Algebra Form and Function

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

Chapter 7

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

Educators

Section 1

Power Functions

Problem 1

In Exercises $1-5,$ identify the exponent and the coefficient for each power function.
The area of a square of side $x$ is $A=x^{2}$.

Nick Johnson

Nick Johnson

Numerade Educator

Problem 2

Identify the exponent and the coefficient for each power function.
The perimeter of a square of side $x$ is $P=4 x$.

Nick Johnson

Numerade Educator

Problem 3

Identify the exponent and the coefficient for each power function.
The side of a cube of volume $V$ is $x=\sqrt[3]{V}$.

Nick Johnson

Numerade Educator

Problem 4

Identify the exponent and the coefficient for each power function.
The circumference of a circle of radius $r$ is $C=2 \pi r$.

Nick Johnson

Numerade Educator

Problem 5

Identify the exponent and the coefficient for each power function.
The surface area of a sphere of radius $r$ is $S=4 \pi r^{2}$.

Nick Johnson

Numerade Educator

Problem 6

The area, $A,$ of a rectangle whose length is 3 times its width is given by $A=3 w^{2}$, where $w$ is its width.
(a) Identify the coefficient and exponent of this power function.
(b) If the width is $5 \mathrm{~cm}$, what is the area of the rectangle?

Nick Johnson

Numerade Educator

Problem 7

The volume, $V$, of a cylinder whose radius is 5 times its height is given by $V=\frac{1}{5} \pi r^{3}$, where $r$ is the radius.
(a) Identify the coefficient and exponent of this power function.
(b) If the radius is $2 \mathrm{~cm},$ what is the volume?
(c) If the height is $0.8 \mathrm{~cm}$, what is the volume?

Nick Johnson

Numerade Educator

Problem 8

A ball dropped into a hole reaches a depth $d=4.9 t^{2}$ meters, where $t$ is the time in seconds since it was dropped.
(a) Identify the coefficient and exponent of this power function.
(b) How deep is the ball after 2 seconds?
(c) If the ball hits the bottom of the hole after 4 seconds, how deep is the hole?

Nick Johnson

Numerade Educator

Problem 9

For the graphs of power functions $f(x)=k x^{p}$ in Exercises $9-13,$ is
(a) $p>1 ?$
(b) $\quad p=1 ?$
(c) $0<p<1 ?$
(d) $\quad p=0$ ?
(e) $\quad p<0 ?$

Nick Johnson

Numerade Educator

Problem 10

For the graphs of power functions $f(x)=k x^{p}$ in Exercises $9-13,$ is
(a) $p>1 ?$
(b) $\quad p=1 ?$
(c) $0<p<1 ?$
(d) $\quad p=0$ ?
(e) $\quad p<0 ?$

Nick Johnson

Numerade Educator

Problem 11

For the graphs of power functions $f(x)=k x^{p}$ in Exercises $9-13,$ is
(a) $p>1 ?$
(b) $\quad p=1 ?$
(c) $0<p<1 ?$
(d) $\quad p=0$ ?
(e) $\quad p<0 ?$

Nick Johnson

Numerade Educator

Problem 12

For the graphs of power functions $f(x)=k x^{p}$ in Exercises $9-13,$ is
(a) $p>1 ?$
(b) $\quad p=1 ?$
(c) $0<p<1 ?$
(d) $\quad p=0$ ?
(e) $\quad p<0 ?$

Nick Johnson

Numerade Educator

Problem 13

For the graphs of power functions $f(x)=k x^{p}$ in Exercises $9-13,$ is
(a) $p>1 ?$
(b) $\quad p=1 ?$
(c) $0<p<1 ?$
(d) $\quad p=0$ ?
(e) $\quad p<0 ?$

Nick Johnson

Numerade Educator

Problem 14

In Exercises $14-17$
(a) Is $y$ proportional, or is it inversely proportional, to a positive power of $x$ ?
(b) Make a table of values showing corresponding values for $y$ when $x$ is $1,10,100,$ and 1000 .
(c) Use your table to determine whether $y$ increases or decreases as $x$ gets larger.
$y=2 x^{2}$

Nick Johnson

Numerade Educator

Problem 15

(a) Is $y$ proportional, or is it inversely proportional, to a positive power of $x$ ?
(b) Make a table of values showing corresponding values for $y$ when $x$ is $1,10,100,$ and 1000 .
(c) Use your table to determine whether $y$ increases or decreases as $x$ gets larger.
$y=3 \sqrt{x}$

Nick Johnson

Numerade Educator

Problem 16

(a) Is $y$ proportional, or is it inversely proportional, to a positive power of $x$ ?
(b) Make a table of values showing corresponding values for $y$ when $x$ is $1,10,100,$ and 1000 .
(c) Use your table to determine whether $y$ increases or decreases as $x$ gets larger.
$y=\frac{1}{x}$

Nick Johnson

Numerade Educator

Problem 17

(a) Is $y$ proportional, or is it inversely proportional, to a positive power of $x$ ?
(b) Make a table of values showing corresponding values for $y$ when $x$ is $1,10,100,$ and 1000 .
(c) Use your table to determine whether $y$ increases or decreases as $x$ gets larger.
$y=\frac{5}{x^{2}}$

Nick Johnson

Numerade Educator

Problem 18

Problems $18-20$ describe power functions of the form $y=$ $k x^{p},$ for $p$ an integer. Is $p$
(a) Even or odd?
(b) Positive or negative?

Nick Johnson

Numerade Educator

Problem 19

Problems $18-20$ describe power functions of the form $y=$ $k x^{p},$ for $p$ an integer. Is $p$
(a) Even or odd?
(b) Positive or negative?

Nick Johnson

Numerade Educator

Problem 20

Problems $18-20$ describe power functions of the form $y=$ $k x^{p},$ for $p$ an integer. Is $p$
(a) Even or odd?
(b) Positive or negative?
The graph of $y=f(x)$ gets closer to the $x$ -axis as $x$ gets large. For $x<0, y<0,$ and for $x>0, y>0$

Nick Johnson

Numerade Educator

Problem 21

In Problems $21-24, a$ and $b$ are positive constants. If $a>b$ then which is larger?
$a^{4}, b^{4}$

Nick Johnson

Numerade Educator

Problem 22

$a$ and $b$ are positive constants. If $a>b$ then which is larger?
$a^{1 / 4}, b^{1 / 4}$

Nick Johnson

Numerade Educator

Problem 23

$a$ and $b$ are positive constants. If $a>b$ then which is larger?
$a^{-4}, b^{-4}$

Nick Johnson

Numerade Educator

Problem 24

$a$ and $b$ are positive constants. If $a>b$ then which is larger?
$a^{-1 / 4}, b^{-1 / 4}$

Nick Johnson

Numerade Educator

Problem 25

(a) In Figure 7.7 , for which $x$ -values is the graph of $y=x^{5}$ above the graph of $y=x^{3},$ and for which $x$ -values is it below?
(b) Express your answers in part (a) algebraically using inequalities.

Nick Johnson

Numerade Educator

Problem 26

(a) In Figure 7.16 , for which $x$ -values is the graph of $y=x^{1 / 2}$ above the graph of $y=x^{1 / 4}$, and for which $x$ -values is it below?
(b) Express your answers in part (a) algebraically using inequalities.

Nick Johnson

Numerade Educator

Problem 27

Figure 7.6 shows that the graph of $y=x^{4}$ is above the graph of $y=x^{2}$ when $x$ is greater than $1,$ and it is below it when $x$ is between 0 and $1 .$ Express these facts algebraically using inequalities.

Nick Johnson

Numerade Educator

Problem 28

When a car's tires are worn, its braking distance increases by $30 \%$. Use this information and Example 1 to find $h(v),$ the braking distance of an Alfa Romeo with worn tires going at a speed of $v$ mph.

Nick Johnson

Numerade Educator

Problem 29

A student takes a part-time job to earn $$\$ 2400$$ for summer travel. The number of hours, $h,$ the student has to work is inversely proportional to the wage, $w$, in dollars per hour, and is given by
$$
h=\frac{2400}{w}
$$
(a) How many hours does the student have to work if the job pays $$\$ 4$$ an hour? What if it pays $$\$ 10$$ an hour?
(b) How do the number of hours change as the wage goes up from $$\$ 4$$ an hour to $$\$ 10$$ an hour? Explain your answer in algebraic and practical terms.
(c) Is the wage, $w$, inversely proportional to the number hours, $h$ ? Express $w$ as a function of $h$.

Nick Johnson

Numerade Educator

Problem 30

If a ball is dropped from a high window, the distance, $D,$ in feet, it falls is proportional to the square of the time, $t,$ in seconds, since it was dropped and is given by
$$
D=16 t^{2}
$$
How far has the ball fallen after three seconds and after five seconds? Which distance is larger? Explain your answer in algebraic terms.

Nick Johnson

Numerade Educator

Problem 31

A city's electricity consumption, $E$, in gigawatt-hours per year, is given by
$$
E=\frac{0.15}{p^{3 / 2}}
$$
where $p$ is the price in dollars per kilowatt-hour charged.
(a) Is $E$ a power function of $p ?$ If so, identify the exponent and the constant of proportionality.
(b) What is the electricity consumption at a price of $\$ 0.16$ per kilowatt-hour? At a price of $\$ 0.25$ per kilowatt hour? Explain the change in electricity consumption in algebraic terms.

Nick Johnson

Numerade Educator

Problem 32

The surface area of a mammal is given by $f(M)=$ $k M^{2 / 3},$ where $M$ is the body mass, and the constant of proportionality $k$ is a positive number that depends on the body shape of the mammal. Is the surface area larger for a mammal of body mass 60 kilograms or for a mammal of body mass 70 kilograms? Explain your answer in algebraic terms.

Nick Johnson

Numerade Educator

Problem 33

The radius, $r$, in $\mathrm{cm}$, of a sphere of volume $V \mathrm{~cm}^{3}$ is approximately $r=0.620 \sqrt[3]{V}$.
(a) Graph the radius function, $r,$ for volumes from 0 to $40 \mathrm{~cm}^{3}$.
(b) Use your graph to estimate the volume of a sphere of radius $2 \mathrm{~cm} .$

Calin Lupas

Numerade Educator

Problem 34

Plot the expressions $x^{2} \cdot x^{3}, x^{5},$ and $x^{6},$ on three separate graphs in the window $-1<x<1,-1<y<1$. Does it appear from the graphs that $x^{2} \cdot x^{3}=x^{5}=$ $x^{2+3}$ or $x^{2} \cdot x^{3}=x^{6}=x^{2 \cdot 3}$ ?

Carson Merrill

Numerade Educator

Problem 35

Plot the expressions $-x^{4}$ and $(-x)^{4}$ on the same graph in the window $-1<x<1,-1<y<1$. Is $-x^{4}=(-x)^{4} ?$

Gregory Higby

Numerade Educator