lesson 1: zooming in on parabolas exercisesrusdmath.weebly.com/uploads/1/1/1/5/11156667/g9_u6... ·...

Copyright © 2014 Pearson Education, Inc. 5

High School: Quadratic Functions

EXERCISES

• Review your Unit Assessment from the previous unit.

• Write your wonderings about working with quadratic functions.

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

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

LESSON 1: ZOOMING IN ON PARABOLAS



EXERCISES

EXERCISES

Use the following graph to answer problems 1–4.

–8–10 –4–6 –2 2 x

y

–4

–2

2

4

1. What are the coordinates of the parabola’s vertex?

2. What is the axis of symmetry?

3. Consider the standard form of a quadratic function f(x) = ax2 + bx + c. What can you say about the constants a, b, and c for the given parabola? Explain.

4. Find an equation for this graph.

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

5. Which constant(s) of the function (a, b, and/or c) determine the vertex of the parabola?

6. Which constant(s) of the function affect the orientation of the parabola?

7. Which constant(s) of the function affect the shape of the parabola?

8. Which constant(s) of the function determine the axis of symmetry of the parabola?

Challenge Problem

9. Suppose you have a quadratic function f(x) = ax2 + bx + c. Describe the graph of g(x) = –ax2 – bx – c with respect to that of f(x).

Hint: Use your answer for problem 1 to set up a system of equations.

LESSON 2: QUADRATIC FUNCTIONS



EXERCISES

EXERCISES

The Frederick Douglass–Susan B. Anthony Memorial Bridge in Rochester, New York, is supported by an arch. Assuming the x-axis represents the bottom of the arch and x and y

are measured in meters, the arch can be described by the equation y x= − +13262

2,178.

1. Use the Graphing tool to draw a graph of the arch.

2. What is the height of the arch?

3. What is the span of the arch?

4. Suppose there are seven braces connecting both sides of the arch that are evenly spaced with respect to the roadway below it. What is the height of each brace?

LESSON 3: REAL-WORLD PROBLEMS



EXERCISESLESSON 3: REAL-WORLD PROBLEMS

5. Emma is volunteering at an animal shelter and has 320 ft of fence she must use to construct 3 equal-sized pens, as shown.

y

x x x x

yy

Explain why the lengths x and y are subject to the constraint 2x + 3y = 160.

6. The area A of each pen is given by A = xy. Use the constraint from problem 5 to write A as a function of x.

7. Use the Graphing tool to draw a graph of A(x).

a. What is the maximum area of each pen?b. What are the dimensions (x, y) that give the maximum area?

Challenge Problem

8. Suppose Emma needs to construct just 2 pens with her 320 ft of fence. What are the new dimensions of each pen if she still wants to maximize the area?



EXERCISESLESSON 4: GRAPHING QUADRATICS

EXERCISES

1. Describe the possible zeros for the quadratic function f(x) = ax2 + bx + c with the following constraints. Illustrate your results graphically, if necessary.

a. a > 0b. a < 0

2. Explain why the formula for a quadratic function’s axis of symmetry is independent of the constant c.

List as much as you can about the vertex of each quadratic function described in problems 3–5.

3. A quadratic function with zeros –13 and 7

4. A quadratic function with a = –2 and b = 3

5. The quadratic function f x x x( ) –= +13

43

53

2

List as much as you can about the constants a, b, and c for the quadratic functions described in problems 6–8.

6. A quadratic function that goes through the origin

7. A quadratic function where the vertex represents the maximum value

8. The quadratic function with vertex (1, 2), containing the point (2, 0)

Challenge Problem

9. Describe the solution(s) (i.e., points of intersection) of the two quadratic functions f(x) = ax2 + bx + c and g(x) = –ax2 + bx + c.



EXERCISES

EXERCISES

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.

1. Find C(12), and explain your solution.

2. What quantity of amplifier cabinets will result in the lowest daily costs of production? How do you know?

3. Find the cost of producing the quantity you determined for problem 2.

4. Use the Graphing tool to draw a graph of the given function. What is the appropriate domain and range?

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 tee box is given by y x x= − +8729

3227

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

5. How far from the tee does the ball reach its maximum height?

6. Find the maximum height of the ball.

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

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

Challenge Problem

9. How long after Miki hits the ball does it reach its maximum height? Try to write the height y as a function of time t, again ignoring air/wind resistance.

LESSON 5: REAL-WORLD PARABOLAS



EXERCISES

EXERCISES

1. Look at these graphs.

a. Label the coordinates for each of the five marked points on each parabola. What function does the green graph represent?

b. How is each point on the orange parabola related to the corresponding point on the green parabola?

c. Write a function to describe the orange parabola.

Write a function for each of the graphs in problems 2–5. Each graph is a translation of the parent function f(x) = x2.

2.

–6 –4 –2

–2

–4

2

4

6

2 4 6 x

y

–6 –4 –2 0 2 4 x

y

2

4

6

8

10

2 4

51

3

2 4

51

3

LESSON 6: TRANSLATIONS



EXERCISES

3.

–6 –4 –2

–2

–4

2

4

6

2 4 6 x

y

4.

–6 –4 –2

–2

–4

2

4

6

2 4 6 x

y

5.

–6 –4 –2

–2

–4

2

4

6

2 4 6 x

y

LESSON 6: TRANSLATIONS



EXERCISESLESSON 6: TRANSLATIONS

Graph each of the functions in problems 6–9 as translations of the parent function f(x) = x2.

6. g(x) = (x – 2)2

7. h(x) = x2 – 2

8. j(x) = (x – 3)2 + 5

9. k(x) = (x + 4)2 – 1

Challenge Problem

10. Use the Graphing tool to draw the graphs of y = a(x + 6)2 – 3 for a = 1, –1, 2, and –2. What is the vertex for each of the given values of a? What can you conclude?



EXERCISES

EXERCISES

1. Look at this graph.

a. Write the coordinates of any three points on the graph.

b. Write the function g(x) that describes the graph.

c. Describe in words the transformation(s) of f(x) = x2 used to obtain g(x).

Find the vertex, axis of symmetry, direction, and stretch/squeeze factor for each of the parabolas in problems 2–6.

2. y = –x2 + 3

3. yx= +2 16

4

4. y x= 12

82( – )

5. y x= +– ( ) –13

7 22

6. y = 2(6 – x)2 + 5

7. 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.

Challenge Problem

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

–5 –4 1–3 2–2 3–1 4 5 x

y

–5

1

–4

2

–3

3

–2

4

–1

5

LESSON 7: TRANSFORMING PARABOLAS



EXERCISES

EXERCISES

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.

1. Draw a graph of the given function.

2. What is the width of the field?

3. What is the maximum height of the field?

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

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.

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

6. The population of another bacteria is given by n(t) = 16(t + 2.5)2 – 16(t + 2.5) + 96. Explain how the two population curves are related.

7. Find the temperature at which the populations of the two bacteria are equal. What is the size of each population at this temperature?

LESSON 8: TRANSFORMATION APPLICATIONS



EXERCISESLESSON 8: TRANSFORMATION APPLICATIONS

Challenge Problem

8. This athletic field consists of a rectangular region with a semicircle at each end.

r

x

Find the dimensions r and x that give the greatest possible area of the rectangular region, given that the perimeter of the entire field is 0.25 mi (1,320 ft).



EXERCISES

• Read through your Self Check and think about your work in this lesson.

• Write down what you have learned during the lesson.

• What would you do differently if you were starting the Self Check task now?

• Which method would you prefer to use if you were doing the task again? Why?

• Compare the new approaches you learned with your original method.

• Record your ideas — keep track of your problem solving strategies.

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

LESSON 9: PUTTING IT TOGETHER



EXERCISES

EXERCISES

1. From this graph, write the function in factored form.

–6 –4 –2 2 x

y

–4

–2

2

4

2. From the following table, write the corresponding function in factored form.

x –1 0 1 2 3 4 5

y 10 4 0 –2 –2 0 4

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

a. What are the coordinates of the x-intercepts?b. What is the axis of symmetry?c. What are the coordinates of the vertex?d. Create a graph of the function.

4. Determine the roots of the equation f(x) = (x + 3)(x – 8).

5. Determine the roots of the equation g(x) = (3x – 9)(x + 2).

Challenge Problem

6. Determine all of the roots for the equation h(x) = x(x – 0.75)(x + 3).

LESSON 12: FACTORED FORM AND ZEROES



EXERCISES

EXERCISES

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

2. Consider a parabola with a maximum at (3, 7). The parabola passes through the point (5, 3). Write an equation for this parabola in vertex form, and graph the parabola.

3. Consider a parabola with a minimum at (–3, –2). The parabola passes through the point (0, 1). Write an equation for this parabola in vertex form, and graph the parabola.

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

a. Will this parabola have a maximum or a minimum at its vertex?b. What are the coordinates of the vertex?c. What is the y-intercept?

Challenge Problem

5. Create two different parabola functions that have the same vertex. Create equations and graphs for both of your functions. How many different parabolas could have the same vertex?

LESSON 13: VERTEX FORM AND THE VERTEX



EXERCISES

EXERCISES

For problems 1–8, refer to the following graph, which shows the height (in meters) of a rising hot air balloon over time (in seconds).

1000 200 300Time (sec)

Hei

ght

(m)

400 500 600

250

0

500

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 (i.e., from point A to point D)?

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

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

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

5. Which section of the balloon’s ascent is the fastest?

6. Which section is the slowest ascent?

7. Describe in words what the different values for rate of change mean in this context.

Challenge Problem

8. On which part of the graph is the average rate of change closest to the overall ascent rate (between points A and D)? On which part of the graph is the average rate of change the least like the overall ascent rate?

LESSON 14: RATE OF CHANGE



EXERCISES

EXERCISES

1. Look at this graph.

a. What is the equation of this parabola?

b. Calculate the average rate of change as x changes from 0 to 1, from 1 to 2, and from 2 to 3.

c. What is the average rate of change from the origin to the vertex of the parabola?

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

x y First Difference

Second Difference

0 -- --

1 --

2

3

4

5

6

3. Create a table similar to the one in problem 2 to show the second difference of this quadratic equation on the domain –3 ≤ x ≤ 3.

y = 3x2

0 2 4 6 8 10 x

y

0

10

2

12

4

6

8

14

16

8 x

y

6420

4

0

6

8

2

10

y = –x2 + 6x

LESSON 15: AVERAGE RATE OF CHANGE



EXERCISESLESSON 15: AVERAGE RATE OF CHANGE

4. What is the second difference for this quadratic equation?

y = x2 + 3x – 2

Challenge Problem

5. In the lesson, it was stated that all quadratic equations have a second difference that is constant. Do all quadratic equations have the same second difference? Compare the second differences of these two equations, and justify your response.

y = x2 + 4 y = 4x2



EXERCISES

EXERCISES

For problems 1–6, refer to the following table.

x –5 –4 –3 –2 –1 0 1f(x) 25 16 9 4 1 0 1

1. Write an equation of the function f(x).

2. g(x) is another function, with y-intercept of 2 and constant slope of 4. Write an equation for g(x).

3. Graph both f(x) and g(x) on the same coordinate plane.

4. On the graph, look at the value of f(–2) and the value of g(–2). What is the value of f(–2) + g(–2)?

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

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

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

LESSON 16: OPERATIONS ON FUNCTIONS



LESSON 16: OPERATIONS ON FUNCTIONS EXERCISES

Challenge Problem

7. This graph shows two quadratic functions—g(x) in green, and h(x) in yellow—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

–8

–6

–4

–2

4

8

2

6



EXERCISES

EXERCISES

1. Which equation correctly 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

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

x 0 1 4 9 16 25 36

f(x)

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

4. How is the graph from problem 3 related to the standard graph of y x= ?

5. Which equation correctly represents this square root 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)

LESSON 17: SQUARE ROOT FUNCTIONS



EXERCISESLESSON 17: SQUARE ROOT FUNCTIONS

Challenge Problem

7. 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

• Read through your Self Check and think about your work in this lesson.

• Write down what you have learned during the lesson.

• What would you do differently if you were starting the Self Check task now?

• Which method would you prefer to use if you were doing the task again? Why?

• Compare the new approaches you learned with your original method.

• Record your ideas — keep track of your problem solving strategies.

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

lesson 1: zooming in on parabolas exercisesrusdmath.weebly.com/uploads/1/1/1/5/11156667/g9_u6... ·...

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