QUADRATIC FUNCTIONSrusdmath.weebly.com/uploads/1/1/1/5/11156667/g9_u6...LESSON 4: GRAPHING...

MATH HIGH SCHOOL

QUADRATIC FUNCTIONS

EXERCISES

Copyright © 2015 Pearson Education, Inc. 3

High School: Quadratic Functions

CONTENTS EXERCISES

EXERCISES

LESSON 1: ZOOMING IN ON PARABOLAS �� 5

LESSON 2: QUADRATIC FUNCTIONS �� 7

LESSON 3: REAL-WORLD PROBLEMS �� 13

LESSON 4: GRAPHING QUADRATICS �� 15

LESSON 5: REAL-WORLD PARABOLAS �� 19

LESSON 6: TRANSLATIONS �� 23

LESSON 7: TRANSFORMING PARABOLAS �� 27

LESSON 8: TRANSFORMATION APPLICATIONS �� 31

LESSON 9: PUTTING IT TOGETHER 1 �� 35

LESSON 12: FACTORED FORM AND ZEROES �� 37

LESSON 13: VERTEX FORM AND THE VERTEX �� 39

LESSON 14: RATE OF CHANGE �� 41

LESSON 15: AVERAGE RATE OF CHANGE �� 43

LESSON 16: OPERATIONS ON FUNCTIONS �� 47

LESSON 17: SQUARE ROOT FUNCTIONS �� 51




CONTENTS ANSWERS

ANSWERS

LESSON 2: QUADRATIC FUNCTIONS �� 59

LESSON 3: REAL-WORLD PROBLEMS �� 61

LESSON 4: GRAPHING QUADRATICS �� 64

LESSON 5: REAL-WORLD PARABOLAS �� 65

LESSON 6: TRANSLATIONS �� 67

LESSON 7: TRANSFORMING PARABOLAS �� 69

LESSON 8: TRANSFORMATION APPLICATIONS �� 71


LESSON 12: FACTORED FORM AND ZEROES �� 74

LESSON 13: VERTEX FORM AND THE VERTEX �� 76

LESSON 14: RATE OF CHANGE �� 78

LESSON 15: AVERAGE RATE OF CHANGE �� 79

LESSON 16: OPERATIONS ON FUNCTIONS �� 81

LESSON 17: SQUARE ROOT FUNCTIONS �� 83




EXERCISES

EXERCISES

1. Write what you already know about quadratic functions or equations.

Share your work with a classmate. Did you write the same things?

2. Write your wonderings about quadratic functions. Share your wonderings with a classmate.

3. Write a goal stating what you plan to accomplish in this unit.

4. Based on your previous work, write three things you will do differently during this unit to increase your success.

For example, consider ways you will participate in classroom discussions, your study habits, how you will organize your time, what you will do when you have a question, and so on.

LESSON 1: ZOOMING IN ON PARABOLAS



EXERCISES

EXERCISES

Use this graph to answer questions 1–2.

This graph shows a quadratic function of the form y = ax2.

–5 0 5 x

y

5

10

1. Which points are on the graph? There may be more than one point on the graph.

A (–3, –8)

B (–2, 4)

C (0, 0)

D (1, 2)

E (2, 8)

2. Which function describes this graph?

A y = –2x2

B y = 2x2

C y = 2x2 + 2

D y = 0.5x2

LESSON 2: QUADRATIC FUNCTIONS



EXERCISESLESSON 2: QUADRATIC FUNCTIONS


This graph shows a quadratic function of the form y = ax2.

–5 0 5 x

y

5

10

3. Write the correct domain and range of this function in the appropriate column.

All real numbers y ≥ 0 y ≤ 0 x ≥ 0 –4 < x < 4

Domain Range

4. This graph opens and is than y = x2.

A upward, wider

B upward, narrower

C downward, wider

D downward, narrower





This graph shows a function of the form y = ax2 + c.

–5 5 x

y

–10

–5

5

5. Which function describes this graph?

A y = –2x2 – 3

B y = 2x2 + 3

C y = –0.5x2 – 3

D y = –0.5x2 + 3

6. Write the correct domain and range of this function in the appropriate column.

All real numbers –3 < x < 3 x ≥ –4 y ≤ 3 y ≤ 0

Domain Range




7. This graph shows a function of the form y = ax2 + c.

–5 5 x

y

–10

–5

5

Which function describes this graph?

A upward, wider

B upward, narrower

C downward, wider


8. Complete the table for the function y x= −13

2

x –5 –4 –3 –2 –1 0 1 2 3 4 5

y

9. Sketch a graph of the function y x= −13

2.



ANSWERSLESSON 2: QUADRATIC FUNCTIONS

10. Select the options that describe the graph of the function y x= −13

2 .

The graph of y x= −13

2 opens _________ and is _________ than y = x2.

A upward, wider

B upward, narrower

C downward, wider


11. The point (–3, –9) does not lie on the graph of y = x2. Explain why.

12. Evan looked at the standard form of a quadratic function: y = ax2 + bx + c. Then he “smiled” and said, “A positive value of a is like a positive attitude, whereas a negative value of a is like a frown.” Explain what Evan meant.

Challenge Problem

13. The standard form of a quadratic function f(x) is given by f(x) = ax2 + bx + c.

a. Which constant(s) of the function affect the direction of the parabola? Explain.

b. Which constant(s) of the function affect the width of the parabola? Explain.



EXERCISES

EXERCISES

Use this information to answer questions 1–4.

A tourist accidentally dropped a cell phone from a building’s observation deck that is 160 ft above the ground. The formula h(t) = –16t2 + 160 represents the height, in feet, of the cell phone above the ground at time t seconds.

1. Complete this table of values.

Time t (sec) 0 1 2 3

Height (ft)h(t) = –16t2 + 160

2. Draw a graph representing the height of the phone and the time in seconds since it was dropped.

3. Approximately when does the cell phone hit the ground?

A Between 2 and 3 seconds

B Between 3 and 4 seconds

C Between 4 and 5 seconds

D Between 5 and 6 seconds

4. The standard formula for a quadratic function is y = ax2 + bx + c.

Explain why a is negative for the dropped cell phone situation and what the value of c means.

LESSON 3: REAL-WORLD PROBLEMS



EXERCISESLESSON 3: REAL-WORLD PROBLEMS


Sarah wants to build a rectangular garden with 180 ft of fencing.

5. a. Make a table with at least five possible widths and lengths for her garden.

b. Then add an extra row that shows the area for those dimensions.

6. Which formula can she use to represent the area of her garden, A, in terms of the width of the rectangular fence, w?

A A(w) = 180 – w

B A(w) = w(180 – w)

C A(w) = w(90 – w)

D A(w) = 90 – w

7. a. Write the formula for the area in terms of the width in standard form.

b. Explain the meaning of the constants a, b, and c for this situation.

8. Find the dimensions that give the greatest area.

Length = ____ ft Width = ____ ft

Challenge Problem

9. Rosa says that as the value of a in y = ax2 gets smaller, the parabola becomes narrower.

a. Draw some graphs of the form y = ax2 for different values of a.

b. Do you agree or disagree with Rosa? Explain.



EXERCISESLESSON 4: GRAPHING QUADRATICS

EXERCISES

1. This is a graph of a quadratic function. Select the point that is the maximum.

0

–2

2

6 x

y

42

4

10

6

12

8

14

2. A quadratic function intersects the x-axis at –3 and 5.

Which line is the axis of symmetry?

A x = –3

B x = –1

C x = 1

D x = 5

3. A quadratic function intersects the x-axis at –13 and 7.

Which statements are true about this quadratic function? There may be more than one true statement.

A The x-coordinate of the vertex is 0.

B The x-coordinate of the vertex is –3.

C The parabola intersects the origin, (0, 0).

D The y-coordinate of the vertex is not 0.

E The parabola opens upward.




4. Consider the quadratic function y = –2x2 + 3x + c.

Which statements are true about this quadratic function? There may be more than one true statement.

A c ≠ 0

B The x-coordinate of the vertex is 34

.

C The x-coordinate of the vertex is −34

.

D The y-coordinate of the vertex is greater than 0.

E The parabola opens downward.

5. Look at the quadratic function f x x x( ) = +13

43

53

2 – . What are the coordinates of the vertex of this function? ( _____, _____ )

6. Which formula is used to determine the x-coordinate of the vertex if you know that y = ax2 + bx + c?

A xab

=2

B xab

= −2

C xba

=2

D xba

= −2

7. A quadratic function y = ax2 + bx + c goes through the origin. List as much information as you can about the constants a, b, and c for this quadratic function.

8. A quadratic function y = ax2 + bx + c has a vertex that is a maximum value. List as much information as you can about the constants a, b, and c for this quadratic function.

9. A quadratic function y = ax2 + bx + c has a vertex (1, 2) and contains the point (2, 0). List as much information as you can about the constants a, b, and c for this quadratic function.




Challenge Problem

10. Consider any quadratic function y = ax2 + bx + c.

a. Explain why the formula for the axis of symmetry of a quadratic function is not dependent on the constant c.

b. For what values of x does the graph of the function intersect the horizontal line y = c?



EXERCISES

EXERCISES

1. Which of these functions are quadratic functions? There may be more than one quadratic function.

A f(x) = 2x + 30

B f(x) = x2 + 30

C f(x) = 2x2

D f x x( ) = +

60

12

E f x x x( ) = +

60

12


A manufacturer of custom amplifier cabinets has daily production costs of

C x x x( ) = − +16

8 7682 , where C gives the cost of producing x cabinets.

2. Find C(12). It costs the manufacturer $ to produce 12 cabinets.

3. Use the graph of the cost function shown to estimate the daily cost of producing 75 cabinets. Select the best estimate.

40Cabinets Produced

Daily Costs for Cabinet Production

Da

ily C

ost

($)

8030 7020100 6050

400

0

500

900

300

800

100

600

200

700

1,000

1,100

1,200

y

x

A $600

B $800

C $1,000

D $1,100

LESSON 5: REAL-WORLD PARABOLAS



EXERCISESLESSON 5: REAL-WORLD PARABOLAS


A manufacturer of custom amplifier cabinets has daily production costs of

C x x x( ) = − +16

8 7682 , where C gives the cost of producing x cabinets.

4. Which statements are true about this cost model? There may be more than one true statement.

A The vertex is a minimum.

B The vertex has an x-value < 0.

C The y-intercept is 768.

D The graph intersects the origin.

E The axis of symmetry is x = 6.

5. What quantity of amplifier cabinets results in the lowest daily costs of production? How do you know?

6. Find the cost of producing the quantity of cabinets that results in the lowest daily cost.

7. What is the appropriate domain and range for this function?


Miki tees off for a par-3 hole at the golf course. Ignoring wind resistance, the height

y of her ball (in meters) above the ground at the tee is given by y x x= − +8

7293227

2 ,

where x represents the horizontal distance (in meters) from the tee.

8. At what distance from the tee does the ball reach its maximum height?

9. Find the maximum height of the ball.

10. At a distance of 18 m from the tee, there is a 12 m tall tree in the golf ball’s path. Does Miki’s ball clear the tree?

11. Miki’s ball travels a horizontal distance of 115 m before touching the ground. What is the elevation difference between the tee and the spot where the ball hits the ground?



EXERCISESLESSON 5: REAL-WORLD PARABOLAS

Challenge Problem

12. A function is given in the form y = ax2 + c (b = 0).

a. Based on this function, what can you say about the graph of the function?

b. Based on this function, what can you say about the symmetry of the parabola?

c. If the graph intersects the x-axis, what can you say about the values of the coefficients a and c?

d. How do you find the values of a and c from the graph?

e. What effect does the value of c have on the domain and the range of the function?



EXERCISES

EXERCISES

1. This function is a translation of the parent function f(x) = x2.

–6 –4 –2

–2

–4

2

4

6

2 4 6 x

y What is the equation for this function?

A f(x) = (x – 1)2

B f(x) = (x + 1)2

C f(x) = x2 + 1

D f(x) = x2 – 1


–6 –4 –2

–2

–4

2

4

6

2 4 6 x

y What is the equation for this function?

A f(x) = (x – 1)2

B f(x) = (x + 1)2

C f(x) = x2 + 1

D f(x) = x2 – 1

LESSON 6: TRANSLATIONS



EXERCISES


–6 –4 –2

–2

–4

2

4

6

2 4 6 x

y

Write an equation for this function.


–6 –4 –2

–2

–4

2

4

6

2 4 6 x

y

Write an equation for this function.

LESSON 6: TRANSLATIONS



EXERCISESLESSON 6: TRANSLATIONS

5. Write each function to the column that describes its translation from f(x) = x2.

f(x) = (x – 4)2 f(x) = (x + 2)2 f(x) = x2 – 2f(x) = x2 + 3.5 f(x) = (x + 3)2 + 4 f(x) = (x – 2)2 + 5

Horizontal Translation Vertical TranslationVertical and Horizontal

Translation

6. Look at these graphs.

–6 –4 –2 0 2 4 x

y

g(x)

h(x)

2

4

6

8

10

2 4

51

3

2 4

51

3

Look at the corresponding points 1–5 in g(x) and h(x).

a. Each point on h(x) is vertically translated _______ __________ units from the points in g(x).

A up 1

B up 2

C up 4

D down 4

E down 3

b. Each point on h(x) is horizontally translated ______ __________ units from the points in g(x).

A left 1

B left 3

C right 4

D right 2

E right 3



EXERCISESLESSON 6: TRANSLATIONS

7. Look at these graphs.

–6 –4 –2 0 2 4 x

y

g(x)

h(x)

2

4

6

8

10

2 4

51

3

2 4

51

3

What function does h(x) represent?

A h(x) =(x – 3)2 + 4

B h(x) = (x + 3)2 + 4

C h(x) = (x – 3)2 – 4

D h(x) = (x – 3)2

8. Sketch a graph of this function as a translation of the parent function f(x) = x2. g(x) = (x – 2)2

9. Sketch a graph of this function as a translation of the parent function f(x) = x2. h(x) = x2 – 2

Challenge Problem

10. The function y = x2 is translated such that the equation of the function is y = (x – h)2 + k.

a. Describe how positive and negative values of k shift the parabola.

b. Describe how positive and negative values of h shift the parabola.

c. Describe how h and k can be used to find the vertex of the parabola.

d. Notice that in the function, h is subtracted while k is added. Write a poem or memorable saying about how to add and subtract constants to y = x2.



EXERCISES

EXERCISES

1. Look at this graph.

–5 –4 1–3 2–2 3

g(x)

–1 4 5 x

y

–5

1

–4

2

–3

3

–2

4

–1

5

Which function describes the graph?

A g(x) = –2x2

B g(x) = 2x2 + 3

C g(x) = –2x2 + 3

D g(x) = –2x2 – 3

2. Quadratic functions are congruent to the function y = x2 when both functions are the same size and shape, because the functions are neither stretched nor squeezed vertically.

Write each quadratic function to the appropriate column to identify whether it is squeezed, stretched, or congruent to y = x2.

y = –x2 + 2 y x= +14

42 y x=12

42 –

y = 3(–x + 4)2 + 2 y x= +– ( ) –13

4 22

Squeezed Vertically Congruent to y = x2 Stretched Vertically

LESSON 7: TRANSFORMING PARABOLAS



EXERCISESLESSON 7: TRANSFORMING PARABOLAS

3. Look at this graph.

–5 –4 1–3 2–2 3

g(x)

–1 4 5 x

y

–5

1

–4

2

–3

3

–2

4

–1

5

Which words describe the transformation(s) of f(x) = x2 to g(x)? There may be more than one correct description.

A Stretch vertically by a factor of 2

B Squeeze vertically by a factor of 1

2C Flip across the x-axis

D Translate up 3 units

E Translate down 3 units

4. Write a function for this graph, which is a translation of the parent function f(x) = x2.

–6 –4 –2

–2

–4

2

4

6

2 4 6 x

y

g(x)



ANSWERSLESSON 7: TRANSFORMING PARABOLAS

5. Which functions are represented by this graph? There may be more than one correct form of the function.

–4 –2

–2

–4

2

4

2 4 x

y A f(x) = –(x – 1)2 + 4

B f(x) = (x – 1)2 + 4

C f(x) = x2 – 4x – 6

D f(x) = x2 + 4x + 1

E f(x) = –x2 + 2x + 3

6. Write a function for this graph, which is a translation of the parent function f(x) = x2.

–6 –4 –2

–2

–4

2

4

6

2 4 6 x

y

g(x)

7. Emma said, “When you order quadratic functions from least to greatest based on the coefficient, you are also ordering quadratic functions from widest to narrowest. For example, these equations are ordered from widest to narrowest.” y = –15x2, y = –0.5x2, y = 0.2x2, y = 5x2

Do you agree with Emma? Explain your reasoning. If you think she is incorrect, tell her the correct way to order the quadratic functions from widest to narrowest.



EXERCISESLESSON 7: TRANSFORMING PARABOLAS

8. This quadratic function is written in vertex form. f(x) = (x + 2)2 – 3

This is the same quadratic function written in standard form. f(x) = x2 + 4x + 1

a. What is the advantage of vertex form?

b. What is the advantage of standard form?

9. Two functions have the following algebraic representations.

f(x) = –13(x – 1)2 + 2 g(x) = –3(x – 1)2

Without graphing, explain how the graphs representing the functions f(x) and g(x) are similar and how they are different.

10. Write an equation for a function h(x) that has been translated up 3 units from f(x) = (x – 1)2 – 4.

Challenge Problem

11. Write the equation y = 2x2 – 12x + 11 in vertex form. Explain your process.



EXERCISES

EXERCISES

Use this information to answer question 1–4.

Two identical balls are dropped from different heights. One ball is dropped from 64 feet, and the other ball is dropped from 144 feet.

The equation for the ball dropped from 64 feet is f(t) = –16t2 + 64.

1. What is the equation for the ball dropped from 144 feet?

A h(t) = –16t2 + 80

B h(t) = –144t2 + 64

C h(t) = –144t2 – 64

D h(t) = –16t2 + 144

2. Graph both functions.

3. Determine how long it takes for each ball to reach the ground.

The ball dropped from 64 feet reaches the ground in seconds.

The ball dropped from 144 feet reaches the ground in seconds.

4. The h(t) function is a of the f(t) function.

A vertical translation

B horizontal translation

C vertical stretch

D vertical squeeze

LESSON 8: TRANSFORMATION APPLICATIONS



EXERCISESLESSON 8: TRANSFORMATION APPLICATIONS

5. The number of bacteria in a refrigerated food is given by n(t) = 16(t – 0.5)2 + 92, where n is the number of bacteria and t the temperature in degrees Celsius. Assume –4 ≤ t ≤ 10.

What is the minimum number of bacteria? At what temperature does this minimum occur?

The minimum number of bacteria is ____. This minimum occurs at ____ °C.

6. The number of bacteria in a refrigerated food is given by n(t) = 16(t – 0.5)2 + 92, which can be rewritten as n(t) =16t2 – 16t + 96, where n is the number of bacteria and t the temperature in degrees Celsius. Assume –4 ≤ t ≤ 10.

The number of bacteria in a different refrigerated food is given by b(t) = 16(t + 2.5)2 – 16(t + 2.5) + 96.

Explain how the two population curves are related. (If helpful, graph the two functions.)

7.

While outdoor playing fields may appear flat, their surfaces are usually parabolic so that rainwater can run off. Suppose the surface of a soccer field can be modeled by f(x) = –0.000556(x – 30)2 + 0.5, where the width x and height y are measured in meters.

Draw a graph of the given function.



EXERCISESLESSON 8: TRANSFORMATION APPLICATIONS

8.

While outdoor playing fields may appear flat, their surfaces are usually parabolic so that rainwater can run off. Suppose the surface of a soccer field can be modeled by f(x) = –0.000556(x – 30)2 + 0.5, where the width x and height y are measured in meters.

What are the appropriate domain and range in the context of this problem?

Challenge Problem

9. For each statement, explain whether you agree or disagree.

a. Evan said, “If a quadratic function has the form y = ax2 (b = 0, c = 0), then its graph goes through the origin and its axis of symmetry is the y-axis.”

b. Miki said, “If the graph of a function is congruent to y = –2x2, then the function is of the form y = –2x2 + bx + c.”



EXERCISES

EXERCISES

1. Read your Self Check and think about your work in this unit. Write three things you have learned.

Share your work with a classmate. Does your classmate understand what you wrote?

2. Consider the quadratic parent function f(x) = x2. Complete the table with examples of the listed transformations to the quadratic parent function. The first example has been done for you.

TransformationBy What Quantity

Vertex Form f(x) = a(x2 – h) + k

Standard Form f(x) = ax2 + bx + c

Translate up 4 f(x) = (x – 0)2 + 4 f(x) = x2 + 4

Translate down

Translate right

Translate left

Reflect, opens downward

Not applicable

Vertical stretch by factor

Vertical squeeze by factor

Translate up and left

Squeeze and translate down

Reflect, opens downward, stretch, and translate up

3. Review the notes you took during the lessons about quadratic functions. Add any additional ideas you have about the topic to your notes.

4. Complete any exercises from this unit you have not finished.

LESSON 9: PUTTING IT TOGETHER 1



EXERCISES

EXERCISES

1. Which equation represents a parabola with zeros at x = 0 and x = –3?

A y = x2 + 3

B y = x2 – 3x

C y = x2 + 3x

D y = 3x2

2. Which graphs represent the function f(x) = (x + 1)(x + 5)? There may be more than one graph that represents the function.

A

–6 –4 –2 2 x

y

–4

–2

2

4

B

–2 4 62 x

y

–4

–2

2

4

C

–6 –4 –2 2 x

y

–4

–2

2

4

D –6 –4 –2–5 –3 –1

–1

x

y

–2

–3

–4

E

2 x

y

–2–4–6

–10

–6

–8

–2

–4

4

2

LESSON 12: FACTORED FORM AND ZEROES



EXERCISESLESSON 12: FACTORED FORM AND ZEROES

3. Write the function represented by this table in factored form.

x –1 0 1 2 3 4 5

y 10 4 0 –2 –2 0 4

4. A quadratic function has the equation f(x) = (x + 3)(x – 8).

What are the zeros of the parabola? x = ____ and x = ____

5. What are the coordinates of the x-intercepts of the function f(x) = (x + 3)(x – 7)?

( ____, ____ ) and ( ____, ____ )

6. What is the axis of symmetry of the function f(x) = (x + 3)(x – 7)?

x = ____

7. What are the coordinates of the vertex of the function f(x) = (x + 3)(x – 7)?

( ____, ____ )

8. Graph the function f(x) = (x + 3)(x – 7).

9. What are the zeros of the equation g(x) = (3x – 9)(x + 2)?

x = ____ and x = ____

Challenge Problem

10. a. Determine all of the zeros for the equation h(x) = x(x – 0.75)(x + 3).

b. Explain why this equation has more zeros than previous equations in this unit.



EXERCISES

EXERCISES

1. What are the coordinates of the vertex for the function f(x) = 4(x – 3)2 + 9?

A (4, 3)

B (3, 9)

C (3, –9)

D (–3, 9)

2. Which equation corresponds to this graph?

8 x

y

642–2

–4

2

–2

4

6

8 A y = –(x + 3)2 – 7

B y = (x – 3)2 + 7

C y = –(x – 3)2 + 7

D y = –(x – 3)2 – 7

3. Consider a parabola with the equation f(x) = –3(x + 7)2 – 8. This parabola will have a ____________ at its vertex.

A maximum

B minimum

4. Consider a parabola with the equation f(x) = –3(x + 7)2 – 8. What are the coordinates of the vertex?

( ____, ____ )

LESSON 13: VERTEX FORM AND THE VERTEX



EXERCISESLESSON 13: VERTEX FORM AND THE VERTEX

5. There are three common forms of quadratic functions: standard, vertex, and factored. Shown in the table are some key features of a parabola that are visible in different forms of the quadratic function.

Write each quadratic function form in the appropriate box to indicate which parabola feature is visible in that form. A form may be used more than once.

Standard formf(x) = ax2 + bx + c

Vertex formf(x) = a(x – h)2 + k

Factored formf(x) = a(x – n)(x – m)

Maximum or Minimum x-intercept y-intercept Roots/Zeros

Maximum or Minimum x-intercept y-intercept Roots/Zeros

6. Consider a parabola with the equation f(x) = –3(x + 7)2 – 8.

What is the y-intercept? ( ____, ____ )

7. Consider a parabola with a minimum point at (–3, –2). The parabola passes through the point (0, 1).

a. Write an equation for this parabola in vertex form.

b. Graph the parabola.

8. Graph this function. f(x) = x2 + 3x – 4

Challenge Problem

9. Create two different parabola functions that have the same vertex.

a. Create equations for each function.

b. Graph each function in the same coordinate plane.

c. Determine how many different parabolas could have the same vertex.



EXERCISES

EXERCISES


This graph shows the height (in meters) of a rising hot air balloon over time (in seconds).

1000 200 300Time (sec)

Rising Hot Air Balloon

Hei

gh

t (m

)

400 500 600

250

0

500

y

x

A (0, 0)

B (180, 300)

C (450, 550)

D (600, 600)

1. What is the average rate of change for the entire ascent of the balloon (that is, from point A to point D)?

The average rate of change is ____ m/sec.

2. What is the rate of change between points A and B?

A 0.6 m/sec B 1 m/sec C 1.22 m/sec D 1.67 m/sec

3. What is the rate of change between points B and C? Show your calculations.

4. What is the rate of change between points C and D? Show your calculations.

5. Describe in words what the different values of rate of change mean in the context of the hot air balloon situation.

LESSON 14: RATE OF CHANGE



EXERCISESLESSON 14: RATE OF CHANGE


A steep hill starts at a vertical height of 4 m above street level. (The horizontal distance for this point is 0.) At a horizontal distance of 10 m, the height is 24 m. This point is the maximum height. The graph representing the hill height h in meters, in terms of the distance traveled d in meters, is a parabola.

6. What is the average rate of change between the two specified points?

7. What is the vertex form for this quadratic function model?

8. Kayla says a model for this hill is: h(d) = –0.2(x + 1)(x – 21)

How does your original function written using vertex form compare with Kayla’s model? In your comparison, consider the vertices, the average rate of change from d = 0 to d = 10, and the shape of each function.

9. The practical domain of Kayla’s function h(d) = –0.2(x + 1)(x – 21) is from to m.

Challenge Problem

10. a. Emma says that quadratic functions grow faster the larger the input value. Do you agree or disagree?

b. David says that even though y = x2 is nonlinear, it still represents a proportional relationship with y directly proportional to x2. Do you agree or disagree?



EXERCISES

EXERCISES


Look at this graph.

1. Which equation corresponds to this parabola?

A y = –x2 + 4x + 16

B y = x2 + 8x

C y = –x2 – 8x

D y = –x2 + 8x

2. What is the average rate of change as x changes from 0 to 1? _____



5. What is the average rate of change from the origin to the vertex of the parabola? Show your calculations.

0 2 4 6 8 10 x

y

0

10

2

12

4

6

8

14

16

LESSON 15: AVERAGE RATE OF CHANGE



EXERCISESLESSON 15: AVERAGE RATE OF CHANGE

6. Complete the table of the first and second differences for this parabola.

x yFirst

DifferenceSecond

Difference

0 — —

1 —

2

3

4

5

6

7. Make a table to show the second difference of the quadratic function y = 3x2 in the domain –3 ≤ x ≤ 3.

x yFirst

DifferenceSecond

Difference

–3 — —

–2 —

–1

0

1

2

3

8. Consider the quadratic equation y = x2 + 3x – 2. The second difference for this quadratic equation is _____ .

8 x

y

6420

4

0

6

8

2

10

y = –x2 + 6x



ANSWERSLESSON 15: AVERAGE RATE OF CHANGE

Challenge Problem

9. In the lesson, it was stated that all quadratic equations have a second difference that is constant.

a. Do all quadratic equations have the same second difference?

b. Compare the second differences of these two equations, and justify your response.

y = x2 + 4 y = 4x2

c. Which of the quadratic function coefficients in standard form affect the second difference? Explain why your answer makes sense.



EXERCISES

EXERCISES

1. Refer to this table.

x –5 –4 –3 –2 –1 0 1

f(x) 25 16 9 4 1 0 1

Write an equation of the function f(x).

2. The function g(x) is defined as having a y-intercept at the point (0, 2) and a constant slope of 4. Write an equation for g(x).

3. The function f(x) is defined by this table.

x –5 –4 –3 –2 –1 0 1

f(x) 25 16 9 4 1 0 1

The function g(x) is defined as having a y-intercept at the point (0, 2) and a constant slope of 4. Graph both f(x) and g(x) in the same coordinate plane.


x –5 –4 –3 –2 –1 0 1

f(x) 25 16 9 4 1 0 1

The function g(x) is defined as having a y-intercept at the point (0, 2) and a constant slope of 4.

a. Find f(–2) and g(–2).

f(–2) = _____

g(–2) = _____

b. Find the sum of f(–2) and g(–2).

f(–2) + g(–2) = _____

LESSON 16: OPERATIONS ON FUNCTIONS



LESSON 16: OPERATIONS ON FUNCTIONS EXERCISES


x –5 –4 –3 –2 –1 0 1

f(x) 25 16 9 4 1 0 1

The function g(x) is defined as having a y-intercept at the point (0, 2) and a constant slope of 4.

Draw the graph of f(x) + g(x).

6. a. Describe the geometrical reasoning behind the graph of f(x) + g(x).

b. How do the graphs of f(x) and g(x) relate to the graph of f(x) + g(x)?

7. f(x) = x2 – 3 g(x) = 5x

What is the value of f(–2) + g(–2)?

A –15

B –9

C 13

D 20

8. Here are three functions. f(x) = x2 – 3 g(x) = 5x h(x) = f(x) + g(x)

What is the value of h(4)?

A 8

B 21

C 33

D 36



EXERCISESLESSON 16: OPERATIONS ON FUNCTIONS

Challenge Problem

9. This graph shows two quadratic functions—g(x) and h(x)—which differ by a linear function. The linear function has a constant rate of change of 2 and goes through the point (1, 7). Find the equations of all three functions.

–4 4–2 82 6 x

y

g(x)

h(x)

–8

–6

–4

–2

4

8

2

6



EXERCISES

EXERCISES

1. Complete this table of values that represents the function f x x( ) = +2 .

x 0 1 4 9 16 25 36

f(x)

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

3. How is the graph of f x x( ) = +2 related to the standard graph of y x= ?

4. Which equation represents this graph?

50 10 15 20 25

5

0

10

15

y

x

A y x= +3

B y x= 3

C y x= 5

D y x= +5

LESSON 17: SQUARE ROOT FUNCTIONS



EXERCISESLESSON 17: SQUARE ROOT FUNCTIONS

5. Which equation represents this graph?

50 10 15 20 25

5

0

y

x

A y x= 3

B y x= + 3

C y x= − 3

D y x= + 3

6. Where does the graph of the equation y x= + −2 4 “start”?

A (–2, –4)

B (2, 4)

C (4, 2)

D (–4, –2)

7.

–5

1

3

y

x

2

–3

–2

–1

–4

5–4 6–3 7–2 8–1 91 2 3 4 10

Which equation represents this graph?

A y x= + +3 4

B y x= +− 4 3

C y x= + −4 3

D y x= − −3 4




8. These are the graphs of f x x( ) = + +3 2 and g x x( ) = +( ) +3 22

, when x > –3.

–5 5 10 15 20 x

y

0

5

10

15

20

f x x( ) = + +3 2

g x x( ) = +( ) +3 22

a. f(x) and g(x) have a domain of x > _____

A –3

B 0

C 2

D 3

b. f(x) and g(x) have a range of y > _____

A –3

B 0

C 2

D 3




9. These are the graphs of f x x( ) = + +3 2 and g x x( ) = +( ) +3 22

, when x > –3.

–5 5 10 15 20 x

y

0

5

10

15

20

f x x( ) = + +3 2

g x x( ) = +( ) +3 22

Which equation is the line of symmetry between functions f(x) and g(x)?

A y = x

B y = x – 3

C y = x + 2

D y = x + 5

Challenge Problem

10. Can you create a square root function that opens to the left? If possible, give an example equation and show the graph of your function.



EXERCISESLESSON 18: PUTTING IT TOGETHER 2

EXERCISES

1. Read through your Self Check and think about your work in this unit.

Write three things you have learned.

Share your work with a classmate. Does your classmate understand what you wrote?

2. Write a summary about the important features of a quadratic function. In your description, include information about the formulas for quadratic functions, the shape of the graph, the rate of change, and the inverse function.

3. Compare quadratic functions in standard, vertex, and factored forms. Give examples of the different forms, and describe what kind of information each form gives you.

4. Review the notes you took during the lessons about quadratic and square root functions. Add any additional ideas you have about the topics to your notes.

5. Complete any exercises from this unit you have not finished.

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