the quadratic function and its graph?

You may be surprised at just how common the parabola and its graph are!


287

Capítulo

8 Hacer gráficas de funciones cuadráticas

Nombre _______________________________________________________ Fecha _________

Estimada familia:

En este capítulo, su hijo aprenderá a hacer gráficas de una función cuadrática. La gráfica de una función cuadrática se llama parábola. La gráfica de una parábola tiene forma de U. Su hijo aprenderá cómo cambiar la forma madre de la función cuadrática mediante la alteración del ancho y el cambio de dirección de la abertura de la parábola.

Prueben esto: Necesitarán un área al aire libre, una pelota de softball y tres personas. Dos personas se paran a una distancia de 10 y 15 pies entre sí. La tercera persona se para detrás, pero aproximadamente en el medio de las otras dos (esta persona observará el lanzamiento). Den la pelota a una persona y pídanle que lance con suavidad la pelota, desde abajo, a la persona que está parada al lado.

La tercera persona debería observar la trayectoria de la pelota. Usen las siguientes preguntas para hablar sobre estas observaciones.

• ¿Qué observaron sobre la trayectoria de la pelota?

• Aproximadamente, ¿cuál alto, desde el suelo, se desplazó la pelota?

• ¿Qué le sucedió a la pelota cuando alcanzó su altura máxima?

• ¿Qué forma tomó la trayectoria de la pelota?

Ahora, párense a 5 pies de distancia entre sí y lancen la pelota lo más alto posible a la persona que está al lado. Hagan las mismas observaciones que antes.

Repitan esto una última vez, pero esta vez párense al menos a 20 pies de distancia entre sí y lancen la pelota. Hagan las mismas observaciones.

• ¿Qué cambió en cada lanzamiento?

Consulten en Internet para investigar la fórmula que usan los científicos/matemáticos para marcar parábolas en los planos de coordenadas. Esta fórmula es una función cuadrática.

En familia, propongan otros usos de las parábolas en el mundo real. ¿Se les ocurren profesiones donde se use la función cuadrática y su gráfica?

¡Quizás les sorpresa saber cuán común son la parábola y su gráfica!


8.1 Start Thinking

Use a graphing calculator to graph the functions in the table. Then complete the table.

How does the value of the coefficient of 2x change the graph of the quadratic equation? Which graph looks the most different from the others? Explain.

Graph the equation.

1. 1y x= − − 2. 32 2y x= + 3. 2y x= − −

4. 3 3y x= + 5. y x= 6. 34 3y x= −

Use the Distributive Property to find the product.

1. ( )( )2 2x x− − 2. ( )( )6 2z z+ − 3. ( )( )8 1g g+ +

4. ( )( )7 3y y− − 5. ( )( )4 10m m − 6. ( )( )4 1x x− −

8.1 Warm Up

8.1 Cumulative Review Warm Up

Quadratic equation Shape Relationship to y x2=22y x= 21

2y x=

2y x= −

( )22y x=


289

8.1 Practice A

Name _________________________________________________________ Date __________

In Exercises 1–6, graph the function. Compare the graph to the graph of ( )f x x2.=

1. ( ) 24g x x= 2. ( ) 21.5h x x= 3. ( ) 213

j x x=

4. ( ) 23g x x= − 5. ( ) 252

k x x= − 6. ( ) 20.5n x x= −

In Exercises 7–9, use a graphing calculator to graph the function. Compare the graph to the graph of y x25 .= −

7. 25y x= 8. 20.5y x= − 9. 20.05y x= −

10. The arch support of a bridge can be modeled by 20.00125 ,y x= − where x and y are measured in feet.

a. The width of the arch is 800 feet. Describe the domain of the function. Explain.

b. Use a graphing calculator to graph the function, using the domain in part (a). Find the height of the arch.

11. Is the y-intercept of the graph of 2y ax= always 0? Explain.

In Exercises 12–15, determine whether the statement is always, sometimes, or never true. Explain your reasoning.

12. The graph of ( ) 2f x ax= is narrower than the graph of ( ) 2g x dx= when .d a= −

13. The graph of ( ) 2f x ax= opens in the same direction as the graph of ( ) 2g x dx=

when .d a=

14. The graph of ( ) 2f x ax= opens in the same direction as the graph of

( ) 2g x dx= when ( ) ( ).g x f x= −

15. The graph of ( ) 2f x ax= opens in the same direction as the graph of

( ) 2g x dx= when ( ) ( ).g x f x= −


8.1 Practice B

Name _________________________________________________________ Date _________

In Exercises 1–6, graph the function. Compare the graph to the graph of ( )f x x2.=

1. ( ) 27g x x= 2. ( ) 20.25h x x= 3. ( ) 272

j x x=

4. ( ) 253

g x x= − 5. ( ) 234

k x x= − 6. ( ) 20.4n x x= −

7. Describe and correct the error in graphing and comparing 2y x= and 22 .y x= −

8. The arch support of a bridge can be modeled by 21300

y x= − , where x and y are

measured in feet.

a. The width of the arch is 900 feet. Describe the domain of the function. Explain.

b. Use a graphing calculator to graph the function, using the domain in part (a). Find the height of the arch.

9. A parabola opens down and passes through the points ( )3, 4− and ( )1, 2− .

How do you know that ( )3, 4− could be the vertex?

10. Given the parabola ( ) 2.f x ax=

a. Find the value of a when the graph passes through ( )3, 1− and 0.a <

b. Find the value of a when the graph passes through ( )3, 1− and 0.a > Explain.

y

−2

2

−2 2 x

y = −2x2

y = x2

The graphs have the same vertex and the same axis of symmetry. The graph of is a reflection in the x-axis of the graph of


291

8.1 Enrichment and Extension

Name _________________________________________________________ Date __________

Working with Quadratic Functions In Exercises 1–14, use your knowledge of quadratic functions.

1. Write the equation of any quadratic function ( )2except .y x=

2. Graph the function.

3. Determine the maximum (or minimum) point of the equation.

4. Determine the domain and range for the function.

5. What is the solution to the equation 2 25?x =

6. Graph the function 2 25.y x= −

7. How does the graph of the function in Exercise 6 help in determining the solution to the equation?

8. What is the solution to the equation 2 16?x =

9. Graph the function 2 16.y x= −

10. How does the graph of the function in Exercise 9 help in determining the solution to the equation?

11. What is the solution to the equation 2 4?x = −

12. Graph the function 2 4.y x= +

13. Why is the graph of this function different from the graphs of the functions in Exercises 6 and 9?

14. Make a conjecture of the possible reasons for your answer to Exercise 13.


Puzzle Time

Name _________________________________________________________ Date _________

Where Does A Squirrel Keep Its Winter Clothes? Write the letter of each answer in the box containing the exercise number.

Compare the graph of the function to the graph of( )f x x 2 .=

1. ( ) 2b x x= − 2. ( ) 25p x x=

3. ( ) 213

q x x= 4. ( ) 24t x x= −

5. ( ) 20.2c x x= − 6. ( ) 26.4h x x=

7. ( ) 20.12r x x= 8. ( ) 28

5d x x= −

9. ( ) 223

s x x= 10. ( ) 219

k x x=

11. The graph of a parabolic bowl can be represented by ( ) 22

5 .g x x= Compare the graph to the graph of

( ) 2.f x x=

12. The decorated archway at the entrance to a craft fair can be represented by ( ) 27 .h x x= − Compare the

graph to the graph of ( ) 2.f x x=

Answers

E. vertical shrink by a factor of 13

T. vertical shrink by a factor of 19

K. reflection in the x-axis; vertical shrink by a factor of 0.2

N. reflection in the x-axis

A. vertical shrink by a factor of 25

T. vertical shrink by a factor of 0.12

R. reflection in the x-axis; vertical stretch by a factor of 4

N. vertical stretch by a factor of 5

R. reflection in the x-axis; vertical stretch by a factor of 8

5

I. vertical stretch by a factor of 6.4

E. vertical shrink by a factor of 23

U. reflection in the x-axis; vertical stretch by a factor of 7

8.1

6 2 11 7 4 9 3 10 8 12 1 5


293

8.2 Start Thinking

Identify the y-intercept of the equation .y mx b= + What is the y-intercept of the equation ?y mx= Explain how different y-intercept values relate to translations of the graph of .y mx=

Use your knowledge of the graph of 2y ax= to describe the effect of the constant c in the equation 2 .y ax c= + Describe the difference in the appearance of the graphs of 2y ax= − and 2 .y x c= −

Find the x- and y-intercepts.

1. 4x y+ = 2. 11y x= −

3. 2 13y x= − 4. 2 5 1x y− = −

5. 6 12x y− = 6. 16 3y x= +

Complete the statement with always, sometimes, or never. Explain your reasoning.

1. If 2 2 ,x y= then x is _______________ equal to .y

2. If x and y are real numbers, then x y+ is _______________ equal to y x+ .

3. For any real number d, the equation 5x d+ = will _______________ have no solution.

8.2 Warm Up



8.2 Practice A

Name _________________________________________________________ Date _________

In Exercises 1–3, graph the function. Compare the graph to the graph of ( )f x x 2 .=

1. ( ) 2 4g x x= + 2. ( ) 2 7h x x= + 3. ( ) 2 2k x x= −


4. ( ) 2 1g x x= − + 5. ( ) 2 3h x x= − − 6. ( ) 23 2j x x= −

In Exercises 7 and 8, describe the transformation from the graph of f to the graph of g. Then graph f and g in the same coordinate plane. Write an equation that represents g in terms of x.

7. ( )( ) ( )

22 1

3

f x x

g x f x

= +

= −

8. ( )( ) ( )

213 1

4

f x x

g x f x

= −

= +

In Exercises 9–12, find the zeros of the function.

9. 2 4y x= − 10. 2 64y x= −

11. ( ) 2 16f x x= − + 12. ( ) 22 50f x x= −

13. You drop a stick from a height of 64 feet. At the same time, your friend drops a stick from a height of 144 feet.

a. After how many seconds does your stick hit the ground?

b. How many seconds later does your friend's stick hit the ground?

In Exercises 14–17, sketch a parabola with the given characteristics.

14. The parabola opens down and the vertex is ( )0, 2 .

15. The vertex is ( )0, 4− and one of the x-intercepts is 3.

16. The related function is decreasing when 0x < and the zeros are 2 and 2.−

17. The lowest point on the parabola is ( )0, 1 .−

18. Your friend claims that in the equation 2 ,y ax c= + the vertex changes when the value of c changes. Is your friend correct? Explain your reasoning.


295

8.2 Practice B

Name _________________________________________________________ Date __________


1. ( ) 2 5g x x= + 2. ( ) 2 10h x x= + 3. ( ) 2 5j x x= −


4. ( ) 22 4g x x= − + 5. ( ) 214 1h x x= − − 6. ( ) 21

3 5k x x= +

In Exercises 7 and 8, describe the transformation from the graph of f to the graph of g. Then graph f and g in the same coordinate plane. Write an equation that represents g in terms of x.

7. ( )( ) ( )

212 4

2

f x x

g x f x

= − −

= −

8. ( )( ) ( )

22 7

9

f x x

g x f x

= +

= −

In Exercises 9–12, find the zeros of the function.

9. 2 81y x= − + 10. 23 75y x= −

11. ( ) 25 20f x x= − + 12. ( ) 212 27f x x= − +

13. The function 216 100y x= − + represents the height y (in feet) of a pencil x seconds after falling out the window of a school building. Find and interpret the x- and y-intercepts.

14. The paths of water from three different waterfalls are given below. Each function gives the height h (in feet) and the horizontal distance d (in feet) of the water.

Waterfall 1: 22.4 1.5h d= − +

Waterfall 2: 22.4 3h d= − +

Waterfall 3: 21.4 3h d= − +

a. Which waterfall drops water from the lowest point?

b. Which waterfall sends water the farthest horizontal distance?

c. What do you notice about the paths of Waterfall 1 and Waterfall 2?



Name ___________________________________________________ Date __________

Even and Odd Functions Contrary to common assumption, even or odd functions are not determined by the degree of the polynomial. The graphs of even functions are symmetric about the y-axis. If ( ) ( )f x f x− = for all values of x, then the function is even. On the other hand, the

graphs of odd functions are symmetric about the origin. If ( ) ( )f x f x− = − for all values of x, then the function is odd.

Even Odd

Example: Determine whether the function ( ) 3f x x x= − is even, odd, or neither.

( )( ) ( ) ( )

( )

3

3

3

3

f x x x

f x x x

x x

x x

= −

− = − − −

= − +

= − −

So, ( )f x is odd.

Determine whether the function is even, odd, or neither.

1. ( ) 32f x x= 2. ( ) 43f x x= −

3. ( ) 5f x x= − 4. ( ) 3 2f x x x= +

5. ( ) 2 3f x x= + 6. ( ) 3 3f x x= +

7. ( ) 3 24 3f x x x x= − − − 8. ( ) 7 5 37 8 3f x x x x x= − + + −

9. ( ) 4 23 5f x x x= + − 10. ( ) 5 7f x x x= − − +

x

y

−4

3−3

2

f(x) = x4 − 4x2

x

y

−4

2−2

4f(x) = x31

3


297

Puzzle Time

Name _________________________________________________________ Date __________

Where Do Birds Relax In Their Houses? Write the letter of each answer in the box containing the exercise number.

Compare the graph of the function to the graph of( )f x x2.=

1. ( ) 2 5j x x= − 2. ( ) 2 4m x x= +

3. ( ) 2 8c x x= − − 4. ( ) 26 7r x x= −

5. ( ) 21 93

g x x= + 6. ( ) 25 1412

p x x= − −

Write an equation that represents g in terms of x.

7. ( )( ) ( )

26 5

3

f x x

g x f x

= +

= +

8. ( )

( ) ( )

23 74

10

f x x

g x f x

= +

= −

9. ( )

( ) ( )

28 139

2

f x x

g x f x

= − −

= −

10. ( )( ) ( )

214 25

18

f x x

g x f x

= −

= +

Find the zeros of the function.

11. 2 4y x= − 12. 2 81y x= −

13. ( ) 2 36f x x= − + 14. ( ) 23 75f x x= −

15. The function 22 98y x= − + represents the height y (in inches) of a penny x seconds after falling to the ground. Find the x-intercept.

Answers

E. 9, 9x x= = −

N. 5, 5x x= − =

H. 7x =

T. 2, 2x x= − =

O. 6, 6x x= − =

B. 10x =

T. ( ) 214 7g x x= −

E. ( ) 234 3g x x= −

C. ( ) 26 8g x x= +

O. ( ) 289 15g x x= − −

R. reflection in the x-axis, translation 8 units down

F. vertical shrink by a factor of 13,

translation 9 units up

H. translation 4 units up

P. reflection in the x-axis, vertical shrink by a factor of 5

12, translation

14 units down

N. vertical stretch by a factor of 6, translation 7 units down

R. translation 5 units down

8.2

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


8.3 Start Thinking

Use a graphing calculator to graph the equation ( ) 23 1.f x x x= − + Find the coordinates of the y-intercept.

Explain how the coordinates of the y-intercept can be determined without graphing the equation.

The x-coordinate of the vertex can be found using the formula

2bxa

−= for any quadratic equation of the form

( ) 2 .f x ax bx c= + + Use the formula to find the x-coordinate of the vertex.

Complete the exercise.

1. Does ( )4, 3 satisfy the equation 23 7?y x x= − +

2. Does ( )0, 1− satisfy the equation 2 122 1?y x x= − + −

3. Does ( )5, 0 satisfy the equation 24 2 4?y x x= − +

4. Does ( )1, 9− − satisfy the equation 22 3 4?y x x= − + −

Solve the inequality. Graph the solution.

1. 4 12y ≥ − 2. 36 6t>

3. 9.35a > 4. 918

2t− ≥

8.3 Warm Up



299

8.3 Practice A

Name _________________________________________________________ Date __________

In Exercises 1 and 2, find the vertex, the axis of symmetry, and the y-intercept of

the graph.

1. 2.

In Exercises 3–6, find (a) the axis of symmetry and (b) the vertex of the graph

of the function.

3. ( ) 23 6f x x x= − 4. 25 3y x x= +

5. 27 14 1y x x= − + + 6. ( ) 24 20 15f x x x= − + +

In Exercises 7–10, graph the function. Describe the domain and range.

7. ( ) 23 12 6f x x x= − + 8. 25 20 9y x x= + −

9. 26 12 5y x x= − − − 10. ( ) 27 28 8f x x x= − + −

11. Describe and correct the error in finding the axis of symmetry of the graph of 22 16 7.y x x= − + +

In Exercises 12 and 13, tell whether the function has a minimum value or

a maximum value. Then find the value.

12. ( ) 25 20 3f x x x= − + 13. 23 12 7y x x= − + −

14. The vertex of a parabola is ( )2, 2 .− Another point on the parabola is ( )5, 7 .

Find another point on the parabola. Justify your answer.

In Exercises 15 and 16, use the minimum or maximum feature of a graphing calculator to approximate the vertex of the graph of the function.

15. 20.2 6 5y x x= + − 16. 25.3 3.6 2y x x= − + +

x

y

4

−6

−2−4

x

y

2

−2

3 5


8.3 Practice B

Name _________________________________________________________ Date _________

In Exercises 1 and 2, find the vertex, the axis of symmetry, and the y-intercept of the graph.

1. 2.

In Exercises 3–6, find (a) the axis of symmetry and (b) the vertex of the graph of the function.

3. ( ) 24 12f x x x= + 4. 25 20 4y x x= − − +

5. 28 24 13y x x= − + + 6. ( ) 223 6 15f x x x= − +

In Exercises 7–10, graph the function. Describe the domain and range.

7. ( ) 24 8 11f x x x= + + 8. 26 12 7y x x= − − −

9. 212 8 3y x x= − + 10. ( ) 22

3 4 2f x x x= − + +

11. Describe and correct the error in finding the vertex of the graph of 2 6 2.y x x= + +

In Exercises 12 and 13, tell whether the function has a minimum value or a maximum value. Then find the value.

12. ( ) 26 24 5f x x x= − + − 13. 213 8 1y x x= + −

In Exercises 14 and 15, use the minimum or maximum feature of a graphing calculator to approximate the vertex of the graph of the function.

14. 22.1 3y x xπ= − + + 15. 2 3 41.25 2 3y x x= − +

x

y

−2

4

2

−2−4−6

x

y

2

5

2−2

( )( )

6 32 2 1

So, the vertex is 3, 2 .

bxa

= − = − = −

−


301


Name _________________________________________________________ Date __________

Finding Maximum Values Maximum values are often needed in different circumstances in mathematics. How would one maximize the area of a rectangular field given 1000 feet of fencing?

Let x and y represent the length and width of the field.

Perimeter: 2 2 1000 500x y y x+ = → = −

Area: ( ) 2500 500A xy x x x x= = − = − +

To solve a problem of maximization, you must find the y-coordinate of the vertex, which is the maximum value of the function. Use the vertex formula for the x-value and substitute the x-value into the area equation to solve for the maximum area.

( )2 500500 , 250

2 2 1bA x x xa

= − + = − = − =−

( ) ( )2 2250 500 250 62,500 ftA = − + =

This is the maximum area possible.

Solve the problem.

1. Find the area of the largest possible rectangular garden that could be surrounded by 600 feet of fencing.

2. A family wants to fence in their lawn using one side of their house as one of the sides. They have 200 feet of fencing. How large is the area they can enclose?

3. A farmer is building a pen for his pigs with 180 feet of fencing. He is going to use a straight rock wall for one of the sides. What is the maximum area possible?

4. An art museum is making a rectangular garden in one corner of its courtyard. It only needs two sides of fencing, and it has 100 feet. What is the maximum area the museum can enclose?

x

y


Puzzle Time

Name _________________________________________________________ Date _________

What Has More Letters Than The Alphabet? Write the letter of each answer in the box containing the exercise number.

Find the axis of symmetry and the vertex of the graph of the function.

1. 22 8 7y x x= + + 2. 26 4 9y x x= − −

3. 23 12 255

y x x= − − − 4. 29 18 604

y x x= − − +

Describe the domain and range of the function.

5. 24 16 7y x x= + −

6. 26 48 40y x x= − + −

7. 23 12 9y x x= − + +

8. 25 40 60y x x= − +

Tell whether the function has a minimum value or a maximum value. Then find the value.

9. 23 12 1y x x= − +

10. 24 48 144y x x= − + −

11. 21 8 72

y x x= − − −

12. 22 2 7y x x= + +

13. The function ( ) 24 20h t t t= − + represents the height (in feet) of an athlete t seconds after pole vaulting. After how many seconds does the athlete reach his or her maximum height?

Answers

H. 1 1 29; ,3 3 3

x = −

S. ( )4; 4, 96x = − −

I. ( )10; 10, 35x = − −

E. ( )2; 2, 1x = − − −

T. 2.5

B. 5

F. maximum; 0

O. maximum; 25

E. minimum; –11

T. minimum; 132

C. all real numbers; 56y ≤

P. all real numbers; 20y ≥ −

F. all real numbers; 23y ≥ −

O. all real numbers; 21y ≤

8.3

13 2 9 8 11 4 12 7 5 10 3 6 1


303

8.4 Start Thinking

Consider the functions ( ) ( ) 23 and 1.f x x g x x= = + Replace x with x− in each function and simplify to find ( )f x− and ( ).g x−

Compare the output values of ( ) ( )and .f x f x− Make the same comparison for ( ) ( ) and .g x g x−

Find the coordinates of the vertex.

1. 2y x= 2. 2 2y x= +

3. 223y x= − 4. 2 5y x x= −

5. 2y x= − 6. 23 2y x x= + +

Tell whether the volume of the solid is a linear or nonlinear function of the missing dimension(s). Explain.

1. 2.

3. 4.

8.4 Warm Up


ss

8 m 3 mm

r

h

2 ft

4 ft

16 in.

h


8.4 Practice A

Name _________________________________________________________ Date _________

In Exercises 1–3, determine whether the function is even, odd, or neither.

1. ( ) 4 1xg x = − 2. ( ) 2 5f x x= − 3. ( ) 22 5h x x= +

In Exercises 4 and 5, determine whether the function represented by the graph is even, odd, or neither.

4. 5.

In Exercises 6–8, find the vertex and the axis of symmetry of the graph of the function.

6. ( ) ( )24 2f x x= + 7. ( ) ( )213 3f x x= − 8. ( )25 7y x= − +

In Exercises 9–11, graph the function. Compare the graph to the graph of ( )f x x2 .=

9. ( ) ( )22 1g x x= + 10. ( ) ( )23 2g x x= − 11. ( ) ( )214 6g x x= +


12. ( )25 3 2y x= − + − 13. ( ) ( )22 2 5f x x= − + 14. ( )23 5 4y x= − + −

In Exercises 15 and 16, graph the function. Compare the graph to the graph of ( )f x x2 .=

15. ( ) ( )23 2g x x= − + 16. ( ) ( )22 4g x x= − + −

In Exercises 17 and 18, rewrite the quadratic function in vertex form.

17. 22 4 1y x x= + − 18. ( ) 23 12 4f x x x= − +

19. The graph of 2y x= is translated 4 units left and 3 units down. Write an equation for the function in vertex form and in standard form. Describe advantages of writing the function in each form.

x

y

2

2−2

x

y

2

4

6

2−2


305

8.4 Practice B

Name _________________________________________________________ Date __________

In Exercises 1–3, determine whether the function is even, odd, or neither.

1. ( ) 23 2f x x x= + 2. ( ) 23g x x= 3. ( ) 21

3 2h x x= −

In Exercises 4 and 5, determine whether the function represented by the graph is even, odd, or neither.

4. 5.


6. ( ) ( )213 6f x x= − + 7. ( ) ( )29 4f x x= − 8. ( )210 9y x= − +

In Exercises 9–11, graph the function. Compare the graph to the graph of ( )f x x2 .=

9. ( ) ( )24 2g x x= + 10. ( ) ( )213 5g x x= − 11. ( ) ( )21

6 1g x x= −


12. ( )26 4 3y x= − − 13. ( ) ( )24 1 5f x x= − + + 14. ( )23 2y x= − + −

In Exercises 15 and 16, graph the function. Compare the graph to the graph of ( )f x x2 .=

15. ( ) ( )23 2 1g x x= + − 16. ( ) ( )212 1 3g x x= − − +

In Exercises 17 and 18, rewrite the quadratic function in vertex form.

17. 25 10 2y x x= − + 18. ( ) 22 8 5f x x x= − + +

19. The graph of 2y x= is reflected in the x-axis and translated 3 units right and 2 units up. Write an equation for the function in vertex form and in standard form. Describe advantages of writing the function in each form.

x

y

2

−2

−2

x

y

2

−3

−3 3



Name _________________________________________________________ Date _________

Cubic Functions Using the Parent Graph Similar to quadratic functions in vertex form, cubic functions can shift up and down and left and right using the same properties. Use the parent function to the right to shift the following function.

Example: Graph ( )31 3 42

y x= − + .

Graph the equation using the parent function as a reference.

1. ( )32y x= + 2. ( )35 1y x= − +

3. 3 2y x= − + 4. ( )31 1 22

y x= + −

5. ( )32 4y x= − 6. ( )32 5y x= − + −

x

y

2

2−2

−2

x

y

2

4

6

2 4 6

y = x3

y = 12(x − 3)3 + 4


307

Puzzle Time

Name _________________________________________________________ Date __________

How Do You Make Sure You Pass A Geometry Test? Write the letter of each answer in the box containing the exercise number.


1. ( ) 25 2f x x= + 2. ( ) 34c x x= −

3. ( ) 6 9g x x= −

Find the vertex and the axis of symmetry of the graph of the function.

4. ( ) ( )24 2d x x= + 5. ( ) ( )27 5 6r x x= − + −

6. ( ) ( )22 8 12h x x= − + 7. ( ) ( )29 3 7s x x= − − +

Compare the graph of the function to the graph of( )f x x2.=

8. ( ) ( )23 4b x x= + 9. ( ) ( )21 9w x x= − − −

10. ( ) ( )218 6k x x= − 11. ( ) ( )27 10m x x= + +

Write a quadratic function in vertex form whose graph has the given vertex and passes through the given point.

12. vertex: ( )4, 2 ;− − passes through ( )7, 7−

13. vertex: ( )2, 3 ; passes through ( )4, 11

14. vertex: ( )4, 6 ;− passes through ( )0, 26−

15. vertex: ( )8, 1 ; passes through ( )10, 13

16. A portion of a ski slope in the shape of a parabola has a vertex of ( )45, 125 and passes through the point ( )70, 0 .

Answers

A. odd G. even

W. neither

E. ( )8, 12 ; 8x =

S. ( )2, 0 ; 2x− = −

A. ( )3, 7 ; 3x =

L. ( )5, 6 ; 5x− − = −

K. ( ) ( )22 2 3f x x= − +

L. ( ) ( )215 45 125f x x= − − +

H. ( ) ( )24 2f x x= + −

O. ( ) ( )23 8 1f x x= − +

N. ( ) ( )22 4 6f x x= − + +

T. reflection in the x-axis, translation 1 unit right and 9 units down

N. translation 7 units left and 10 units up

L. translation 4 units left, and a vertical stretch by a factor of 3

E. translation 6 units right, and a vertical shrink by a factor of 1

8

8.4

13 11 15 3 2 16 5 9 12 6 7 14 1 8 10 4


8.5 Start Thinking

Solve the equation ( ) 22 1f x x x= − − by following these steps.

Step 1: Replace ( ) with 0.f x

Step 2: Factor 22 1.x x− −

Step 3: Set up two equations such that each factor is equal to zero.

Step 4: Solve each equation separately.

Factor the expression.

1. 24 49x − 2. 2 6 8x x+ +

3. 22 9 5a a− − 4. 2 3x x−

5. 2 5 4a a+ + 6. 22 7 4t t+ −

Write an equation in slope-intercept form of the line shown.

1. 2.

8.5 Warm Up


x

y

2

6

−4

x

y

2

2

−2

−4

−2


309

8.5 Practice A

Name _________________________________________________________ Date __________

In Exercises 1 and 2, find the x-intercepts and axis of symmetry of the graph of the function.

1. 2.

In Exercises 3–6, graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts. Describe the domain and range of the function.

3. ( ) ( )( )3 1f x x x= + − 4. ( )( )5 1y x x= − − +

5. ( ) 22 16f x x x= − 6. 2 8 7y x x= + +

In Exercises 7–10, find the zero(s) of the function.

7. ( )( )4 5 9y x x= − − − 8. ( ) ( )( )14 3 2f x x x= + −

9. ( ) 2 7 30g x x x= − − 10. 22 10y x x= − −

In Exercises 11–14, use zeros to graph the function.

11. ( )( )1 3y x x= + − 12. ( ) ( )( )2 2 6f x x x= − + +

13. ( ) 2 10 21g x x x= − + 14. 2 6y x x= − −

In Exercises 15–19, write a quadratic function in standard form whose graph satisfies the given conditions.

15. vertex: ( )5, 4−

16. x-intercepts: 2 and 7

17. passes through ( ) ( ) ( )3, 0 , 1, 0 , and 1, 8− −

18. axis of symmetry: 3x = −

19. passes through: ( ) ( )4, 0 and 4, 0−

x

y2

−2

y = (x + 2)(x − 2)

−3

x

y4

−8

−2−6

y = −2(x + 1)(x + 4)


8.5 Practice B

Name _________________________________________________________ Date _________

In Exercises 1 and 2, find the x-intercepts and axis of symmetry of the graph of the function.

1. ( ) ( )13 5f x x x= − + 2. ( ) ( )( )9 6 4g x x x= + −

In Exercises 3–6, graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts. Describe the domain and range of the function.

3. ( ) ( )( )4 3 2f x x x= + + 4. ( )( )3 4 2y x x= − − +

5. ( ) 2 7 12p x x x= − + 6. 22 20 42y x x= + +

In Exercises 7–10, find the zero(s) of the function.

7. ( ) ( )( )23 8 5f x x x= + − 8. ( ) 23 13 4g x x x= + +

9. ( )( )2 25 7y x x= − + 10. 3 81y x x= −

In Exercises 11–14, use zeros to graph the function.

11. ( ) ( )( )2 5 3f x x x= − − − 12. ( ) 2 2 24g x x x= + −

13. 24 16 20y x x= − − + 14. ( ) 23 12f x x= −

In Exercises 15–19, write a quadratic function in standard form whose graph satisfies the given conditions.

15. vertex: ( )6, 2−

16. x-intercepts: 5 and 8−

17. passes through ( ) ( ) ( )4, 0 , 2, 0 , and 0, 4− −

18. y decreases as x increases when 1;x < y increases as x increases when 1x >

19. range: 6y ≤

20. The cross section of a satellite dish can be modeled by the function ( )216 9 ,y x= −

where x and y are measured in feet. The x-axis represents the top of the opening of the dish.

a. How wide is the satellite dish?

b. How deep is the satellite dish?


311


Name _________________________________________________________ Date __________

Explore Higher Degree Polynomial Functions 1. Explain the word simplify in your own words.

2. Simplify the following and name the polynomial by degree and number of terms.

a. ( )( )2 7 5x x x− − −

b. ( )21 31 52 4

x x x+ − −

c. ( )3 4 2 2x x− − − +

3. Explain the word factor in your own words.

4. Factor the following after you name the polynomial by degree and number of terms.

a. 9 6 3 1x x x+ − −

b. 4 81x −

c. 23 20 12x x− − −

5. What are the roots of an equation? What are the zeros of a function?

6. Solve the equations by factoring if ( ) 0.f x = State how many zeros each function has.

a. ( ) 24 36 80f x x x= + +

b. ( ) 2 9f x x= −

c. ( ) 2 10 25f x x x= − + −

7. What do you notice about the degree of each polynomial function and the number of zeros?

8. What does it mean when a function has one zero? two zeros? no zeros ? You may use a drawing to explain.


Puzzle Time

Name _________________________________________________________ Date _________

What Did One Wall Say To The Other Wall? Write the letter of each answer in the box containing the exercise number.


1. 2 16y x= − 2. ( ) 2 10f x x x= −

3. ( ) 2 7 12r x x x= + + 4. 23 18 24y x x= − +


5. ( ) ( )( )3 3 9s x x x= − − − 6. ( ) ( )( )16 4 12h x x x= + −

7. 2 17 30y x x= − + 8. ( ) 24 12 72g x x x= − + +

9. 3 144y x x= − 10. ( ) ( )( )214 49c x x x= + −

11. ( ) 3 2 16 16v x x x x= − − +

12. ( ) 3 25 4 20k x x x x= + − −

Write a quadratic function in standard form whose graph satisfies the given condition(s).

13. vertex: ( )9, 4− − 14. x-intercepts: 8 and 5−

15. passes through ( ) ( ) ( )3, 0 , 4, 0 , 2, 20−

16. passes through ( ) ( ) ( )3, 0 , 7, 0 , 6, 36− −

17. Write a cubic function in standard form whose graph has x-intercepts of 4, 2, and 6.− −

18. Write a cubic function in standard form whose graph has x-intercepts of 8, 1, and 5.−

Answers

T. ( )5, 25 ; 5x− =

O. ( )7 712 4 2, ; x− − = −

M. ( )3, 3 ; 3x− =

E. ( )0, 16 ; 0x− =

R. ( ) ( )4, 0 , 12, 0−

Y. ( ) ( )2, 0 , 15, 0

O. ( ) ( )3, 0 , 6, 0−

H. ( ) ( )3, 0 , 9, 0

U. ( ) ( ) ( )4, 0 , 1, 0 , 4, 0−

E. ( ) ( ) ( )12, 0 , 0, 0 , 12, 0−

C. ( ) ( ) ( )5, 0 , 2, 0 , 2, 0− −

T. ( ) ( ) ( )14, 0 , 7, 0 , 7, 0− −

A. ( ) 22 2 24f x x x= − + +

N. ( ) 3 22 43 40f x x x x= + − +

E. ( ) 2 3 40f x x x= + −

R. ( ) 3 28 48f x x x= − −

E. ( ) 2 18 77f x x x= + +

T. ( ) 24 16 84f x x x= − −

8.5

4 13 9 16 7 3 11 15 2 10 5 14 12 8 6 18 1 17


313

8.6 Start Thinking

Complete the table.

Sketch a graph for each type of function.

Describe one example of a real-life situation for each type of function.

Tell which quadrant or axis the point lies on.

1. ( )1, 0− 2. ( )4, 6− 3. ( )1, 3− 4. ( )1, 2

5. ( )3, 4− 6. ( )2, 0− 7. ( )4, 5− 8. ( )6, 1−

9. ( )3, 3 10. ( )5, 1− 11. ( )3, 0− 12. ( )1, 3− −

Write a system of linear equations that has the ordered pair as its solution.

1. ( )4, 4 2. ( )3 13− − 3. ( )1, 7−

4. ( )16, 26− 5. ( )1, 3 6. ( )3, 2−

8.6 Warm Up


Type of function General form Graph characteristics linear

exponential quadratic


8.6 Practice A

Name _________________________________________________________ Date _________

In Exercises 1 and 2, tell whether the points appear to represent a linear, an exponential, or a quadratic function.

1. 2.

In Exercises 3–6, plot the points. Tell whether the points appear to represent a linear, an exponential, or a quadratic function.

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

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

5. ( ) ( ) ( ) ( ) ( )3, 6 , 2, 1 , 1, 2 , 0, 3 , 1, 2− − − − −

6. ( ) ( ) ( ) ( ) ( )1 19 32, , 1, , 0, 1 , 1, 3 , 2, 9− −

7. The table shows the demand for a certain commodity (measured in thousands), where x is the number of the month of the year.

a. During what month is the demand at a minimum?

b. Plot the points. Let x be the independent variable. Then determine the type of function that best represents this situation.

c. Write a function in standard form that models the data.

d. Use the function from part (c) to find the demand for the commodity (measured in thousands) during August.

x

y

4

2

2−2

x

y

−2

−5

2 4−2

Number of month, x 1 2 3 4 5 6

Demand, y 5 2 1 2 5 10


315

8.6 Practice B

Name _________________________________________________________ Date __________

In Exercises 1 and 2, tell whether the points appear to represent a linear, an exponential, or a quadratic function.

1. 2.

In Exercises 3–6, plot the points. Tell whether the points appear to represent a linear, an exponential, or a quadratic function.

3. ( ) ( ) ( ) ( ) ( )1 19 32, , 1, , 0, 1 , 1, 3 , 2, 9− −

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

5. ( ) ( ) ( ) ( ) ( )4, 2 , 2, 1 , 0, 0 , 2, 1 , 4, 2− − − −

6. ( ) ( ) ( ) ( ) ( )3, 2 , 2, 1 , 1, 0 , 0, 1 , 1, 2− − − − −

In Exercises 7–10, tell whether the table of values represents a linear, an exponential, or a quadratic function.

7. 8.

9. 10.

11. Write a function that has constant second differences of 4.

x

y

4

2

2 4 x

y

4

2

2−2

x −3 −2 −1 0 1 2

y 0.9 0.4 0.1 0 0.1 0.4

x 1 2 3 4 5 6

y 1 −1 −3 −5 −7 −9

x 1 2 3 4 5 6

y 9 4 1 0 1 4 x −1 0 1 2 3

y 6 3 32 3

4 38



Name _________________________________________________________ Date _________

Increasing and Decreasing Functions When the y-values of a function increase as the x-values increase, the function is increasing, and when the y-values decrease as the x-values increase, the function is decreasing. However, at the maxima or minima of the graph, the function neither increases nor decreases. It is not a simple task to find the intervals on which a function increases or decreases, but the task is easier when given a graph.

Example: Consider the graph shown above.

The intervals on which the function increases are ( ) ( ), 1 2, −∞ − ∪ ∞ .

The interval on which the function decreases is ( )1, 2 .− Notice the use of parentheses, because the endpoints of the intervals are not included. Also be aware that when giving increasing and decreasing intervals, only use the x-values.

Determine on what interval(s) the function is increasing or decreasing.

1. 2.

3. 4.

x

y

−8

2−2

(2, −6)

(−1, −3)

x

y

−4

2 4

x

y2

−4

2−2

(2, −4)

(0, 0)

(−2, −4)

x

y3

−2

2

(−1, 2)

(1, −2)

x

y

4

−4

4

(1, −5)

(−3, 4)


317

Puzzle Time

Name _________________________________________________________ Date __________

Where Does A Snake Go To Get A New Skin? Write the letter of each answer in the box containing the exercise number.

Tell whether the points represent a linear, an exponential, or a quadratic model.

1. ( ) ( ) ( ) ( ) ( )2, 6 , 0, 4 , 2, 6 , 4, 0 , 6, 14− − −

F. Linear G. Exponential H. Quadratic

2. ( ) ( ) ( ) ( ) ( )2, 14 , 1, 10 , 0, 6 , 1, 2 , 2, 2− − −

A. Linear B. Exponential C. Quadratic

3. ( ) ( ) ( ) ( ) ( )231, , 0, 2 , 1, 6 , 2, 18 , 3, 54−

S. Linear T. Exponential U. Quadratic

Write the function represented by the points.

4. ( ) ( ) ( ) ( ) ( )4, 7 , 2, 3 , 0, 1 , 2, 5 , 4, 9− − − −

E. 2 1y x= + F. 2xy = G. 23 3 4y x x= + +

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

M. 8 6y x= + N. ( )8 6xy = O. 2 6 8y x x= − +

6. ( ) ( ) ( ) ( ) ( )521, , 0, 5 , 1, 10 , 2, 20 , 3, 40−

R. 5 2y x= + S. ( )5 2xy = T. 25 2 1y x x= + +

7. ( ) ( ) ( ) ( ) ( )0, 3 , 1, 0 , 2, 1 , 3, 0 , 4, 3−

B. 4 3y x= + C. ( )4 3xy = D. 2 4 3y x x= − +

8.6

3 5 2 6 1 4 7


Chapter

8 Cumulative Review

Name _________________________________________________________ Date _________

Solve the equation.

1. 1 5 10 7 9 2x x x+ − = − − 2. 4 4 20 2y y+ = +

Solve the inequality.

3. ( )8 3 2 4 1h h+ ≥ + 4. 2 3 5 4y− − − ≥ − 5. 2 9 1 6x + + >

6. Your local bank offers free checking for accounts with a balance of at least $500. You have a balance of $516.46, and you write a check for $31.96. How much do you need to deposit to avoid being charged a service fee?

7. A number x plus 32 is no more than 38. Write this sentence as an inequality.

Graph the linear equation or linear inequality.

8. 2 3 6x y− = 9. 4.5y > −

Write an equation of the line in slope-intercept form that passes through the given point and is parallel to the given line.

10. ( )3, 2 ; 1y x− = − 11. ( ) 230, 5 ; 1 ( 1)y x− − = − 12. ( )4, 6 ; 6 3 9x y− − − =

Solve the system of linear equations by graphing, substitution, or elimination.

13. 5 32

y xy

= += −

14. 2 9 254 9 23

x yx y

− − = −− − = −

15. 3 13 3 15

x yx y+ =

− − = −

16. Your school is selling tickets for a musical. On the first day, three children’s tickets and nine adult tickets are sold for a total of $75. On the second day, eight children’s tickets and five adult tickets are sold for a total of $67. How much does one children’s ticket and one adult ticket cost?

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

17. 4 4 3

2 3 423x y zx y z

− −

− 18. 4

33mm

− 19.

3 1 1

4 0 03x y zx y z

− −

−

Evaluate the expression.

20. 3 64 21. 4 532 22. 5 243−

Evaluate the function for the given value of x.

23. 4 ; 1xy x= = − 24. ( )7 2 ; 0xy x= − − = 25. ( ) 74(7) ; 2xf x x= = −


319

Chapter

8 Cumulative Review (continued)

Name _________________________________________________________ Date __________

Write a function that represents the situation.

26. A $1000 investment increases in value by 5% every year.

27. A $60 video game decreases in value by 80% every year.

Solve the equation. Check your solution.

28. 5 122 2x − = 29. 5 3 284 4x + = 30. 4 8 83 9x x− +=

Find the sum or the difference.

31. ( ) ( )2 5 4 9g g− − + + 32. ( ) ( )3 4 11 2h h+ + − −

33. ( ) ( )1 5 14x x− − + 34. ( ) ( )9 8 7 6y y+ − − −

Find the product.

35. ( )( )6 4x x− +

36. ( )( )8 5y y− +

37. ( )22x +

38. ( )( )3 5 3 5m m− −

39. ( )211 5x y− −

Solve the equation.

40. ( )( )4 12 7 28 0x x− + = 41. 224 3 0g g− =

Factor the polynomial.

42. 2 14 15m m+ − 43. 2 5 24z z+ − 44. 2 16 17x x− −

45. 25 30 40x x− + 46. 22 2 4y y+ − 47. 24 4 8w w− −

48. A rock is thrown from the top of a tall building. The distance d (in feet) between the rock and the ground t seconds after it is thrown is given by 216 4 382.d t t= − − + How long after the rock is thrown is it 370 feet from the ground?

Factor the polynomial.

49. 2 121x − 50. 2 24 144a a− + 51. 24 36b −

Solve the equation.

52. 2 81 0z − = 53. 2 26 169 0y y− + =


Chapter

8 Cumulative Review (continued)

Name _________________________________________________________ Date _________

Factor the polynomial completely.

54. 3 22 8 3 12x x x+ − − 55. 3 25 10 7 14y y y− + −

Graph the function. Compare the graph to the graph of f x x2( ) .=

56. ( ) 24h x x= 57. ( ) 20.2t x x= 58. ( ) 225

n x x= −

59. ( ) 27a x x= − 60. ( ) 20.625r x x= − 61. ( ) 212

m x x=

62. ( ) 2 3g x x= + 63. ( ) 2 10h x x= + 64. ( ) 2 10p x x= −

65. ( ) 2 2s x x= − − 66. ( ) 24 2p x x= + 67. ( ) 21 55

q x x= − −


68. 2 4y x= − 69. ( ) 29 36f x x= − + 70. ( ) 250 18f x x= −

71. The function ( ) 2016f t t s= − + represents the approximate height (in feet) of

an object falling t seconds after it is dropped from an initial height 0s (in feet). A watermelon is dropped from a height of 100 feet.

a. After how many seconds does the watermelon hit the ground?

b. Suppose the initial height is adjusted by k feet. How will this affect the answer for part (a)?

Find (a) the axis of symmetry and (b) the vertex of the graph of the function.

72. 210 40 9y x x= − − − 73. ( ) 24 24 30f x x x= − −

Graph the function. Describe the domain and range.

74. ( ) 22 16 9f x x x= − − + 75. ( ) 2 18 1f x x x= − + −

Tell whether the function has a minimum value or a maximum value. Then find the value.

76. ( ) 23 24 5f x x x= − − + 77. ( ) 25 40 14f x x x= + −

78. ( ) 27 28 10f x x x= − + − 79. ( ) 29 36 21f x x x= − +


321

Cumulative Review (continued) Chapter

8

Name _________________________________________________________ Date __________


80. ( ) 4f x x= 81. ( ) 2 5g x x= + 82. ( ) 24 8 5h x x x= + −


83. ( ) 21 ( 2)4

f x x= −

84. ( ) 23( 1)g x x= −

85. ( ) 2( 3)h x x= +

86. ( ) 23( 7) 8f x x= − − −

87. ( ) ( )28 2 9g x x= + +

Graph the function. Compare the graph to the graph of f x x2( ) .=

88. ( ) ( )22 3g x x= − 89. ( ) ( )24 1 5g x x= + +

Graph the quadratic function.

90. ( ) ( )( )2 5 1f x x x= − + 91. ( )( )3 2 7y x x= − + −

92. ( ) 2 36f x x= − 93. ( ) 2 2 15h x x x= − −

Find the zero(s) of the function.

94. ( )( )3 7 1y x x= − + − 95. ( ) 2 15 26g x x x= + +

96. ( ) ( )( )23 9f x x x= + − 97. ( ) 22 6 20h x x x= − −

Tell whether the data represents a linear, an exponential, or a quadratic function. Then write the function.

98. ( ) ( ) ( ) ( ) ( )2, 4 , 1, 7 , 0, 10 , 1, 13 , 2, 16− −

99. ( ) ( ) ( ) ( ) ( )2, 5 , 1, 8 , 0, 9 , 1, 8 , 2, 5− − − − − − −

100. ( ) ( ) ( ) ( ) ( )0, 1 , 1, 3 , 2, 9 , 3, 27 , 4, 81

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