Evaluate and Graph Exponential Functions

The functions we have studied so far do not give us a model for many naturally occurring phenomena. From the growth of populations and the spread of viruses to radioactive decay and compounding interest, the models are very different from what we have studied so far. These models involve exponential functions.

Notice that in this function, the variable is the exponent. In our functions so far, the variables were the base.

By graphing a few exponential functions, we will be able to see their unique properties.

If we look at the graphs from the previous Example and Try Its, we can identify some of the properties of exponential functions.

all have the same basic shape. This is the shape we expect from an exponential function where

a > 1 .

What is the domain for each function? From the graphs we can see that the domain is the set of all real numbers. There is no restriction on the domain. We write the domain in interval notation as

(− \infty, \infty) .

Look at each graph. What is the range of the function? The graph never hits the

x

-axis. The range is all positive numbers. We write the range in interval notation as

(0, \infty) .

Whenever a graph of a function approaches a line but never touches it, we call that line an asymptote. For the exponential functions we are looking at, the graph approaches the

x

Now let’s look at the graphs from the previous Example and Try Its so we can now identify some of the properties of exponential functions where

0 < a < 1 .

all have the same basic shape. While this is the shape we expect from an exponential function where

0 < a < 1,

We notice that for each function, the graph still contains the point (0, 1). This make sense because

a^{0} = 1

What is the domain and range for each function? From the graphs we can see that the domain is the set of all real numbers and we write the domain in interval notation as

(− \infty, \infty) .

-axis. The range is all positive numbers. We write the range in interval notation as

(0, \infty) .

We will summarize these properties in the chart below. Which also include when

a > 1 .

It is important for us to notice that both of these graphs are one-to-one, as they both pass the horizontal line test. This means the exponential function will have an inverse. We will look at this later.

When we graphed quadratic functions, we were able to graph using translation rather than just plotting points. Will that work in graphing exponential functions?

in the last example, we see that adding one in the exponent caused a horizontal shift of one unit to the left. Recognizing this pattern allows us to graph other functions with the same pattern by translation.

Let’s now consider another situation that might be graphed more easily by translation, once we recognize the pattern.

in the last example, we see that subtracting 2 caused a vertical shift of down two units. Notice that the horizontal asymptote also shifted down 2 units. Recognizing this pattern allows us to graph other functions with the same pattern by translation.

All of our exponential functions have had either an integer or a rational number as the base. We will now look at an exponential function with an irrational number as the base.

when $a > 1$	when $0 < a < 1$
Domain	$(− \infty, \infty)$	Domain	$(− \infty, \infty)$
Range	$(0, \infty)$	Range	$(0, \infty)$
$x$ -intercept	none	$x$ -intercept	none
$y$ -intercept	$(0, 1)$	$y$ -intercept	$(0, 1)$
Contains	$(1, a), (−1, \frac{1}{a})$	Contains	$(1, a), (−1, \frac{1}{a})$
Asymptote	$x$ -axis, the line $y = 0$	Asymptote	$x$ -axis, the line $y = 0$
Basic shape	increasing	Basic shape	decreasing

Before we can look at this exponential function, we need to define the irrational number, e. This number is used as a base in many applications in the sciences and business that are modeled by exponential functions. The number is defined as the value of

{(1 + \frac{1}{n})}^{n}

as n gets larger and larger. We say, as n approaches infinity, or increases without bound. The table shows the value of

{(1 + \frac{1}{n})}^{n}

| {: valign=”top”}|———- | 1 | 2 | {: valign=”top”}| 2 | 2.25 | {: valign=”top”}| 5 | 2.48832 | {: valign=”top”}| 10 | 2.59374246 | {: valign=”top”}| 100 | 2.704813829… | {: valign=”top”}| 1,000 | 2.716923932… | {: valign=”top”}| 10,000 | 2.718145927… | {: valign=”top”}| 100,000 | 2.718268237… | {: valign=”top”}| 1,000,000 | 2.718280469… | {: valign=”top”}| 1,000,000,000 | 2.718281827… | {: valign=”top”}{: summary=”This table has 11 rows and two columns. In the first row, which is the header row, we have n and the quantity 1 plus 1 over n to the n power. Below the n in the first column we have 1, 2, 5, 10, 100, 1000, 10,000, 100,000, 1,000,000, and 1,000,000,000. Below the quantity 1 plus 1 over n to the n power in the second column we have 2, 2.25, 2.48832, 2.59374246, 2.704813829…, 2.716923932…, 2.718145927…, 2.718268237…, 2.718280469…, and 2.718281827….”}

in that we use a symbol to represent it because its decimal representation never stops or repeats. The irrational number e is called the natural base.

Solve Exponential Equations

are called exponential equations. To solve them we use a property that says as long as

a > 0

In other words, in an exponential equation, if the bases are equal then the exponents are equal.

To use this property, we must be certain that both sides of the equation are written with the same base.

Use Exponential Models in Applications

Exponential functions model many situations. If you own a bank account, you have experienced the use of an exponential function. There are two formulas that are used to determine the balance in the account when interest is earned. If a principal, P, is invested at an interest rate, r, for t years, the new balance, A, will depend on how often the interest is compounded. If the interest is compounded n times a year we use the formula

A = P {(1 + \frac{r}{n})}^{n t} .

If the interest is compounded continuously, we use the formula

A = P e^{r t} .

As you work with the Interest formulas, it is often helpful to identify the values of the variables first and then substitute them into the formula.

Other topics that are modeled by exponential functions involve growth and decay. Both also use the formula

A = P e^{r t}

we used for the growth of money. For growth and decay, generally we use

A_{0},

the principal. We see that exponential growth has a positive rate of growth and exponential decay has a negative rate of growth.

Exponential growth is typically seen in the growth of populations of humans or animals or bacteria. Our next example looks at the growth of a virus.

Key Concepts

when $a > 1$	when $0 < a < 1$
Domain	$(− \infty, \infty)$	Domain	$(− \infty, \infty)$
Range	$(0, \infty)$	Range	$(0, \infty)$
$x$ -intercept	none	$x$ -intercept	none
$y$ -intercept	$(0, 1)$	$y$ -intercept	$(0, 1)$
Contains	$(1, a), (−1, \frac{1}{a})$	Contains	$(1, a), (−1, \frac{1}{a})$
Asymptote	$x$ -axis, the line $y = 0$	Asymptote	$x$ -axis, the line $y = 0$
Basic shape	increasing	Basic shape	decreasing

Practice Makes Perfect

Graph Exponential Functions

In the following exercises, graph each exponential function.

f (x) = 2^{x}

![This figure shows a curve that passes through (negative 1, 1 over 2) through (0, 1) to (1, 2).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_309_img.jpg)

g (x) = 3^{x}

f (x) = 6^{x}

![This figure shows a curve that passes through (negative 1, 1 over 6) through (0, 1) to (1, 6).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_311_img.jpg)

g (x) = 7^{x}

f (x) = {(1.5)}^{x}

![This figure shows a curve that passes through (negative 1, 2 over 3) through (0, 1) to (1, 3 over 2).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_313_img.jpg)

g (x) = {(2.5)}^{x}

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

![This figure shows a curve that passes through (negative 1, 2) through (0, 1) to (1, 1 over 2).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_315_img.jpg)

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

f (x) = {(\frac{1}{6})}^{x}

![This figure shows a curve that passes through (negative1, 6) through (0, 1) to (1, 1 over 6).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_317_img.jpg)

g (x) = {(\frac{1}{7})}^{x}

f (x) = {(0.4)}^{x}

![This figure shows a curve that passes through (negative 1, 5 over 2) through (0, 1) to (1, 2 over 5).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_319_img.jpg)

g (x) = {(0.6)}^{x}

In the following exercises, graph each function in the same coordinate system.

f (x) = 4^{x}, g (x) = 4^{x - 1}

![This figure shows two functions. The first function f of x equals 4 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 4), (0, 1) and (1, 4). The second function g of x equals 4 to the x minus 1 power is marked in red and passes through the points (0, 1 over 4), (1, 1) and (2, 4).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_321_img.jpg)

f (x) = 3^{x}, g (x) = 3^{x - 1}

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

![This figure shows two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1) and (1, 2). The second function g of x equals 2 to the x minus 2 power is marked in red and passes through the points (0, 1 over 4), (1, 1 over 2), and (2, 1).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_323_img.jpg)

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

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

![This figure shows two functions. The first function f of x equals 3 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 3), (0, 1), and (1, 3). The second function g of x equals 3 to the x power plus 2 is marked in red and passes through the points (negative 2, 1), (negative 1, 3), and (0, 5).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_325_img.jpg)

f (x) = 4^{x}, g (x) = 4^{x} + 2

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

![This figure shows two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1), and (1, 2). The second function g of x equals 2 to the x power plus 1 is marked in red and passes through the points (negative 1, 1), (0, 2), and (1, 4).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_327_img.jpg)

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

In the following exercises, graph each exponential function.

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

![This figure shows an exponential curve that passes through (negative 3, 1 over 3), (negative 2, 1), and (0, 9).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_329_img.jpg)

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

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

![This figure shows an exponential that passes through (negative 1, 7 over 2), (0, 4), and (1, 5).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_331_img.jpg)

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

f (x) = {(\frac{1}{2})}^{x - 4}

![This figure shows an exponential that passes through (2, 4), (3, 2), and (4, 1).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_333_img.jpg)

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

f (x) = e^{x} + 1

![This figure shows an exponential that passes through (1, 1 plus 1 over e), (0, 2), and (1, e).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_335_img.jpg)

f (x) = e^{x - 2}

f (x) = − 2^{x}

![This figure shows an exponential that passes through (negative 1, negative 1 over 2), (0, negative 1), and (1, 2).](/algebra-intermediate-book/resources/CNX_IntAlg_Figure_10_02_337_img.jpg)

f (x) = 3^{x}

Solve Exponential Equations

In the following exercises, solve each equation.

2^{3 x - 8} = 16

x = 4

2^{2 x - 3} = 32

3^{x + 3} = 9

x = −1

3^{x^{2}} = 81

4^{x^{2}} = 4

x = −1, x = 1

4^{x} = 32

4^{x + 2} = 64

x = 1

4^{x + 3} = 16

2^{x^{2} + 2 x} = \frac{1}{2}

x = −1

3^{x^{2} - 2 x} = \frac{1}{3}

e^{3 x} \cdot e^{4} = e^{10}

x = 2

e^{2 x} \cdot e^{3} = e^{9}

\frac{e^{x^{2}}}{e^{2}} = e^{x}

x = −1, x = 2

\frac{e^{x^{2}}}{e^{3}} = e^{2 x}

In the following exercises, match the graphs to one of the following functions: ⓐ $2^{x}$

ⓑ $2^{x + 1}$

ⓓ $2^{x} + 2$

ⓔ $2^{x} - 2$

ⓕ $3^{x}$

This figure shows an exponential that passes through (1, 1 over 3), (0, 1), and (1, 3).

ⓕ

This figure shows an exponential that passes through (negative 2, 1 over 2), (negative 1, 1), and (0, 2).

This figure shows an exponential that passes through (1, 1 over 2), (0, 1), and (1, 2).

ⓐ

This figure shows an exponential that passes through (0, 1 over 2), (1, 1), and (2, 2).

This figure shows an exponential that passes through (negative 1, 3 over 2), (0, negative 1), and (1, 0).

ⓔ

This figure shows an exponential that passes through (negative 1, 5 over 2), (0, 3), and (1, 4).

Use exponential models in applications

In the following exercises, use an exponential model to solve.

Edgar accumulated $$ 5,000$

in credit card debt. If the interest rate is $20 %$

per year, and he does not make any payments for 2 years, how much will he owe on this debt in 2 years by each method of compounding?* * *

ⓐ compound quarterly* * *

ⓑ compound monthly* * *

ⓐ $$ 7,387.28$

ⓑ $$ 7,434.57$

Cynthia invested $$ 12,000$

in a savings account. If the interest rate is $6 %,$

how much will be in the account in 10 years by each method of compounding?* * *

ⓐ compound quarterly* * *

ⓑ compound monthly* * *

Rochelle deposits $$ 5,000$

in an IRA. What will be the value of her investment in 25 years if the investment is earning $8 %$

per year and is compounded continuously?

$ 36,945.28

Nazerhy deposits $$ 8,000$

in a certificate of deposit. The annual interest rate is $6 %$

and the interest will be compounded quarterly. How much will the certificate be worth in 10 years?

A researcher at the Center for Disease Control and Prevention is studying the growth of a bacteria. He starts his experiment with 100 of the bacteria that grows at a rate of $6 %$

per hour. He will check on the bacteria every 8 hours. How many bacteria will he find in 8 hours?

223 bacteria

A biologist is observing the growth pattern of a virus. She starts with 50 of the virus that grows at a rate of $20 %$

per hour. She will check on the virus in 24 hours. How many viruses will she find?

In the last ten years the population of Indonesia has grown at a rate of $1.12 %$

per year to 258,316,051. If this rate continues, what will be the population in 10 more years?

288,929,825

In the last ten years the population of Brazil has grown at a rate of $0.9 %$

per year to 205,823,665. If this rate continues, what will be the population in 10 more years?

Writing Exercises

Explain how you can distinguish between exponential functions and polynomial functions.

Answers will vary.

Compare and contrast the graphs of $y = x^{2}$

and $y = 2^{x}$

What happens to an exponential function as the values of $x$

decreases? Will the graph ever cross the* * *

y

-axis? Explain.

Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four rows and four columns. The first row, which serves as a header, reads I can…, Confidently, With some help, and No—I don’t get it. The first column below the header row reads Graph exponential functions, solve exponential equations, and use exponential models in applications. ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

	$\frac{e^{x^{2}}}{e^{3}} = e^{2 x}$
Use the Property of Exponents: $\frac{a^{m}}{a^{n}} = a^{m - n} .$	$e^{x^{2} - 3} = e^{2 x}$
Write a new equation by setting the exponents equal.	$x^{2} - 3 = 2 x$
Solve the equation.	$x^{2} - 2 x - 3 = 0$
	$(x - 3) (x + 1) = 0$
	$x = 3, x = - 1$
Check the solutions.


{: valign=”top”}	x-intercept	None
{: valign=”top”}	y-intercept	$(0, 1)$