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Precalculus with Limits (2010)

Ron Larson, David C. Falvo

Chapter 2

Polynomial and Rational Functions

Educators


Problem 1

Linear, constant, and squaring functions are examples of___functions.

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Problem 2

A polynomial function of degree $n$ and leading coefficient $a_{n}$ is a function of 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

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Problem 3

$A$______function is a second-degree polynomial function, and its graph is called a_____

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Problem 4

The graph of a quadratic function is symmetric about its________

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Problem 5

If the graph of a quadratic function opens upward, then its leading coefficient is______and the vertex of the graph is a_____

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Problem 6

If the graph of a quadratic function opens downward, then its leading coefficient is_______and the vertex of the graph is a _____

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Problem 7

In Exercises $7-12,$ match the quadratic function with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).]
$$
f(x)=(x-2)^{2}
$$

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Problem 8

In Exercises $7-12,$ match the quadratic function with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).]
$$
f(x)=(x+4)^{2}
$$

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Problem 9

In Exercises $7-12,$ match the quadratic function with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).]
$$
f(x)=x^{2}-2
$$

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Problem 10

In Exercises $7-12,$ match the quadratic function with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).]
$$
f(x)=(x+1)^{2}-2
$$

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Problem 11

In Exercises $7-12,$ match the quadratic function with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).]
$$
f(x)=4-(x-2)^{2}
$$

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Problem 12

In Exercises $7-12,$ match the quadratic function with its graph. IThe graphs are labeled (a), (b), (c), (d), (e), and (f).]
$$
f(x)=-(x-4)^{2}
$$

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Problem 13

In Exercises $13-16,$ graph each function. Compare the graph of each function with the graph of $y=x^{2} .$
$$
\begin{array}{ll}{\text { (a) } f(x)=\frac{1}{2} x^{2}} & {\text { (b) } g(x)=-\frac{1}{8} x^{2}} \\ {\text { (c) } h(x)=\frac{3}{2} x^{2}} & {\text { (d) } k(x)=-3 x^{2}}\end{array}
$$

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Problem 14

In Exercises $13-16,$ graph each function. Compare the graph of each function with the graph of $y=x^{2} .$
$$
\begin{array}{ll}{\text { (a) } f(x)=x^{2}+1} & {\text { (b) } g(x)=x^{2}-1} \\ {\text { (c) } h(x)=x^{2}+3} & {\text { (d) } k(x)=x^{2}-3}\end{array}
$$

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Problem 15

In Exercises $13-16,$ graph each function. Compare the graph of each function with the graph of $y=x^{2} .$
$$
\begin{array}{ll}{\text { (a) } f(x)=(x-1)^{2}} & {\text { (b) } g(x)=(3 x)^{2}+1} \\ {\text { (c) } h(x)=\left(\frac{1}{3} x\right)^{2}-3} & {\text { (d) } k(x)=(x+3)^{2}}\end{array}
$$

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Problem 16

In Exercises $13-16,$ graph each function. Compare the graph of each function with the graph of $y=x^{2} .$
$$
\begin{array}{l}{\text { (a) } f(x)=-\frac{1}{2}(x-2)^{2}+1} \\ {\text { (b) } g(x)=\left[\frac{1}{2}(x-1)\right]^{2}-3} \\ {\text { (c) } h(x)=-\frac{1}{2}(x+2)^{2}-1} \\ {\text { (d) } k(x)=[2(x+1)]^{2}+4}\end{array}
$$

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Problem 17

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=1-x^{2}
$$

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Problem 18

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
g(x)=x^{2}-8
$$

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Problem 19

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=x^{2}+7
$$

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Problem 20

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
h(x)=12-x^{2}
$$

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Problem 21

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=\frac{1}{2} x^{2}-4
$$

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Problem 22

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=16-\frac{1}{4} x^{2}
$$

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Problem 23

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=(x+4)^{2}-3
$$

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Problem 24

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=(x-6)^{2}+8
$$

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Problem 25

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
h(x)=x^{2}-8 x+16
$$

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Problem 26

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
g(x)=x^{2}+2 x+1
$$

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Problem 27

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=x^{2}-x+\frac{5}{4}
$$

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Problem 28

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=x^{2}+3 x+\frac{1}{4}
$$

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Problem 29

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=-x^{2}+2 x+5
$$

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Problem 30

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=-x^{2}-4 x+1
$$

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Problem 31

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
h(x)=4 x^{2}-4 x+21
$$

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Problem 32

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=2 x^{2}-x+1
$$

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Problem 33

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=\frac{1}{4} x^{2}-2 x-12
$$

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Problem 34

In Exercises $17-34$ , sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and $x$ -intercept(s).
$$
f(x)=-\frac{1}{3} x^{2}+3 x-6
$$

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Problem 35

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

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Problem 36

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

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Problem 37

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

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Problem 38

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

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Problem 39

In Exercises $35-42,$ use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercepts. Then check your results algebraically by writing the quadratic function in standard form.
$$
f(x)=2 x^{2}-16 x+31
$$

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Problem 40

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

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Problem 41

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

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Problem 42

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

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Problem 43

In Exercises $43-46,$ write an equation for the parabola in standard form.

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Problem 44

In Exercises $43-46,$ write an equation for the parabola in standard form.

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Problem 45

In Exercises $43-46,$ write an equation for the parabola in standard form.

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Problem 46

In Exercises $43-46,$ write an equation for the parabola in standard form.

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Problem 47

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

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Problem 48

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

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Problem 49

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

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Problem 50

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

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Problem 51

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

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Problem 52

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

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Problem 53

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

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Problem 54

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

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Problem 55

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

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Problem 56

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

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Problem 57

GRAPHICAL REASONING 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.
$$
y=x^{2}-4 x-5
$$

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Problem 58

GRAPHICAL REASONING 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.
$$
y=2 x^{2}+5 x-3
$$

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Problem 59

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

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Problem 60

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

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Problem 61

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

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Problem 62

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

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Problem 63

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

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Problem 64

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

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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.)
$(-1,0),(3,0)$

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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.)
$(-5,0),(5,0)$

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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.)
$(0,0),(10,0)$

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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.)
$(4,0),(8,0)$

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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.)
$(-3,0),\left(-\frac{1}{2}, 0\right)$

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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.)
$\left(-\frac{5}{2}, 0\right),(2,0)$

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Problem 71

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

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Problem 72

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

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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

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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 $42 .$

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Problem 75

PATH OF A DIVER The path of a diver is given by
$$
y=-\frac{4}{9} x^{2}+\frac{24}{9} x+12
$$
where $y$ 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?

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Problem 76

HEIGHT OF A BALL The heighty (in feet) of a punted football is given by
$$
y=-\frac{16}{2025} x^{2}+\frac{9}{5} x+1.5
$$
where $x$ 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?

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Problem 77

MINIMUM COST A manulacturer 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?

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Problem 78

MAXIMUM PROFIT 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?

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Problem 79

MAXIMUM REVENUE 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 price per unit is $\$ 20$ , S25, and $\$ 30$ .
(b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain
your results.

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Problem 80

MAXIMUM REVENUE 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 price per pet is $\$ 4, \$ 6,$ and $\$ 8$ .
(b) Find the price that will yield a maximum revenue. What is the maximum revenue? Fxplain your results.

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Problem 81

NUMERICAL, GRAPHICAL, AND ANALYTICAL ANALYSIS A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals (see figure).
(a) Write the area $A$ of the corrals as a function of $x$ .
(b) Create 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), (c), and (d).

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Problem 82

GEOMETRY 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 illustrates 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?

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Problem 83

MAXIMUM REVENUE 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 miximum revenue?

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Problem 84

MAXIMUM AREA 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.
(a) Write the area $A$ of the window as a function of $x$ .
(b) What dimensions will produce a window of maximum area?

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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 this 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 cigaretle consumption per smoker?

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Problem 86

DATA ANALYSIS: SALES The sales $y$ (in billions of dollars) for Harley-Davidson from 2000 through 2007 are shown in the table. (Source: U.S. Harley- Davidson, Inc.)
(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) algebraicully.
(f) Use the model to predict the sales for Harley- Davidson in 2010 .

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Problem 87

TRUE OR FALSE? In Exercises $87-90$ , determine whether the statement is true or false. Justify your answer.
The function given by $f(x)=-12 x^{2}-1$ has no $x$ -intercepts.

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Problem 88

TRUE OR FALSE? In Exercises $87-90$ , 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.

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Problem 89

TRUE OR FALSE? In Exercises $87-90$ , determine whether the statement is true or false. Justify your answer.
The graph of a quadratic function with a negative leading coefficient will have a maximum value at its vertex.

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Problem 90

TRUE OR FALSE? In Exercises $87-90$ , determine whether the statement is true or false. Justify your answer.
The graph of a quadratic function with a positive leading mcoefficient will have a minimum value at its vertex.

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Problem 91

THINK ABOUT IT In Exercises $91-94$ , 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

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Problem 92

THINK ABOUT IT In Exercises $91-94$ , 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

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Problem 93

THINK ABOUT IT In Exercises $91-94$ , 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

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Problem 94

THINK ABOUT IT In Exercises $91-94$ , 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: $-50$

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Problem 95

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

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Problem 96

CAPSTONE The profit $P$ (in millions of dollars) for a recreational vehicle retailer is modeled by a quadratic function of the foorm
$$P=a t^{2}+b t+c$$
where $t$ represents the year. If you were president of the company, which of the models below would you prefer? Explain your reasoning.
(a) $a$ is positive and $-b /(2 a) \leq t$
(b) $a$ is positive and $t \leq-b /(2 a)$ .
(c) a is negative and - $b /(2 a) \leq t$
(d) a is negative and $t \leq-b /(2 a)$

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Problem 97

GRAPHICAL ANALYSIS
(a) Graph $y=a x^{2}$ for $a=-2,-1,-0.5,0.5,1$ and 2 How does changing the value of $a$ affect the graph?
(b) Graph $y=(x-h)^{2}$ for $h=-4,-2,2,$ and 4 . How does changing the value of $h$ affect the graph?
(c) Graph $y=x^{2}+k$ for $k=-4,-2,2,$ and 4 . How does changing the value of $k$ affect the graph?

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Problem 98

Describe the sequence of transformation from $f$ to $g$ given that $f(x)=x^{2}$ and $g(x)=a(x-h)^{2}+k$ (Assume $a, h,$ and $k$ are positive.)

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Problem 99

Is it possible for a quadratic equation to have only one $x$ -intercept? Explain.

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Problem 100

Assume that the function given by
$$
f(x)=a x^{2}+b x+c, \quad a \neq 0
$$
has two real zeros. Show that the $x$ -coordinate of the vertex of the graph is the average of the zeros of $f .$ (Hint: Use the Quadratic Formula.)

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