Algebra Form and Function

William G. McCallum, Eric Connally, Deborah Hughes-Hallett

Chapter 5

Linear Functions, Expressions, and Equations - all with Video Answers

Educators

Section 1

Linear Functions

Problem 1

A homing pigeon starts 1000 miles from home and flies 50 miles toward home each day. Express distance from home in miles, $D,$ as a function of number of days, $d$.

Nick Johnson

Nick Johnson

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Problem 2

You buy a saguaro cactus $5 \mathrm{ft}$ high and it grows at a rate of 0.2 inches each year. Express its height in inches, $h$, as a function of time $t$ in years since the purchase.

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Problem 3

The temperature of the soil is $30^{\circ} \mathrm{C}$ at the surface and decreases by $0.04^{\circ} \mathrm{C}$ for each centimeter below the surface. Express temperature $T$ as a function of depth $d$, in centimeters, below the surface.

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Problem 4

Give the values for $b$ and $m$ for the linear functions in Exercises 4-9.
$$
f(x)=3 x+12
$$

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Problem 5

Give the values for $b$ and $m$ for the linear functions.
$$
g(t)=250 t-5300
$$

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Problem 6

Give the values for $b$ and $m$ for the linear functions.
$$
h(n)=0.01 n+100
$$

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Problem 7

Give the values for $b$ and $m$ for the linear functions.
$$
v(z)=30
$$

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Problem 8

Give the values for $b$ and $m$ for the linear functions.
$$
w(c)=0.5 c
$$

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Problem 9

Give the values for $b$ and $m$ for the linear functions.
$$
u(k)=0.007-0.003 k
$$

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Problem 10

The cost, $\$ C$, of hiring a repairman for $h$ hours is given by $C=50+25 h$
(a) What does the repairman charge to walk in the door?
(b) What is his hourly rate?

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Problem 11

The cost, $\$ C$, of renting a limousine for $h$ hours above the 4 hour minimum is given by $C=300+100 h$.
(a) What does the 300 represent?
(b) What is the hourly rate?

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Problem 12

The population of a town, $t$ years after it is founded, is given by $P(t)=5000+350 t$.
(a) What is the population when it is founded?
(b) What is the population of the town one year after it is founded? How much does it increase by during the first year? During the second year?

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Problem 13

In Exercises $13-23,$ identify the initial value and the rate of change, and explain their meanings in practical terms.
An orbiting spaceship releases a probe that travels directly away from Earth. The probe's distance $s$ (in km) from Earth after $t$ seconds is given by $s=600+5 t$.

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Problem 14

Identify the initial value and the rate of change, and explain their meanings in practical terms.
After a rain storm, the water in a trough begins to evaporate. The amount in gallons remaining after $t$ days is given by $V=50-1.2 t$.

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Problem 15

Identify the initial value and the rate of change, and explain their meanings in practical terms.
The monthly charge of a cell phone is $25+0.06 n$ dollars, where $n$ is the number of minutes used.

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Problem 16

Identify the initial value and the rate of change, and explain their meanings in practical terms.
The number of people enrolled in Mathematics 101 is $200-5 y,$ where $y$ is the number of years since 2004 .

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Problem 17

Identify the initial value and the rate of change, and explain their meanings in practical terms.
On a spring day the temperature in degrees Fahrenheit is $50+1.2 h,$ where $h$ is the number of hours since noon.

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Problem 18

Identify the initial value and the rate of change, and explain their meanings in practical terms.
The value of an antique is $2500+80 n$ dollars, where $n$ is the number of years since the antique is purchased.

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Problem 19

The value of an antique is $2500+80 n$ dollars, where $n$ is the number of years since the antique is purchased.
A professor calculates a homework grade of $100-3 n$ for $n$ missing homework assignments.

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Problem 20

Identify the initial value and the rate of change, and explain their meanings in practical terms.
The cost, $C$, in dollars, of a high school dance attended by $n$ students is given by $C=500+20 n$.

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Problem 21

Identify the initial value and the rate of change, and explain their meanings in practical terms.
The total amount, $C$, in dollars, spent by a company on a piece of heavy machinery after $t$ years in service is given by $C=20,000+1500 t$.

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Problem 22

Identify the initial value and the rate of change, and explain their meanings in practical terms.
The population, $P,$ of a city is predicted to be $P=$ $9000+500 t$ in $t$ years from now.

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Numerade Educator

Problem 23

Identify the initial value and the rate of change, and explain their meanings in practical terms.
The distance, $d$, in meters from the shore, of a surfer riding a wave is given by $d=120-5 t,$ where $t$ is the number of seconds since she caught the wave.

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Problem 24

In Exercises 24-29, identify the slope and $y$ -intercept and graph the function.
$$
f(x)=2 x+3
$$

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Problem 25

Identify the slope and $y$ -intercept and graph the function.
$$
f(x)=4-x
$$

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Problem 26

Identify the slope and $y$ -intercept and graph the function.
$$
f(x)=-2+0.5 x
$$

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Problem 27

Identify the slope and $y$ -intercept and graph the function.
$$
f(x)=3 x-2
$$

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Problem 28

Identify the slope and $y$ -intercept and graph the function.
$$
f(x)=-2 x+5
$$

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Problem 29

Identify the slope and $y$ -intercept and graph the function.
$$
f(x)=-0.5 x-0.2
$$

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Problem 30

If the tickets for a concert cost $p$ dollars each, the number of people who will attend is $2500-80 p .$ Which of the following best describes the meaning of the 80 in this expression?
(i) The price of an individual ticket.
(ii) The slope of the graph of attendance against ticket price.
(iii) The price at which no one will go to the concert.
(iv) The number of people who will decide not to go if the price is raised by one dollar.

Nick Johnson

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Problem 31

Long Island Power Authority charges its residential customers a monthly service charge plus an energy charge based on the amount of electricity used. ${ }^{2}$ The monthly cost of electricity is approximated by the function: $C=f(h)=36.60+0.14 h,$ where $h$ represents the number of kilowatt hours $(\mathrm{kWh})$ of electricity used in excess of $250 \mathrm{kWh}$.
(a) What does the coefficient 0.14 mean in terms of the cost of electricity?
(b) Find $f(50)$ and interpret its meaning.

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Problem 32

The following functions describe four different collections of baseball cards. The collections begin with different numbers of cards and cards are bought and sold at different rates. The number, $B$, of cards in each collection is a function of the number of years, $t,$ that the collection has been held. Describe each of these collec-
tions in words.
(a) $B=200+100 t$
(b) $\quad B=100+200 t$
(c) $\quad B=2000-100 t$
(d) $\quad B=100-200 t$

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Problem 33

Match the functions in (a)-(e) with the graphs in Figure 5.8 . The constants $s$ and $k$ are the same in each function.
(a) $f(x)=s$
(b) $f(x)=k x$
(c) $f(x)=k x-s$
(d) $f(x)=2 s-k x$
(e) $\quad f(x)=2 s-2 k x$

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Problem 34

The velocity of an object tossed up in the air is modeled by the function $v(t)=48-32 t,$ where $t$ is measured in seconds, and $v(t)$ is measured in feet per second.
(a) Create a table of values for the function.
(b) Graph the function.
(c) Explain what the constants 48 and -32 tell you about the velocity.
(d) What does a positive velocity indicate? A negative velocity?

Swati Agarwal

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Problem 35

If $a$ is a constant, does the equation $y=a x+5 a$ define $y$ as a linear function of $x ?$ If so, identify the slope and vertical intercept.

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Problem 36

The graphs of two linear functions have the same slope, but different $x$ -intercepts. Can they have the same $y$ intercept?

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Problem 37

The graphs of two linear functions have the same $x$ intercept, but different slopes. Can they have the same $y$ -intercept?

Nick Johnson

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Problem 38

Give the slope and $y$ -intercept for the graphs of the functions in Problems $38-43$.
$$
f(x)=220-12 x
$$

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Problem 39

Give the slope and $y$ -intercept for the graphs of the functions.
$$
f(x)=\frac{1}{3} x-11
$$

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Problem 40

Give the slope and $y$ -intercept for the graphs of the functions.
$$
f(x)=\frac{x}{7}-12
$$

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Problem 41

Give the slope and $y$ -intercept for the graphs of the functions.
$$
f(x)=\frac{20-2 x}{3}
$$

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Problem 42

Give the slope and $y$ -intercept for the graphs of the functions.
$$
f(x)=15-2(3-2 x)
$$

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Problem 43

Give the slope and $y$ -intercept for the graphs of the functions.
$$
f(x)=\pi x
$$

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Problem 44

If $n$ birds eating continuously consume $V$ in $^{3}$ of seed in $T$ hours, how much does one bird consume per hour?

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Problem 45

If $n$ birds eating continuously consume $W$ ounces of seed in $T$ hours, what are the units of $W /(n T)$ ? What does $W /(n T)$ represent in practical terms?

Nick Johnson

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