College Algebra

Ron Larson

Chapter 3

Polynomial Functions - all with Video Answers

Educators

+ 3 more educators

Section 1

Quadratic Functions and Models

Problem 1

Fill in the blanks.
Linear, constant, and squaring functions are examples of ________ functions.

JH

J Hardin

Numerade Educator

Problem 2

Fill in the blanks.
A polynomial function of $x$ with degree $n$ has the form $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$ $\left(a_{n} \neq 0\right),$ where $n$ is a _______ _______ and $a_{n}, a_{n-1}, \ldots, a_{1}, a_{0}$ are ________ numbers.

Vanessa Lamar

Numerade Educator

Problem 3

Fill in the blanks.
A ________ function is a second-degree polynomial function, and its graph is called a ________.

JH

J Hardin

Numerade Educator

Problem 4

Fill in the blanks.
The graph of a quadratic function is symmetric about its ________.

Suzanne W.

Numerade Educator

Problem 5

Fill in the blanks.
When the graph of a quadratic function opens upward, its leading coefficient is ________ and the vertex of the graph is a ________ .

JH

J Hardin

Numerade Educator

Problem 6

Fill in the blanks.
When the graph of a quadratic function opens downward, its leading coefficient is ________ and the vertex of the graph is a ________ .

JH

J Hardin

Numerade Educator

Problem 7

In Exercises 7–12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
GRAPH CAN'T COPY
$$f(x)=(x-2)^{2}$$

James Kiss

Numerade Educator

Problem 8

In Exercises 7–12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
GRAPH CAN'T COPY
$$f(x)=(x+4)^{2}$$

James Kiss

Numerade Educator

Problem 9

In Exercises 7–12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
GRAPH CAN'T COPY
$$f(x)=x^{2}-2$$

James Kiss

Numerade Educator

Problem 10

In Exercises 7–12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
GRAPH CAN'T COPY
$$f(x)=(x+1)^{2}-2$$

James Kiss

Numerade Educator

Problem 11

In Exercises 7–12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
GRAPH CAN'T COPY
$$f(x)=4-(x-2)^{2}$$

James Kiss

Numerade Educator

Problem 12

In Exercises 7–12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]
GRAPH CAN'T COPY
$$f(x)=-(x-4)^{2}$$

James Kiss

Numerade Educator

Problem 13

In Exercises $13-16,$ sketch the graph of each quadratic function and compare it with the graph of $y=x^{2}$.
(a) $f(x)=\frac{1}{2} x^{2}$
(b) $g(x)=-\frac{1}{8} x^{2}$
(c) $h(x)=\frac{3}{2} x^{2}$
(d) $k(x)=-3 x^{2}$

James Kiss

Numerade Educator

Problem 14

In Exercises $13-16,$ sketch the graph of each quadratic function and compare it with the graph of $y=x^{2}$.
(a) $f(x)=x^{2}+1$
(b) $g(x)=x^{2}-1$
(c) $h(x)=x^{2}+3$
(d) $k(x)=x^{2}-3$

James Kiss

Numerade Educator

Problem 15

In Exercises $13-16,$ sketch the graph of each quadratic function and compare it with the graph of $y=x^{2}$.
(a) $f(x)=(x-1)^{2}$
(b) $g(x)=(3 x)^{2}+1$
(c) $h(x)=\left(\frac{1}{3} x\right)^{2}-3$
(d) $k(x)=(x+3)^{2}$

James Kiss

Numerade Educator

Problem 16

In Exercises $13-16,$ sketch the graph of each quadratic function and compare it with the graph of $y=x^{2}$.
(a) $f(x)=-\frac{1}{2}(x-2)^{2}+1$
(b) $g(x)=\left[\frac{1}{2}(x-1)\right]^{2}-3$
(c) $h(x)=-\frac{1}{2}(x+2)^{2}-1$
(d) $k(x)=[2(x+1)]^{2}+4$

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 17

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}-6 x$$

James Kiss

Numerade Educator

Problem 18

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$g(x)=x^{2}-8 x$$

James Kiss

Numerade Educator

Problem 19

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$h(x)=x^{2}-8 x+16$$

James Kiss

Numerade Educator

Problem 20

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$g(x)=x^{2}+2 x+1$$

James Kiss

Numerade Educator

Problem 21

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}+8 x+13$$

James Kiss

Numerade Educator

Problem 22

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}-12 x+44$$

James Kiss

Numerade Educator

Problem 23

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}-14 x+54$$

James Kiss

Numerade Educator

Problem 24

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$h(x)=x^{2}+16 x-17$$

James Kiss

Numerade Educator

Problem 25

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}+34 x+289$$

James Kiss

Numerade Educator

Problem 26

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}-30 x+225$$

James Kiss

Numerade Educator

Problem 27

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}-x+\frac{5}{4}$$

James Kiss

Numerade Educator

Problem 28

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=x^{2}+3 x+\frac{1}{4}$$

James Kiss

Numerade Educator

Problem 29

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=-x^{2}+2 x+5$$

James Kiss

Numerade Educator

Problem 30

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=-x^{2}-4 x+1$$

James Kiss

Numerade Educator

Problem 31

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$h(x)=4 x^{2}-4 x+21$$

James Kiss

Numerade Educator

Problem 32

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=2 x^{2}-x+1$$

Erika Bustos

Numerade Educator

Problem 33

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=\frac{1}{4} x^{2}-2 x-12$$

Allison Knapp

Numerade Educator

Problem 34

In Exercises $17-34$, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$f(x)=-\frac{1}{3} x^{2}+3 x-6$$

Allison Knapp

Numerade Educator

Problem 35

In Exercises $35-42,$ use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=-\left(x^{2}+2 x-3\right)$$

James Kiss

Numerade Educator

Problem 36

In Exercises $35-42,$ use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=-\left(x^{2}+x-30\right)$$

James Kiss

Numerade Educator

Problem 37

In Exercises $35-42,$ use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$g(x)=x^{2}+8 x+11$$

James Kiss

Numerade Educator

Problem 38

In Exercises $35-42,$ use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=x^{2}+10 x+14$$

James Kiss

Numerade Educator

Problem 39

In Exercises $35-42,$ use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=2 x^{2}-16 x+32$$

James Kiss

Numerade Educator

Problem 40

In Exercises $35-42,$ use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=-4 x^{2}+24 x-41$$

James Kiss

Numerade Educator

Problem 41

In Exercises $35-42,$ use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$g(x)=\frac{1}{2}\left(x^{2}+4 x-2\right)$$

James Kiss

Numerade Educator

Problem 42

In Exercises $35-42,$ use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=\frac{3}{5}\left(x^{2}+6 x-5\right)$$

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 43

In Exercises $43-46,$ write an equation for the parabola in standard form.
GRAPH CAN'T COPY

James Kiss

Numerade Educator

Problem 44

In Exercises $43-46,$ write an equation for the parabola in standard form.
GRAPH CAN'T COPY

James Kiss

Numerade Educator

Problem 45

In Exercises $43-46,$ write an equation for the parabola in standard form.
GRAPH CAN'T COPY

James Kiss

Numerade Educator

Problem 46

In Exercises $43-46,$ write an equation for the parabola in standard form.
GRAPH CAN'T COPY

James Kiss

Numerade Educator

Problem 47

In Exercises 47–56, write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point.
Vertex: $(-2,5) ;$ point: $(0,9)$

James Kiss

Numerade Educator

Problem 48

In Exercises 47–56, write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point.
Vertex: $(4,-1) ;$ point: $(2,3)$

James Kiss

Numerade Educator

Problem 49

In Exercises 47–56, write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point.
Vertex: $(1,-2) ;$ point: $(-1,14)$

James Kiss

Numerade Educator

Problem 50

In Exercises 47–56, write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point.
Vertex: $(2,3) ;$ point: $(0,2)$

James Kiss

Numerade Educator

Problem 51

In Exercises 47–56, write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point.
Vertex: $(5,12) ;$ point: $(7,15)$

James Kiss

Numerade Educator

Problem 52

In Exercises 47–56, write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point.
Vertex: $(-2,-2) ;$ point: $(-1,0)$

James Kiss

Numerade Educator

Problem 53

In Exercises 47–56, write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point.
Vertex: $\left(-\frac{1}{4}, \frac{3}{2}\right) ;$ point: $(-2,0)$

Erika Bustos

Numerade Educator

Problem 54

In Exercises 47–56, write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point.
Vertex: $\left(\frac{5}{2},-\frac{3}{4}\right) ;$ point: $(-2,4)$

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 55

In Exercises 47–56, write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point.
Vertex: $\left(-\frac{5}{2}, 0\right) ;$ point: $\left(-\frac{7}{2},-\frac{16}{3}\right)$

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 56

In Exercises 47–56, write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point.
Vertex: $(6,6) ;$ point: $\left(\frac{61}{10}, \frac{3}{2}\right)$

Yujie Wang

College of San Mateo

Problem 57

In Exercises 57 and 58 , determine the $x$-intercept(s) of the graph visually. Then find the $x$-intercept(s) algebraically to confirm your results.
$$y=x^{2}-4 x-5$$

JH

J Hardin

Numerade Educator

Problem 58

In Exercises 57 and 58 , determine the $x$-intercept(s) of the graph visually. Then find the $x$-intercept(s) algebraically to confirm your results.
$$y=2 x^{2}+5 x-3$$

JH

J Hardin

Numerade Educator

Problem 59

In Exercises $59-64,$ use a graphing utility to graph the quadratic function. Find the $x$-intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation $f(x)=0$.
$$f(x)=x^{2}-4 x$$

James Kiss

Numerade Educator

Problem 60

In Exercises $59-64,$ use a graphing utility to graph the quadratic function. Find the $x$-intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation $f(x)=0$.
$$f(x)=-2 x^{2}+10 x$$

James Kiss

Numerade Educator

Problem 61

In Exercises $59-64,$ use a graphing utility to graph the quadratic function. Find the $x$-intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation $f(x)=0$.
$$f(x)=x^{2}-9 x+18$$

James Kiss

Numerade Educator

Problem 62

In Exercises $59-64,$ use a graphing utility to graph the quadratic function. Find the $x$-intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation $f(x)=0$.
$$f(x)=x^{2}-8 x-20$$

James Kiss

Numerade Educator

Problem 63

In Exercises $59-64,$ use a graphing utility to graph the quadratic function. Find the $x$-intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation $f(x)=0$.
$$f(x)=2 x^{2}-7 x-30$$

James Kiss

Numerade Educator

Problem 64

In Exercises $59-64,$ use a graphing utility to graph the quadratic function. Find the $x$-intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation $f(x)=0$.
$$f(x)=\frac{7}{10}\left(x^{2}+12 x-45\right)$$

James Kiss

Numerade Educator

Problem 65

In Exercises $65-70$, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given $x$-intercepts. (There are many correct answers.)
$$(-1,0),(3,0)$$

Chris Trentman

Numerade Educator

Problem 66

In Exercises $65-70$, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given $x$-intercepts. (There are many correct answers.)
$$(-5,0),(5,0)$$

Chris Trentman

Numerade Educator

Problem 67

In Exercises $65-70$, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given $x$-intercepts. (There are many correct answers.)
$$(0,0),(10,0)$$

Chris Trentman

Numerade Educator

Problem 68

In Exercises $65-70$, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given $x$-intercepts. (There are many correct answers.)
$$(4,0),(8,0)$$

Chris Trentman

Numerade Educator

Problem 69

In Exercises $65-70$, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given $x$-intercepts. (There are many correct answers.)
$$(-3,0),\left(-\frac{1}{2}, 0\right)$$

Chris Trentman

Numerade Educator

Problem 70

In Exercises $65-70$, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given $x$-intercepts. (There are many correct answers.)
$$\left(-\frac{5}{2}, 0\right),(2,0)$$

Allison Knapp

Numerade Educator

Problem 71

In Exercises 71–74, find two positive real numbers whose product is a maximum.
The sum is 110.

Yujie Wang

College of San Mateo

Problem 72

In Exercises 71–74, find two positive real numbers whose product is a maximum.
The sum is $S$.

Yujie Wang

College of San Mateo

Problem 73

In Exercises 71–74, find two positive real numbers whose product is a maximum.
The sum of the first and twice the second is 24.

Yujie Wang

College of San Mateo

Problem 74

In Exercises 71–74, find two positive real numbers whose product is a maximum.
The sum of the first and three times the second is 42.

Yujie Wang

College of San Mateo

Problem 75

The path of a diver is given by the function
$$f(x)=-\frac{4}{9} x^{2}+\frac{24}{9} x+12$$
where $f(x)$ is the height (in feet) and $x$ is the horizontal distance from the end of the diving board (in feet). What is the maximum height of the diver?

Yujie Wang

College of San Mateo

Problem 76

The path of a punted football is given by the function
$$f(x)=-\frac{16}{2025} x^{2}+\frac{9}{5} x+1.5$$
where $f(x)$ is the height (in feet) and $x$ is the horizontal distance (in feet) from the point at which the ball is punted.
(a) How high is the ball when it is punted?
(b) What is the maximum height of the punt?
(c) How long is the punt?

Allison Knapp

Numerade Educator

Problem 77

A manufacturer of lighting fixtures has daily production costs of $C=800-10 x+0.25 x^{2}$ where $C$ is the total cost (in dollars) and $x$ is the number of units produced. How many fixtures should be produced each day to yield a minimum cost?

Vanessa Lamar

Numerade Educator

Problem 78

The profit $P$ (in hundreds of dollars) that a company makes depends on the amount $x$ (in hundreds of dollars) the company spends on advertising according to the model $P=230+20 x-0.5 x^{2} .$ What expenditure for advertising will yield a maximum profit?

Vanessa Lamar

Numerade Educator

Problem 79

The total revenue $R$ earned (in thousands of dollars) from manufacturing handheld video games is given by
$$R(p)=-25 p^{2}+1200 p$$
where $p$ is the price per unit (in dollars).
(a) Find the revenues when the prices per unit are $\$ 20$, $\$ 25,$ and $\$ 30$
(b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results.

Yujie Wang

College of San Mateo

Problem 80

The total revenue $R$ earned per day (in dollars) from a pet-sitting service is given by $R(p)=-12 p^{2}+150 p,$ where $p$ is the price charged per pet (in dollars).
(a) Find the revenues when the prices per pet are $\$ 4$ $\$ 6,$ and $\$ 8$
(b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results.

Chris Trentman

Numerade Educator

Problem 81

A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals (see figure).
FIGURE CANNOT BE COPY
(a) Write the area $A$ of the corrals as a function of $x .$
(b) Construct a table showing possible values of $x$ and the corresponding areas of the corral. Use the table to estimate the dimensions that will produce the maximum enclosed area.
(c) Use a graphing utility to graph the area function. Use the graph to approximate the dimensions that will produce the maximum enclosed area.
(d) Write the area function in standard form to find analytically the dimensions that will produce the maximum area.
(e) Compare your results from parts (b), $(\mathrm{c}),$ and $(\mathrm{d})$

Chapman Howard

Numerade Educator

Problem 82

An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200-meter single-lane running track.
(a) Draw a diagram that gives a visual representation of the problem. Let $x$ and $y$ represent the length and width of the rectangular region, respectively.
(b) Determine the radius of each semicircular end of the room. Determine the distance, in terms of $y,$ around the inside edge of each semicircular part of the track.
(c) Use the result of part (b) to write an equation, in terms of $x$ and $y,$ for the distance traveled in one lap around the track. Solve for $y$
(d) Use the result of part (c) to write the area $A$ of the rectangular region as a function of $x .$ What dimensions will produce a rectangle of maximum area?

Chris Trentman

Numerade Educator

Problem 83

$A$ small theater has a seating capacity of $2000 .$ When the ticket price is $\$ 20$, attendance is $1500 .$ For each $\$ 1$ decrease in price, attendance increases by $100 .$
(a) Write the revenue $R$ of the theater as a function of ticket price $x$
(b) What ticket price will yield a maximum revenue? What is the maximum revenue?

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 84

A Norman window is constructed by adjoining a semicircle to the top of an ordinary
rectangular window (see figure). The perimeter of the window is 16 feet.
Write the area $A$ of the window as a function of $x .$
What dimensions will produce a window of maximum area?

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 85

Graphical Analysis From 1950 through $2005,$ the per capita consumption $C$ of cigarettes by Americans (age 18 and older) can be modeled by $C=3565.0+60.30 t-1.783 t^{2}, 0 \leq t \leq 55,$ where $t$ is the year, with $t=0$ corresponding to 1950 . (Source: Tobacco Outlook Report)
(a) Use a graphing utility to graph the model.
(b) Use the graph of the model to approximate the maximum average annual consumption. Beginning in $1966,$ all cigarette packages were required by law to carry a health warning. Do you think the warning had any effect? Explain.
(c) In $2005,$ the U.S. population (age 18 and over) was $296,329,000 .$ Of those, about $59,858,458$ were smokers. What was the average annual cigarette consumption per smoker in $2005 ?$ What was the average daily cigarette consumption per smoker?

Chris Trentman

Numerade Educator

Problem 86

The sales $Y$ (in billions of dollars) for Harley-Davidson from 2000 through 2010 are
shown in the table. (Source: U.S. Harley-Davidson, Inc.)
$$\begin{array}{|l|l|}
\hline \text { Year } & \text { Sale, } y\\
\hline 2000 & 2.91 \\
2001 & 3.36 \\
2002 & 4.09 \\
2003 & 4.62 \\
2004 & 5.02 \\
2005 & 5.34 \\
2006 & 5.80 \\
2007 & 5.73 \\
2008 & 5.59 \\
2009 & 4.78 \\
2010 & 4.86 \\ \hline
\end{array}$$
(a) Use a graphing utility to create a scatter plot of the data. Let $x$ represent the year, with $x=0$ corresponding to 2000
(b) Use the regression feature of the graphing utility to find a quadratic model for the data.
(c) Use the graphing utility to graph the model in the same viewing window as the scatter plot. How well does the model fit the data?
(d) Use the trace feature of the graphing utility to approximate the year in which the sales for Harley-Davidson were the greatest.
(e) Verify your answer to part (d) algebraically.
(f) Use the model to predict the sales for Harley-Davidson in 2013

Sheryl Ezze

Numerade Educator

Problem 87

In Exercises 87 and 88, determine whether the statement is true or false. Justify your answer.
The graph of $f(x)=-12 x^{2}-1$ has no $x$ -intercepts.

Chris Trentman

Numerade Educator

Problem 88

In Exercises 87 and 88, determine whether the statement is true or false. Justify your answer.
The graphs of
$$f(x)=-4 x^{2}-10 x+7$$
and
$$g(x)=12 x^{2}+30 x+1$$
have the same axis of symmetry.

Erika Bustos

Numerade Educator

Problem 89

In Exercises 89–92, find the values of $b$ such that the function has the given maximum or
minimum value.
$f(x)=-x^{2}+b x-75 ;$ Maximum value: 25

Yujie Wang

College of San Mateo

Problem 90

In Exercises 89–92, find the values of $b$ such that the function has the given maximum or
minimum value.
$f(x)=-x^{2}+b x-16 ;$ Maximum value: 48

Yujie Wang

College of San Mateo

Problem 91

In Exercises 89–92, find the values of $b$ such that the function has the given maximum or
minimum value.
$f(x)=x^{2}+b x+26 ;$ Minimum value: 10

Yujie Wang

College of San Mateo

Problem 92

In Exercises 89–92, find the values of $b$ such that the function has the given maximum or
minimum value.
$f(x)=x^{2}+b x-25 ;$ Minimum value: $-50$

Yujie Wang

College of San Mateo

Problem 93

Write the quadratic function
$$f(x)=a x^{2}+b x+c$$
in standard form to verify that the vertex occurs at
$$\left(-\frac{b}{2 a}, f\left(-\frac{b}{2 a}\right)\right).$$

Vanessa Lamar

Numerade Educator

Problem 94

The graph shows a quadratic function of the form
$$P(t)=a t^{2}+b t+c$$
which represents the yearly profits for a company, where $P(t)$ is the profit in year $t$.
(a) Is the value of $a$ positive, negative, or zero? Explain.
(b) Write an expression in terms of $a$ and $b$ that represents the year $t$ when the company made the least profit.
(c) The company made the same yearly profits in 2004 and $2012 .$ Estimate the year in which the company made the least profit.
(d) Assume that the model is still valid today. Are the yearly profits currently increasing, decreasing, or constant? Explain.

Sheryl Ezze

Numerade Educator

Problem 95

Proof Assume that the function
$$f(x)=a x^{2}+b x+c, a \neq 0$$
has two real zeros. Prove that the $x$ -coordinate of the vertex of the graph is the average of the zeros of $f$ (Hint: Use the Quadratic Formula.)
Project: Height of a Basketball To work an extended application analyzing the height of a basketball after it has been dropped, visit this text's website at LarsonPrecalculus.com.

Derek Follett

Numerade Educator