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Algebra 2 Copyright © Big Ideas Learning, LLC Resources by Chapter All rights reserved. 28

Chapter 2 Family and Community Involvement (English) ............................................ 29

Family and Community Involvement (Spanish) ........................................... 30

Section 2.1 ..................................................................................................... 31

Section 2.2 ..................................................................................................... 36

Section 2.3 ..................................................................................................... 41

Section 2.4 ..................................................................................................... 46

Cumulative Review ....................................................................................... 51

Copyright © Big Ideas Learning, LLC Algebra 2 All rights reserved. Resources by Chapter

29

Chapter

2 Quadratic Functions

Name _________________________________________________________ Date __________

Dear Family,

How warm does the temperature get during the summer months in the city or town where you live? How cold does the temperature get during the winter months? It may surprise you how often you can use a quadratic function to model naturally occurring data such as average monthly high and low temperatures in a city.

Use an almanac or the Internet to research information about the weather in the city or town where you live. Then complete each table below. For each table, let = 1x represent January, = 2x represent February, and so on.

• Make a scatter plot of each data set. Do you notice any patterns? Does the data show a quadratic relationship? How do you know?

• If possible, use a graphing calculator to find a quadratic function that models each set of data. Graph the function on your scatter plot. Is it a good fit? Explain.

• Why do you think average monthly temperature data usually follows a quadratic pattern?

• If average global temperatures are going to increase over time, then how do you think these changes will affect your graphs?

Choose a city you would like to visit in the United States. Then complete the tables and answer the questions above for the city you chose. Compare the graphs to the ones that represent your city. Do the greatest average monthly temperatures occur in the same month?

Think of other naturally occurring data that may follow a quadratic pattern. Does average monthly precipitation or average monthly snowfall follow a quadratic pattern?

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

Average high temperature (oF),y

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

Average low temperature (oF),y


Capítulo

2 Funciones cuadráticas

Nombre _______________________________________________________ Fecha ________

Estimada familia:

¿Cuán cálida es la temperatura durante los meses de verano en la ciudad donde viven? ¿Cuán fría es la temperatura durante los meses de invierno? Quizás les sorprenda saber con qué frecuencia usan una función cuadrática para representar datos que surgen naturalmente, tal como el promedio de temperaturas altas y bajas en una ciudad.

Consulten en un almanaque o Internet para investigar sobre el tiempo en la ciudad donde viven. Luego, completen la siguiente tabla. Para cara tabla, imaginen que

= 1x representa enero, = 2x representa febrero, etc.

• Hagan un diagrama de dispersión para cada conjunto de datos. ¿Observan algún patrón? ¿Los datos muestran una relación cuadrática? ¿Cómo lo saben?

• Si es posible, usen una calculadora gráfica para hallar una función cuadrática que represente cada conjunto de datos. Hagan una gráfica de la función en su diagrama de dispersión. ¿Es un buen ajuste? Expliquen.

• ¿Por qué creen que los datos sobre la temperatura mensual promedio sigue un patrón cuadrático?

• Si las temperaturas promedio globales van a aumentar con el transcurso del tiempo, entonces, ¿cómo creen que estos cambios afectarán a sus gráficas?

Elijan una ciudad de Estados Unidos que les gustaría visitar. Luego, completen las tablas y respondan las preguntas mencionadas anteriormente sobre la ciudad que eligieron. Comparen las gráficas con las gráficas que representan a su ciudad. ¿Las temperaturas promedio más altas ocurren en el mismo mes?

Piensen en otros datos que surgen naturalmente que tal vez sigan un patrón cuadrático. ¿La precipitación mensual promedio o la nevada mensual promedio siguen un patrón cuadrático?

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

Temperatura alta promedio °( F), y

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

Temperatura baja promedio °( F), y


31

2.1 Start Thinking

Make a table of values and use it to graph the following functions on the same coordinate plane. Use the same x-values for each function.

( )( ) ( )( )( )

2

2

2

2

1

2

1

f x x

f x x

f x x

f x x

=

= −

=

= +

Describe how the graphs of the last three functions differ from the graph of ( ) 2.f x x=

Multiply.

1. ( )( )3 2 2 4x x− − 2. ( )( )5 2 4 1x x+ +

3. ( )( )4 2 3x y x y+ − 4. ( )3 4 1a a +

5. ( )( )4 1 5 2x x+ − 6. ( )( )5 4 3 2y y+ +

Write a function g whose graph represents the indicated transformation of the graph f.

1. ( ) 6;f x x= + translation 3 units right

2. ( ) 3;f x x= − translation 1 unit left

3. ( ) 5 2 3;f x x= − − translation 1 unit down

2.1 Warm Up

2.1 Cumulative Review Warm Up


2.1 Practice A

Name _________________________________________________________ Date _________

In Exercises 1–6, describe the transformation of 2( )f x x= represented by g. Then graph each function.

1. ( ) 2 2g x x= − 2. ( ) 2 1g x x= + 3. ( ) ( )21g x x= +

4. ( ) ( )22g x x= − 5. ( ) ( )25g x x= − 6. ( ) ( )22 1g x x= + −


7. ( ) 22g x x= − 8. ( ) ( )22g x x= − 9. ( ) 214g x x=

10. Describe and correct the error in analyzing the graph of ( ) 213 .f x x= −

In Exercises 11 and 12, describe the transformation of the graph of the parent quadratic function. Then identify the vertex.

11. ( ) ( )22 3 2f x x= + + 12. ( ) 25 1f x x= − −

In Exercises 13 and 14, write a rule for g described by the transformations of the graph of f. Then identify the vertex.

13. ( ) 2;f x x= vertical stretch by a factor of 3 and a reflection in the x-axis, followed by a translation 3 units down

14. ( ) 24 5;f x x= + horizontal stretch by a factor of 2 and a translation 2 units up, followed by a reflection in the x-axis

15. Let the graph of g be a translation 4 units down and 3 units right, followed by a horizontal shrink by a factor of 1

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

a. Identify the values of a, h, and k. Write the transformed function in vertex form.

b. Suppose the horizontal shrink was performed first, followed by the translations. Identify the values of a, h, and k, and write the transformed function in vertex form.

x

g

y

−8

−4

4−4 The graph of g is a reflection in the x-axis, followed by a vertical stretch by a factor of of the graph of the

parent quadratic function.


33

2.1 Practice B

Name _________________________________________________________ Date __________


1. ( ) 2 3g x x= + 2. ( ) ( )25g x x= + 3. ( ) ( )26 4g x x= + −

4. ( ) ( )21 5g x x= − + 5. ( ) ( )24 3g x x= − + 6. ( ) ( )28 2g x x= + −


7. ( ) ( )212g x x= − 8. ( ) 21

3 2g x x= + 9. ( ) ( )213 1g x x= +

In Exercises 10 and 11, describe the transformation of the graph of the parent quadratic function. Then identify the vertex.

10. ( ) ( )23 6 4f x x= − + − 11. ( ) ( )213 2 1f x x= − +

In Exercises 12 and 13, write a rule for g described by the transformations of the graph of f. Then identify the vertex.

12. ( ) 2;f x x= vertical shrink by a factor of 12 and a reflection in the y-axis,

followed by a translation 2 units left

13. ( ) ( )24 2;f x x= + + horizontal shrink by a factor of 13 and a translation

2 units up, followed by a reflection in the x-axis

14. Justify each step in writing a function g based on the transformations of ( ) 24 3 .f x x x= −

translation 3 units up followed by a reflection in the y-axis

( ) ( ) 3h x f x= +

24 3 3x x= − +

( ) ( )g x h x= − 24 3 3x x= + +


2.1 Enrichment and Extension

Name _________________________________________________________ Date _________

Transformations of Quadratic Functions Displayed below are 10 parabolas.

The equations for three of the graphs are as follows.

( )

( )

2

2

2

3 1 1

3

3 4 2

y x

y x

y x

= − − −

=

= + −

Find the equations of the other seven parabolas with the help of a graphing calculator.

x

y

4

4 6 82−2−6−8


35

Puzzle Time

Name _________________________________________________________ Date __________

What Is The Most Densely Populated Country On The Mainland Of The Americas? Write the letter of each answer in the box containing the exercise number.

Describe the transformation of ( )f x x2= represented

by g.

1. ( ) 22g x x= − 2. ( ) ( )21g x x= −

3. ( ) 2 1g x x= − 4. ( ) ( )21g x x= +

5. ( ) 212g x x= − 2 6. ( ) ( )22 1g x x= − −

Write a rule for g described by the transformations of the graph of f.

7. ( ) 2;f x x= vertical stretch by a factor of 2 and a reflection in the x-axis, followed by a translation 3 units down

8. ( ) 2;f x x= vertical shrink by a factor of 12, followed

by a translation 3 units left

9. ( ) 24 10;f x x= + horizontal stretch by a factor of 2, followed by a translation 3 units up

10. ( ) ( )22 8;f x x= − − horizontal shrink by a factor

of 12 and a translation 5 units down, followed by a

reflection in the x-axis

Answers

O. ( ) 2 13g x x= +

L. translation 1 unit right

R. ( ) ( )22 2 13g x x= − − +

S. translation 1 unit down

V. translation 2 units right followed by a translation 1 unit down

L. vertical shrink by a factor of 12

followed by a translation 2 units down

E. reflection in the x-axis and a vertical stretch by a factor of 2

A. ( ) 22 3g x x= − −

D. ( ) ( )212 3g x x= +

A. translation 1 unit left

2.1

1 2 3 4 5 6 7 8 9 10


2.2 Start Thinking

Complete the table for the function ( ) .f x x= Graph the values from the table on a piece of graph paper.

What is the shape of the graph? Are opposite integers the same distance from the y-axis on the graph? Is this graph symmetric? Why or why not?

Give the coordinates of the image of point ( )5, 3P − after each reflection.

1. reflection in the y-axis 2. reflection in the x-axis

3. reflection in the line through ( ) ( )5, 6 and 8, 6− −

4. reflection in the line through ( ) ( )1, 1 and 1, 2− − − −

Determine if the data show a linear relationship. If so, write an equation of a line of fit. Estimate y when 20x = and explain its context in the situation.

1.

2.

2.2 Warm Up


x −2 −1 0 1 2

f (x)

Minutes jogging, x 2 5 10 15

Calories burned, y 22 55 110 165

Years, x 10 12 17 21

Height (feet), y 4.2 5.0 6.0 6.1


37

2.2 Practice A

Name _________________________________________________________ Date __________

In Exercises 1–12, graph the function. Label the vertex and axis of symmetry.

1. ( ) ( )22f x x= − 2. ( ) ( )21f x x= +

3. ( ) ( )22 4g x x= + + 4. ( ) ( )23 2h x x= − −

5. ( )23 1 3y x= − − + 6. ( ) ( )24 2 1f x x= + −

7. 2 2 1y x x= − + 8. 23 6 1y x x= + +

9. 23 6 4y x x= − + + 10. ( ) 2 6 3f x x x= − + −

11. ( ) 2 2g x x= − + 12. ( ) 25 4f x x= −

13. Explain why you cannot use the axes of symmetry to distinguish between the quadratic functions 23 12 1y x x= + + and 2 4 5.y x x= + +

14. Which function represents the parabola with the narrowest graph? Explain your reasoning.

A. 2 3y x= + B. 20.5 2y x= −

C. ( )23 2y x= + D. 22 1y x= − +

In Exercises 15–18, find the minimum or maximum value of the function. Describe the domain and range of the function, and where the function is increasing and decreasing.

15. 25 2y x= + 16. 24 3y x= −

17. 2 4 1y x x= − + − 18. ( ) 22 4 9f x x x= − + +

19. The number of customers in a grocery store is modeled by the function 2 10 50,y x x= − + + where y is the number of customers in the store

and x is the number of hours after 7:00 A.M.

a. At what time is the maximum number of customers in the store?

b. How many customers are in the store at the time in part (a)?


2.2 Practice B

Name _________________________________________________________ Date _________

In Exercises 1–12, graph the function. Label the vertex and axis of symmetry.

1. ( ) ( )23 2 4f x x= − − − 2. ( ) ( )23 1 5f x x= + +

3. ( ) ( )212 3 2g x x= − + + 4. ( ) ( )21

2 2 1h x x= − −

5. ( )20.6 2y x= − 6. ( ) 20.25 1f x x= −

7. 2 8y x= − + 8. 27 2y x= +

9. 21.5 6 3y x x= − + 10. ( ) 20.5 3 1f x x x= + −

11. 252 5 1y x x= − + 12. ( ) 23

2 6 4f x x x= − − −

13. A quadratic function is decreasing to the left of 3x = and increasing to the right of 3.x = Will the vertex be the highest or lowest point on the graph of the parabola? Explain.

14. The graph of which function has the same axis of symmetry as the graph of 22 8 3?y x x= − + Explain your reasoning.

A. 24 16 5y x x= − + − B. 22 8 7y x x= + +

C. 23 6 7y x x= − + D. 26 10 1y x x= − + −

In Exercises 15–18, find the minimum or maximum value of the function. Describe the domain and range of the function, and where the function is increasing and decreasing.

15. 23 12y x= + 16. 2 6y x x= − −

17. 213 2 3y x x= − − + 18. ( ) 21

2 3 7f x x x= + +

19. The height of a bridge is given by 23 ,y x x= − + where y is the height of the bridge (in miles) and x is the number of miles from the base of the bridge.

a. How far from the base of the bridge does the maximum height occur?

b. What is the maximum height of the bridge?


39


Name _________________________________________________________ Date __________

Characteristics of Quadratic Functions Example: Write the quadratic function in standard form that has a vertex at ( )2, 5 and

passes through the point ( )3, 7 .

Solution:

( )2y a x h k= − + Write the vertex form of a quadratic function.

( )22 5y a x= − + Substitute in the vertex for h and k.

( )27 3 2 5a= − + Substitute the other point for x and y.

2a = Solve for a.

( )22 2 5y x= − + Substitute h, k, and a.

22 8 13y x x= − + Simplify.

In Exercises 1–6, write the quadratic function in standard form.

1. vertex ( )1, 2− and passes through point ( )3, 10

2. vertex ( )1, 2− − and passes through point ( )4, 7−

3. vertex ( )2, 9− and passes through point ( )1, 9−

4. vertex ( )1, 0− and passes through point ( )3, 12− −

5. vertex ( )1, 6 and passes through point ( )2, 5

6. vertex ( )2, 0− and passes through point ( )2, 8

7. Could there be a quadratic function that has an undefined axis of symmetry? Why or why or not?

8. The graph of a quadratic function has a vertex at ( )3, 6 .− One point on the

graph is ( )7, 10 . What is another point on the graph? Explain how you found the other point.


Puzzle Time

Name _________________________________________________________ Date _________

What Is Roz Savage Famous For?

1 2 3 4 5 6

7 8 9 10 11 12

Complete each exercise. Find the answer in the answer column. Write the word under the answer in the box containing the exercise number.

2.2

Find the vertex and axis of symmetry of the function.

1. ( ) 29 3f x x= − 2. 2 2 5y x x= − + −

3. ( ) 20.5 10g x x x= − − − 4. ( ) 22 8 1f x x x= − + −

Find the minimum or maximum value of the function.

5. ( ) 23 12 10f x x x= − + − 6. 2 8y x= − +

7. ( ) 2 2 1g x x x= − + 8. 22 20y x x= −

Match the graph with its function.

9. 10.

11. 12.

8maximum

FIRST

( )( )22 4

f xx

=− − +

THREE

50minimum

−

TO

( ) 212f x x=

ACROSS

( )0, 30x

−=

THIS

( )2, 72x =

WAS

0minimum

WOMAN

2maximum

THE

( )1, 9.51x

− −= −

ROWER

( )( )22 1 1

f xx

=− + −

OCEANS

( ) 2 2f x x= −

ROW

( )x1, 4

1−=

BRITISH

x

y

4

6

2

42−2−4

x

y

4

6

8

2

42−2−4

x

y4

2

62−2

x

y

−4

−6

−8

2−2−4


41

2.3 Start Thinking

On a piece of graph paper, sketch the graph of 212 .y x=

Draw the line 12y = − and mark point ( )1

20, .P

Mark any point A on the graph of 212 .y x= Use a ruler to

measure the distance ( )14to the nearest -inch from point A

to point P. Can you find a spot on the line 12y = − that is

the same distance from point A? Try any three additional points B, C, and D for verification. Do you believe this method will always work? Why or why not?

Find the distance between the points. If necessary, round to the nearest tenth.

1. ( ) ( )7, 3 , 13, 7− 2. ( ) ( )1, 5 , 4, 4− − − −

3. ( ) ( )6, 11 , 6, 7− 4. ( ) ( )3, 0 , 4, 2− −

5. ( ) ( )15, 8 , 3, 4− − − − 6. ( ) ( )5, 7 , 2, 7− − −

Use the elimination method to solve the system.

1. 4 2 02 3 3 9

6 2 0

x y zx y z

x y z

+ − =− + =

− − + =

2. 3 3 84 3 18

6 6 2 3

x y zx y zx y z

− + =− − = −− − =

2.3 Warm Up



2.3 Practice A

Name _________________________________________________________ Date _________

In Exercises 1–6, use the Distance Formula to write an equation of the parabola.

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

directrix: 2y = − directrix: 3y = directrix: 6y =

4. vertex: ( )0, 0 5. vertex: ( )0, 0 6. vertex: ( )0, 0

directrix: 4y = focus: ( )0, 1− directrix: 2y =

7. Which of the given characteristics describe parabolas that open up? Explain your reasoning.

A. focus: ( )0, 3 B. focus: ( )0, 5− C. focus: ( )0, 10−

directrix: 3y = − directrix: 5y = directrix: 10y =

In Exercises 8–10, identify the focus, directrix, and axis of symmetry of the parabola. Graph the equation.

8. 2112y x= 9. 21

16y x= − 10. 218x y=

11. The cross section (with units in inches) of a parabolic satellite dish can be modeled by the equation 21

48 .y x= How far is the receiver from the vertex

of the cross section? Explain.

In Exercises 12–17, write an equation of the parabola with the given characteristics.

12. focus: ( )2, 0 13. focus: ( )4, 0− 14. focus: ( )340,

directrix: 2x = − directrix: 4x = directrix: 34y = −

15. directrix: 6x = − 16. focus: ( )0, 2 17. directrix: 1x =

vertex: ( )0, 0 vertex: ( )0, 0 vertex: ( )0, 0

In Exercises 18–21, identify the vertex, focus, directrix, and axis of symmetry of the parabola. Describe the transformations of the graph of the standard equation with vertex ( )0, 0 and 1.p =

18. ( )2112 1 3y x= − + 19. ( )21

8 5 2y x= − + −

20. ( )214 4 2x y= + + 21. ( )21

28 6 10y x= − + +


43

2.3 Practice B

Name _________________________________________________________ Date __________

In Exercises 1–6, use the Distance Formula to write an equation of the parabola.

1. focus: ( )0, 5 2. focus: ( )0, 6− 3. focus: ( )0, 4

directrix: 5y = − directrix: 6y = directrix: 4y = −


directrix: 8y = focus: ( )0, 7− directrix: 2y = −

In Exercises 7–12, identify the focus, directrix, and axis of symmetry of the parabola. Graph the equation.

7. 2132y x= − 8. 21

4x y= 9. 2 12y x=

10. 2 36x y− = 11. 28 2 0x y+ = 12. 22 0x y− =

In Exercises 13 and 14, write an equation of the parabola shown.

13. 14.

In Exercises 15–20, write an equation of the parabola with the given characteristics.

15. focus: ( )140, − 16. focus: ( )12, 0− 17. focus: ( )3

5, 0

directrix: 14y = directrix: 12x = directrix: 3

5x = −


directrix: 23y = focus: ( )3

4, 0− directrix: 13x = −

In Exercises 21–24, identify the vertex, focus, directrix, and axis of symmetry of the parabola. Describe the transformations of the graph of the standard equation with vertex ( )0, 0 and 1.p =

21. ( )2116 2 3x y= − − − 22. ( )28 2 1y x= + −

23. ( )25 3 6x y= + + 24. ( )2132 1 9y x= − + +

x

y

vertex

directrix

y = −3x

y

vertex

directrixy = 2



Name _________________________________________________________ Date _________

Focus of a Parabola Write an equation of the parabola with vertex at (0, 0) and the given directrix or focus.

1. focus: 21 , 0a

2. directrix: 2yn

= −

3. focus: 30, b

4. directrix: 16

xn

= −

5. Given: Equation 2y x= and parallel line segments RS and ,OT where

( ) ( ) ( )2 2 2, , , , , ,r r s s t t= = =R S T and O is the origin. Prove that .r s t+ =

6. Create another parallel line segment UV, where U and V are two other points on the parabola.

Prove that the midpoints of all three line segments lie on the same line.

x

4

6

8

10

2

42−2−4

y

O

T

R

S

y = x2


45

Puzzle Time

Name _________________________________________________________ Date __________

What National Park In The United States Is Known For Its 10,000 Hot Springs And Geysers? Write the letter of each answer in the box containing the exercise number.

Use the Distance Formula to write an equation of the parabola.

1. focus: ( )4, 0 2. directrix: 2y =

directrix: 4x = − vertex: ( )0, 0

3. focus: ( )0, 8− 4. directrix: 5y = −

directrix: 8y = vertex: ( )0, 5

5. focus: ( )0, 1− 6. focus: ( )0, 1.5−

vertex: ( )0, 0 vertex: ( )0, 0

Identify the focus, directrix, and axis of symmetry of the parabola.

7. 216y x= 8. 21

9y x= −

9. 213x y= 10. 21

16x y= −

11. 210 5 0x y− = 12. 220 0x y− =

13. ( )21 4y x= + −

14. ( )212 5 1x y= − − +

15. 2 24x y− =

2.3

Answers

W. 2

6xy = −

A. focus: ( )1, 3.75− − directrix: 4.25y = − axis of symmetry: 1x = −

Y. 2

16yx =

T. focus: ( )0, 2.25− − directrix: 2.25y = axis of symmetry: 0x =

E. 2

8xy = − O.

2

4xy = −

S. focus: ( )0, 1.5 directrix: 1.5y = − axis of symmetry: 0x =

K. focus: ( )0, 6− directrix: 6y = axis of symmetry: 0x =

O. focus: ( )0.75, 0 directrix: 0.75x = − axis of symmetry: 0y =

L. 2

540xy = +

N. focus: ( )4, 0− directrix: 4x = axis of symmetry: 0y =

E. focus: ( )0, 0.125 directrix: 0.125y = − axis of symmetry: 0x =

L. 2

32xy = −

P. focus: ( )0, 0.0125 directrix: 0.0125y = −axis of symmetry: 0x =

R. focus: ( )0.5, 5 directrix: 1.5x = axis of symmetry: 5y =

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


2.4 Start Thinking

Use the graph of the parabola ( )214 4y x= −

to label the x- and y-axis and give the graph an appropriate title for a real-life situation. Then find y when 25x = and describe what this signifies in terms of the labels you created.

Write an equation of a line in point-slope form with the information given.

1. passes through: ( )6, 0 ; slope: 16−

2. passes through: ( )1, 3 ; slope: 12

3. passes through: ( )4, 1 ;− slope: 2−

4. passes through: ( )3, 3 ;− slope: 3



Use a graphing calculator to graph the function and its parent function. Then describe the transformation.

1. ( ) 4 1f x x= − 2. ( ) 2h x x= −

3. ( ) 22 7g x x= + 4. ( ) ( )2 232f x x= − − −

2.4 Warm Up


x

y

4

2

0

8

6

420 86

y = (x − 4)214


47

2.4 Practice A

Name _________________________________________________________ Date __________

In Exercises 1–3, write an equation of the parabola in vertex form.

1. passes through ( )6, 4 and has vertex ( )2, 3−

2. passes through ( )3, 10− − and has vertex ( )3, 8−

3. passes through ( )0, 5− and has vertex ( )1, 4−

In Exercises 4–6, write an equation of the parabola in intercept form.

4. x-intercepts of 10 and 6; passes through ( )11, 8


6. x-intercepts of 14 and 2;− − passes through ( )16, 8− −

7. Use the parabola shown.

a. Write an equation of the parabola in vertex form.

b. Expand the equation in part (a) to the form 2 .y ax bx c= + +

c. Write an equation of the parabola in intercept form.

d. Expand the equation in part (c) to the form 2 .y ax bx c= + +

e. Do both methods give an equation that represents the parabola? Which method did you find easier? Explain.

8. A basketball is thrown up in the air toward the hoop. The table shows the heights y (in feet) of the basketball after x seconds. Find the height of the basketball after 5 seconds. Round your answer to the nearest hundredth.

Time, x 0 9 18

Basketball height, y 6 10 6

x

y

2

−4

−2

62−4−6

(1, 5)

(4, 0)(−2, 0)


2.4 Practice B

Name _________________________________________________________ Date _________

In Exercises 1–3, write an equation of the parabola in vertex form.



3. passes through ( )2, 2 and has vertex ( )1, 1− −

In Exercises 4–6, write an equation of the parabola in intercept form.


5. x-intercepts of 7 and 1;− − passes through ( )1, 1

6. x-intercepts of 9 and 9;− passes through ( )0, 4

7. Describe and correct the error in writing an equation of the parabola.

8. The graph shows the area y (in square feet) of rectangles that have a perimeter of 200 feet and a length of x feet.

a. Interpret the meaning of the vertex in this situation.

b. Write an equation for the parabola to predict the area of the rectangle when the length is 2 feet.

c. Compare the average rates of change in the area from 0 to 50 feet and 50 to 100 feet.

Vertex:

Passes through

The equation is

Area and Perimeter of Rectangles

Are

a (s

qu

are

feet

)

Length (feet)

x

y

1000

500

0

2000

2500

1500

40200 80 10060(0, 0)

(50, 2500)

(100, 0)


49


Name _________________________________________________________ Date __________

Modeling with Quadratic Functions In Exercises 1–5, analyze the differences in the outputs. Determine whether the data are linear or quadratic. Write an equation that fits the data. If quadratic, write the equation in (a) standard form and (b) vertex form, and (c) state the transformation from the parent function 2 .x

1.

2.

3.

4.

5.

Altitude (1000 feet), x 1 1.5 2 2.5 3

Boiling water temperature ( F), y° 210.3 209.4 208.5 207.6 206.7

Time (seconds), x 1 2 3 4 5

Height (feet), y 73.5 78.4 73.5 58.8 34.3

Units sold, x 1 2 3 4 5

Profit (thousands of dollars), y 39 60 75 84 87

Depth (feet), x 0 10 20 30 40

Pressure (pounds per square inch), y 14.7 19.03 23.36 27.69 32.02

Time (seconds), x 1 1.5 2 2.5 3

Height (feet), y 12 12.75 11 6.75 0


Puzzle Time

Name _________________________________________________________ Date _________

Don Featherstone Is Famous For Being The First To Make This Plastic Lawn Ornament In 1957. Write the letter of each answer in the box containing the exercise number.

Write an equation of the parabola in vertex form.


2. passes through ( )4, 10 and has vertex ( )0, 0


4. passes through ( )5, 3− − and has vertex ( )7, 12

5. passes through ( )9, 15 and has vertex ( )6, 21−

6. passes through ( )0, 0 and has vertex ( )10, 4− −

Write an equation of the parabola in intercept form.



9. x-intercepts of 5− and 1;− passes through ( )10, 2−


11. x-intercepts of 10− and 8; passes through ( )6, 4− −

12. x-intercepts of 0 and 4; passes through ( )12, 8−

Answers

I. 258y x=

I. ( )( )245 5 1y x x= + +

L. ( )2125 10 4y x= + −

N. ( )21964 14 20y x= − −

M. ( )( )19 9 3y x x= − − +

F. ( )2275 6 21y x= − + +

A. ( )( )427 10 2y x x= − − +

N. ( )( )38 4 6y x x= − −

G. ( )( )114 10 8y x x= + −

K. ( )2548 7 12y x= − − +

O. ( )112 4y x x= − −

P. ( )219 2 1y x= − + +

2.4

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


51

Chapter

2 Cumulative Review

Name _________________________________________________________ Date __________

In Exercises 1–16, solve the proportion.

1. 64 12

x− = 2. 53 9

x=−

3. 19 386 x

− = 4. 7 1413 x

−=

5. 7 3515 x− −= 6. 1

4 2x −= 7. 2 6

9x− =

− 8. 24 8

36 x− −=

9. 35 35

x=− −

10. 12 6045x

= 11. 802 20

x −=−

12. 410 15

x=

13. 1317 34

x=− −

14. 5 4056x

−= 15. 215 60

x− =−

16. 81 95x

−=

17. Your chemistry test has 64 questions. Your teacher rounds to the nearest whole percent.

a. You have 57 correct answers. What percent of your answers are correct?

b. You have 61 correct answers. What percent of your answers are correct?

c. You want to earn at least an 85%. How many correct answers must you have?

d. You want to earn at least a 93%. How many correct answers must you have?

18. Your English literature test has 28 questions. Your teacher rounds to the nearest whole percent.

a. You have 25 correct answers. What percent of your answers are correct?

b. You have 21 correct answers. What percent of your answers are correct?

c. You want to earn at least an 85%. How many correct answers must you have?

d. You want to earn at least a 93%. How many correct answers must you have?

e. You want to earn at least a 77%. How many correct answers must you have?

19. You want to mix leftover yellow and blue paint to make green paint. The ratio to make the green paint is 2 parts yellow paint to 1 part blue paint. You have 6 cups of yellow paint. How many cups of blue paint do you need to make the green paint?

20. You want to make an environmentally friendly carpet cleaner using salt and white vinegar. The recipe ratio of white vinegar to salt is 8:2. You have one cup of white vinegar. How many tablespoons of salt are needed to make the carpet cleaner?


Chapter

2 Cumulative Review (continued)

Name _________________________________________________________ Date _________

In Exercises 21–35, find the distance between the two points. Round your answer to the nearest hundredth.

21. ( )4, 3− and ( )1, 2− − 22. ( )3, 7 and ( )4, 6− 23. ( )9, 8− and ( )11, 10−

24. ( )2, 5 and ( )1, 5− − 25. ( )9, 10− − and ( )1, 6 26. ( )2, 3 and ( )1, 10

27. ( )8, 6 and ( )4, 3− 28. ( )2, 9− − and ( )4, 5− 29. ( )12, 7− and ( )9, 3

30. ( )5, 8− and ( )2, 11 31. ( )5, 10 and ( )1, 2− 32. ( )3, 2− − and ( )5, 4

33. ( )7, 3− and ( )4, 8− − 34. ( )8, 1 and ( )9, 2− 35. ( )2, 1− − and ( )7, 7

In Exercises 36–44, find the x-intercept of the graph of the linear equation.

36. 37 5y x= − 37. 12 27y x= + 38. 7 39y x= − +

39. ( )5 2y x= + 40. ( )7 10y x= − + 41. ( )4 8y x= −

42. 6 5 30x y− + = − 43. 11 7 33x y+ = − 44. 14 8 28x y− =

In Exercises 45–53, solve the equation for x.

45. 8 24y x= + 46. 12 3y x= + 47. 5 35y x= − +

48. 74

xy −= 49. 3 86

xy − +=−

50. 4 56

xy +=

51. 3 6 30x y− = 52. 8 2 40x y− − = 53. 5 15 40x y− = −

In Exercises 54–65, solve the equation. Check for extraneous solutions.

54. 4 2 0x − = 55. 7 5 0x− − = 56. 6 10 3 0x + + =

57. 9 8 0x− − − = 58. 3 8 6 4x + − = 59. 7 5 4 3x − − =

60. 2 9 0x − = 61. 3 10 0x − = 62. 4 1 1x − =

63. 9 7 3x − = 64. 2 4 4x x− = + 65. 5 10 6x x+ = −

66. You are riding in a car and traveling at an average speed of 65 miles per hour. The destination is 325 miles away. How long does it take you to get there?

67. You are riding in a car and traveling at an average speed of 48 miles per hour. The destination is 108 miles away. How long does it take you to get there?

68. You are on a cruise ship that travels at an average speed of 24 knots per hour. The first port of call is 1449 miles away from where you set sail (1 knot is about 1.15 miles). How long does it take you to get there?


53

Chapter


Name _________________________________________________________ Date __________

In Exercises 69–83, write an equation for the line that passes through the points.

69. ( )11, 2− and ( )3, 1− 70. ( )12, 7 and ( )11, 3 71. ( )4, 6 and ( )4, 2−

72. ( )5, 8− and ( )1, 6− − 73. ( )8, 6− and ( )2, 7− 74. ( )12, 8− and ( )2, 2

75. ( )1, 8 and ( )9, 2− 76. ( )2, 10− − and ( )6, 9 77. ( )3, 5 and ( )6, 7−

78. ( 4− , 3) and ( )8, 12− 79. ( )12, 5 and ( )4, 6− 80. ( )11, 3− and ( )3, 3

81. ( )5, 4− and ( )9, 1 82. ( )6, 3− and ( )1, 4− − 83. ( )9, 6− and ( )5, 1−

In Exercises 84–104, factor the trinomial.

84. 2 6x x+ − 85. 2 42x x− − 86. 2 8 48x x+ −

87. 2 4 77x x− − 88. 2 13 12x x+ + 89. 2 6 16x x+ −

90. 2 2 24x x− − 91. 2 4 5x x+ − 92. 2 4 3x x− +

93. 2 11 30x x+ + 94. 2 20 96x x− + 95. 2 13 40x x+ +

96. 22 3 35x x− − 97. 23 2 40x x− − 98. 22 19 10x x+ −

99. 23 32 20x x− + 100. 24 36 144x x+ − 101. 25 7 6x x+ −

102. 26 10 44x x+ − 103. 24 27 81x x+ − 104. 22 2 40x x+ −

In Exercises 105–122, write the quadratic function in standard form.

105. ( )( )8 4y x x= − + − 106. ( )( )4 4 3y x x= + − 107. ( )( )2 1 2y x x= − − +

108. ( )( )6 3 4y x x= − + − 109. ( )( )5 3 8y x x= − − − 110. ( )( )3 4 7y x x= + −

111. ( )26 3y x= − 112. ( )27 1y x= − + 113. ( )22 4y x= −

114. ( )24 1y x= − + 115. ( )25 7y x= + − 116. ( )28 9y x= − +

117. ( )23 1 4y x= + + 118. ( )27 5 3y x= − + − 119. ( )29 3 7y x= − − +

120. ( )24 8 5y x= − + − 121. ( )22 5 1y x= − + 122. ( )23 1 1y x= + −

123. You painted a picture that is 18 inches by 24 inches. You want to put wooden pieces around the outside as a frame. There are 648 square inches of wooden pieces. You are using all the material, and you want an even boarder around the entire picture. What should be the width of the border?

124. You want to create a decorative border for your garden that measures 12 feet by 7 feet. You decide to put a 1-foot border around the garden. How many square feet of rocks do you need?


Chapter


Name _________________________________________________________ Date _________

125. You have a piece of canvas that you want to paint. You have enough paint to cover 252 square inches. You know you want one side to be 14 inches. How long should the other side be if you want to use up all the paint?

126. You have a square table and want to create a tablecloth for it using a piece of fabric. The side length of the table is 72 inches. How much fabric do you need?

In Exercises 127–146, use the quadratic formula to solve the equation.

127. 2 4 58 0x x− + = 128. 2 2 14 0x x+ − =

129. 2 3 16 0x x+ − = 130. 2 5 18 0x x− + =

131. 2 6 21 0x x+ − = 132. 2 8 18 0x x− − =

133. 22 14 45 0x x+ − = 134. 23 5 6 0x x− + =

135. 24 7 5 0x x− − + = 136. 25 4 32 0x x− + =

137. 26 7 4 0x x− + = 138. 24 13 21 0x x− + − =

139. 22 3 8x x+ = − 140. 22 5 18x x− + =

141. 23 16 34x x− + = 142. 24 15 28x x− − =

143. 22 12 6x x− = − 144. 26 9 12x x+ = −

145. 2 25 4 23 3 14 34x x x x− + = + + 146. 2 23 4 14 9 7 19x x x x− + = − +

In Exercises 147–152, graph the function and its parent function. Then describe the transformation.

147. ( ) 2g x x= 148. ( ) 12h x x= − 149. ( ) 4c x x= −

150. ( ) 22d x x= 151. ( ) 4k x x= + 152. ( ) 3 2m x x= − +

In Exercises 153–160, write a function g whose graph represents the indicated transformation of the graph of f.

153. ( ) 2 ;f x x= − translation 3 units down 154. ( ) 47 ;f x x= translation 1 unit up

155. ( ) 4;f x x= + translation 2 units left 156. ( ) 12 7;f x x= − translation 6 units right

157. ( ) 3 5;f x x= − + reflection in the x-axis

158. ( ) 1 2;f x x= + − reflection in the x-axis

159. ( ) 3 1;f x x= − reflection in the y-axis 160. ( ) 34 5;f x x= + reflection in the y-axis

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