Graphing Exponential Functions

Evaluating Exponential Functions

The rule for an exponential function includes a variable as an exponent.

An exponential function is a function, where

a

and

b

are real numbers,

a\neq 0

b>0

, and

b\ne1

, written in the form:

f(x)=ab^x

The base of an exponential function is the value of

b

in the function rule. To graph an exponential function, start by evaluating the function for several values of

x

, and use corresponding values of

x

and

f(x)

to write ordered pairs. Then plot the points, and connect them with a smooth curve.

Exponential functions may involve the base $e$ . The number $e$ is a mathematical constant that is an irrational number with an approximate value of 2.71828. Values of the exponential function $f(x)=e^x$ can be approximated by using technology such as a graphing calculator.

For an exponential function, an increase of 1 in the value of $x$ results in a change in the value of the function by a factor of the base, $b$ .

Evaluating Exponential Functions with Different Bases

Value	Base $3$	Base $\frac{1}{2}$	Base $e$
$x$	$f(x)=3^x$	$f(x)=\left(\frac{1}{2}\right)^x$	$f(x)=e^x$
$-1$	$3^{-1}=\frac{1}{3}$	$\left(\frac{1}{2}\right)^{-1}=2$	$e^{-1}\approx0.36788$
$0$	$3^0=1$	$\left(\frac{1}{2}\right)^{0}=1$	$e^0=1$
1	$3^1=3$	$\left(\frac{1}{2}\right)^1=\frac{1}{2}$	$e^1\approx2.71828$
$2$	$3^2=9$	$\left(\frac{1}{2}\right)^2=\frac{1}{4}$	$e^2\approx7.38906$

As the value of $x$ increases by 1, the value of each exponential function changes by a factor of its base. For example, as each $x$ -value increases by 1, the value of $f(x)=3^x$ changes by a factor of 3. The value of each exponential function is 1 when $x=0$ .

Properties of Exponential Functions and Their Graphs

The base of an exponential parent function determines whether the function is increasing or decreasing.

An asymptote is a line that a graph approaches as one of the variables approaches infinity or negative infinity. An asymptote is often shown on a graph as a dotted line. The graph of an exponential function has a horizontal asymptote. The equation for a horizontal asymptote has the form $y=c$ , where $c$ is a real number.

A parent function is a function of a certain type that has the simplest algebraic rule. The parent exponential function is $f(x)=b^x$ , where $b>0$ and $b\ne1$ . The domain is all real numbers, and the range is $y>0$ . This means that the graph never touches or goes below the $x$ -axis, where $y=0$ . Using interval notation, the domain is $(-\infty, \infty)$ , and the range is $(0, \infty)$ . The horizontal asymptote is $y=0$ . Two points on the graph are $(0,1)$ and $(1,b)$ .

An increasing function is a function in which the

y

-values increase as the

x

-values increase. A decreasing function is a function in which the

y

-values decrease as the

x

-values increase. For

f(x)=b^x

, the function is increasing if

b>1

and decreasing if

0<b<1

Determining whether the graph of an exponential function in the form

f(x)=b^x

is increasing or decreasing depends on the relationship between the

x

- and

y

-values and the value of

b

. For example, the graph of

f(x)=e^x

is increasing because the

y

-values increase as the

x

-values increase and

e>1

. The graph of

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

is decreasing because the

y

-values decrease as the

x

-values increase and

0\lt \frac{1}{2}\lt 1

Transformations of Exponential Functions

Transformations can be used to graph exponential functions.

A transformation is a change of the graph of a function. Transformations of exponential functions are performed in the same way as transformations of other parent functions. An exponential parent function of the form

f(x)=b^{x}

can be translated vertically

k

units and horizontally

h

units.

Vertical Translations	Horizontal Translations
For the graph of $f(x)=b^x$ and $k>0$ : The graph of $f(x)+k=b^x+k$ is translated up by $k$ units. The graph of $f(x)-k=b^x-k$ is translated down by $k$ units.	For the graph of $f(x)=b^x$ and $h>0$ : The graph of $f(x-h)=b^{x-h}$ is translated right by $h$ units. The graph of $f(x+h)=b^{x+h}$ is translated left by $h$ units.

An exponential function

f(x)=b^x

can be stretched or compressed by a factor of

a

. It can also be reflected across the

x

-axis or

y

-axis.

Stretches and Compressions	Reflections
For the graph of $f(x)=b^x$ and $a>0$ : The graph of $af(x)=ab^x$ is a vertical stretch of the graph of $f(x)$ by a factor of $a$ if $a>1$ . The graph of $af(x)=ab^x$ is a vertical compression of the graph of $f(x)$ by a factor of $a$ if $0\lt a\lt 1$ .	For the graph of $f(x)=b^x$ : The graph of $-f(x)=-b^x$ is a reflection of the graph of $f(x)$ across the $x$ -axis. The graph of $f(-x)=b^{-x}$ is a reflection of the graph of $f(x)$ across the $y$ -axis.

Graphing a Translation of an Exponential Function

Graph the given exponential function:

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

Identify the parent function. Sketch its graph.

The base of the function is 3:

f(x)={\color{Red}{3}}^{x+3}-2

So, the parent function is:

f(x)={\color{Red}{3}}^x

Identify and perform any stretches, compressions, or reflections.

The given function can be rewritten as:

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

When the parent function is multiplied by a constant,

a

, the graph is stretched if

a>1

or compressed if

0<a<1

. If

a<0

, the graph is reflected across the

x

-axis.

In this case, the value of $a$ is 1, so the graph of the parent function is not vertically stretched or compressed, and is not reflected across the $x$ -axis.

Perform the translations. If a value $h$ is added to the $x$ -value in a function, the graph is translated $h$ units to the left. If a value $k$ is subtracted from the entire function, the graph is translated down by $k$ units.

Since $h=3$ and it is added, translate the parent function 3 units to the left.
Since $k=2$ and it is subtracted, translate the parent function 2 units down.

Graphing a Transformation of an Exponential Function

Graph the given function:

f(x)=-2\cdot3^x+1

Identify the parent function. Sketch its graph.

The base of the given function is 3:

f(x)=-2\cdot{\color{Red}{3}}^{x}+1

So, the parent function is:

f(x)=3^x

Identify and perform any vertical stretches or compressions.

\begin{aligned}f(x)&=-2\cdot 3^x+1\\&=-1\cdot2\cdot 3^x+1\end{aligned}

When the parent function is multiplied by a constant,

a

, the graph is stretched if

a>1

or compressed if

0<a<1

. If

a<0

, the graph is reflected across the

x

-axis. In this case, the function is multiplied by both -1 and 2. The result of multiplying the parent function by 2 is a vertical stretch by a factor of 2.

Identify and perform any reflections.

The result of multiplying the parent function by -1 is a reflection across the

x

-axis. So, the graph from Step 2 is reflected across the

x

-axis.

Perform the translations. If a value $h$ is added or subtracted from the $x$ -value in a function, the graph is translated $h$ units to the left or right. If a value $k$ is added to the entire function, the graph is translated up by $k$ units.

Since $h=0$ , the parent function is not translated to the left or right.
Since $k=1$ and it is added, translate the graph 1 unit up.

<Vocabulary >Applications of Exponential Functions

Exponential Functions

Contents

Graphing Exponential Functions

Evaluating Exponential Functions

Evaluating Exponential Functions with Different Bases

Properties of Exponential Functions and Their Graphs

Transformations of Exponential Functions