Algebra and Trigonometry

Robert Blitzer

Chapter 3

Polynomial and Rational Functions - all with Video Answers

Educators

CC

TB

Section 1

Quadratic Functions

Problem 1

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
$$
\begin{array}{ll}
{f(x)=(x+1)^{2}-1} & {g(x)=(x+1)^{2}+1} \\
{h(x)=(x-1)^{2}+1} & {j(x)=(x-1)^{2}-1}
\end{array}
$$
(Graph can't copy)

Ashley Hanson

Ashley Hanson

Numerade Educator

Problem 2

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
$$
\begin{array}{ll}
{f(x)=(x+1)^{2}-1} & {g(x)=(x+1)^{2}+1} \\
{h(x)=(x-1)^{2}+1} & {j(x)=(x-1)^{2}-1}
\end{array}
$$
(Graph can't copy)

Ashley Hanson

Numerade Educator

Problem 3

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
$$
\begin{array}{ll}
{f(x)=(x+1)^{2}-1} & {g(x)=(x+1)^{2}+1} \\
{h(x)=(x-1)^{2}+1} & {j(x)=(x-1)^{2}-1}
\end{array}
$$
(Graph can't copy)

Ashley Hanson

Numerade Educator

Problem 4

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
$$
\begin{array}{ll}
{f(x)=(x+1)^{2}-1} & {g(x)=(x+1)^{2}+1} \\
{h(x)=(x-1)^{2}+1} & {j(x)=(x-1)^{2}-1}
\end{array}
$$
(Graph can't copy)

Ashley Hanson

Numerade Educator

Problem 5

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
$$
\begin{array}{ll}
{f(x)=x^{2}+2 x+1} & {g(x)=x^{2}-2 x+1} \\
{h(x)=x^{2}-1} & {j(x)=-x^{2}-1}
\end{array}
$$
(Graph can't copy)

Ashley Hanson

Numerade Educator

Problem 6

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
$$
\begin{array}{ll}
{f(x)=x^{2}+2 x+1} & {g(x)=x^{2}-2 x+1} \\
{h(x)=x^{2}-1} & {j(x)=-x^{2}-1}
\end{array}
$$
(Graph can't copy)

Ashley Hanson

Numerade Educator

Problem 7

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
$$
\begin{array}{ll}
{f(x)=x^{2}+2 x+1} & {g(x)=x^{2}-2 x+1} \\
{h(x)=x^{2}-1} & {j(x)=-x^{2}-1}
\end{array}
$$
(Graph can't copy)

Ashley Hanson

Numerade Educator

Problem 8

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
$$
\begin{array}{ll}
{f(x)=x^{2}+2 x+1} & {g(x)=x^{2}-2 x+1} \\
{h(x)=x^{2}-1} & {j(x)=-x^{2}-1}
\end{array}
$$
(Graph can't copy)

Ashley Hanson

Numerade Educator

Problem 9

Find the coordinates of the vertex for the parabola defined by the given quadratic function.
$$
f(x)=2(x-3)^{2}+1
$$

Ashley Hanson

Numerade Educator

Problem 10

Find the coordinates of the vertex for the parabola defined by the given quadratic function.
$$
f(x)=-3(x-2)^{2}+12
$$

Ashley Hanson

Numerade Educator

Problem 11

Find the coordinates of the vertex for the parabola defined by the given quadratic function.
$$
f(x)=-2(x+1)^{2}+5
$$

Ashley Hanson

Numerade Educator

Problem 12

Find the coordinates of the vertex for the parabola defined by the given quadratic function.
$$
f(x)=-2(x+4)^{2}-8
$$

Ashley Hanson

Numerade Educator

Problem 13

Find the coordinates of the vertex for the parabola defined by the given quadratic function.
$$
f(x)=2 x^{2}-8 x+3
$$

Ashley Hanson

Numerade Educator

Problem 14

Find the coordinates of the vertex for the parabola defined by the given quadratic function.
$$
f(x)=3 x^{2}-12 x+1
$$

Ashley Hanson

Numerade Educator

Problem 15

Find the coordinates of the vertex for the parabola defined by the given quadratic function.
$$
f(x)=-x^{2}-2 x+8
$$

Ashley Hanson

Numerade Educator

Problem 16

Find the coordinates of the vertex for the parabola defined by the given quadratic function.
$$
f(x)=-2 x^{2}+8 x-1
$$

Ashley Hanson

Numerade Educator

Problem 17

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=(x-4)^{2}-1
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 18

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=(x-1)^{2}-2
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 19

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=(x-1)^{2}+2
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 20

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=(x-3)^{2}+2
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 21

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
y-1=(x-3)^{2}
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 22

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
y-3=(x-1)^{2}
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 23

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=2(x+2)^{2}-1
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 24

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=\frac{5}{4}-\left(x-\frac{1}{2}\right)^{2}
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 25

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=4-(x-1)^{2}
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 26

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=1-(x-3)^{2}
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 27

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=x^{2}-2 x-3
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 28

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=x^{2}-2 x-15
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 29

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=x^{2}+3 x-10
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 30

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=2 x^{2}-7 x-4
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 31

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=2 x-x^{2}+3
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 32

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=5-4 x-x^{2}
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 33

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=x^{2}+6 x+3
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 34

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=x^{2}+4 x-1
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 35

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=2 x^{2}+4 x-3
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 36

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=3 x^{2}-2 x-4
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 37

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=2 x-x^{2}-2
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 38

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
$$
f(x)=6-4 x+x^{2}
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 39

An equation of a quadratic function is given.
a. Determine, without graphing, whether the function has a minimum value or a maximum value.
b. Find the minimum or maximum value and determine where it occurs.
c. Identify the function's domain and its range.
$$
f(x)=3 x^{2}-12 x-1
$$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 40

An equation of a quadratic function is given.
a. Determine, without graphing, whether the function has a minimum value or a maximum value.
b. Find the minimum or maximum value and determine where it occurs.
c. Identify the function's domain and its range.
$$
f(x)=2 x^{2}-8 x-3
$$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 41

An equation of a quadratic function is given.
a. Determine, without graphing, whether the function has a minimum value or a maximum value.
b. Find the minimum or maximum value and determine where it occurs.
c. Identify the function's domain and its range.
$$
f(x)=-4 x^{2}+8 x-3
$$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 42

An equation of a quadratic function is given.
a. Determine, without graphing, whether the function has a minimum value or a maximum value.
b. Find the minimum or maximum value and determine where it occurs.
c. Identify the function's domain and its range.
$$
f(x)=-2 x^{2}-12 x+3
$$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 43

An equation of a quadratic function is given.
a. Determine, without graphing, whether the function has a minimum value or a maximum value.
b. Find the minimum or maximum value and determine where it occurs.
c. Identify the function's domain and its range.
$$
f(x)=5 x^{2}-5 x
$$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 44

An equation of a quadratic function is given.
a. Determine, without graphing, whether the function has a minimum value or a maximum value.
b. Find the minimum or maximum value and determine where it occurs.
c. Identify the function's domain and its range.
$$
f(x)=6 x^{2}-6 x
$$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 45

Give the domain and the range of each quadratic function whose graph is described.
The vertex is $(-1,-2)$ and the parabola opens up.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 46

Give the domain and the range of each quadratic function whose graph is described.
The vertex is $(-3,-4)$ and the parabola opens down.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 47

Give the domain and the range of each quadratic function whose graph is described.
Maximum $=-6$ at $x=10$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 48

Give the domain and the range of each quadratic function whose graph is described.
Minimum $=18$ at $x=-6$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 49

Write an equation in standard form of the parabola that has the same shape as the graph of $f(x)=2 x^{2},$ but with the given point as the vertex.
$$
(5,3)
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 50

Write an equation in standard form of the parabola that has the same shape as the graph of $f(x)=2 x^{2},$ but with the given point as the vertex.
$$
(7,4)
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 51

Write an equation in standard form of the parabola that has the same shape as the graph of $f(x)=2 x^{2},$ but with the given point as the vertex.
$$
(-10,-5)
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 52

Write an equation in standard form of the parabola that has the same shape as the graph of $f(x)=2 x^{2},$ but with the given point as the vertex.
$$
(-8,-6)
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 53

Write an equation in standard form of the parabola that has the same shape as the graph of $f(x)=3 x^{2}$ or $g(x)=-3 x^{2},$ but with the given maximum or minimum.
Maximum $=4$ at $x=-2$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 54

Write an equation in standard form of the parabola that has the same shape as the graph of $f(x)=3 x^{2}$ or $g(x)=-3 x^{2},$ but with the given maximum or minimum.
Maximum $=-7$ at $x=5$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 55

Write an equation in standard form of the parabola that has the same shape as the graph of $f(x)=3 x^{2}$ or $g(x)=-3 x^{2},$ but with the given maximum or minimum.
Minimum $=0$ at $x=11$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 56

Write an equation in standard form of the parabola that has the same shape as the graph of $f(x)=3 x^{2}$ or $g(x)=-3 x^{2},$ but with the given maximum or minimum.
Minimum $=0$ at $x=9$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 57

An athlete whose event is the shot put releases the shot with the same initial velocity but at different angles. The figure shows the parabolic paths for shots released at angles of $35^{\circ}$ and $65^{\circ} .$ Exercises $57-58$ are based on the functions that model the parabolic paths.
When the shot whose path is shown by the blue graph is released at an angle of $35^{\circ},$ its height, $f(x),$ in feet, can be modeled by
$$
f(x)=-0.01 x^{2}+0.7 x+6.1
$$
where $x$ is the shot's horizontal distance, in feet, from its point of release. Use this model to solve parts (a) through (c) and verify your answers using the blue graph.
a. What is the maximum height of the shot and how far from its point of release does this occur?
b. What is the shot's maximum horizontal distance, to the nearest tenth of a foot, or the distance of the throw?
c. From what height was the shot released?

Linh Vu

Numerade Educator

Problem 58

An athlete whose event is the shot put releases the shot with the same initial velocity but at different angles. The figure shows the parabolic paths for shots released at angles of $35^{\circ}$ and $65^{\circ} .$ Exercises $57-58$ are based on the functions that model the parabolic paths.
When the shot whose path is shown by the red graph on the previous page is released at an angle of $65^{\circ},$ its height, $g(x),$ in feet, can be modeled by
$$
g(x)=-0.04 x^{2}+2.1 x+6.1
$$
where $x$ is the shot's horizontal distance, in feet, from its point of release. Use this model to solve parts (a) through (c) and verify your answers using the red graph.
a. What is the maximum height, to the nearest tenth of a foot, of the shot and how far from its point of release does this occur?
b. What is the shot's maximum horizontal distance, to the nearest tenth of a foot, or the distance of the throw?
c. From what height was the shot released?

Linh Vu

Numerade Educator

Problem 59

A ball is thrown upward and outward from a height of 6 feet. The height of the ball, $f(x),$ in feet, can be modeled by
$$
f(x)=-0.8 x^{2}+2.4 x+6
$$
where $x$ is the ball's horizontal distance, in feet, from where it was thrown.
a. What is the maximum height of the ball and how far from where it was thrown does this occur?
b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of a foot.
c. Graph the function that models the ball's parabolic path.

Linh Vu

Numerade Educator

Problem 60

A ball is thrown upward and outward from a height of 6 feet. The height of the ball, $f(x),$ in feet, can be modeled by
$$
f(x)=-0.8 x^{2}+3.2 x+6
$$
where $x$ is the ball's horizontal distance, in feet, from where it was thrown.
a. What is the maximum height of the ball and how far from where it was thrown does this occur?
b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of a foot.
c. Graph the function that models the ball's parabolic path.

James Kiss

Numerade Educator

Problem 61

Among all pairs of numbers whose sum is $16,$ find a pair whose product is as large as possible. What is the maximum product?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 62

Among all pairs of numbers whose sum is $20,$ find a pair whose product is as large as possible. What is the maximum product?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 63

Among all pairs of numbers whose difference is $16,$ find a pair whose product is as small as possible. What is the minimum product?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 64

Among all pairs of numbers whose difference is $24,$ find a pair whose product is as small as possible. What is the minimum product?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 65

You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 66

You have 200 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 67

You have 50 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 68

You have 80 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 69

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Six hundred feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?

Linh Vu

Numerade Educator

Problem 70

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Four hundred feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?

Linh Vu

Numerade Educator

Problem 71

A rain gutter is made from sheets of aluminum that are 20 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 72

A rain gutter is made from sheets of aluminum that are 12 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 73

Hunky Beef, a local sandwich store, has a fixed weekly cost of $\$ 525.00,$ and variable costs for making a roast beef sandwich are $\$ 0.55$
a. Let $x$ represent the number of roast beef sandwiches made and sold each week. Write the weekly cost function, C, for Hunky Beef. (Hint: The cost function is the sum of fixed and variable costs.
b. The function $R(x)=-0.001 x^{2}+3 x$ describes the money, in dollars, that Hunky Beef takes in each week from the sale of $x$ roast beef sandwiches. Use this revenue function and the cost function from part (a) to write the store's weekly profit function, $P .$ (Hint: The profit function is the difference between the revenue and cost functions.)
c. Use the store's profit function to determine the number of roast beef sandwiches it should make and sell each week to maximize profit. What is the maximum weekly profit?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 74

Explaining the Concepts
What is a quadratic function?

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 75

Explaining the Concepts
What is a parabola? Describe its shape.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 76

Explaining the Concepts
Explain how to decide whether a parabola opens upward or downward.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 77

Explaining the Concepts
Describe how to find a parabola's vertex if its equation is expressed in standard form. Give an example.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 78

Explaining the Concepts
Describe how to find a parabola's vertex if its equation is in the form $f(x)=a x^{2}+b x+c .$ Use $f(x)=x^{2}-6 x+8$ as an example.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 79

Explaining the Concepts
A parabola that opens upward has its vertex at $(1,2)$ Describe as much as you can about the parabola based on this information. Include in your discussion the number of $x$ -intercepts (if any) for the parabola.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 80

Use a graphing utility to verify any five of your hand-drawn graphs in Exercises $17-38$

James Kiss

Numerade Educator

Problem 81

a. Use a graphing utility to graph $y=2 x^{2}-82 x+720$ in a standard viewing rectangle. What do you observe?
b. Find the coordinates of the vertex for the given quadratic function.
c. The answer to part (b) is $(20.5,-120.5) .$ Because the leading coefficient, $2,$ of the given function is positive, the vertex is a minimum point on the graph. Use this fact to help find a viewing rectangle that will give a relatively complete picture of the parabola. With an axis of symmetry at $x=20.5,$ the setting for $x$ should extend past this, so try $\mathrm{X} \min =0$ and $\mathrm{Xmax}=30 .$ The setting for $y$ should include (and probably go below) the $y$ -coordinate of the graph's minimum y-value, so try Ymin = -130. Experiment with Ymax until your utility shows the parabola's major features.
d. In general, explain how knowing the coordinates of a parabola's vertex can help determine a reasonable viewing rectangle on a graphing utility for obtaining a complete picture of the parabola.

James Kiss

Numerade Educator

Problem 82

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function.
$$
y=-0.25 x^{2}+40 x
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 83

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function.
$$
y=-4 x^{2}+20 x+160
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 84

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function.
$$
y=5 x^{2}+40 x+600
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 85

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function.
$$
y=0.01 x^{2}+0.6 x+100
$$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 86

The bar graph shows the ratings of American Idol from season 1 ( 2002) through season (12 2013)
(Bar Graph can't copy)
a. Let $x$ represent American Idol's season number and let $y$ represent the average number of viewers, in millions. Use a graphing utility to draw a scatter plot of the data. Explain why a quadratic function is appropriate for modeling these data.
b. Use the quadratic regression feature to find the quadratic function that best fits the data. Round all numbers to two decimal places.
c. Use the model in part (b) to determine the season in which American Idol had the greatest number of viewers. Round to the nearest whole number. According to the model, how many millions of viewers were there in that season? Round to one decimal place.
d. How do the results obtained from the model in part (c) compare with the data displayed by the graph?
e. Use a graphing utility to draw a scatter plot of the data and graph the quadratic function of best fit on the scatter plot. Can you see why projections based on the graph had the producers of American Idol looking for a shake-up? No shake-up was found and the show's final season aired in 2016 .

Teresa Fuston

Numerade Educator

Problem 87

Determine whether each statement makes sense or does not make sense, and explain your reasoning.
I must have made an error when graphing this parabola because its axis of symmetry is the $y$ -axis.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 88

Determine whether each statement makes sense or does not make sense, and explain your reasoning.
I like to think of a parabola's vertex as the point where it intersects its axis of symmetry.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 89

Determine whether each statement makes sense or does not make sense, and explain your reasoning.
I threw a baseball vertically upward and its path was a parabola.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 90

Determine whether each statement makes sense or does not make sense, and explain your reasoning.
Figure 3.7 on page 354 shows that a linear function provides a better description of the football's path than a quadratic function.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 91

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
No quadratic functions have a range of $(-\infty, \infty)$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 92

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The vertex of the parabola described by $f(x)=2(x-5)^{2}-1$ is at $(5,1)$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 93

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The graph of $f(x)=-2(x+4)^{2}-8$ has one $y$ -intercept and two $x$ -intercepts.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 94

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The maximum value of $y$ for the quadratic function $f(x)=-x^{2}+x+1$ is 1

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 95

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose $y$ -coordinate is the same as the given point.
$$
f(x)=3(x+2)^{2}-5 ;(-1,-2)
$$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 96

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose $y$ -coordinate is the same as the given point.
$$
f(x)=(x-3)^{2}+2 ; \quad(6,11)
$$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 97

Write the equation of each parabola in standard form.
Vertex: $(-3,-4) ;$ The graph passes through the point $(1,4)$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 98

Write the equation of each parabola in standard form.
Vertex: $(-3,-1) ;$ The graph passes through the point $(-2,-3)$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 99

A rancher has 1000 feet of fencing to construct six corrals, as shown in the figure. Find the dimensions that maximize the enclosed area. What is the maximum area?

TB

Timothy Behan

Numerade Educator

Problem 100

The annual yield per lemon tree is fairly constant at 320 pounds when the number of trees per acre is 50 or fewer. For each additional tree over $50,$ the annual yield per tree for all trees on the acre decreases by 4 pounds due to overcrowding. Find the number of trees that should be planted on an acre to produce the maximum yield. How many pounds is the maximum yield?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 101

Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that initially increase and then decrease, or vice versa, and therefore can be modeled by a quadratic function. Group members should select the two sets of data that are most interesting and relevant. For each data set selected,
a. Use the quadratic regression feature of a graphing utility to find the quadratic function that best fits the data.
b. Use the equation of the quadratic function to make a prediction from the data. What circumstances might affect the accuracy of your prediction?
c. Use the equation of the quadratic function to write and solve a problem involving maximizing or minimizing the function.

Lourence Gonhovi

Lourence Gonhovi

Numerade Educator

Problem 102

Does the equation $3 x+y^{2}=10$ define $y$ as a function of $x ?$ (Section $2.1,$ Example 3 )

Jason Gerber

Numerade Educator

Problem 103

Use the following graph to solve this exercise.
(Graph can't copy)
a. Determine the function's domain.
b. Determine the function's range.
c. What are the $x$ -intercepts?
d. What is the $y$ -intercept?
e. Determine $f(-4)$

Yujie Wang

College of San Mateo

Problem 104

If $f(x)=4 x^{2}-2 x+7,$ find
$$
\frac{p(x+h)-f(x)}{h}, h \neq 0
$$
and simplify.

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 105

Exercises 105–107 will help you prepare for the material covered in the next section.
$$
\text { Factor: } x^{3}+3 x^{2}-x-3
$$

Linh Vu

Numerade Educator

Problem 106

Exercises 105–107 will help you prepare for the material covered in the next section.
If $f(x)=x^{3}-2 x-5,$ find $f(2)$ and $f(3) .$ Then explain why the continuous graph of $f$ must cross the $x$ -axis between 2 and 3

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 107

Exercises 105–107 will help you prepare for the material covered in the next section.
Determine whether $f(x)=x^{4}-2 x^{2}+1$ is even, odd, or neither. Describe the symmetry, if any, for the graph of $f$

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator