4.2 Exponential Functions

Introduction

In Exponential Growth and Decay, we studied functions that describe exponential growth or decay. More formally, we define an exponential function as follows.

Some examples of exponential functions are

$f (x) = 5^{x}, P (t) = 250 (1.7)^{t}, and g (n) = 2.4 (0.3)^{n}$

The constant $a$ is the $y$ -intercept of the graph because

$f (0) = a \cdot b^{0} = a \cdot 1 = a$

For the examples above, we find that the $y$ -intercepts are

$\begin{aligned} f (0) & = 5^{0} = 1, \\ P (0) & = 250 (1.7)^{0} = 250, and \\ g (0) & = 2.4 (0.3)^{0} = 2.4 \end{aligned}$

The positive constant $b$ is called the base of the exponential function.

Which of the following is an exponential function?

_____

$f (x) = 3 (4)^{x}$

Which of the following is an exponential function?

$f (x) = 3 x^{4}$
$f (x) = 3 (4)^{x}$
$f (x) = 2 x^{3 / 4}$
$f (x) = \frac{4}{x^{3}}$

Graphs of Exponential Functions

The graphs of exponential functions have two characteristic shapes, depending on whether the base, $b$ , is greater than $1$ or less than $1$ . As typical examples, consider the graphs of $f (x) = 2^{x}$ and $g (x) = {(\frac{1}{2})}^{x}$ shown below. Some values for $f$ and $g$ are recorded in the tables.

Function graph showing y = b^x and y = 1. Adjustable parameter: Base b (b) = 2. Viewing window: x from -5.07 to 5.07, y from -0.63 to 5.63. — The two characteristic shapes are one family: y = bˣ with the base on a slider. For b > 1 the function is increasing — and the larger the base, the faster it climbs; try b = 2 against b = 3. Slide b below 1 and the graph flips to the decreasing shape: the smaller the base, the steeper the decay. Right at b = 1 the “growth” degenerates into the horizontal line y = 1. Two things never change as you drag: every graph passes through (0, 1) (the dashed line marks height 1), because b⁰ = 1 for any base, and the graph never touches the x-axis — a positive base to any power stays positive.

$x$	$f (x)$
$minus 3$	$\frac{1}{8}$
$minus 2$	$\frac{1}{4}$
$minus 1$	$\frac{1}{2}$
$0$	$1$
$1$	$2$
$2$	$4$
$3$	$8$

$x$	$g (x)$
$minus 3$	$8$
$minus 2$	$4$
$minus 1$	$2$
$0$	$1$
$1$	$\frac{1}{2}$
$2$	$\frac{1}{4}$
$3$	$\frac{1}{8}$

increasing and decreasing exponential graphs

Notice that $f (x) = 2^{x}$ is an increasing function and $g (x) = {(\frac{1}{2})}^{x}$ is a decreasing function. Both are concave up. In general, exponential functions have the following properties.

In the table for $f (x)$ , you can see that as the $x$ -values decrease toward negative infinity, the corresponding $y$ -values decrease toward zero. As a result, the graph of $f$ decreases toward the $x$ -axis as we move to the left. Thus, the negative $x$ -axis is a horizontal asymptote for exponential functions with $b greater than 1$ , as shown in figure (a).

For exponential functions with $0 less than b less than 1$ , the positive $x$ -axis is an asymptote, as illustrated in figure (b). (See Some Basic Functions to review asymptotes.)

Which statement is true?

_____

The outputs of an exponential function cannot be negative.

Which statement is true?

An exponential function is not defined for negative inputs.
The outputs of an exponential function cannot be negative.
The $y$ -intercept of the function $f (x) = 2 (3)^{x}$ is $open parenthesis 0, 6 close parenthesis$ .
The function $f (x) = 16 (0.5)^{x}$ decreases by $8$ each time we increase $x$ by $1$ .

In Example, we compare two increasing exponential functions. The larger the value of the base, $b$ , the faster the function grows. In this example, both functions have $a equals 1$ .

Example 1

Compare the graphs of $f (x) = 3^{x}$ and $g (x) = 4^{x}$ .

We evaluate each function for several convenient values, as shown in the table.

Then we plot the points for each function and connect them with smooth curves. For positive $x$ -values, $g (x)$ is always larger than $f (x)$ , and is increasing more rapidly. In the figure, we can see that $g (x) = 4^{x}$ climbs more rapidly than $f (x) = 3^{x}$ . Both graphs cross the $y$ -axis at (0, 1).

$x$	$f (x)$	$g (x)$
$minus 2$	$\frac{1}{9}$	$\frac{1}{16}$
$minus 1$	$\frac{1}{3}$	$\frac{1}{4}$
$0$	$1$	$1$
$1$	$3$	$4$
$2$	$9$	$16$

State the ranges of the functions $f$ and $g$ from the previous Example on the domain $[ minus 2, 2 ]$ .
$f :$ _____
$g :$ _____
State the ranges of the functions $p$ and $q$ shown in the Note above on the domain $[ minus 2, 2 ]$ . Round your answers to two decimal places.
$p :$ _____
$q :$ _____

$f : [\frac{1}{9}, 9]$ ; $g : [\frac{1}{16}, 16]$
$p: [ 0.64, 1.56 ]$ ; $q: [ 0.25, 4 ]$

State the ranges of the functions $f$ and $g$ from the previous Example on the domain $[ minus 2, 2 ]$ .
State the ranges of the functions $p$ and $q$ shown in the Note above on the domain $[ minus 2, 2 ]$ . Round your answers to two decimal places.

$f : [\frac{1}{9}, 9]$ ; $g : [\frac{1}{16}, 16]$
$p: [ 0.64, 1.56 ]$ ; $q: [ 0.25, 4 ]$

Transformations of Exponential Functions

In Modeling with Functions, we considered transformations of the basic graphs. For instance, the graphs of the functions $y = x^{2} - 4$ and $y = (x - 4)^{2}$ are shifts of the basic parabola, $y = x^{2}$ . In a similar way, we can shift or stretch the graph of an exponential function while the basic shape is preserved.

Example 2

Use your calculator to graph the following functions. Describe how these graphs compare with the graph of $h (x) = 2^{x}$ .

$f (x) = 2^{x} + 3$
$g (x) = 2^{x + 3}$

Enter the formulas for the three functions as shown below. Note the parentheses around the exponent in the keying sequence for $Y sub 3 equals g open parenthesis x close parenthesis.$

$Y sub 1 equals 2$ ^ $X$

$Y sub 2 equals 2$ ^ $X$ + 3

$Y sub 3 equals 2$ ^ ( $X$ + 3 )

The graphs of $h (x) = 2^{x}$ , $f (x) = 2^{x} + 3$ , and $g (x) = 2^{x + 3}$ in the standard window are shown below.

The graph of $f (x) = 2^{x} + 3$ , shown in figure (b), has the same basic shape as that of $h (x) = 2^{x}$ , but it has a horizontal asymptote at $y equals 3$ instead of at $y equals 0$ (the $x$ -axis). In fact, $f (x) = h (x) + 3$ , so the graph of $f$ is a vertical translation of the graph of $h$ by $3$ units. If every point on the graph of $h (x) = 2^{x}$ is moved $3$ units upward, the result is the graph of $f (x) = 2^{x} + 3$ .
First note that $g (x) = 2^{x} + 3 = h (x + 3)$ . In fact, the graph of $g (x) = 2^{x + 3}$ shown in figure (c) has the same basic shape as $h (x) = 2^{x}$ but has been translated $3$ units to the left.

Which function translates the graph of $y = 8^{x}$ two units to the right?

_____

$f (x) = 8^{x - 2}$

Which function translates the graph of $y = 8^{x}$ two units to the right?

$f (x) = 8^{x - 2}$
$f (x) = 8^{2 x}$
$f (x) = 2 + (8)^{x}$
$f (x) = 8^{x} - 2$

What about reflections? Recall that the graph of $y = - f (x)$ is the reflection about the $x$ -axis of the graph of $y = f (x)$ . The graphs of $y = 2^{x}$ and $y = - 2^{x}$ are shown at left below.

vertical reflection of an exponential function

horizontal reflection of an exponential function

You may have also noticed a relationship between the graphs of $f (x) = 2^{x}$ and $g (x) = {(\frac{1}{2})}^{x}$ , which are shown at right above. The graph of $g$ is the reflection of the graph of $f$ about the $y$ -axis. We can see why this is true by writing the formula for $g (x)$ in another way:

$g (x) = {(\frac{1}{2})}^{x} = {(2^{- 1})}^{x} = 2^{- x}$

We see that $g (x)$ is the same function as $f (- x)$ . Replacing $x$ by $- x$ in the formula for a function switches every point $(p, q)$ on the graph with the point $(- p, q)$ and thus reflects the graph about the $y$ -axis.

Which of the functions below have the same graph? Explain why.

_____

$f (x) = {(\frac{1}{4})}^{x}$
$g (x) = - 4^{x}$
$h (x) = 4^{- x}$

(a) and (c) are the same function.

Which of the functions below have the same graph? Explain why.

$f (x) = {(\frac{1}{4})}^{x}$
$g (x) = - 4^{x}$
$h (x) = 4^{- x}$

(a) and (c) are the same function.

How are the graphs of $f (x) = b^{x}$ and $g (x) = {(\frac{1}{b})}^{x}$ related?

_____

How are the graphs of $f (x) = b^{x}$ and $g (x) = {(\frac{1}{b})}^{x}$ related?

Comparing Exponential and Power Functions

Exponential functions are not the same as the power functions we studied in Power Functions. Although both involve expressions with exponents, it is the location of the variable that makes the difference.

Note

Power Functions vs Exponential Functions

	Power Functions	Exponential Functions
General formula	$h (x) = k x^{p}$	$f (x) = a b^{x}$
Description	variable base and constant exponent	constant base and variable exponent
Example	$h open parenthesis x close parenthesis equals 2 x cubed$	$f (x) = 2 (3^{x})$

These two families of functions have very different properties, as well.

Example 3

Compare the power function $h open parenthesis x close parenthesis equals 2 x cubed$ and the exponential function $f (x) = 2 (3^{x})$ .

First, compare the values for these two functions shown in the table.

The scaling exponent for $h (x)$ is $3$ , so that when $x$ doubles, say, from $1$ to $2$ , the output is multiplied by $2 cubed$ , or $8$ .

On the other hand, we can tell that $f$ is exponential because its values increase by a factor of $3$ for each unit increase in $x$ . (To see this, divide any function value by the previous one.)

$x$	$h open parenthesis x close parenthesis equals 2 x cubed$	$f (x) = 2 (3^{x})$
$minus 3$	$minus 54$	$\frac{2}{27}$
$minus 2$	$minus 16$	$\frac{1}{4}$
$minus 1$	$minus 2$	$\frac{2}{3}$
$0$	$0$	$2$
$1$	$2$	$6$
$2$	$16$	$18$
$3$	$54$	$54$

As you would expect, the graphs of the two functions are also quite different. For starters, note that the power function goes through the origin, while the exponential function has $y$ -intercept $open parenthesis 0, 2 close parenthesis$ as shown at left below.

From the table, we see that $h (3) = f (3) = 54$ , so the two graphs intersect at $x equals 3$ . (They also intersect at approximately $x equals 2.48$ .) However, if you compare the values of $h open parenthesis x close parenthesis equals 2 x cubed$ and $f (x) = 2 (3^{x})$ for larger values of $x$ , you will see that eventually the exponential function overtakes the power function, as shown at right above.

power function vs exponential from 0 to 50

The relationship in Example holds true for all increasing power and exponential functions: For large enough values of $x$ , the exponential function will always be greater than the power function, regardless of the parameters in the functions. The figure at left shows the graphs of $f (x) = x^{6}$ and $g (x) = {1.8}^{x}$ . At first, $f (x) > g (x)$ , but at around $x equals 37$ , $g (x)$ overtakes $f (x)$ , and $g (x) > f (x)$ for all $x greater than 37$ .

Which function grows fastest in the long run?

_____

$f (x) = 2 (3^{x})$

Which function grows fastest in the long run?

$f open parenthesis x close parenthesis equals 5 x squared$
$f (x) = 3 x^{5}$
$f (x) = 5 (2^{x})$
$f (x) = 2 (3^{x})$

Which of the following functions are exponential functions, and which are power functions?

$F (x) = {1.5}^{x}$ _____
$G (x) = 3 x^{1.5}$ _____
$H (x) = 3^{1.5 x}$ _____
$K (x) = (3 x)^{1.5}$ _____

Exponential: (a) and (c); power: (b) and (d)

Which of the following functions are exponential functions, and which are power functions?

$F (x) = {1.5}^{x}$
$G (x) = 3 x^{1.5}$
$H (x) = 3^{1.5 x}$
$K (x) = (3 x)^{1.5}$

Exponential: (a) and (c); power: (b) and (d)

Discuss the differences between a power function and an exponential function.

_____

Discuss the differences between a power function and an exponential function.

Exponential Equations

An exponential equation is one in which the variable is part of an exponent. For example, the equation

$3^{x} = 81$

is exponential.

Many exponential equations can be solved by writing both sides of the equation as powers with the same base. To solve the equation above, we write

$3^{x} = 3^{4}$

which is true if and only if $x equals 4$ .

In general, if two equivalent powers have the same base, then their exponents must be equal also, as long as the base is not $0$ or $\pm 1$ .

Sometimes the laws of exponents can be used to express both sides of an equation as single powers of a common base.

Example 4

Solve the following equations.

$3^{x - 2} = 9^{3}$
$27 \cdot 3^{- 2 x} = 9^{x + 1}$

Using the fact that $9 equals 3 squared$ , we write each side of the equation as a power of $3$ :
$\begin{aligned} 3^{x - 2} & = {(3^{2})}^{3} \\ 3^{x - 2} & = 3^{6} \end{aligned}$
Now we equate the exponents to obtain
$\begin{aligned} x - 2 & = 6 \\ x & = 8 \end{aligned}$
We write each factor as a power of $3$ .
$3^{3} \cdot 3^{- 2 x} = {(3^{2})}^{x + 1}$
We use the laws of exponents to simplify each side:
$3^{3 - 2 x} = 3^{2 x + 2}$
Now we equate the exponents to obtain
$\begin{aligned} 3 - 2 x & = 2 x + 2 \\ - 4 x = & - 1 \end{aligned}$
The solution is $x = \frac{1}{4}$ .

Which is a good strategy for solving $3^{x - 2} = 81$ ?

_____

Write the right side as a power of 3.

Which is a good strategy for solving $3^{x - 2} = 81$ ?

Divide both sides by 3.
Add $3 squared$ to both sides.
Simplify the left side.
Write the right side as a power of 3.

Solve the equation $2^{x + 2} = 128$ .

$x =$ _____

$Write each side as a power of 2.$

$Equate exponents.$

$x equals 5$

Solve the equation $2^{x + 2} = 128$ .

Write each side as a power of 2, then equate exponents to find $x equals 5$

Exponential equations arise frequently in the study of exponential growth.

Example 5

During the summer a population of fleas doubles in number every $5$ days. If a population starts with $10$ fleas, how long will it be before there are $10, 240$ fleas?

Let $P$ represent the number of fleas present after $t$ days. The original population of $10$ is multiplied by a factor of $2$ every $5$ days, or

$P (t) = 10 \cdot 2^{t / 5}$

We set $P equals 10, 240$ and solve for $t$ :

$\begin{aligned} 10, 240 & = 10 \cdot 2^{t / 5} & Divide both sides by 10. \\ 1024 & = 2^{t / 5} & Write 1024 as a power of 2. \\ 2^{10} & = 2^{t / 5} \end{aligned}$

We equate the exponents to get $10 = \frac{t}{5}$ , or $t equals 50$ . The population will grow to $10, 240$ fleas in $50$ days.

During an advertising campaign in a large city, the makers of Chip-O's corn chips estimate that the number of people who have heard of Chip-O's increases by a factor of $8$ every 4 days.

If 100 people are given trial bags of Chip-O's to start the campaign, write a function, $N (t)$ , for the number of people who have heard of Chip-O's after $t$ days of advertising.
$N (t) =$ _____
Use your calculator to graph the function $N (t)$ on the domain $0 less than or equal to t less than or equal to 15$ .
How many days should the makers run the campaign in order for Chip-O's to be familiar to $51, 200$ people? Use algebraic methods to find your answer and verify on your graph.
Answer: _____ days

$N (t) = 100 \cdot 8^{t / 4}$
A graph is below.
12 days

A graph for part (b):

During an advertising campaign in a large city, the makers of Chip-O's corn chips estimate that the number of people who have heard of Chip-O's increases by a factor of $8$ every 4 days.

If 100 people are given trial bags of Chip-O's to start the campaign, write a function, $N (t)$ , for the number of people who have heard of Chip-O's after $t$ days of advertising.
Use your calculator to graph the function $N (t)$ on the domain $0 less than or equal to t less than or equal to 15$ .
How many days should the makers run the campaign in order for Chip-O's to be familiar to $51, 200$ people? Use algebraic methods to find your answer and verify on your graph.

$N (t) = 100 \cdot 8^{t / 4}$
12 days

Suppose $g$ is an exponential function, with $g open parenthesis 0 close parenthesis equals 48$ and $g open parenthesis 1 close parenthesis equals 36$ . What is $g open parenthesis 2 close parenthesis$ ?

_____

$24$
$27$
$12$
$18$

Technology

Graphical Solution of Exponential Equations

It is not always so easy to express both sides of the equation as powers of the same base. In the following sections, we will develop more general methods for finding exact solutions to exponential equations. But we can use a graphing utility to obtain approximate solutions.

Example 6

Use the graph of $y = 2^{x}$ to find an approximate solution to the equation $2^{x} = 5$ accurate to the nearest hundredth.

Enter $Y sub 1 equals 2$ ^ X and use the standard graphing window (ZOOM 6) to obtain the graph shown in figure (a). We are looking for a point on this graph with $y$ -coordinate $5$ .

Using the TRACE feature, we see that the $y$ -coordinates are too small when $x less than 2.1$ and too large when $x greater than 2.4$ . The solution we want lies somewhere between $x equals 2.1$ and $x equals 2.4$ , but this approximation is not accurate enough.

To improve our approximation, we will use the intersect feature. Set $Y sub 2 equals 5$ and press GRAPH. The $x$ -coordinate of the intersection point of the two graphs is the solution of the equation $2^{x} = 5$ Activating the intersect command results in figure (b), and we see that, to the nearest hundredth, the solution is $2.32$ .

We can verify that our estimate is reasonable by substituting into the equation:

$2^{2.32} \overset{?}{=} 5$

We enter 2 ^ 2.32 ENTER to get $4.993322196$ . This number is not equal to $5$ , but it is close, so we believe that $x equals 2.32$ is a reasonable approximation to the solution of the equation $2^{x} = 5$ .

Use the graph of $y = 5^{x}$ to find an approximate solution to $5^{x} = 285$ , accurate to two decimal places.

Answer: $x \approx$ _____

The point on the graph where $y equals 285$ has $x approximately equals 3.51$

Use the graph of $y = 5^{x}$ to find an approximate solution to $5^{x} = 285$ , accurate to two decimal places.

The point on the graph where $y equals 285$ has $x approximately equals 3.51$

Give an example of an exponential equation, and describe how to solve it.

_____

Give an example of an exponential equation, and describe how to solve it.

Section Summary

Vocabulary

Look up the definitions of new terms in the Glossary.

Exponential function
Base
Exponential equation

CONCEPTS

An exponential function has the form
$f (x) = a b^{x}, where b > 0 and b \neq 1, a \neq 0$
Quantities that increase or decrease by a constant percent in each time period grow or decay exponentially.
Note
Properties of Exponential Functions $f (x) = a b^{x}, a > 0$
1. Domain: all real numbers.
2. Range: all positive numbers.
3. If $b greater than 1$ , the function is increasing and concave up; if $0 less than b less than 1$ , the function is decreasing and concave up.
4. The $y$ -intercept is $open parenthesis 0, a close parenthesis$ . There is no $x$ -intercept.
The graphs of exponential functions can be transformed by shifts, stretches, and reflections.
Note
Reflections of Graphs
1. The graph of $y = - f (x)$ is the reflection of the graph of $y = f (x)$ about the $x$ -axis.
2. The graph of $y = f (- x)$ is the reflection of the graph of $y = f (x)$ about the $y$ -axis.
Exponential functions $f (x) = a b^{x}$ have different properties than power functions $f (x) = k x^{p}$ .
We can solve some exponential equations by writing both sides with the same base and equating the exponents.
We can use graphs to find approximate solutions to exponential equations.

STUDY QUESTIONS

Give the general form for an exponential function. What restrictions do we place on the base of the function?
Explain why the output of an exponential function $f (x) = b^{x}$ is always positive, even if $x$ is negative.
How are the graphs of the functions $f (x) = b^{x}$ and $g (x) = {(\frac{1}{b})}^{x}$ related?
How is an exponential function different from a power function?
Delbert says that $8 {(\frac{1}{2})}^{x}$ is equivalent to $4 to the power x$ . Convince him that he is mistaken.
Explain the algebraic technique for solving exponential equations described in this section.

SKILLS

Practice each skill in the Homework problems listed.

Describe the graph of an exponential function: #1–14
Graph transformations of exponential functions: #15–18, 53–60
Evaluate exponential functions: #19–22
Find the equation of an exponential function from its graph: #23–26
Solve exponential equations: #27–44
Distinguish between power and exponential functions: #45–52, 65, and 66

Homework 4.2

For Problems 1 and 2, find the $y$ -intercept of each exponential function and decide whether the graph is increasing or decreasing.

$f (x) = 26 (1.4)^{x}$
$g (x) = 1.2 (0.84)^{x}$
$h (x) = 75 {(\frac{4}{5})}^{x}$
$k (x) = \frac{2}{3} {(\frac{9}{8})}^{x}$

$26$ ; increasing
$1.2$ ; decreasing
$75$ ; decreasing
$\frac{2}{3}$ ; increasing

$M (x) = 1.5 (0.05)^{x}$
$N (x) = 0.05 (1.05)^{x}$
$P (x) = {(\frac{5}{8})}^{x}$
$Q (x) = {(\frac{4}{3})}^{x}$

For Problems 3–6, make a table of values and graph each pair of functions by hand on the domain $[ minus 3, 3 ]$ . Describe the similarities and differences between the two graphs.

$f (x) = 3^{x}$
$g (x) = {(\frac{1}{3})}^{x}$

$x$	$minus 3$	$minus 2$	$minus 1$	$0$	$1$	$2$	$3$
$f (x) = 3^{x}$	$\frac{1}{27}$	$\frac{1}{9}$	$\frac{1}{3}$	$1$	$3$	$9$	$27$
$g (x) = {(\frac{1}{3})}^{x}$	$27$	$9$	$3$	$1$	$\frac{1}{3}$	$\frac{1}{9}$	$\frac{1}{27}$

The two graphs are reflections of each other across the $y$ -axis. $f$ is increasing, $g$ is decreasing. $f$ has the negative $x$ -axis as an asymptote, and $g$ has the positive $x$ -axis as its asymptote.

$F (x) = {(\frac{1}{10})}^{x}$
$G (x) = 10^{x}$

$h (t) = 4^{- t}$
$q (t) = - 4^{t}$

$t$	$minus 3$	$minus 2$	$minus 1$	$0$	$1$	$2$	$3$
$h (t) = 4^{- t}$	$64$	$16$	$4$	$1$	$\frac{1}{4}$	$\frac{1}{16}$	$\frac{1}{64}$
$q (t) = - 4^{t}$	$\frac{- 1}{64}$	$\frac{- 1}{16}$	$\frac{- 1}{4}$	$minus 1$	$minus 4$	$minus 16$	$minus 64$

exponential decay and negative of growth

The graphs are reflections of each other across the origin. Both are decreasing, but $h$ has the negative $t$ -axis as an asymptote, and $q$ has the positive t-axis as its asymptote.

$P (t) = - 5^{t}$
$R (t) = 5^{- t}$

For Problems 7–12, match each function with its graph.

$f (x) = 3 (2^{x})$
$f (x) = 3 {(\frac{1}{2})}^{x}$
$f (x) = 3 {(\frac{1}{3})}^{x}$
$f (x) = 3 (3^{x})$

$g (x) = 2 ({1.5}^{x})$
$g (x) = 2 {(1.25)}^{x}$
$g (x) = 2 {(0.75)}^{x}$
$g (x) = 2 (0.25)^{x}$

For Problems 9–12,

Use a graphing calculator to graph the functions on the domain $[ minus 5, 5 ]$ .
Give the range of the function on that domain, accurate to hundredths.

$g (t) = 4 ({1.3}^{t})$

$[ 1.08, 14.85 ]$

$h (t) = 3 ({2.4}^{t})$

$N (x) = 50 ({0.8}^{x})$

$[ 16.38, 152.59 ]$

$P (x) = 80 ({0.7}^{x})$

For Problems 13 and 14, in each group of functions, which have identical graphs? Explain why.

$h (x) = 6^{x}$
$k (x) = {(\frac{1}{6})}^{x}$
$m (x) = 6^{- x}$
$n (x) = \frac{1}{6^{x}}$

Because they are defined by equivalent expressions, (b), (c), and (d) have identical graphs

$Q (t) = 5^{t}$
$R (t) = {(\frac{1}{5})}^{t}$
$F (t) = {(\frac{1}{5})}^{- t}$
$G (t) = \frac{1}{5^{- t}}$

For Problems 15–18,

Use the order of operations to explain why the two functions are different.
Complete the table of values and graph both functions in the same window.
Describe each as a transformation of $y = 2^{x}$ or $y = 3^{x}$ .

$f (x) = 2^{x - 1}$ , $g (x) = 2^{x} - 1$

$x$	$y = 2^{x}$	$f (x)$	$g (x)$
$minus 2$
$minus 1$
$0$
$1$
$2$

To evaluate $f$ we subtract $1$ from the input before evaluating the exponential function; to evaluate $g$ we subtract $1$ from the output of the exponential function.

$x$	$y = 2^{x}$	$f (x)$	$g (x)$
$minus 2$	$\frac{1}{4}$	$\frac{1}{8}$	$\frac{- 3}{4}$
$minus 1$	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{- 1}{2}$
$0$	$1$	$\frac{1}{2}$	$0$
$1$	$2$	$1$	$1$
$2$	$4$	$2$	$3$

The graph of $f$ is translated $1$ unit to the right; the graph of $g$ is shifted $1$ unit down.

$f (x) = 3^{x} + 2$ , $g (x) = 3^{x + 2}$

$x$	$y = 3^{x}$	$f (x)$	$g (x)$
$minus 2$
$minus 1$
$0$
$1$
$2$

$f (x) = - 3^{x}$ , $g (x) = 3^{- x}$

$x$	$y = 3^{x}$	$f (x)$	$g (x)$
$minus 2$
$minus 1$
$0$
$1$
$2$

To evaluate $f$ we take the negative of the output of the exponential function; to evaluate $g$ we take the negative of the input.

$x$	$y = 3^{x}$	$f (x)$	$g (x)$
$minus 2$	$\frac{1}{9}$	$\frac{- 1}{9}$	$9$
$minus 1$	$\frac{1}{3}$	$\frac{- 1}{3}$	$3$
$0$	$1$	$minus 1$	$1$
$1$	$3$	$minus 3$	$\frac{1}{3}$
$2$	$9$	$minus 9$	$\frac{1}{9}$

The graph of $f$ is reflected about the $x$ -axis; the graph of $g$ is reflected about the $y$ -axis.

$f (x) = 2^{- x}$ , $g (x) = - 2^{x}$

$x$	$y = 2^{x}$	$f (x)$	$g (x)$
$minus 2$
$minus 1$
$0$
$1$
$2$

In Problems 19–22, for the given function, evaluate each pair of expressions. Are they equivalent?

$f (x) = 3 (5^{x})$

$f open parenthesis a plus 2 close parenthesis$ and $9 f open parenthesis a close parenthesis$
$f open parenthesis 2 a close parenthesis$ and $2 f open parenthesis a close parenthesis$

$3 (5^{a + 2})$ is not equivalent to $9 \cdot 3 (5^{a})$ .
$3 open parenthesis 5 to the power 2 a close parenthesis$ is not equivalent to $2 \cdot 3 (5^{a})$ .

$g (x) = {1.8}^{x}$

$g open parenthesis h plus 3 close parenthesis$ and $g open parenthesis h close parenthesis g open parenthesis 3 close parenthesis$
$g open parenthesis 2 h close parenthesis$ and $[ g open parenthesis h close parenthesis ] squared$

$P (t) = 8^{t}$

$P (w) - P (z)$ and $P (w - z)$
$P (- x)$ and $\frac{1}{P (x)}$

$8^{w} - 8^{z}$ is not equivalent to $8^{w - z}$ .
$8^{- x}$ is equivalent to $\frac{1}{8^{x}}$ .

$Q (t) = 5 (0.2)^{t}$

$Q open parenthesis b minus 1 close parenthesis$ and $5 Q open parenthesis b close parenthesis$
$Q (a) Q (b)$ and $5 Q open parenthesis a plus b close parenthesis$

The graph of $f (x) = P_{0} b^{x}$ is shown in the figure.

Read the value of $P sub 0$ from the graph.
Make a short table of values for the function by reading values from the graph. Does your table confirm that the function is exponential?
Use your table to calculate the growth factor, $b$ .
Using your answers to parts (a) and (c), write a formula for $f (x)$ .

$P sub 0 equals 300$
$x$ $0$ $1$ $2$
$f (x)$ $300$ $600$ $1200$
$b equals 2$
$f (x) = 300 (2)^{x}$

$x$	$0$	$1$	$2$
$f (x)$	$300$	$600$	$1200$

The graph of $g (x) = P_{0} b^{x}$ is shown in the figure.

Read the value of $P sub 0$ from the graph.
Make a short table of values for the function by reading values from the graph. Does your table confirm that the function is exponential?
Use your table to calculate the decay factor, $b$ .
Using your answers to parts (a) and (c), write a formula for $g (x)$ .

For several days after the Northridge earthquake on January 17, 1994, the area received a number of significant aftershocks. The red graph shows that the number of aftershocks decreased exponentially over time. The graph of the function $S (d) = S_{0} b^{d}$ , shown in black, approximates the data. (Source: Los Angeles Times, June 27, 1995)

Read the value of $S sub 0$ from the graph.
Find an approximation for the decay factor, $b$ , by comparing two points on the graph. (Some of the points on the graph of $S (d)$ are approximately $open parenthesis 1, 82 close parenthesis$ , $open parenthesis 2, 45 close parenthesis$ , $open parenthesis 3, 25 close parenthesis$ , and $open parenthesis 4, 14 close parenthesis$ .)
Using your answers to (a) and (b), write a formula for $S (d)$ .

$S sub 0 equals 150$
$b approximately equals 0.55$
$S (d) = 150 (0.55)^{d}$

The frequency of a musical note depends on its pitch. The graph shows that the frequency increases exponentially. The function $F (p) = F_{0} b^{p}$ gives the frequency as a function of the number of half-tones, $p$ , above the starting point on the scale.

Read the value of $F sub 0$ from the graph. (This is the frequency of the note A above middle C.)
Find an approximation for the growth factor, $b$ , by comparing two points on the graph. (Some of the points on the graph of $F (p)$ are approximately $open parenthesis 1, 466 close parenthesis$ , $open parenthesis 2, 494 close parenthesis$ , $open parenthesis 3, 523 close parenthesis$ , and $open parenthesis 4, 554 close parenthesis$ .)
Using your answers to (a) and (b), write a formula for $F (p)$ .
The frequency doubles when you raise a note by one octave, which is equivalent to $12$ half-tones. Use this information to find an exact value for $b$ .

Solve the equation algebraically.

$5^{x + 2} = 25^{4 / 3}$

$\frac{2}{3}$

$3^{x - 1} = 27^{1 / 2}$

$3^{2 x - 1} = \frac{\sqrt{3}}{9}$

$\frac{- 1}{4}$

$2^{3 x - 1} = \frac{\sqrt{2}}{16}$

$4 \cdot 2^{x - 3} = 8^{- 2 x}$

$\frac{1}{7}$

$9 \cdot 3^{x + 2} = 81^{- x}$

$27^{4 x + 2} = 81^{x - 1}$

$\frac{- 5}{4}$

$16^{2 - 3 x} = 64^{x + 5}$

$10^{x^{2} - 1} = 1000$

$\pm 2$

$5^{x^{2} - x - 4} = 25$

Before the advent of antibiotics, an outbreak of cholera might spread through a city so that the number of cases doubled every $6$ days.

Twenty-six cases were discovered on July 5. Write a function for the number of cases of cholera $t$ days later.
Use your calculator to graph your function on the interval $0 less than or equal to t less than or equal to 90$ .
When should hospitals expect to be treating $106, 496$ cases? Use algebraic methods to find your answer, and verify it on your graph.

$N (t) = 26 (2)^{t / 6}$
$72$ days later

An outbreak of ungulate fever can sweep through the livestock in a region so that the number of animals affected triples every $4$ days.

A rancher discovers $4$ cases of ungulate fever among his herd. Write a function for the number of cases of ungulate fever $t$ days later.
Use your calculator to graph your function on the interval $0 less than or equal to t less than or equal to 20$ .
If the rancher does not act quickly, how long will it be until $324$ head are affected? Use algebraic methods to find your answer, and verify it on your graph.

A smart television set loses $30 %$ of its value every $2$ years.

Write a function for the value of a television set $t$ years after it was purchased if it cost $$ 700$ originally.
Use your calculator to graph your function on the interval $0 less than or equal to t less than or equal to 20$ .
How long will it be before a $$ 700$ television set depreciates to $$ 343$ ? Use algebraic methods to find your answer, and verify it on your graph.

$V (t) = 700 (0.7)^{t / 2}$
$4$ yr

A mobile home loses $20 %$ of its value every $3$ years.

A certain mobile home costs $$ 20, 000$ . Write a function for its value after $t$ years.
Use your calculator to graph your function on the interval $0 less than or equal to t less than or equal to 30$ .
How long will it be before a $$ 20, 000$ mobile home depreciates to $$ 12, 800$ ? Use algebraic methods to find your answer, and verify it on your graph.

For Problems 41–44, use a graph to find an approximate solution accurate to the nearest hundredth.

$3^{x - 1} = 4$

$x equals 2.26$

$2^{x + 3} = 5$

$4^{- x} = 7$

$x = - 1.40$

$6^{- x} = 3$

For Problems 45 and 46, decide whether each function is an exponential function, a power function, or neither.

$g (t) = 3 t^{0.4}$
$h (t) = 4 (0.3)^{t}$
$D (x) = 6 x^{1 / 2}$
$E (x) = 4 x + x^{4}$

Power
Exponential
Power
Neither

$R (w) = 5 (5)^{w - 1}$
$Q (w) = 2^{w} - w^{2}$
$M (z) = 0.2 z^{1.3}$
$N (z) = z^{- 3}$

For Problems 47–50, decide whether the table could describe a linear function, a power function, an exponential function, or none of these. Find a formula for each linear, power, or exponential function.

$x$ $y$
$0$ $3$
$1$ $6$
$2$ $12$
$3$ $24$
$4$ $48$
$t$ $P$
$0$ $0$
$1$ $0.5$
$2$ $2$
$3$ $4.5$
$4$ $8$

$x$	$y$
$0$	$3$
$1$	$6$
$2$	$12$
$3$	$24$
$4$	$48$

$t$	$P$
$0$	$0$
$1$	$0.5$
$2$	$2$
$3$	$4.5$
$4$	$8$

Exponential $y = 3 \cdot 2^{x}$
Power $P equals 0.5 t squared$

$x$ $N$
$0$ $0$
$1$ $2$
$2$ $16$
$3$ $54$
$4$ $128$
$p$ $R$
$0$ $405$
$1$ $135$
$2$ $45$
$3$ $15$
$4$ $5$

$x$	$N$
$0$	$0$
$1$	$2$
$2$	$16$
$3$	$54$
$4$	$128$

$p$	$R$
$0$	$405$
$1$	$135$
$2$	$45$
$3$	$15$
$4$	$5$

$t$ $y$
$1$ $100$
$2$ $50$
$3$ $33 \frac{1}{3}$
$4$ $25$
$5$ $20$
$x$ $P$
$1$ $\frac{1}{2}$
$2$ $1$
$3$ $2$
$4$ $4$
$5$ $8$

$t$	$y$
$1$	$100$
$2$	$50$
$3$	$33 \frac{1}{3}$
$4$	$25$
$5$	$20$

$x$	$P$
$1$	$\frac{1}{2}$
$2$	$1$
$3$	$2$
$4$	$4$
$5$	$8$

Power $y = 100 x^{- 1}$
Exponential $P = \frac{1}{4} \cdot 2^{x}$

$h$ $a$
$0$ $70$
$1$ $7$
$2$ $0.7$
$3$ $0.07$
$4$ $0.007$
$t$ $Q$
$0$ $0$
$1$ $\frac{1}{4}$
$2$ $1$
$3$ $\frac{9}{4}$
$4$ $4$

$h$	$a$
$0$	$70$
$1$	$7$
$2$	$0.7$
$3$	$0.07$
$4$	$0.007$

$t$	$Q$
$0$	$0$
$1$	$\frac{1}{4}$
$2$	$1$
$3$	$\frac{9}{4}$
$4$	$4$

For Problems 51 and 52, fill in the tables. Graph each pair of functions in the same window. Then answer the questions below.

Give the range of $f$ and the range of $g$ .
For how many values of $x$ does $f (x) = g (x)$ ?
Estimate the value(s) of $x$ for which $f (x) = g (x)$ .
For what values of $x$ is $f (x) < g (x)$ ?
Which function grows more rapidly for large values of $x$ ?

$x$	$f (x) = x^{2}$	$g (x) = 2^{x}$
$minus 2$
$minus 1$
$0$
$1$
$2$
$3$
$4$
$5$

$x$	$f (x) = x^{2}$	$g (x) = 2^{x}$
$minus 2$	$4$	$\frac{1}{4}$
$minus 1$	$1$	$\frac{1}{2}$
$0$	$0$	1
$1$	$1$	$2$
$2$	$4$	$4$
$3$	$9$	$8$
$4$	$16$	$16$
$5$	$25$	$32$

Range of $f$ : $[ 0, ∞ close parenthesis$ ; Range of $g$ : $open parenthesis 0, ∞ close parenthesis$
$3$
$minus 0.7667$ , $2$ , $4$
$open parenthesis minus 0.7667, 2 close parenthesis$ and $open parenthesis 4, ∞ close parenthesis$
g

$x$	$f (x) = x^{3}$	$g (x) = 3^{x}$
$minus 2$
$minus 1$
$0$
$1$
$2$
$3$
$4$
$5$

For Problems 53–60, sketch the graph of each transformation of the given function, then write a formula and check your sketch with a graphing calculator. State the domain and range of each transformation, its intercept(s), and any asymptotes.

$f (x) = 3^{x}$

$y = f (x) - 4$
$y = f (x - 4)$
$y = - 4 f (x)$

$y = 3^{x} - 4$

Domain: $open parenthesis minus ∞, ∞ close parenthesis$ ; range: $open parenthesis minus 4, ∞ close parenthesis$ , $x$ -intercept $open parenthesis 1.26, 0 close parenthesis$ ; $y$ -intercept $open parenthesis 0, minus 3 close parenthesis$ ; horizontal asymptote $y = - 4$
$y = 3^{x - 4}$ ,

Domain: $open parenthesis minus ∞, ∞ close parenthesis$ ; range: $open parenthesis 0, ∞ close parenthesis$ , no $x$ -intercept; $y$ -intercept $(0, \frac{1}{81})$ ; the $x$ -axis is the horizontal asymptote.
$y = - 4 \cdot 3^{x}$ ,

Domain: $open parenthesis minus ∞, ∞ close parenthesis$ ; range: $open parenthesis minus ∞, 0 close parenthesis$ , no $x$ -intercept; $y$ -intercept $open parenthesis 0, minus 4 close parenthesis$ ; the $x$ -axis is the horizontal asymptote.

$g (x) = 4^{x}$

$y = g (x) + 2$
$y = g (x + 2)$
$y equals 2 g open parenthesis x close parenthesis$

$h (t) = 6^{t}$

$y = - h (t)$
$y = h (- t)$
$y = - h (- t)$

$y = - 6^{t}$

Domain: $open parenthesis minus ∞, ∞ close parenthesis$ ; range: $open parenthesis minus ∞, 0 close parenthesis$ , no $t$ -intercept; $y$ -intercept $open parenthesis 0, minus 1 close parenthesis$ ; the $t$ -axis is the horizontal asymptote.
$y = 6^{- t}$ ,

Domain: $open parenthesis minus ∞, ∞ close parenthesis$ ; range: $open parenthesis 0, ∞ close parenthesis$ , no $t$ -intercept; $y$ -intercept $open parenthesis 0, 1 close parenthesis$ ; the $t$ -axis is the horizontal asymptote.
$y = - 6^{- t}$ ,

Domain: $open parenthesis minus ∞, ∞ close parenthesis$ ; range: $open parenthesis minus ∞, 0 close parenthesis$ , no $t$ -intercept; $y$ -intercept $open parenthesis 0, minus 1 close parenthesis$ ; the $t$ -axis is the horizontal asymptote.

$j (t) = {(\frac{1}{3})}^{t}$

$y = j (- t)$
$y = - j (t)$
$y = - j (- t)$

$g (x) = 2^{x}$

$y = g (x - 3)$
$y = g (x - 3) + 4$

$y = 2^{x - 3}$

Domain: $open parenthesis minus ∞, ∞ close parenthesis$ ; range: $open parenthesis 0, ∞ close parenthesis$ , no $x$ -intercept; $y$ -intercept $(0, \frac{1}{8})$ ; the $x$ -axis is the horizontal asymptote.
$y = 2^{x - 3} + 4$ ,

Domain: $open parenthesis minus ∞, ∞ close parenthesis$ ; range: $open parenthesis 4, ∞ close parenthesis$ , no $x$ -intercept; $y$ -intercept $(0, \frac{33}{8})$ ; horizontal asymptote $y equals 4$

$f (x) = 10^{x}$

$y = f (x + 5)$
$y = f (x + 5) - 20$

$N (t) = {(\frac{1}{2})}^{t}$

$y = - N (t)$
$y = 6 - N (t)$

$y = - {(\frac{1}{2})}^{t}$

Domain: $open parenthesis minus ∞, ∞ close parenthesis$ ; range: $open parenthesis minus ∞, 0 close parenthesis$ , no $t$ -intercept; $y$ -intercept $open parenthesis 0, minus 1 close parenthesis$ ; the $t$ -axis is the horizontal asymptote.
$y = 6 - {(\frac{1}{2})}^{t}$ ,

Domain: $open parenthesis minus ∞, ∞ close parenthesis$ ; range: $open parenthesis minus ∞, 6 close parenthesis$ , $t$ -intercept approximately $open parenthesis minus 2.58, 0 close parenthesis$ ; $y$ -intercept $open parenthesis 0, 5 close parenthesis$ ; horizontal asymptote is $y equals 6$

$P (t) = {0.4}^{t}$

$y = - P (t)$
$y = 8 - P (t)$

For Problems 61–64,

Describe the graph as a transformation of $y = 2^{x}$ .
Give an equation for the function graphed.

The graph of $y = 2^{x}$ has been reflected about the $y$ -axis and shifted up $2$ units.
$y = 2^{- x} + 2$

The graph of $y = 2^{x}$ has been reflected about the $x$ -axis and shifted up $10$ units.
$y = - 2^{x} + 10$

For Problems 65 and 66, match the graph of each function to its formula. In each formula, $a greater than 0$ and $b greater than 1$ .

$y = a b^{x}$
$y = a b^{- x}$
$y = a x^{b}$

$y = a x^{- b}$
$y = - a b^{x}$
$y = a x^{1 / b}$

The function $f (t)$ describes a volunteer's heart rate during a treadmill test.

$f (t) = {\begin{cases} 100 & 0 \leq t < 3 \\ 56 t - 68 & 3 \leq t < 4 \\ 186 - 500 (0.5)^{t} & 4 \leq t < 9 \\ 100 + 6.6 (0.6)^{t - 14} & 9 \leq t < 20 \end{cases}$

The heart rate is given in beats per minute and $t$ is in minutes. (See Some Basic Functions to review functions defined piecewise.) (Source: Davis, Kimmet, and Autry, 1986)

Evaluate the function to complete the table.

$t$	$3.5$	$4$	$8$	$10$	$15$
$f (t)$

Sketch the graph of the function.
The treadmill test began with walking at $5.5$ kilometers per hour, then jogging, starting at $12$ kilometers per hour and increasing to $14$ kilometers per hour, and finished with a cool-down walking period. Identify each of these activities on the graph and describe the volunteer's heart rate during each phase.

$t$ $3.5$ $4$ $8$ $10$ $15$
$f (t)$ $128$ $154.75$ $184.05$ $150.93$ $103.96$
From $0$ to $3$ minutes, the volunteer is walking with heart rate $100$ beats per minute. The volunteer jogged at a steady pace from $3$ to $4$ minutes, and the heart rate increased to about $155$ beats per minutes. From $4$ to $9$ minutes, the jogging pace increased, and the heart rate rose to about $185$ beats per minute. The cooldown started at $9$ minutes, and the heart rate decreased rapidly and leveled off to about $100$ beats per minute.

$t$	$3.5$	$4$	$8$	$10$	$15$
$f (t)$	$128$	$154.75$	$184.05$	$150.93$	$103.96$

Carbon dioxide ( $CO sub 2$ ) is called a greenhouse gas because it traps part of the Earth's outgoing energy. Animals release $CO sub 2$ into the atmosphere, and plants remove $CO sub 2$ through photosynthesis. In modern times, deforestation and the burning of fossil fuels both contribute to $CO sub 2$ levels. The figure shows atmospheric concentrations of $CO sub 2$ , in parts per million, measured at the Mauna Loa Observatory in Hawaii.

The red curve shows annual oscillations in $CO sub 2$ levels. Can you explain why $CO sub 2$ levels vary throughout the year? Hint: Why would photosynthesis vary throughout the year?
The blue curve shows the average annual $CO sub 2$ readings. By approximately how much does the $CO sub 2$ level vary from its average value during the year?
In 1960, the average $CO sub 2$ level was $316.75$ parts per million, and the average level has been rising by $0.4 %$ per year. If the level continues to rise at this rate, what $CO sub 2$ readings can we expect in the year 2100?

Modeling, Functions, and Graphs by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.