Graphs of Exponential Functions

Matthew Schurmann

Learning Objectives

Graph exponential functions.
Graph exponential functions using transformations.

As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.

Graphing Exponential Functions

Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form $f\left(x\right)={b}^{x}$ whose base is greater than one. We’ll use the function $f\left(x\right)={2}^{x}.$ Observe how the output values in the table below change as the input increases by $1.$

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$f\left(x\right)={2}^{x}$	$frac{1}{8}$	$frac{1}{4}$	$frac{1}{2}$	$1$	$2$	$4$	$8$

Each output value is the product of the previous output and the base $2.$ We call the base $2$ the constant ratio. In fact, for any exponential function with the form $f\left(x\right)=a{b}^{x}$ $b$ is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of $a.$

Notice from the table that

the output values are positive for all values of $x;$
as $x$ increases, the output values increase without bound; and
as $x$ decreases, the output values grow smaller, approaching zero.

The graph below shows the exponential growth function $f\left(x\right)={2}^{x}.$

Graph of the exponential function, 2^(x), with labeled points at (-3, 1/8), (-2, ¼), (-1, ½), (0, 1), (1, 2), (2, 4), and (3, 8). The graph notes that the x-axis is an asymptote.

The domain of $f\left(x\right)={2}^{x}$ is all real numbers, the range is $\left(0,\infty \right)$ and the horizontal asymptote is $y=0.$

To get a sense of the behavior of exponential decay, we can create a table of values for a function of the form $f\left(x\right)={b}^{x}$ whose base is between zero and one. We’ll use the function $g\left(x\right)={\left(frac{1}{2}\right)}^{x}.$ Observe how the output values change as the input increases by $1.$

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$g(x)=\left(frac{1}{2}\right)^{x}$	$8$	$4$	$2$	$1$	$frac{1}{2}$	$frac{1}{4}$	$frac{1}{8}$

Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio $frac{1}{2}.$

Notice from the table that

the output values are positive for all values of $x;$
as $x$ increases, the output values grow smaller, approaching zero; and
as $x$ decreases, the output values grow without bound.

The graph below shows the exponential decay function $g\left(x\right)={\left(frac{1}{2}\right)}^{x}.$

Graph of decreasing exponential function, (1/2)^x, with labeled points at (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1/2), (2, 1/4), and (3, 1/8). The graph notes that the x-axis is an asymptote.

The domain of $g\left(x\right)={\left(frac{1}{2}\right)}^{x}$ is all real numbers, the range is $\left(0,\infty \right)$ and the horizontal asymptote is $y=0.$

Characteristics of the Graph of the Parent Function f(x) = b^x

An exponential function with the form $f\left(x\right)={b}^{x}$ $b>0$ $b \ne 1$ has these characteristics:

one-to-one function
horizontal asymptote: $y=0$
domain: $\left(-\infty , \infty \right)$
range: $\left(0,\infty \right)$
x-intercept: none
y-intercept: $\left(0,1\right)$
increasing if $b>1$
decreasing if $b<1$

The graphs below compare the graphs of exponential growth and decay functions.

How To

Given an exponential function of the form $f\left(x\right)={b}^{x}$ graph the function.

Create a table of points.
Plot at least $3$ point from the table, including the y-intercept $\left(0,1\right).$
Draw a smooth curve through the points.
State the domain $\left(-\infty ,\infty \right)$ the range $\left(0,\infty \right)$ and the horizontal asymptote, $y=0.$

Sketching the Graph of an Exponential Function of the Form f(x) = b^x

Sketch a graph of $f\left(x\right)={0.25}^{x}.$ State the domain, range, and asymptote.

Show Solution

Before graphing, identify the behavior and create a table of points for the graph.

Since $b=0.25$ is between zero and one, we know the function is decreasing. The \left tail of the graph will increase without bound, and the right tail will approach the asymptote $y=0.$
Create a table of points.

$x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$

$f\left(x\right)={0.25}^{x}$ $64$ $16$ $4$ $1$ $0.25$ $0.0625$ $0.015625$
Plot the y-intercept $\left(0,1\right)$ along with two other points. We can use $\left(-1,4\right)$ and $\left(1,0.25\right).$

Draw a smooth curve connecting the points.

Graph of the decaying exponential function f(x) = 0.25^x with labeled points at (-1, 4), (0, 1), and (1, 0.25).

The domain is $\left(-\infty ,\infty \right);$ the range is $\left(0,\infty \right);$ the horizontal asymptote is $y=0.$

Try It

Sketch the graph of $f\left(x\right)={4}^{x}.$ State the domain, range, and asymptote.

Show Solution

The domain is $\left(-\infty ,\infty \right);$ the range is $\left(0,\infty \right);$ the horizontal asymptote is $y=0.$

Graph of the increasing exponential function f(x) = 4^x with labeled points at (-1, 0.25), (0, 1), and (1, 4).

Graphing Transformations of Exponential Functions

Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent function $f\left(x\right)={b}^{x}$ without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.

Graphing a Vertical Shift

The first transformation occurs when we add a constant $d$ to the parent function $f\left(x\right)={b}^{x}$ giving us a vertical shift $d$ units in the same direction as the sign. For example, if we begin by graphing a parent function $f\left(x\right)={2}^{x}$ we can then graph two vertical shifts alongside it, using $d=3:$ the upward shift $g\left(x\right)={2}^{x}+3$ and the downward shift $h\left(x\right)={2}^{x}-3.$

Graph of three functions, g(x) = 2^x+3 in blue with an asymptote at y=3, f(x) = 2^x in orange with an asymptote at y=0, and h(x)=2^x-3 with an asymptote at y=-3. Note that each functions’ transformations are described in the text.

Observe the results of shifting $f\left(x\right)={2}^{x}$ vertically:

The domain $\left(-\infty ,\infty \right)$ remains unchanged.
When the function is shifted up units to
- The y-intercept shifts up $3$ units to $\left(0,4\right).$
- The asymptote shifts up $3$ units to $y=3.$
- The range becomes $\left(3,\infty \right).$
When the function is shifted down units to
- The y-intercept shifts down $3$ units to $\left(0,-2\right).$
- The asymptote also shifts down $3$ units to $y=-3.$
- The range becomes $\left(-3,\infty \right).$

Graphing a Horizontal Shift

The next transformation occurs when we add a constant $c$ to the input of the parent function $f\left(x\right)={b}^{x}$ giving us a horizontal shift $c$ units in the opposite direction of the sign. For example, if we begin by graphing the parent function $f\left(x\right)={2}^{x}$ we can then graph two horizontal shifts alongside it, using $c=3:$ the shift \left $g\left(x\right)={2}^{x+3}$ and the shift right $h\left(x\right)={2}^{x-3}.$ Both horizontal shifts are shown below.

Graph of three functions, g(x) = 2^(x+3) in blue, f(x) = 2^x in orange, and h(x)=2^(x-3). Each functions’ asymptotes are at y=0Note that each functions’ transformations are described in the text.

Observe the results of shifting $f\left(x\right)={2}^{x}$ horizontally:

The domain $\left(-\infty ,\infty \right)$ remains unchanged.
The asymptote $y=0$ remains unchanged.
The y-intercept shifts such that:
- When the function is shifted \left $3$ units to $g\left(x\right)={2}^{x+3}$ the y-intercept becomes $\left(0,8\right).$ This is because ${2}^{x+3}=\left(8\right){2}^{x}$ so the initial value of the function is $8.$
- When the function is shifted right $3$ units to $h\left(x\right)={2}^{x-3}$ the y-intercept becomes $\left(0,\dfrac{1}{8}\right).$ Again, see that ${2}^{x-3}=\left(\dfrac{1}{8}\right){2}^{x}$ so the initial value of the function is $\dfrac{1}{8}.$

Shifts of the Parent Function f(x) = b^x

For any constants $c$ and $d$ the function $f\left(x\right)={b}^{x+c}+d$ shifts the parent function $f\left(x\right)={b}^{x}$

vertically $d$ units, in the same direction of the sign of $d.$
horizontally $c$ units, in the opposite direction of the sign of $c.$
The y-intercept becomes $\left(0,{b}^{c}+d\right).$
The horizontal asymptote becomes $y=d.$
The range becomes $\left(d,\infty \right).$
The domain $\left(-\infty ,\infty \right)$ remains unchanged.

How To

Given an exponential function with the form $f\left(x\right)={b}^{x+c}+d$ graph the translation.

Draw the horizontal asymptote $y=d.$
Identify the shift as $\left(-c,d\right).$ Shift the graph of $f\left(x\right)={b}^{x}$ \left $c$ units if $c$ is positive, and right $c$ units if $c$ is negative.
Shift the graph of $f\left(x\right)={b}^{x}$ up $d$ units if $d$ is positive, and down $d$ units if $d$ is negative.
State the domain $\left(-\infty ,\infty \right)$ the range $\left(d,\infty \right)$ and the horizontal asymptote $y=d.$

Graphing a Shift of an Exponential Function

Graph $f\left(x\right)={2}^{x+1}-3.$ State the domain, range, and asymptote.

Show Solution

We have an exponential equation of the form $f\left(x\right)={b}^{x+c}+d$ with $b=2$ $c=1$ and $d=-3.$

Draw the horizontal asymptote $y=d$ so draw $y=-3.$

Identify the shift as $\left(-c, d\right)$ so the shift is $\left(-1,-3\right).$

Shift the graph of $f\left(x\right)={b}^{x}$ \left 1 units and down 3 units.

Graph of the function, f(x) = 2^(x+1)-3, with an asymptote at y=-3. Labeled points in the graph are (-1, -2), (0, -1), and (1, 1).

The domain is $\left(-\infty ,\infty \right);$ the range is $\left(-3,\infty \right);$ the horizontal asymptote is $y=-3.$

Try It

Graph $f\left(x\right)={2}^{x-1}+3.$ State domain, range, and asymptote.

Show Solution

The domain is $\left(-\infty ,\infty \right);$ the range is $\left(3,\infty \right);$ the horizontal asymptote is $y=3.$

Graph of the function, f(x) = 2^(x-1)+3, with an asymptote at y=3. Labeled points in the graph are (-1, 3.25), (0, 3.5), and (1, 4).

Try It

Solve $4=7.85{\left(1.15\right)}^{x}-2.27$ graphically. Round to the nearest thousandth.

Show Solution

$x \approx -1.608$

Graphing a Stretch or Compression

While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function $f\left(x\right)={b}^{x}$ by a constant $|a|>0.$ For example, if we begin by graphing the parent function $f\left(x\right)={2}^{x}$ we can then graph the stretch, using $a=3$ to get $g\left(x\right)=3{\left(2\right)}^{x}$ as shown on the left and the compression, using $a=frac{1}{3}$ to get $h\left(x\right)=frac{1}{3}{\left(2\right)}^{x}$ as shown on the right.

Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression. — (a) $g\left(x\right)=3{\left(2\right)}^{x}$ stretches the graph of $f\left(x\right)={2}^{x}$ vertically by a factor of $3.$ (b) $h\left(x\right)=frac{1}{3}{\left(2\right)}^{x}$ compresses the graph of $f\left(x\right)={2}^{x}$ vertically by a factor of $frac{1}{3}.$

(a) $g\left(x\right)=3{\left(2\right)}^{x}$ stretches the graph of $f\left(x\right)={2}^{x}$ vertically by a factor of $3.$ (b) $h\left(x\right)=frac{1}{3}{\left(2\right)}^{x}$ compresses the graph of $f\left(x\right)={2}^{x}$ vertically by a factor of $frac{1}{3}.$

Stretches and Compressions of the Parent Function f(x) = b^x

For any factor $a>0$ the function $f\left(x\right)=a{\left(b\right)}^{x}$

is stretched vertically by a factor of $a$ if $|a|>1.$
is compressed vertically by a factor of $a$ if $|a|<1.$
has a y-intercept of $\left(0,a\right).$
has a horizontal asymptote at $y=0$ a range of $\left(0,\infty \right)$ and a domain of $\left(-\infty ,\infty \right)$ which are unchanged from the parent function.

Graphing the Stretch of an Exponential Function

Sketch a graph of $f\left(x\right)=4{\left(frac{1}{2}\right)}^{x}.$ State the domain, range, and asymptote.

Show Solution

[hidden-answer a=”274741″]

Before graphing, identify the behavior and key points on the graph.

Since $b=frac{1}{2}$ is between zero and one, the \left tail of the graph will increase without bound as $x$ decreases, and the right tail will approach the x-axis as $x$ increases.
Since $a=4$ the graph of $f\left(x\right)={\left(frac{1}{2}\right)}^{x}$ will be stretched by a factor of $4.$
Create a table of points.

$x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$

$f\left(x\right)=4\left(frac{1}{2}\right)^{x}$ $32$ $16$ $8$ $4$ $2$ $1$ $0.5$
Plot the y-intercept $\left(0,4\right)$ along with two other points. We can use $\left(-1,8\right)$ and $\left(1,2\right).$

Draw a smooth curve connecting the points.

Graph of the function, f(x) = 4(1/2)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 8), (0, 4), and (1, 2).

The domain is $\left(-\infty ,\infty \right);$ the range is $\left(0,\infty \right);$ the horizontal asymptote is $y=0.$

Try It

Sketch the graph of $f\left(x\right)=frac{1}{2}{\left(4\right)}^{x}.$ State the domain, range, and asymptote.

Show Solution

The domain is $\left(-\infty ,\infty \right);$ the range is $\left(0,\infty \right);$ the horizontal asymptote is $y=0.$ Graph of the function, f(x) = (1/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).

Key Equations

General Form for the Translation of the Parent Function $text{ }f\left(x\right)={b}^{x}$

$f\left(x\right)=a{b}^{x+c}+d$

Key Concepts

The graph of the function $f\left(x\right)={b}^{x}$ has a y-intercept at $\left(0, 1\right)$ domain $\left(-\infty , \infty \right)$ range $\left(0, \infty \right)$ and horizontal asymptote $y=0$ .
If $b>1$ the function is increasing. The \left tail of the graph will approach the asymptote $y=0$ and the right tail will increase without bound.
If $0<b<1$ the function is decreasing. The \left tail of the graph will increase without bound, and the right tail will approach the asymptote $y=0.$
The equation $f\left(x\right)={b}^{x}+d$ represents a vertical shift of the parent function $f\left(x\right)={b}^{x}.$
The equation $f\left(x\right)={b}^{x+c}$ represents a horizontal shift of the parent function $f\left(x\right)={b}^{x}.$
Approximate solutions of the equation $f\left(x\right)={b}^{x+c}+d$ can be found using a graphing calculator.
The equation $f\left(x\right)=a{b}^{x}$ where $a>0$ represents a vertical stretch if $|a|>1$ or compression if $0<|a|<1$ of the parent function $f\left(x\right)={b}^{x}.$
When the parent function $f\left(x\right)={b}^{x}$ is multiplied by $-1$ the result $f\left(x\right)=-{b}^{x}$ is a reflection about the x-axis. When the input is multiplied by $-1$ the result $f\left(x\right)={b}^{-x}$ is a reflection about the y-axis.
All translations of the exponential function can be summarized by the general equation $f\left(x\right)=a{b}^{x+c}+d.$
Using the general equation $f\left(x\right)=a{b}^{x+c}+d$ we can write the equation of a function given its description.