Big Ideas Math Algebra 1 Answers Chapter 8 Graphing Quadratic Functions

Big Ideas Math Algebra 1 Answers Chapter 8 Graphing Quadratic Functions Pdf of all exercises along with extra practice sections, quizzes, chapter tests, cumulative assessment, etc. are provided here. Students who are searching for the perfect study guide to enhance their math skills can refer to this Big ideas math algebra 1 ch 8 answer key.

Graphing quadratic functions concepts can be pretty easy by solving from the Big ideas math book answers of Algebra 1 Chapter 8 Graphing Quadratic Functions. The main aim of providing this Ch 8 Graphing Quadratic Functions Big Ideas Math Algebra 1 Answers is to offer quality education to the students and support them to grow high & become pro in math concepts.

Big Ideas Math Book Algebra 1 Answer Key Ch 8 Graphing Quadratic Functions

By using the BIM Textbook Solutions Algebra 1 Chapter 8 Graphing Quadratic Functions, you can solve all questions at the time of homework & assignments. Here is the list of topic-wise Big Ideas math Algebra 1 ch 8 Graphing Quadratic Functions Answer Key that helps you learn and understand every single concept of algebra 1 maths.

BigIdeasMath Algebra 1 Graphing Quadratic Functions Chapter 8 Exercise questions and answers are prepared by subject experts based on the Common Core Standards. So, students can easily solve all exercise questions, Questions from Practice Test, Chapter Test, Cumulative Practice, Performance Test, etc. covered in BIM Algebra 1 Ch 8 Answer key

• Graphing Quadratic Functions Maintaining Mathematical Proficiency – Page 417
• Graphing Quadratic Functions Mathematical Practices – Page 418
• Lesson 8.1 Graphing f(x) = ax2 – Page (419-424)
• Graphing f(x) = ax2 8.1 Exercises – Page (423-424)
• Lesson 8.2 Graphing f(x) = ax2 + c – Page (425-430)
• Graphing f(x) = ax2 + c 8.2 Exercises – Page (429-430)
• Lesson 8.3 Graphing f(x) = ax2 + bx + c – Page (431-438)
• Graphing f(x) = ax2 + bx + c 8.3 Exercises – Page (436-438)
• Graphing Quadratic Functions Study Skills: Learning Visually – Page 439
• Graphing Quadratic Functions 8.1 – 8.3 Quiz – Page 440
• Lesson 8.4 Graphing f(x) = a(x – h)2 + k – Page (441-448)
• Graphing f(x) = a(x – h)2 + k 8.4 Exercises – Page (446-448)
• Lesson 8.5 Using Intercept Form – Page (449-458)
• Using Intercept Form 8.5 Exercises – Page (455-458)
• Lesson 8.6 Comparing Linear, Exponential, and Quadratic Functions – Page (459-468)
• Comparing Linear, Exponential, and Quadratic Functions 8.6 Exercises – Page (465-468)
• Graphing Quadratic Functions Performance Task: Asteroid Aim – Page 469
• Graphing Quadratic Functions Chapter Review – Page (470-472)
• Graphing Quadratic Functions Chapter Test – Page 473 
• Graphing Quadratic Functions Cumulative Assessment – Page (474-475)

Graphing Quadratic Functions Maintaining Mathematical Proficiency

Graph the linear equation.

Question 1.
y = 2x – 3
Answer:

Question 2.
y = -3x + 4
Answer:

Question 3.
y = – \(\frac{1}{2}\)x – 2
Answer:

Question 4.
y = x + 5
Answer:

Evaluate the expression when x = −2.
Question 5.
5x² – 9
Answer:

Question 6.
3x² + x – 2
Answer:

Question 7.
-x² + 4x + 1
Answer:

Question 8.
x² + 8x + 5
Answer:

Question 9.
-2x² – 4x + 3
Answer:

Question 10.
-4x² + 2x – 6
Answer:

Question 11.
ABSTRACT REASONING
Complete the table. Find a pattern in the differences of consecutive y-values. Use the pattern to write an expression for y when x = 6.

Answer:

Graphing Quadratic Functions Mathematical Practices

Mathematically proficient students try special cases of the original problem to gain insight into its solution.

Monitoring Progress

Graph the quadratic function. Then describe its graph.
Question 1.
y = -x²
Answer:

Question 2.
y = 2x²
Answer:

Question 3.
f(x) = 2x² + 1
Answer:

Question 4.
f(x) = 2x² – 1
Answer:

Question 5.
f(x) = \(\frac{1}{2}\)x² + 4x + 3
Answer:

Question 6.
f(x) = \(\frac{1}{2}\) x² – 4x + 3
Answer:

Question 7.
y = -2(x + 1)² + 1
Answer:

Question 8.
y = -2(x – 1)² + 1
Answer:

Question 9.
How are the graphs in Monitoring Progress Questions 1-8 similar? How are they different?
Answer:

Lesson 8.1 Graphing f(x) = ax²

Essential Question What are some of the characteristics of the graph of a quadratic function of the form f(x) = ax²?

EXPLORATION 1

Graphing Quadratic Functions
Work with a partner. Graph each quadratic function. Compare each graph to the graph of f(x) = x².

Communicate Your Answer

Question 2.
What are some of the characteristics of the graph of a quadratic function of the form f(x) = ax²?
Answer:

Question 3.
How does the value of a affect the graph of f(x) = ax²? Consider 0 < a < 1, a> 1, -1 < a < 0, and a < -1. Use a graphing calculator to verify your answers.

Answer:

Question 4.
The figure shows the graph of a quadratic function of the form y = ax². Which of the intervals in Question 3 describes the value of a? Explain your reasoning.

Answer:

Monitoring Progress

Identify characteristics of the quadratic function and its graph.
Question 1.

Answer:

Question 2.

Answer:

Graph the function. Compare the graph to the graph of f(x) = x².
Question 3.
g(x) = 5x²
Answer:

Question 4.
h(x) = \(\frac{1}{3}\)x²
Answer:

Question 5.
n(x) = \(\frac{3}{2}\)x²
Answer:

Question 6.
p(x) = -3x²
Answer:

Question 7.
q(x) = -0.1x²
Answer:

Question 8.
g(x) = –\(\frac{1}{4}\)x²
Answer:

Question 9.
The cross section of a spotlight can be modeled by the graph of y = 0.5x², where x and y are measured in inches and -2 ≤ x ≤ 2. Find the width and depth of the spotlight.
Answer:

Graphing f(x) = ax² 8.1 Exercises

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
What is the U-shaped graph of a quadratic function called?
Answer:

Question 2.
WRITING
When does the graph of a quadratic function open up? open down?
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3 and 4, identify characteristics of the quadratic function and its graph.
Question 3.

Answer:

Question 4.

Answer:

In Exercises 5–12, graph the function. Compare the graph to the graph of f(x) = x².
Question 5.
g(x) = 6x²
Answer:

Question 6.
b(x) = 2.5x²
Answer:

Question 7.
h(x) = \(\frac{1}{4}\)x²
Answer:

Question 8.
j(x) = 0.75x²
Answer:

Question 9.
m(x) = -2x²
Answer:

Question 10.
q(x) = –\(\frac{9}{2}\)x²
Answer:

Question 11.
k(x) = -0.2x²
Answer:

Question 12.
p(x) = –\(\frac{2}{3}\)x²
Answer:

In Exercises 13–16, use a graphing calculator to graph the function. Compare the graph to the graph of y = −4x².
Question 13.
y = 4x²
Answer:

Question 14.
y = -0.4x²
Answer:

Question 15.
y = -0.04x²
Answer:

Question 16.
y = -0.004x²
Answer:

Question 17.
ERROR ANALYSIS
Describe and correct the error in graphing and comparing y = x² and y = 0.5x².

Answer:

Question 18.
MODELING WITH MATHEMATICS
The arch support of a bridge can be modeled by y = -0.0012x², where x and y are measured in feet. Find the height and width of the arch.

Answer:

Question 19.
PROBLEM SOLVING
The breaking strength z (in pounds) of a manila rope can be modeled by z = 8900d², where d is the diameter (in inches) of the rope.

a. Describe the domain and range of the function.
b. Graph the function using the domain in part (a).
c. A manila rope has four times the breaking strength of another manila rope. Does the stronger rope have four times the diameter? Explain.
Answer:

Question 20.
HOW DO YOU SEE IT?
Describe the possible values of a.

Answer:

ANALYZING GRAPHS In Exercises 21–23, use the graph.

Question 21.
When is each function increasing?
Answer:

Question 22.
When is each function decreasing?
Answer:

Question 23.
Which function could include the point (-2, 3)? Find the value of a when the graph passes through (-2, 3).
Answer:

Question 24.
REASONING
Is the x-intercept of the graph of y = x² always 0? Justify your answer.
Answer:

Question 25.
REASONING
A parabola opens up and passes through (-4, 2) and (6, -3). How do you know that (-4, 2) is not the vertex?
Answer:

ABSTRACT REASONING In Exercises 26–29, determine whether the statement is always, sometimes, or never true. Explain your reasoning.
Question 26.
The graph of f(x) = x² is narrower than the graph of g(x) = x² when a > 0.
Answer:

Question 27.
The graph of f(x) = x² is narrower than the graph of g(x) = x² when |a| > 1.
Answer:

Question 28.
The graph of f(x) = x² is wider than the graph of g(x) = x² when 0 < |a| < 1.
Answer:

Question 29.
The graph of f(x) = x² is wider than the graph of g(x) = dx² when |a | > |d| .
Answer:

Question 30.
THOUGHT PROVOKING
Draw the isosceles triangle shown. Divide each leg into eight congruent segments. Connect the highest point of one leg with the lowest point of the other leg. Then connect the second highest point of one leg to the second lowest point of the other leg. Continue this process. Write a quadratic equation whose graph models the shape that appears.

Answer:

Question 31.
MAKING AN ARGUMENT
The diagram shows the parabolic cross section of a swirling glass of water, where x and y are measured in centimeters.

a. About how wide is the mouth of the glass?
b. Your friend claims that the rotational speed of the water would have to increase for the cross section to be modeled by y = 0.1x². Is your friend correct? Explain your reasoning.
Answer:

Maintaining Mathematical Proficiency

Evaluate the expression when n = 3 and x = −2.
Question 32.
n² + 5
Answer:

Question 33.
3x² – 9
Answer:

Question 34.
-4n² + 11
Answer:

Question 35.
n + 2x²
Answer:

Lesson 8.2 Graphing f(x) = ax² + c

Essential Question How does the value of c affect the graph of f(x) = -ax² + c?

EXPLORATION 1

Graphing y = ax² + c
Work with a partner. Sketch the graphs of the functions in the same coordinate plane. What do you notice?

EXPLORATION 2

Finding x-Intercepts of Graphs
Work with a partner. Graph each function. Find the x-intercepts of the graph. Explain how you found the x-intercepts.

Communicate Your Answer

Question 3.
How does the value of c affect the graph of f(x) = ax² + c?
Answer:

Question 4.
Use a graphing calculator to verify your answers to Question 3.

Answer:

Question 5.
The figure shows the graph of a quadratic function of the form y = ax² + c. Describe possible values of a and c. Explain your reasoning.

Answer:

Monitoring Progress

Graph the function. Compare the graph to the graph of f(x) = x².
Question 1.
g(x) = x² – 5
Answer:

Question 2.
h(x) = x² + 3
Answer:

Graph the function. Compare the graph to the graph of f(x) = x².
Question 3.
g(x) = 2x² – 5
Answer:

Question 4.
h(x) = – \(\frac{1}{4}\)x² + 4
Answer:

Question 5.
Let f(x) = 3x² – 1 and g(x) = f (x) + 3.
a. Describe the transformation from the graph of f to the graph of g. Then graph f and g in the same coordinate plane.
b. Write an equation that represents g in terms of x.
Answer:

Question 6.
Explain why only nonnegative values of t are used in Example 4.
Answer:

Question 7.
WHAT IF?
The egg is dropped from a height of 100 feet. After how many seconds does the egg hit the ground?
Answer:

Graphing f(x) = ax² + c 8.2 Exercises

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
State the vertex and axis of symmetry of the graph of y = ax² + c.
Answer:

Question 2.
WRITING
How does the graph of y = ax² + c compare to the graph of y = ax²?
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, graph the function. Compare the graph to the graph of f(x) = x².
Question 3.
g(x) = x² + 6
Answer:

Question 4.
h(x) = x² + 8
Answer:

Question 5.
p(x) = x² – 3
Answer:

Question 6.
q(x) = x² – 1
Answer:

In Exercises 7–12, graph the function. Compare the graph to the graph of f(x) = x².
Question 7.
g(x) = -x² + 3
Answer:

Question 8.
h(x) = -x² – 7
Answer:

Question 9.
s(x) = 2x² – 4
Answer:

Question 10.
t(x) = -3x² + 1
Answer:

Question 11.
p(x) = – \(\frac{1}{3}\)x² – 2
Answer:

Question 12.
q(x) = \(\frac{1}{2}\)x² + 6
Answer:

In Exercises 13–16, 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.
Question 13.
f(x) = 3x² + 4
g(x) = f(x) + 2
Answer:

Question 14.
f(x) = \(\frac{1}{2}\)x² + 1
g(x) = f(x) – 4
Answer:

Question 15.
f(x) = – \(\frac{1}{4}\)x² – 6
g(x) = f(x) – 3
Answer:

Question 16.
f(x) = 4x² – 5
g(x) = f(x) + 7
Answer:

Question 17.
ERROR ANALYSIS
Describe and correct the error in comparing the graphs.

Answer:

Question 18.
ERROR ANALYSIS
Describe and correct the error in graphing and comparing f(x) = x² and g(x) = x² – 10.

Answer:

In Exercises 19–26, find the zeros of the function.
Question 19.
y = x² – 1
Answer:

Question 20.
y = x² – 36
Answer:

Question 21.
f(x) = -x² + 25
Answer:

Question 22.
f(x) = -x² + 49
Answer:

Question 23.
f(x) = 4x² – 16
Answer:

Question 24.
f(x) = 3x² – 27
Answer:

Question 25.
f(x) = -12x² + 3
Answer:

Question 26.
f(x) = -8x² + 98
Answer:

Question 27.
MODELING WITH MATHEMATICS
A water balloon is dropped from a height of 144 feet.
a. After how many seconds does the water balloon hit the ground?
b. Suppose the initial height is adjusted by k feet. How does this affect part (a)?
Answer:

Question 28.
MODELING WITH MATHEMATICS
The function y = -16x² + 36 represents the height y (in feet) of an apple x seconds after falling from a tree. Find and interpret the x- and y-intercepts.
Answer:

In Exercises 29–32, sketch a parabola with the given characteristics.
Question 29.
The parabola opens up, and the vertex is (0, 3).
Answer:

Question 30.
The vertex is (0, 4), and one of the x-intercepts is 2.
Answer:

Question 31.
The related function is increasing when x < 0, and the zeros are -1 and 1.
Answer:

Question 32.
The highest point on the parabola is (0, -5).
Answer:

Question 33.
DRAWING CONCLUSIONS
You and your friend both drop a ball at the same time. The function h(x) = -16x² + 256 represents the height (in feet) of your ball after x seconds. The function g(x) = -16x² + 300 represents the height (in feet) of your friend’s ball after x seconds.
a. Write the function T(x) = h(x) – g(x). What does T(x) represent?
b. When your ball hits the ground, what is the height of your friend’s ball? Use a graph to justify your answer.
Answer:

Question 34.
MAKING AN ARGUMENT
Your friend claims that in the equation y = ax² + c, the vertex changes when the value of a changes. Is your friend correct? Explain your reasoning.
Answer:

Question 35.
MATHEMATICAL CONNECTIONS
The area A (in square feet) of a square patio is represented by A = x², where x is the length of one side of the patio. You add 48 square feet to the patio, resulting in a total area of 192 square feet. What are the dimensions of the original patio? Use a graph to justify your answer.
Answer:

Question 36.
HOW DO YOU SEE IT?
The graph of f(x) = ax² + c is shown. Points A and B are the same distance from the vertex of the graph of f. Which point is closer to the vertex of the graph of f as c increases?

Answer:

Question 37.
REASONING
Describe two algebraic methods you can use to find the zeros of the function f(t) = -16t² + 400. Check your answer by graphing.
Answer:

Question 38.
PROBLEM SOLVING
The paths of water from three different garden waterfalls are given below. Each function gives the height h (in feet) and the horizontal distance d (in feet) of the water.

Waterfall 1 h = -3.1d² + 4.8
Waterfall 2 h = -3.5d² + 1.9
Waterfall 3 h = -1.1d² + 1.6
a. Which waterfall drops water from the highest point?
b. Which waterfall follows the narrowest path?
c. Which waterfall sends water the farthest?
Answer:

Question 39.
WRITING EQUATIONS
Two acorns fall to the ground from an oak tree. One falls 45 feet, while the other falls 32 feet.
a. For each acorn, write an equation that represents the height h (in feet) as a function of the time t (in seconds).
b. Describe how the graphs of the two equations are related.
Answer:

Question 40.
THOUGHT PROVOKING
One of two classic problems in calculus is to find the area under a curve. Approximate the area of the region bounded by the parabola and the x-axis. Show your work.

Answer:

Question 41.
CRITICAL THINKING
A cross section of the parabolic surface of the antenna shown can be modeled by y = 0.012x², where x and y are measured in feet. The antenna is moved up so that the outer edges of the dish are 25 feet above x-axis. Where is the vertex of the cross section located? Explain.

Answer:

Maintaining Mathematical Proficiency

Evaluate the expression when a = 4 and b = −3.
Question 42.
\(\frac{a}{4b}\)
Answer:

Question 43.
–\(\frac{b}{2a}\)
Answer:

Question 44.
\(\frac{a-b}{3 a+b}\)
Answer:

Question 45.
–\(\frac{b+2 a}{a b}\)
Answer:

Lesson 8.3 Graphing f(x) = ax² + bx + c

Essential Question How can you find the vertex of the graph of f(x) = ax² + bx + c?

EXPLORATION 1

Comparing x-Intercepts with the Vertex
Work with a partner.
a. Sketch the graphs of y = 2x² – 8x and y = 2x² – 8x + 6.
b. What do you notice about the x-coordinate of the vertex of each graph?
c. Use the graph of y = 2x² – 8x to find its x-intercepts. Verify your answer by solving 0 = 2x² – 8x.
d. Compare the value of the x-coordinate of the vertex with the values of the x-intercepts.

EXPLORATION 2

Finding x-Intercepts
Work with a partner.
a. Solve 0 = ax² + bx for x by factoring.
b. What are the x-intercepts of the graph of y = ax² + bx?
c. Copy and complete the table to verify your answer.

EXPLORATION 3

Deductive Reasoning
Work with a partner. Complete the following logical argument.

Communicate Your Answer

Question 4.
How can you find the vertex of the graph of f(x) = ax² + bx + c?
Answer:

Question 5.
Without graphing, find the vertex of the graph of f(x) = x² – 4x + 3. Check your result by graphing.
Answer:

Monitoring Progress

Find (a) the axis of symmetry and (b) the vertex of the graph of the function.
Question 1.
f(x) = 3x² – 2x
Answer:

Question 2.
g(x) = x² + 6x + 5
Answer:

Question 3.
h(x) = – \(\frac{1}{2}\)x² + 7x – 4
Answer:

Graph the function. Describe the domain and range.
Question 4.
h(x) = 2x² + 4x + 1
Answer:

Question 5.
k(x) = x² – 8x + 7
Answer:

Question 6.
p(x) = -5x² – 10x – 2
Answer:

Tell whether the function has a minimum value or a maximum value. Then find the value.
Question 7.
g(x) = 8x² – 8x + 6
Answer:

Question 8.
h(x) = – \(\frac{1}{4}\)x² + 3x + 1
Answer:

Question 9.
The cables between the two towers of the Tacoma Narrows Bridge in Washington form a parabola that can be modeled by y = 0.00016x² – 0.46x + 507, where x and y are measured in feet. What is the height of the cable above the water at its lowest point?
Answer:

Question 10.
Which balloon is in the air longer? Explain your reasoning.
Answer:

Question 11.
Which balloon reaches its maximum height faster? Explain your reasoning.
Answer:

Graphing f(x) = ax² + bx + c 8.3 Exercises

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Explain how you can tell whether a quadratic function has a maximum value or a minimum value without graphing the function.
Answer:

Question 2.
DIFFERENT WORDS, SAME QUESTION
Consider the quadratic function f(x) = -2x² + 8x + 24. Which is different? Find “both” answers.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, find the vertex, the axis of symmetry, and the y-intercept of the graph.
Question 3.

Answer:

Question 4.

Answer:

Question 5.

Answer:

Question 6.

Answer:

In Exercises 7–12, find (a) the axis of symmetry and (b) the vertex of the graph of the function.
Question 7.
f(x) = 2x² – 4x
Answer:

Question 8.
y = 3x² + 2x
Answer:

Question 9.
y = -9x² – 18x – 1
Answer:

Question 10.
f(x) = -6x² + 24x – 20
Answer:

Question 11.
f(x) = \(\frac{2}{5}\)x² – 4x + 14
Answer:

Question 12.
y = – \(\frac{3}{4}\) x² + 9x – 18
Answer:

In Exercises 13–18, graph the function. Describe the domain and range.
Question 13.
f(x) = 2x² + 12x + 4
Answer:

Question 14.
y = 4x² + 24x + 13
Answer:

Question 15.
y = -8x² – 16x – 9
Answer:

Question 16.
f(x) = -5x² + 20x – 7
Answer:

Question 17.
y = \(\frac{2}{3}\)x² – 6x + 5
Answer:

Question 18.
f(x) = – \(\frac{1}{2}\)x² – 3x – 4
Answer:

Question 19.
ERROR ANALYSIS
Describe and correct the error in finding the axis of symmetry of the graph of y = 3x² – 12x + 11.

Answer:

Question 20.
ERROR ANALYSIS
Describe and correct the error in graphing the function f(x) = -x²+ 4x + 3.

Answer:

In Exercises 21–26, tell whether the function has a minimum value or a maximum value. Then find the value.
Question 21.
y = 3x² – 18x + 15
Answer:

Question 22.
f(x) = -5x² + 10x + 7
Answer:

Question 23.
f(x) = -4x² + 4x – 2
Answer:

Question 24.
y = 2x² – 10x + 13
Answer:

Question 25.
y = – \(\frac{1}{2}\)x² – 11x + 6
Answer:

Question 26.
f(x) = \(\frac{1}{5}\)x² – 5x + 27
Answer:

Question 27.
MODELING WITH MATHEMATICS
The function shown represents the height h (in feet) of a firework t seconds after it is launched. The firework explodes at its highest point.

a. When does the firework explode?
b. At what height does the firework explode?
Answer:

Question 28.
MODELING WITH MATHEMATICS
The function h(t) = -16t² + 16t represents the height (in feet) of a horse t seconds after it jumps during a steeplechase.
a. When does the horse reach its maximum height?
b. Can the horse clear a fence that is 3.5 feet tall? If so, by how much?
c. How long is the horse in the air?
Answer:

Question 29.
MODELING WITH MATHEMATICS
The cable between two towers of a suspension bridge can be modeled by the function shown, where x and y are measured in feet. The cable is at road level midway between the towers.

a. How far from each tower shown is the lowest point of the cable?
b. How high is the road above the water?
c. Describe the domain and range of the function shown.
Answer:

Question 30.
REASONING
Find the axis of symmetry of the graph of the equation y = ax² + bx + c when b = 0. Can you find the axis of symmetry when a = 0? Explain.
Answer:

Question 31.
ATTENDING TO PRECISION
The vertex of a parabola is (3, -1). One point on the parabola is (6, 8). Find another point on the parabola. Justify your answer.
Answer:

Question 32.
MAKING AN ARGUMENT
Your friend claims that it is possible to draw a parabola through any two points with different x-coordinates. Is your friend correct? Explain.
Answer:

USING TOOLS In Exercises 33–36, use the minimum or maximum feature of a graphing calculator to approximate the vertex of the graph of the function.
Question 33.
y = 0.5x² + \(\sqrt{2x}\) x – 3
Answer:

Question 34.
y = -6.2x² + 4.8x – 1
Answer:

Question 35.
y = -πx²+ 3x
Answer:

Question 36.
y = 0.25x² – 5^2/3x + 2
Answer:

Question 37.
MODELING WITH MATHEMATICS
The opening of one aircraft hangar is a parabolic arch that can be modeled by the equation y = -0.006x²+ 1.5x, where x and y are measured in feet. The opening of a second aircraft hangar is shown in the graph.

a. Which aircraft hangar is taller?
b. Which aircraft hangar is wider?
Answer:

Question 38.
MODELING WITH MATHEMATICS
An office supply store sells about 80 graphing calculators per month for $120 each. For each $6 decrease in price, the store expects to sell eight more calculators. The revenue from calculator sales is given by the function R(n) = (unit price)(units sold), or R(n) = (120 – 6n)(80 + 8n), where n is the number of $6 price decreases.

a. How much should the store charge to maximize monthly revenue?
b. Using a different revenue model, the store expects to sell five more calculators for each $4 decrease in price. Which revenue model results in a greater maximum monthly revenue? Explain.
Answer:

MATHEMATICAL CONNECTIONS In Exercises 39 and 40, (a) find the value of x that maximizes the area of the figure and (b) find the maximum area.
Question 39.

Answer:

Question 40

Answer:

Question 41.
WRITING
Compare the graph of g(x) = x² + 4x + 1 with the graph of h(x) = x² – 4x + 1.
Answer:

Question 42.
HOW DO YOU SEE IT?
During an archery competition, an archer shoots an arrow. The arrow follows the parabolic path shown, where x and y are measured in meters.

a. What is the initial height of the arrow?
b. Estimate the maximum height of the arrow.
c. How far does the arrow travel?
Answer:

Question 43.
USING TOOLS
The graph of a quadratic function passes through (3, 2), (4, 7), and (9, 2). Does the graph open up or down? Explain your reasoning.
Answer:

Question 44.
REASONING
For a quadratic function f, what does f(-\(\frac{b}{2a}\)) represent? Explain your reasoning.
Answer:

Question 45.
PROBLEM SOLVING
Write a function of the form y = ax² + bx whose graph contains the points (1, 6) and (3, 6).
Answer:

Question 46.
CRITICAL THINKING
Parabolas A and B contain the points shown. Identify characteristics of each parabola, if possible. Explain your reasoning.

Answer:

Question 47.
MODELING WITH MATHEMATICS
At a basketball game, an air cannon launches T-shirts into the crowd. The function y = – \(\frac{1}{8}\)x² + 4x represents the path of a T-shirt. The function 3y = 2x – 14 represents the height of the bleachers. In both functions, y represents vertical height (in feet) and x represents horizontal distance (in feet). At what height does the T-shirt land in the bleachers?
Answer:

Question 48.
THOUGHT PROVOKING
One of two classic problems in calculus is finding the slope of a tangent line to a curve. An example of a tangent line, which just touches the parabola at one point, is shown.

Approximate the slope of the tangent line to the graph of y = x² at the point (1, 1). Explain your reasoning.
Answer:

Question 49.
PROBLEM SOLVING
The owners of a dog shelter want to enclose a rectangular play area on the side of their building. They have k feet of fencing. What is the maximum area of the outside enclosure in terms of k? (Hint: Find the y-coordinate of the vertex of the graph of the area function.)

Answer:

Maintaining Mathematical Proficiency

Describe the transformation(s) from the graph of f(x) = |x| to the graph of the given function.
Question 50.
q(x) = |x + 6|
Answer:

Question 51.
h(x) = -0.5|x|
Answer:

Question 52.
g(x) = |x – 2| + 5
Answer:

Question 53.
p(x) = 3|x + 1|
Answer:

Graphing Quadratic Functions Study Skills: Learning Visually

8.1– 8.3 What Did You Learn?

Core Vocabulary

Core Concepts
Section 8.1
Characteristics of Quadratic Functions, p. 420
Graphing f(x) = ax² When a > 0, p. 421
Graphing f (x) = ax²When a < 0, p. 421

Section 8.2
Graphing f(x) = ax² + c, p. 426

Section 8.3
Graphing f(x) = ax² + bx + c, p. 432
Maximum and Minimum Values, p. 433

Mathematical Practices

Question 1.
Explain your plan for solving Exercise 18 on page 423.
Answer:

Question 2.
How does graphing a function in Exercise 27 on page 429 help you answer the questions?
Answer:

Question 3.
What definition and characteristics of the graph of a quadratic function did you use to answer Exercise 44 on page 438?
Answer:

Study Skills: Learning Visually

• Draw a picture of a word problem before writing a verbal model. You do not have to be an artist.
• When making a review card for a word problem, include a picture. This will help you recall the information while taking a test.
• Make sure your notes are visually neat for easy recall
  

Graphing Quadratic Functions 8.1 – 8.3 Quiz

Identify characteristics of the quadratic function and its graph.
Question 1.

Answer:

Question 2.

Answer:

Graph the function. Compare the graph to the graph of f(x) = x².
Question 3.
h(x) = -x²
Answer:

Question 4.
p(x) = 2x² + 2
Answer:

Question 5.
r(x) = 4x² – 16
Answer:

Question 6.
b(x) = 8x²
Answer:

Question 7.
g(x) = \(\frac{2}{5}\)x²
Answer:

Question 8.
m(x) = – \(\frac{1}{2}\)x² – 4
Answer:

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.
Question 9.
f(x) = 2x² + 1; g(x) = f(x) + 2
Answer:

Question 10.
f(x) = -3x² + 12; g(x) = f(x) – 9
Answer:

Question 11.
f(x) = \(\frac{1}{2}\)x² – 2; g(x) = f(x) – 6
Answer:

Question 12.
f(x) = 5x² – 3; g(x) = f(x) + 1
Answer:

Graph the function. Describe the domain and range.
Question 13.
f(x) = -4x² – 4x + 7
Answer:

Question 14.
f(x) = 2x² + 12x + 5
Answer:

Question 15.
y = x² + 4x – 5
Answer:

Question 16.
y = -3x² + 6x + 9
Answer:

Tell whether the function has a minimum value or a maximum value. Then find the value.
Question 17.
f(x) = 5x² + 10x – 3
Answer:

Question 18.
f(x) = – \(\frac{1}{2}\)x² + 2x + 16
Answer:

Question 19.
y = -x² + 4x + 12
Answer:

Question 20.
y = 2x² + 8x + 3
Answer:

Question 21.
The distance y (in feet) that a coconut falls after t seconds is given by the function y = 16t². Use a graph to determine how many seconds it takes for the coconut to fall 64 feet.
Answer:

Question 22.
The function y = -16t² + 25 represents the height y (in feet) of a pinecone t seconds after falling from a tree.
a. After how many seconds does the pinecone hit the ground?
b. A second pinecone falls from a height of 36 feet. Which pinecone hits the ground in the least amount of time? Explain.
Answer:

Question 23.
The function shown models the height (in feet) of a softball t seconds after it is pitched in an underhand motion. Describe the domain and range. Find the maximum height of the softball.

Answer:

Lesson 8.4 Graphing f(x) = a(x – h)² + k

Essential Question How can you describe the graph of f(x) = a(x – h)²?

EXPLORATION 1

Graphing y = a(x − h)² When h > 0
Work with a partner. Sketch the graphs of the functions in the same coordinate plane. How does the value of h affect the graph of y = a(x – h)²?

EXPLORATION 2

Graphing y = a(x − h)² When h < 0
Work with a partner. Sketch the graphs of the functions in the same coordinate plane. How does the value of h affect the graph of y = a(x – h)²?

Communicate Your Answer

Question 3.
How can you describe the graph of f(x) = a(x – h)²?
Answer:

Question 4.
Without graphing, describe the graph of each function. Use a graphing calculator to check your answer.

a. y = (x – 3)²
b. y = (x + 3)²
c. y = -(x – 3)²
Answer:

Monitoring Progress

Determine whether the function is even, odd, or neither.
Question 1.
f(x) = 5x
Answer:

Question 2.
g(x) = 2x
Answer:

Question 3.
h(x) = 2x² + 3
Answer:

Graph the function. Compare the graph to the graph of f(x) = x².
Question 4.
g(x) = 2(x + 5)²
Answer:

Question 5.
h(x) = -(x – 2)²
Answer:

Graph the function. Compare the graph to the graph of f(x) = x².
Question 6.
g(x) = 3(x – 1)² + 6
Answer:

Question 7.
h(x) = \(\frac{1}{2}\)(x + 4)² – 2
Answer:

Question 8.
Consider function g in Example 3. Graph f(x) = g(x) – 3
Answer:

Question 9.
WHAT IF?
The vertex is (3, 6). Write and graph a quadratic function that models the path.
Answer:

Graphing f(x) = a(x – h)² + k 8.4 Exercises

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Compare the graph of an even function with the graph of an odd function.
Answer:

Question 2.
OPEN-ENDED
Write a quadratic function whose graph has a vertex of (1, 2).
Answer:

Question 3.
WRITING
Describe the transformation from the graph of f(x) = ax² to the graph of g(x) = a(x – h)² + k.
Answer:

Question 4.
WHICH ONE DOESN’T BELONG?
Which function does not belong with the other three? Explain your reasoning.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 5–12, determine whether the function is even, odd, or neither.
Question 5.
f(x) = 4x + 3
Answer:

Question 6.
g(x) = 3x²
Answer:

Question 7.
h(x) = 5^x + 2
Answer:

Question 8.
m(x) = 2x² – 7x
Answer:

Question 9.
p(x) = -x² + 8
Answer:

Question 10.
f(x) = – \(\frac{1}{2}\)x
Answer:

Question 11.
n(x) = 2x² – 7x + 3
Answer:

Question 12.
r(x) = -6x² + 5
Answer:

In Exercises 13–18, determine whether the function represented by the graph is even, odd, or neither.
Question 13.

Answer:

Question 14.

Answer:

Question 15.

Answer:

Question 16.

Answer:

Question 17.

Answer:

Question 18.

Answer:

In Exercises 19–22, find the vertex and the axis of symmetry of the graph of the function.
Question 19.
f(x) = 3(x + 1)²
Answer:

Question 20.
f(x) = \(\frac{1}{4}\)(x – 6)²
Answer:

Question 21.
y = – \(\frac{1}{8}\)(x – 4)²
Answer:

Question 22.
y = -5(x + 9)²
Answer:

In Exercises 23–28, graph the function. Compare the graph to the graph of f(x) = x².
Question 23.
g(x) = 2(x + 3)²
Answer:

Question 24.
p(x) = 3(x – 1)²
Answer:

Question 25.
r(x) = \(\frac{1}{4}\)(x + 10)²
Answer:

Question 26.
n(x) = \(\frac{1}{4}\)(x – 6)²
Answer:

Question 27.
d(x) = \(\frac{1}{5}\)(x – 5)²
Answer:

Question 28.
q(x) = 6(x + 2)²
Answer:

Question 29.
ERROR ANALYSIS
Describe and correct the error in determining whether the function f(x) = x² + 3 is even, odd, or neither.

Answer:

Question 30.
ERROR ANALYSIS
Describe and correct the error in finding the vertex of the graph of the function.

Answer:

In Exercises 31–34, find the vertex and the axis of symmetry of the graph of the function.
Question 31.
y = -6(x + 4)² – 3
Answer:

Question 32.
f(x) = 3(x – 3)² + 6
Answer:

Question 33.
f(x) = -4(x + 3)² + 1
Answer:

Question 34.
y = -(x – 6)² – 5
Answer:

In Exercises 35–38, match the function with its graph.
Question 35.
y = -(x + 1)² – 3
Answer:

Question 36.
y = – \(\frac{1}{2}\)(x – 1)² + 3
Answer:

Question 37.
y = \(\frac{1}{3}\)(x – 1)² + 3
Answer:

Question 38.
y = 2(x + 1)² – 3
Answer:

In Exercises 39–44, graph the function. Compare the graph to the graph of f(x) = x².
Question 39.
h(x) = (x – 2)² + 4
Answer:

Question 40.
g(x) = (x + 1)² – 7
Answer:

Question 41.
r(x) = 4(x – 1)² – 5
Answer:

Question 42.
n(x) = -(x + 4)² + 2
Answer:

Question 43.
g(x) = – \(\frac{1}{3}\)(x + 3)² – 2
Answer:

Question 44.
r(x) = \(\frac{1}{2}\)(x – 2)² – 4
Answer:

In Exercises 45–48, let f(x) = (x − 2)² + 1. Match the function with its graph.
Question 45.
g(x) = f(x – 1)
Answer:

Question 46.
r(x) = f(x + 2)
Answer:

Question 47.
h(x) = f(x) + 2
Answer:

Question 48.
p(x) = f(x) – 3
Answer:

In Exercises 49–54, graph g.
Question 49.
f(x) = 2(x – 1)² + 1; g(x) = f(x + 3)
Answer:

Question 50.
f(x) = -(x + 1)² + 2; g(x) = \(\frac{1}{2}\)f(x)
Answer:

Question 51.
f(x) = -3(x + 5)² – 6; g(x) = 2f(x)
Answer:

Question 52.
f(x) = 5(x – 3)² – 1; g(x) = (x) – 6
Answer:

Question 53.
f(x) = (x + 3)² + 5; g(x) = f(x – 4)
Answer:

Question 54.
f(x) = -2(x – 4)² – 8; g(x) = -f(x)
Answer:

Question 55.
MODELING WITH MATHEMATICS
The height (in meters) of a bird diving to catch a fish is represented by h(t) = 5(t – 2.5)², where t is the number of seconds after beginning the dive.

a. Graph h.
b. Another bird’s dive is represented by r(t) = 2h(t). Graph r.
c. Compare the graphs. Which bird starts its dive from a greater height? Explain.
Answer:

Question 56.
MODELING WITH MATHEMATICS
A kicker punts a football. The height (in yards) of the football is represented by f(x) = – \(\frac{1}{9}\)(x – 30)² + 25, where x is the horizontal distance (in yards) from the kicker’s goal line.
a. Graph f. Describe the domain and range.
b. On the next possession, the kicker punts the football. The height of the football is represented by g(x) = f (x + 5). Graph g. Describe the domain and range.
c. Compare the graphs. On which possession does the kicker punt closer to his goal line? Explain.
Answer:

In Exercises 57–62, write a quadratic function in vertex form whose graph has the given vertex and passes through the given point.
Question 57.
vertex: (1, 2); passes through (3, 10)
Answer:

Question 58.
vertex: (-3, 5); passes through (0, -14)
Answer:

Question 59.
vertex: (-2, -4); passes through (-1, -6)
Answer:

Question 60.
vertex: (1, 8); passes through (3, 12)
Answer:

Question 61.
vertex: (5, -2); passes through (7, 0)
Answer:

Question 62.
vertex: (-5, -1); passes through (-2, 2)
Answer:

Question 63.
MODELING WITH MATHEMATICS
A portion of a roller coaster track is in the shape of a parabola. Write and graph a quadratic function that models this portion of the roller coaster with a maximum height of 90 feet, represented by a vertex of (25, 90), passing through the point (50, 0).

Answer:

Question 64.
MODELING WITH MATHEMATICS
A flare is launched from a boat and travels in a parabolic path until reaching the water. Write and graph a quadratic function that models the path of the are with a maximum height of 300 meters, represented by a vertex of (59, 300), landing in the water at the point (119, 0).
Answer:

In Exercises 65–68, rewrite the quadratic function in vertex form.
Question 65.
y = 2x)² – 8x + 4
Answer:

Question 66.
y = 3x)² + 6x – 1
Answer:

Question 67.
f(x) = -5x)² + 10x + 36
Answer:

Question 68.
f(x) = -x)² + 4x + 2
Answer:

Question 69.
REASONING
Can a function be symmetric about the x-axis? Explain.
Answer:

Question 70.
HOW DO YOU SEE IT?
The graph of a quadratic function is shown. Determine which symbols to use to complete the vertex form of the quadratic function. Explain your reasoning.

Answer:

In Exercises 71–74, describe the transformation from the graph of f to the graph of h. Write an equation that represents h in terms of x.
Question 71.
f(x) = -(x + 1))² – 2
h(x) = f(x) + 4
Answer:

Question 72.
f(x) = 2(x – 1))² + 1
h(x) = f(x – 5)
Answer:

Question 73.
f(x) = 4(x – 2))² + 3
h(x) = 2f(x)
Answer:

Question 74.
f(x) = -(x + 5))² – 6
h(x) = \(\frac{1}{3}\)f(x)
Answer:

Question 75.
REASONING
The graph of y = x² is translated 2 units right and 5 units down. Write an equation for the function in vertex form and in standard form. Describe advantages of writing the function in each form.
Answer:

Question 76.
THOUGHT PROVOKING
Which of the following are true? Justify your answers.
a. Any constant multiple of an even function is even.
b. Any constant multiple of an odd function is odd.
c. The sum or difference of two even functions is even.
d. The sum or difference of two odd functions is odd.
e. The sum or difference of an even function and an odd function is odd.
Answer:

Question 77.
COMPARING FUNCTIONS
A cross section of a birdbath can be modeled by y = \(\frac{1}{81}\)(x – 18)² – 4, where x and y are measured in inches. The graph shows the cross section of another birdbath.

a. Which birdbath is deeper? Explain.
b. Which birdbath is wider? Explain.
Answer:

Question 78.
REASONING
Compare the graphs of y = 2x² + 8x +8 and y = x² without graphing the functions. How can factoring help you compare the parabolas? Explain.
Answer:

Question 79.
MAKING AN ARGUMENT
Your friend says all absolute value functions are even because of their symmetry. Is your friend correct? Explain.
Answer:

Maintaining Mathematical Proficiency

Solve the equation.
Question 80.
x(x – 1) = 0
Answer:

Question 81.
(x + 3)(x – 8) = 0
Answer:

Question 82.
(3x – 9)(4x + 12) = 0
Answer:

Lesson 8.5 Using Intercept Form

Essential Question What are some of the characteristics of the graph of f(x) = a(x – p)(x – q)?

EXPLORATION 1

Using Zeros to Write Functions
Work with a partner. Each graph represents a function of the form f(x) = (x – p)(x – q) or f(x) = -(x – p)(x – q). Write the function represented by each graph. Explain your reasoning.

Communicate Your Answer

Question 2.
What are some of the characteristics of the graph of f(x) = a(x – p)(x – q)?
Answer:

Question 3.
Consider the graph of f(x) = a(x – p)(x – q).

a. Does changing the sign of a change the x-intercepts? Does changing the sign of a change the y-intercept? Explain your reasoning.
b. Does changing the value of p change the x-intercepts? Does changing the value of p change the y-intercept? Explain your reasoning.
Answer:

Monitoring Progress

Graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts. Describe the domain and range of the function.
Question 1.
f(x) = (x + 2)(x – 3)
Answer:

Question 2.
g(x) = -2(x – 4)(x + 1)
Answer:

Question 3.
h(x) = 4x² – 36
Answer:

Find the zero(s) of the function.
Question 4.
f(x) = (x – 6)(x – 1)
Answer:

Question 5.
g(x) = 3x² – 12x + 12
Answer:

Question 6.
h(x) = x(x² – 1)
Answer:

Use zeros to graph the function.
Question 7.
f(x) = (x – 1)(x – 4)
Answer:

Question 8.
g(x) = x² + x – 12
Answer:

Write a quadratic function in standard form whose graph satisfies the given condition(s).
Question 9.
x-intercepts: -1 and 1
Answer:

Question 10.
vertex: (8, 8) 11. passes through (0, 0), (10, 0), and (4, 12)
Answer:

Question 12.
passes through (-5, 0), (4, 0), and (3, -16)
Answer:

Use zeros to graph the function.
Question 13.
g(x) = (x – 1)(x – 3)(x + 3)
Answer:

Question 14.
h(x) = x³ – 6x² + 5x
Answer:

Question 15.
The zeros of a cubic function are -3, -1, and 1. The graph of the function passes through the point (0, -3). Write the function.
Answer:

Using Intercept Form 8.5 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
The values p and q are __________ of the graph of the function f(x) = a(x – p)(x – q).
Answer:

Question 2.
WRITING
Explain how to find the maximum value or minimum value of a quadratic function when the function is given in intercept form.
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, find the x-intercepts and axis of symmetry of the graph of the function.
Question 3.

Answer:

Question 4.

Answer:

Question 5.
f(x) = -5(x + 7)(x – 5)
Answer:

Question 6.
g(x) = \(\frac{2}{3}\) x(x + 8)
Answer:

In Exercises 7–12, graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts. Describe the domain and range of the function.
Question 7.
f(x) = (x + 4)(x + 1)
Answer:

Question 8.
y = (x – 2)(x + 2)
Answer:

Question 9.
y = -(x + 6)(x – 4)
Answer:

Question 10.
h(x) = -4(x – 7)(x – 3)
Answer:

Question 11.
g(x) = 5(x + 1)(x + 2)
Answer:

Question 12.
y = -2(x – 3)(x + 4)
Answer:

In Exercises 13–20, graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts. Describe the domain and range of the function.
Question 13.
y = x² – 9
Answer:

Question 14.
f(x) = x² – 8x
Answer:

Question 15.
h(x) = -5x² + 5x
Answer:

Question 16.
y = 3x² – 48
Answer:

Question 17.
q(x) = x² + 9x + 14
Answer:

Question 18.
p(x) = x² + 6x – 27
Answer:

Question 19.
y = 4x² – 36x + 32
Answer:

Question 20.
y = -2x² – 4x + 30
Answer:

In Exercises 21–30, find the zero(s) of the function.
Question 21.
y = -2(x – 2)(x – 10)
Answer:

Question 22.
f(x) = \(\frac{1}{3}\)(x + 5)(x – 1)
Answer:

Question 23.
g(x) = x² + 5x – 24
Answer:

Question 24.
y = x² – 17x + 52
Answer:

Question 25.
y = 3x² – 15x – 42
Answer:

Question 26.
g(x) = -4x² – 8x – 4
Answer:

Question 27.
f(x) = (x + 5)(x² – 4)
Answer:

Question 28.
h(x) = (x² – 36)(x – 11)
Answer:

Question 29.
y = x³ – 49x
Answer:

Question 30.
y = x³ – x² – 9x + 9
Answer:

In Exercises 31–36, match the function with its graph.
Question 31.
y = (x + 5)(x + 3)
Answer:

Question 32.
y = (x + 5)(x – 3)
Answer:

Question 33.
y = (x – 5)(x + 3)
Answer:

Question 34.
y = (x – 5)(x – 3)
Answer:

Question 35.
y = (x + 5)(x – 5)
Answer:

Question 36.
y = (x + 3)(x – 3)
Answer:

In Exercises 37–42, use zeros to graph the function.
Question 37.
f(x) = (x + 2)(x – 6)
Answer:

Question 38.
g(x) = -3(x + 1)(x + 7)
Answer:

Question 39.
y = x² – 11x + 18
Answer:

Question 40.
y = x² – x – 30
Answer:

Question 41.
y = -5x² – 10x + 40
Answer:

Question 42.
h(x) = 8x² – 8
Answer:

ERROR ANALYSIS In Exercises 43 and 44, describe and correct the error in finding the zeros of the function.
Question 43.

Answer:

Question 44.

Answer:

In Exercises 45–56, write a quadratic function in standard form whose graph satisfies the given condition(s).
Question 45.
vertex: (7, -3)
Answer:

Question 46.
vertex: (4, 8)
Answer:

Question 47.
x-intercepts: 1 and 9
Answer:

Question 48.
x-intercepts: -2 and -5
Answer:

Question 49.
passes through (-4, 0), (3, 0), and (2, -18)
Answer:

Question 50.
passes through (-5, 0), (-1, 0), and (-4, 3)
Answer:

Question 51.
passes through (7, 0)
Answer:

Question 52.
passes through (0, 0) and (6, 0)
Answer:

Question 53.
axis of symmetry: x = -5
Answer:

Question 54.
y increases as x increases when x < 4; y decreases as x increases when x > 4.
Answer:

Question 55.
range: y ≥ -3
Answer:

Question 56.
range: y ≤ 10
Answer:

In Exercises 57–60, write the quadratic function represented by the graph.
Question 57.

Answer:

Question 58.

Answer:

Question 59.

Answer:

Question 60.

Answer:

In Exercises 61–68, use zeros to graph the function.
Question 61.
y = 5x(x + 2)(x – 6)
Answer:

Question 62.
f(x) = -x(x + 9)(x + 3)
Answer:

Question 63.
h(x) = (x – 2)(x + 2)(x + 7)
Answer:

Question 64.
y = (x + 1)(x – 5)(x – 4)
Answer:

Question 65.
f(x) = 3x³ – 48x
Answer:

Question 66.
y = -2x³ + 20x² – 50x
Answer:

Question 67.
y = -x³ – 16x² – 28x
Answer:

Question 68.
g(x) = 6x³ + 30x² – 36x
Answer:

In Exercises 69–72, write the cubic function represented by the graph.
Question 69.

Answer:

Question 70.

Answer:

Question 71.

Answer:

Question 72.

Answer:

In Exercises 73–76, write a cubic function whose graph satisfies the given condition(s).
Question 73.
x-intercepts: -2, 3, and 8
Answer:

Question 74.
x-intercepts: -7, -5, and 0
Answer:

Question 75.
passes through (1, 0) and (7, 0)
Answer:

Question 76.
passes through (0, 6)
Answer:

In Exercises 77–80, all the zeros of a function are given. Use the zeros and the other point given to write a quadratic or cubic function represented by the table.
Question 77.

Answer:

Question 78.

Answer:

Question 79.

Answer:

Question 80.

Answer:

In Exercises 81–84, sketch a parabola that satisfies the given conditions.
Question 81.
x-intercepts: -4 and 2; range: y ≥ -3
Answer:

Question 82.
axis of symmetry: x = 6; passes through (4, 15)
Answer:

Question 83.
range: y ≤ 5; passes through (0, 2)
Answer:

Question 84.
x-intercept: 6; y-intercept: 1; range: y ≥ -4
Answer:

Question 85.
MODELING WITH MATHEMATICS
Satellite dishes are shaped like parabolas to optimally receive signals. The cross section of a satellite dish can be modeled by the function shown, 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?
c. Write a quadratic function in standard form that models the cross section of a satellite dish that is 6 feet wide and 1.5 feet deep.
Answer:

Question 86.
MODELING WITH MATHEMATICS
A professional basketball player’s shot is modeled by the function shown, where x and y are measured in feet.

a. Does the player make the shot? Explain.
b. The basketball player releases another shot from the point (13, 0) and makes the shot. The shot also passes through the point (10, 1.4). Write a quadratic function in standard form that models the path of the shot.
Answer:

USING STRUCTURE In Exercises 87–90, match the function with its graph.
Question 87.
y = -x² + 5x
Answer:

Question 88.
y = x² – x – 12
Answer:

Question 89.
y = x³ – 2x² – 8x
Answer:

Question 90.
y = x³ – 4x² – 11x + 30
Answer:

Question 91.
CRITICAL THINKING
Write a quadratic function represented by the table, if possible. If not, explain why.

Answer:

Question 92.
HOW DO YOU SEE IT?
The graph shows the parabolic arch that supports the roof of a convention center, where x and y are measured in feet.

a. The arch can be represented by a function of the form f(x) = a(x – p)(x – q). Estimate the values of p and q.
b. Estimate the width and height of the arch. Explain how you can use your height estimate to calculate a.
Answer:

ANALYZING EQUATIONS In Exercises 93 and 94,
(a) rewrite the quadratic function in intercept form and
(b) graph the function using any method. Explain the method you used.
Question 93.
f(x) = -3(x + 1)² + 27
Answer:

Question 94.
g(x) = 2(x – 1)² – 2
Answer:

Question 95.
WRITING
Can a quadratic function with exactly one real zero be written in intercept form? Explain.
Answer:

Question 96.
MAKING AN ARGUMENT
Your friend claims that any quadratic function can be written in standard form and in vertex form. Is your friend correct? Explain.
Answer:

Question 97.
PROBLEM SOLVING
Write the function represented by the graph in intercept form.

Answer:

Question 98.
THOUGHT PROVOKING
Sketch the graph of each function. Explain your procedure.
a. f(x) = (x² – 1)(x² – 4)
b. g(x) = x(x² – 1)(x² – 4)
Answer:

Question 99.
REASONING
Let k be a constant. Find the zeros of the function f(x) = kx² – k²x – 2k³ in terms of k.
Answer:

PROBLEM SOLVING In Exercises 100 and 101, write a system of two quadratic equations whose graphs intersect at the given points. Explain your reasoning.
Question 100.
(-4, 0) and (2, 0)
Answer:

Question 101.
(3, 6) and (7, 6)
Answer:

Maintaining Mathematical Proficiency

The scatter plot shows the amounts x (in grams) of fat and the numbers y of calories in 12 burgers at a fast-food restaurant.

Question 102.
How many calories are in the burger that contains 12 grams of fat?
Answer:

Question 103.
How many grams of fat are in the burger that contains 600 calories?
Answer:

Question 104.
What tends to happen to the number of calories as the number of grams of fat increases?
Answer:

Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.
Question 105.
3, 11, 21, 33, 47, . . .
Answer:

Question 106.
-2, -6, -18, -54, . . .
Answer:

Question 107.
26, 18, 10, 2, -6, . . .
Answer:

Question 108.
4, 5, 9, 14, 23, . . .
Answer:

Lesson 8.6 Comparing Linear, Exponential, and Quadratic Functions

Essential Question How can you compare the growth rates of linear, exponential, and quadratic functions?

EXPLORATION 1

Comparing Speeds
Work with a partner. Three cars start traveling at the same time. The distance traveled in t minutes is y miles. Complete each table and sketch all three graphs in the same coordinate plane. Compare the speeds of the three cars. Which car has a constant speed? Which car is accelerating the most? Explain your reasoning.

EXPLORATION 2

Comparing Speeds

Work with a partner. Analyze the speeds of the three cars over the given time periods. The distance traveled in t minutes is y miles. Which car eventually overtakes the others?

Communicate Your Answer

Question 3.
How can you compare the growth rates of linear, exponential, and quadratic functions?
Answer:

Question 4.
Which function has a growth rate that is eventually much greater than the growth rates of the other two functions? Explain your reasoning.
Answer:

Monitoring Progress

Plot the points. Tell whether the points appear to represent a linear, an exponential, or a quadratic function.
Question 1.
(-1, 5), (2, -1), (0, -1), (3, 5), (1, -3)
Answer:

Question 2.
(-1, 2), (-2, 8), (-3, 32), (0, \(\frac{1}{2}\)), (1, \(\frac{1}{8}\))
Answer:

Question 3.
(-3, 5), (0, -1), (2, -5), (-4, 7), (1, -3)
Answer:

Question 4.
Tell whether the table of values represents a linear, an exponential, or a quadratic function.

Answer:

Question 5.
Tell whether the table of values represents a linear, an exponential, or a quadratic function. Then write the function.

Answer:

Question 6.
Compare the websites in Example 4 by calculating and interpreting the average rates of change from Day 0 to Day 10.
Answer:

Question 7.
WHAT IF?
Tinyville’s population increased by 8% each year. In what year were the populations about equal?
Answer:

Comparing Linear, Exponential, and Quadratic Functions 8.6 Exercises

Vocabulary and Core Concept Check

Question 1.
WRITING
Name three types of functions that you can use to model data. Describe the equation and graph of each type of function.
Answer:

Question 2.
WRITING
How can you decide whether to use a linear, an exponential, or a quadratic function to model a data set?
Answer:

Question 3.
VOCABULARY
Describe how to find the average rate of change of a function y = f(x) between x = a and x = b.
Answer:

Question 4.
WHICH ONE DOESN’T BELONG?
Which graph does not belong with the other three? Explain your reasoning.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 5–8, tell whether the points appear to represent a linear, an exponential, or a quadratic function.
Question 5.

Answer:

Question 6.

Answer:

Question 7.

Answer:

Question 8.

Answer:

In Exercises 9–14, plot the points. Tell whether the points appear to represent a linear, an exponential, or a quadratic function.
Question 9.
(-2, -1), (-1, 0), (1, 2), (2, 3), (0, 1)
Answer:

Question 10.
( 0, \(\frac{1}{4}\)), (1, 1), (2, 4), (3, 16), (-1, \(\frac{1}{16}\))
Answer:

Question 11.
(0, -3), (1, 0), (2, 9), (-2, 9), (-1, 0)
Answer:

Question 12.
(-1, -3), (-3, 5), (0, -1), (1, 5), (2, 15)
Answer:

Question 13.
(-4, -4), (-2, -3.4), (0, -), (2, -2.6), (4, -2)
Answer:

Question 14.
(0, 8), (-4, 0.25), (-3, 0.4), (-2, 1), (-1, 3)
Answer:

In Exercises 15–18, tell whether the table of values represents a linear, an exponential, or a quadratic function.
Question 15.

Answer:

Question 16.

Answer:

Question 17.

Answer:

Question 18.

Answer:

Question 19.
MODELING WITH MATHEMATICS
A student takes a subway to a public library. The table shows the distances d (in miles) the student travels in t minutes. Let the time t represent the independent variable. Tell whether the data can be modeled by a linear, an exponential, or a quadratic function. Explain.

Answer:

Question 20.
MODELING WITH MATHEMATICS
A store sells custom circular rugs. The table shows the costs c (in dollars) of rugs that have diameters of d feet. Let the diameter d represent the independent variable. Tell whether the data can be modeled by a linear, an exponential, or a quadratic function. Explain.

Answer:

In Exercises 21–26, tell whether the data represent a linear, an exponential, or a quadratic function. Then write the function.
Question 21.
(-2, 8), (-1, 0), (0, -4), (1, -4), (2, 0), (3, 8)
Answer:

Question 22.
(-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 0.5)
Answer:

Question 23.

Answer:

Question 24.

Answer:

Question 25.

Answer:

Question 26.

Answer:

Question 27.
ERROR ANALYSIS
Describe and correct the error in determining whether the table represents a linear, an exponential, or a quadratic function.

Answer:

Question 28.
ERROR ANALYSIS
Describe and correct the error in writing the function represented by the table.

Answer:

Question 29.
REASONING
The table shows the numbers of people attending the first five football games at a high school.

a. Plot the points. Let the game g represent the independent variable.
b. Can a linear, an exponential, or a quadratic function represent this situation? Explain.
Answer:

Question 30.
MODELING WITH MATHEMATICS
The table shows the breathing rates y (in liters of air per minute) of a cyclist traveling at different speeds x (in miles per hour).

a. Plot the points. Let the speed x represent the independent variable. Then determine the type of function that best represents this situation.
b. Write a function that models the data.
c. Find the breathing rate of a cyclist traveling 18 miles per hour. Round your answer to the nearest tenth.
Answer:

Question 31.
ANALYZING RATES OF CHANGE
The function f(t) = -16t² + 48t + 3 represents the height (in feet) of a volleyball t seconds after it is hit into the air.
a. Copy and complete the table.

b. Plot the ordered pairs and draw a smooth curve through the points.
c. Describe where the function is increasing and decreasing.
d. Find the average rate of change for each 0.5-second interval in the table. What do you notice about the average rates of change when the function is increasing? decreasing?
Answer:

Question 32.
ANALYZING RELATIONSHIPS
The population of Town A in 1970 was 3000. The population of Town A increased by 20% every decade. Let x represent the number of decades since 1970. The graph shows the population of Town B.

a. Compare the populations of the towns by calculating and interpreting the average rates of change from 1990 to 2010.
b. Predict which town will have a greater population after 2020. Explain.
Answer:

Question 33.
ANALYZING RELATIONSHIPS
Three organizations are collecting donations for a cause. Organization A begins with one donation, and the number of donations quadruples each hour. The table shows the numbers of donations collected by Organization B. The graph shows the numbers of donations collected by Organization C.

a. What type of function represents the numbers of donations collected by Organization A? B? C?
b. Find the average rates of change of each function for each 1-hour interval from t = 0 to t = 6.
c. For which function does the average rate of change increase most quickly? What does this tell you about the numbers of donations collected by the three organizations?
Answer:

Question 34.
COMPARING FUNCTIONS
The room expenses for two different resorts are shown.

a. For what length of vacation does each resort cost about the same?
b. Suppose Blue Water Resort charges $1450 for the first three nights and $105 for each additional night. Would Sea Breeze Resort ever be more expensive than Blue Water Resort? Explain.
c. Suppose Sea Breeze Resort charges $1200 for the first three nights. The charge increases 10% for each additional night. Would Blue Water Resort ever be more expensive than Sea Breeze Resort? Explain.
Answer:

Question 35.
REASONING
Explain why the average rate of change of a linear function is constant and the average rate of change of a quadratic or exponential function is not constant.
Answer:

Question 36.
HOW DO YOU SEE IT?
Match each graph with its function. Explain your reasoning.

Answer:

Question 37.
CRITICAL THINKING
In the ordered pairs below, the y-values are given in terms of n. Tell whether the ordered pairs represent a linear, an exponential, or a quadratic function. Explain.
(1, 3n – 1), (2, 10n + 2), (3, 26n),
(4, 51n – 7), (5, 85n – 19)
Answer:

Question 38.
USING STRUCTURE
Write a function that has constant second differences of 3.
Answer:

Question 39.
CRITICAL THINKING
Is the graph of a set of points enough to determine whether the points represent a linear, an exponential, or a quadratic function? Justify your answer.
Answer:

Question 40.
THOUGHT PROVOKING
Find four different patterns in the figure. Determine whether each pattern represents a linear, an exponential, or a quadratic function. Write a model for each pattern.

Answer:

Question 41.
MAKING AN ARGUMENT
Function p is an exponential function and function q is a quadratic function. Your friend says that after about x = 3, function q will always have a greater y-value than function p. Is your friend correct? Explain.

Answer:

Question 42.
USING TOOLS
The table shows the amount a (in billions of dollars) United States residents spent on pets or pet-related products and services each year for a 5-year period. Let the year x represent the independent variable. Using technology, find a function that models the data. How did you choose the model? Predict how much residents will spend on pets or pet-related products and services in Year 7.

Answer:

Maintaining Mathematical Proficiency

Evaluate the expression.
Question 43.
\(\sqrt{121}\)
Answer:

Question 44.
\(\sqrt [ 3 ]{ 125 }\)
Answer:

Question 45.
\(\sqrt [ 3 ]{ 512 }\)
Answer:

Question 46.
\(\sqrt [ 5 ]{ 243 }\)
Answer:

Find the product.
Question 47.
(x + 8)(x – 8)
Answer:

Question 48.
(4y + 2)(4y – 2)
Answer:

Question 49.
(3a – 5b)(3a + 5b)
Answer:

Question 50.
(-2r + 6s)(-2r – 6s)
Answer:

Graphing Quadratic Functions Performance Task: Asteroid Aim

8.4–8.6 What Did You Learn?

Core Vocabulary

Core Concepts

Mathematical Practices

Question 1.
How can you use technology to confirm your answer in Exercise 64 on page 448?
Answer:

Question 2.
How did you use the structure of the equation in Exercise 85 on page 457 to solve the problem?
Answer:

Question 3.
Describe why your answer makes sense considering the context of the data in Exercise 20 on page 466.
Answer:

Performance Task: Asteroid Aim

Apps take a long time to design and program. One app in development is a game in which players shoot lasers at asteroids. They score points based on the number of hits per shot. The designer wants your feedback. Do you think students will like the game and want to play it? What changes would improve it?
To explore the answers to this question and more, go to

Answer:

Graphing Quadratic Functions Chapter Review

8.1 Graphing f(x) = ax² (pp. 419–424)

Graph the function. Compare the graph to the graph of f(x) = x².
Question 1.
p(x) = 7x²
Answer:

Question 2.
q(x) = \(\frac{1}{2}\)x²
Answer:

Question 3.
g(x) = – \(\frac{3}{4}\)x²
Answer:

Question 4.
h(x) = -6x²
Answer:

Question 5.
Identify characteristics of the quadratic function and its graph.

Answer:

8.2 Graphing f(x) = ax² + c (pp. 425–430)

Graph the function. Compare the graph to the graph of f(x) = x².
Question 6.
g(x) = x² + 5
Answer:

Question 7.
h(x) = -x² – 4
Answer:

Question 8.
m(x) = -2x² + 6
Answer:

Question 9.
n(x) = \(\frac{1}{3}\)x² – 5
Answer:

8.3 Graphing f(x) = ax² + bx + c (pp. 431–438)

Graph the function. Describe the domain and range.
Question 10.
y = x² – 2x + 7
Answer:

Question 11.
f(x) = -3x² + 3x – 4
Answer:

Question 12.
y = \(\frac{1}{2}\)x² – 6x + 10
Answer:

Question 13.
The function f(t) = -16t² + 88t + 12 represents the height (in feet) of a pumpkin t seconds after it is launched from a catapult. When does the pumpkin reach its maximum height? What is the maximum height of the pumpkin?
Answer:

8.4 Graphing f(x) = a(x − h)² + k    (pp. 441–448)

Determine whether the function is even, odd, or neither.
Question 14.
w(x) = 5^x
Answer:

Question 15.
r(x) = -8x
Answer:

Question 16.
h(x) = 3x² – 2x
Answer:

Graph the function. Compare the graph to the graph of f(x) = x².
Question 17.
h(x) = 2(x – 4)²
Answer:

Question 18.
g(x) = \(\frac{1}{2}\)(x – 1)² + 1
Answer:

Question 19.
q(x) = -(x + 4)² + 7
Answer:

Question 20.
Consider the function g(x) = -3(x + 2)² – 4. Graph h(x) = g(x = 1).
Answer:

Question 21.
Write a quadratic function whose graph has a vertex of (3, 2) and passes through the point (4, 7).
Answer:

8.5 Using Intercept Form (pp. 449–458)

Graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts. Describe the domain and range of the function.
Question 22.
y = (x – 4)(x + 2)
Answer:

Question 23.
f(x) = -3(x + 3)(x + 1)
Answer:

Question 24.
y = x² – 8x + 15
Answer:

Use zeros to graph the function.
Question 25.
y = -2x² + 6x + 8
Answer:

Question 26.
f(x) = x² + x – 2
Answer:

Question 27.
f(x) = 2x² – 18x
Answer:

Question 28.
Write a quadratic function in standard form whose graph passes through (4, 0) and (6, 0).
Answer:

8.6 Comparing Linear, Exponential, and Quadratic Functions (pp. 459−468)

Question 29.
Tell whether the table of values represents a linear, an exponential, or a quadratic function. Then write the function.

Answer:

Question 30.
The balance y (in dollars) of your savings account after t years is represented by y = 200(1.1)t. The beginning balance of your friend’s account is $250, and the balance increases by $20 each year. (a) Compare the account balances by calculating and interpreting the average rates of change from t = 2 to t = 7. (b) Predict which account will have a greater balance after 10 years. Explain.
Answer:

Graphing Quadratic Functions Chapter Test

Graph the function. Compare the graph to the graph of f(x) = x².
Question 1.
h(x) = 2x² – 3
Answer:

Question 2.
g(x) = –\(\frac{1}{2}\)x²
Answer:

Question 3.
p(x) = \(\frac{1}{2}\)(x + 1)² – 1
Answer:

Question 4.
Consider the graph of the function f.

a. Find the domain, range, and zeros of the function.
b. Write the function f in standard form.
c. Compare the graph of f to the graph of g(x) = x².
d. Graph h(x) = f (x – 6).
Answer:

Use zeros to graph the function. Describe the domain and range of the function.
Question 5.
f(x) = 2x² – 8x + 8
Answer:

Question 6.
y = -(x + 5)(x – 1)
Answer:

Question 7.
h(x) = 16x² – 4
Answer:

Tell whether the table of values represents a linear, an exponential, or a quadratic function. Explain your reasoning. Then write the function.
Question 8.

Answer:

Question 9.

Answer:

Write a quadratic function in standard form whose graph satisfies the given conditions. Explain the process you used.
Question 10.
passes through (-8, 0), (-2, 0), and (-6, 4)
Answer:

Question 11.
passes through (0, 0), (10, 0), and (9, -27)
Answer:

Question 12.
is even and has a range of y ≥ 3
Answer:

Question 13.
passes through (4, 0) and (1, 9)
Answer:

Question 14.
The table shows the distances d (in miles) that Earth moves in its orbit around the Sun after t seconds. Let the time t be the independent variable. Tell whether the data can be modeled by a linear, an exponential, or a quadratic function. Explain. Then write a function that models the data.

Answer:

Question 15.
You are playing tennis with a friend. The path of the tennis ball after you return a serve can be modeled by the function y = -0.005x² + 0.17x + 3, where x is the horizontal distance (in feet) from where you hit the ball and y is the height (in feet) of the ball.
a. What is the maximum height of the tennis ball?
b. You are standing 30 feet from the net, which is 3 feet high. Will the ball clear the net? Explain your reasoning.
Answer:

Question 16.
Find values of a, b, and c so that the function f(x) = ax² + bx + c is (a) even, (b) odd, and (c) neither even nor odd.
Answer:

Question 17.
Consider the function f(x) = x² + 4. Find the average rate of change from x = 0 to x = 1, from x = 1 to x = 2, and from x = 2 to x = 3. What do you notice about the average rates of change when the function is increasing?
Answer:

Graphing Quadratic Functions Cumulative Assessment

Question 1.
Which function is represented by the graph?

Answer:

Question 2.
Find all numbers between 0 and 100 that are in the range of the function defined below.(HSF-IF.A.3)
f(1) = 1, f(2) = 1, f(n) = f(n – 1) + f(n – 2)
Answer:

Question 3.
The function f(t) = -16t² + v₀t + s₀ represents the height (in feet) of a ball t seconds after it is thrown from an initial height s0 (in feet) with an initial vertical velocity v0 (in feet per second). The ball reaches its maximum height after \(\frac{7}{8}\) second when it is thrown with an initial vertical velocity of ______ feet per second.
Answer:

Question 4.
Classify each system of equations by the number of solutions.

Answer:

Question 5.
Your friend claims that quadratic functions can have two, one, or no real zeros. Do you support your friend’s claim? Use graphs to justify your answer.
Answer:

Question 6.
Which polynomial represents the area (in square feet) of the shaded region of the figure?

Answer:

Question 7.
Consider the functions represented by the tables.

a. Classify each function as linear, exponential, or quadratic.
b. Order the functions from least to greatest according to the average rates of change between x = 1 and x = 3.
Answer:

Question 8.
Complete each function using the symbols + or – , so that the graph of the quadratic function satisfies the given conditions.

Answer:

Question 9.
The graph shows the amounts y (in dollars) that a referee earns for refereeing x high school volleyball games.

a. Does the graph represent a linear or nonlinear function? Explain.
b. Describe the domain of the function. Is the domain discrete or continuous?
c. Write a function that models the data.
d. Can the referee earn exactly $500? Explain.
Answer:

Question 10.
Which expressions are equivalent to (b^-5)^-4?

Answer: