College Algebra

Michael Sullivan

Chapter 4

Linear and Quadratic Functions

Educators


Problem 1

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
Graph $y=2 x-3 .(\mathrm{pp} .157-164)$

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Problem 2

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
Find the slope of the line joining the points $(2,5)$ and $(-1,3) .(\mathrm{pp} .167-175)$

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Problem 3

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
Find the average rate of change of $f(x)=3 x^{2}-2,$ from 2 to $4 .(\mathrm{pp} .222-230)$

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Problem 4

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
Solve: $60 x-900=-15 x+2850 .(\mathrm{pp} .82-87)$

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Problem 5

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
If $f(x)=x^{2}-4,$ find $f(-2) \cdot(\mathrm{pp} .200-208)$

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Problem 6

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
True or False The graph of the function $f(x)=x^{2}$ is increasing on the interval $(0, \infty) .(\mathrm{pp} .214-217)$

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Problem 7

For the graph of the linear function $f(x)=m x+b, m$ is the _____ and $b$ is the ______.

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Problem 8

For the graph of the linear function $H(z)=-4 z+3,$ the slope is _____ and the $y$ -intercept is ______.

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Problem 9

If the slope $m$ of the graph of a linear function is _______ ,the function is increasing over its domain.

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Problem 10

True or False The slope of a nonvertical line is the average rate of change of the linear function.

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Problem 11

True or False If the average rate of change of a linear function is $\frac{2}{3},$ then if $y$ increases by $3, x$ will increase by 2

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Problem 12

True or False The average rate of change of $f(x)=2 x+8$ is 8

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Problem 13

A linear function is given.
(a) Determine the slope and y-intercept of each function.
(b) Use the slope and y-intercept to graph the linear function.
(c) Determine the average rate of change of each function.
(d) Determine whether the linear function is increasing, decreasing, or constant.
$$f(x)=2 x+3$$

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Problem 14

A linear function is given.
(a) Determine the slope and y-intercept of each function.
(b) Use the slope and y-intercept to graph the linear function.
(c) Determine the average rate of change of each function.
(d) Determine whether the linear function is increasing, decreasing, or constant.
$$g(x)=5 x-4$$

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Problem 15

A linear function is given.
(a) Determine the slope and y-intercept of each function.
(b) Use the slope and y-intercept to graph the linear function.
(c) Determine the average rate of change of each function.
(d) Determine whether the linear function is increasing, decreasing, or constant.
$$h(x)=-3 x+4$$

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Problem 16

A linear function is given.
(a) Determine the slope and y-intercept of each function.
(b) Use the slope and y-intercept to graph the linear function.
(c) Determine the average rate of change of each function.
(d) Determine whether the linear function is increasing, decreasing, or constant.
$$p(x)=-x+6$$

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Problem 17

A linear function is given.
(a) Determine the slope and y-intercept of each function.
(b) Use the slope and y-intercept to graph the linear function.
(c) Determine the average rate of change of each function.
(d) Determine whether the linear function is increasing, decreasing, or constant.
$$f(x)=\frac{1}{4} x-3$$

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Problem 18

A linear function is given.
(a) Determine the slope and y-intercept of each function.
(b) Use the slope and y-intercept to graph the linear function.
(c) Determine the average rate of change of each function.
(d) Determine whether the linear function is increasing, decreasing, or constant.
$$h(x)=-\frac{2}{3} x+4$$

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Problem 19

A linear function is given.
(a) Determine the slope and y-intercept of each function.
(b) Use the slope and y-intercept to graph the linear function.
(c) Determine the average rate of change of each function.
(d) Determine whether the linear function is increasing, decreasing, or constant.
$$F(x)=4$$

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Problem 20

A linear function is given.
(a) Determine the slope and y-intercept of each function.
(b) Use the slope and y-intercept to graph the linear function.
(c) Determine the average rate of change of each function.
(d) Determine whether the linear function is increasing, decreasing, or constant.
$$G(x)=-2$$

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Problem 21

Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.
(TABLE CANNOT COPY)

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Problem 22

Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.
(TABLE CANNOT COPY)

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Problem 23

Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.
(TABLE CANNOT COPY)

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Problem 24

Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.
(TABLE CANNOT COPY)

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Problem 25

Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.
(TABLE CANNOT COPY)

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Problem 26

Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.
(TABLE CANNOT COPY)

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Problem 27

Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.
(TABLE CANNOT COPY)

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Problem 28

Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.
(TABLE CANNOT COPY)

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Problem 29

Suppose that $f(x)=4 x-1$ and $g(x)=-2 x+5$
(a) Solve $f(x)=0$
(b) Solve $f(x)>0$
(d) Solve $f(x) \leq g(x)$
(c) Solve $f(x)=g(x)$
(e) Graph $y=f(x)$ and $y=g(x)$ and label the point that represents the solution to the equation $f(x)=g(x)$

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Problem 30

Suppose that $f(x)=3 x+5$ and $g(x)=-2 x+15$
(a) Solve $f(x)=0$
(b) Solve $f(x)<0$
(c) Solve $f(x)=g(x)$
(d) Solve $f(x) \geq g(x)$
(e) Graph $y=f(x)$ and $y=g(x)$ and label the point that represents the solution to the equation $f(x)=g(x)$

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Problem 31

In parts (a)-(f), use the following figure.
(FIGURE CANNOT COPY)
(a) Solve $f(x)=50$
(b) Solve $f(x)=80$
(c) Solve $f(x)=0$
(d) Solve $f(x)>50$
(e) Solve $f(x) \leq 80$
(f) Solve $0<f(x)<80$

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Problem 32

In parts (a)-(f), use the following figure.
(FIGURE CANNOT COPY)
(a) Solve $g(x)=20$
(b) Solve $g(x)=60$
(c) Solve $g(x)=0$
(d) Solve $g(x)>20$
(e) Solve $g(x) \leq 60$
(f) Solve $0<g(x)<60$

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Problem 33

In parts (a) and (b) use the following figure.
(FIGURE CANNOT COPY)
(a) Solve the equation: $f(x)=g(x)$
(b) Solve the inequality: $f(x)>g(x)$

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Problem 34

In parts (a) and (b), use the following figure.
(FIGURE CANNOT COPY)
(a) Solve the equation: $f(x)=g(x)$
(b) Solve the inequality: $f(x) \leq g(x)$

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Problem 35

In parts (a) and (b), use the following figure.
(FIGURE CANNOT COPY)
(a) Solve the equation: $f(x)=g(x)$
(b) Solve the inequality: $g(x) \leq f(x)<h(x)$

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Problem 36

In parts (a) and (b), use the following figure.
(FIGURE CANNOT COPY)
(a) Solve the equation: $f(x)=g(x)$
(b) Solve the inequality: $g(x)<f(x) \leq h(x)$

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Problem 37

Car Rentals The cost $C$, in dollars, of renting a moving truck for a day is modeled by the function $C(x)=0.25 x+35$ where $x$ is the number of miles driven.
(a) What is the cost if you drive $x=40$ miles?
(b) If the cost of renting the moving truck is $\$ 80,$ how many miles did you drive?
(c) Suppose that you want the cost to be no more than $\$ 100$. What is the maximum number of miles that you can drive?
(d) What is the implied domain of $C ?$

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Problem 38

Phone Charges The monthly cost $C,$ in dollars, for international calls on a certain cellular phone plan is modeled by the function $C(x)=0.38 x+5,$ where $x$ is the number of minutes used.
(a) What is the cost if you talk on the phone for $x=50$ minutes?
(b) Suppose that your monthly bill is $\$ 29.32 .$ How many minutes did you use the phone?
(c) Suppose that you budget yourself $\$ 60$ per month for the phone. What is the maximum number of minutes that you can talk?
(d) What is the implied domain of $C$ if there are 30 days in the month?

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Problem 39

Supply and Demand Suppose that the quantity supplied $S$ and quantity demanded $D$ of T-shirts at a concert are given by the following functions:
$$\begin{aligned}S(p) &=-200+50 p \\D(p) &=1000-25 p\end{aligned}$$
where $p$ is the price of a T-shirt.
(a) Find the equilibrium price for T-shirts at this concert. What is the equilibrium quantity?
(b) Determine the prices for which quantity demanded is greater than quantity supplied.
(c) What do you think will eventually happen to the price of T-shirts if quantity demanded is greater than quantity supplied?

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Problem 40

Supply and Demand Suppose that the quantity supplied $S$ and quantity demanded $D$ of hot dogs at a baseball game are given by the following functions:
$$\begin{aligned}S(p) &=-2000+3000 p \\D(p) &=10,000-1000 p\end{aligned}$$
where $p$ is the price of a hot dog.
(a) Find the equilibrium price for hot dogs at the baseball game. What is the equilibrium quantity?
(b) Determine the prices for which quantity demanded is less than quantity supplied.
(c) What do you think will eventually happen to the price of hot dogs if quantity demanded is less than quantity supplied?

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Problem 41

Taxes The function $T(x)=0.15(x-8350)+835$ represents the tax bill $T$ of a single person whose adjusted gross income is $x$ dollars for income between 8350 dollars and 33,95 dollars inclusive, in 2009
(a) What is the domain of this linear function?
(b) What is a single filer's tax bill if adjusted gross income is 20,000 dollars ?
(c) Which variable is independent and which is dependent?
(d) Graph the linear function over the domain specified in part (a).
(e) What is a single filer's adjusted gross income if the tax bill is 3707.50 dollars ?

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Problem 42

Luxury Tax In $2002,$ major league baseball signed a labor agreement with the players. In this agreement, any team whose payroll exceeded 136.5million dollars in 2006 had to pay a luxury tax of $40 \%$ (for second offenses). The linear function $T(p)=0.40(p-136.5)$ describes the luxury tax $T$ of a team whose payroll was $p$ (in millions of dollars).
(a) What is the implied domain of this linear function?
(b) What was the luxury tax for the New York Yankees whose 2006 payroll was 171.1 dollars million?
(c) Graph the linear function.
(d) What is the payroll of a team that pays a luxury tax of 11.7 million dollars ?

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Problem 43

The point at which a company's profits equal zero is called the company's break-even point. For Problems 43 and $44,$ let $R$ represent a company's revenue, let $C$ represent the company's costs, and let $x$ represent the number of units produced and sold each day.
(a) Find the firm's break-even point; that is, find $x$ so that $R=C$.
(b) Find the values of $x$ such that $R(x)>C(x) .$ This represents the number of units that the company must sell to earn a profit.
$$\begin{array}{l}R(x)=8 x \\C(x)=4.5 x+17,500\end{array}$$

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Problem 44

The point at which a company's profits equal zero is called the company's break-even point. For Problems 43 and $44,$ let $R$ represent a company's revenue, let $C$ represent the company's costs, and let $x$ represent the number of units produced and sold each day.
(a) Find the firm's break-even point; that is, find $x$ so that $R=C$.
(b) Find the values of $x$ such that $R(x)>C(x) .$ This represents the number of units that the company must sell to earn a profit.
$$\begin{array}{l}R(x)=12 x \\C(x)=10 x+15,000\end{array}$$

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Problem 45

Straight-line Depreciation Suppose that a company has just purchased a new computer for 3000 dollars . The company chooses to depreciate the computer using the straight-line method over 3 years.
(a) Write a linear model that expresses the book value $V$ of the computer as a function of its age $x$.
(b) What is the implied domain of the function found in part (a)?
(c) Graph the linear function.
(d) What is the book value of the computer after 2 years?
(e) When will the computer have a book value of 2000 dollars ?

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Problem 46

Straight-line Depreciation Suppose that a company has just purchased a new machine for its manufacturing facility for 120,000 dollars . The company chooses to depreciate the machine using the straight-line method over 10 years.
(a) Write a linear model that expresses the book value $V$ of the machine as a function of its age $x$
(b) What is the implied domain of the function found in part (a)?
(c) Graph the linear function.
(d) What is the book value of the machine after 4 years?
(e) When will the machine have a book value of 72,000 dollars ?

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Problem 47

cost Function The simplest cost function is the linear cost function, $C(x)=m x+b,$ where the $y$ -intercept $b$ represents the fixed costs of operating a business and the slope $m$ represents the cost of each item produced. Suppose that a small bicycle manufacturer has daily fixed costs of 1800 dollars and each bicycle costs 90 dollars to manufacture.
(a) Write a linear model that expresses the cost $C$ of manufacturing $x$ bicycles in a day.
(b) Graph the model.
(c) What is the cost of manufacturing 14 bicycles in a day?
(d) How many bicycles could be manufactured for 3780 dollars ?

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Problem 48

cost Function Refer to Problem $47 .$ Suppose that the landlord of the building increases the bicycle manufacturer's rent by 100 dollars per month.
(a) Assuming that the manufacturer is open for business 20 days per month, what are the new daily fixed costs?
(b) Write a linear model that expresses the cost $C$ of manufacturing $x$ bicycles in a day with the higher rent.
(c) Graph the model.
(d) What is the cost of manufacturing 14 bicycles in a day?
(e) How many bicycles can be manufactured for 3780 dollars ?

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Problem 49

Truck Rentals $A$ truck rental company rents a truck for one day by charging 29 dollars plus 0.07 dollars per mile.
(a) Write a linear model that relates the cost $C,$ in dollars, of renting the truck to the number $x$ of miles driven.
(b) What is the cost of renting the truck if the truck is driven 110 miles? 230 miles?

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Problem 50

Long Distance A phone company offers a domestic long distance package by charging 5 dollars plus 0.05 dollars per minute.
(a) Write a linear model that relates the cost $C$, in dollars, of talking $x$ minutes.
(b) What is the cost of talking 105 minutes? 180 minutes?

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Problem 51

Developing a Linear Model from Data The following data represent the price $p$ and quantity demanded per day $q$ of $24^{n}$ LCD monitor.
(TABLE CANNOT COPY)
(a) Plot the ordered pairs $(p, q)$ in a Cartesian plane.
(b) Show that quantity demanded $q$ is a linear function of the price $p$
(c) Determine the linear function that describes the relation between $p$ and $q$
(d) What is the implied domain of the linear function?
(e) Graph the linear function in the Cartesian plane drawn in part (a).
(f) Interpret the slope.
(g) Interpret the values of the intercepts.

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Problem 52

Developing a Linear Model from Data The following data represent the various combinations of soda and hot dogs that Yolanda can buy at a baseball game with 60 dollars.
(TABLE CANNOT COPY)
(a) Plot the ordered pairs $(s, h)$ in a Cartesian plane.
(b) Show that the number of hot dogs purchased $h$ is a linear function of the number of sodas purchased $s$.
(c) Determine the linear function that describes the relation between $s$ and $h$
(d) What is the implied domain of the linear function?
(e) Graph the linear function in the Cartesian plane drawn in part (a).
(f) Interpret the slope.
(g) Interpret the values of the intercepts.

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Problem 53

Which of the following functions might have the graph shown? (More than one answer is possible.)
(GRAPH CANNOT COPY)
(a) $f(x)=2 x-7$
(b) $g(x)=-3 x+4$
(c) $H(x)=5$
(d) $F(x)=3 x+4$
(e) $G(x)=\frac{1}{2} x+2$

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Problem 54

Which of the following functions might have the graph shown? (More than one answer is possible.)
(GRAPH CANNOT COPY)
(a) $f(x)=3 x+1$
(b) $g(x)=-2 x+3$
(c) $H(x)=3$
(d) $F(x)=-4 x-1$
(e) $G(x)=-\frac{2}{3} x+3$

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Problem 55

Under what circumstances is a linear function $f(x)=m x+b$ odd? Can a linear function ever be even?

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Problem 56

Explain how the graph of $f(x)=m x+b$ can be used to solve $m x+b>0$

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