12.2: Graphs of Exponential Functions

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Roy Simpson, Cosumnes River College
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Learning Objectives

Graph exponential functions.
Graph exponential functions using transformations.

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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 $f(x)= b^x$ whose base is greater than one. We'll use the function $f(x)= 2^x $. Observe how the output values in Table $ \PageIndex{ 1 } $ change as the input increases by $1$.

**Table $ \PageIndex{ 1 } $**
$x$	$−3$	$−2$	$−1$	$0$	$1$	$2$	$3$
$f(x)= 2^x$	$ \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(x) = 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 $a$. If this sounds familiar, it's because we gave this property a name in the previous section - the Base Multiplier Property.

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.

Figure $ \PageIndex{ 1 } $ shows the exponential growth function $f(x)= 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. — **Figure $ \PageIndex{ 1 } $**

*Notice that the graph gets close to the $ x $-axis, but never touches it.*

The domain of $f(x)= 2^x$ is all real numbers, the range is $ (0, \infty) $, 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 $f(x)= b^x$ whose base is between zero and one. We'll use the function $g(x)= \left( \frac{1}{2} \right)^x $. Observe how the output values in Table $ \PageIndex{ 2 } $ change as the input increases by $1$.

**Table $ \PageIndex{ 2 } $**
$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$ 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.

Figure $ \PageIndex{ 2 } $ shows the exponential decay function, $g(x)= \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. — **Figure $ \PageIndex{ 2 } $**

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

Theorem: Properties of the Graphs of Exponential Functions

Example $ \PageIndex{ 1 } $: Sketching the Graph of an Exponential Function of the Form $f( x ) = b^x $

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

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 as in Table $ \PageIndex{ 3 } $.

**Table $ \PageIndex{ 3 } $**
$x$	$−3$	$−2$	$−1$	$0$	$1$	$2$	$3$
$ f(x) = 0.25^x $	$64$	$16$	$4$	$1$	$0.25$	$0.0625$	$0.015625$

Plot the $ y $-intercept, $(0,1)$, along with two other points. We can use $ (-1,4) $ and $ (1, 0.25) $.

Draw a smooth curve connecting the points as in Figure $ \PageIndex{ 3 } $.

Graph of the decaying exponential function f(x) = 0.25^x with labeled points at (-1, 4), (0, 1), and (1, 0.25). — **Figure $ \PageIndex{ 3 } $**

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

Checkpoint $ \PageIndex{ 1 } $

Sketch the graph of $ f(x) = 4^x $. State the domain, range, and asymptote.

Graphing Transformations of Exponential Functions

Transformations of exponential graphs behave similarly to those of other functions. Just as with other toolkit functions, we can apply the four types of transformations - shifts, reflections, stretches, and compressions - to the function $f(x) = b^x$ 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 function $ f(x) = b^x $, giving us a vertical shift $d$ units in the same direction as the sign. For example, if we begin by graphing a function, $ f(x) = 2^x $, we can then graph two vertical shifts alongside it, using $d=3$: the upward shift, $g(x)= 2^x + 3$ and the downward shift, $h(x)= 2^x − 3$. Both vertical shifts are shown in Figure $ \PageIndex{ 4 } $.

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. — **Figure $ \PageIndex{ 4 } $**

Observe the results of shifting $ f(x) = 2^x $ vertically:

The domain, $ (-\infty, \infty) $ remains unchanged.
When the function is shifted up $3$ units to $g(x)= 2^x + 3$:
- The $ y $-intercept shifts up $3$ units to $ (0,4) $.
- The asymptote shifts up $3$ units to $y=3$.
- The range becomes $ (3, \infty) $.
When the function is shifted down $3$ units to $h(x)= 2^x − 3$:
- The $ y $-intercept shifts down $3$ units to $ (0,-2) $.
- The asymptote also shifts down $3$ units to $y=−3$.
- The range becomes $ (-3, \infty) $.

Graphing a Horizontal Shift

The next transformation occurs when we add a constant $c$ to the input of the function $ f(x) = b^x $, giving us a horizontal shift $c$ units in the opposite direction of the sign. For example, if we begin by graphing the function $ f(x) = 2^x $, we can then graph two horizontal shifts alongside it, using $c=3$: the shift left, $ g(x) = 2^{x + 3} $, and the shift right, $ h(x) = 2^{x - 3} $. Both horizontal shifts are shown in Figure $ \PageIndex{ 5 } $.

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. — **Figure $ \PageIndex{ 5 } $**

Observe the results of shifting $ f(x) = 2^x $ horizontally:

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

Example $ \PageIndex{ 2 } $: Graphing a Shift of an Exponential Function

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

Solution

We have an exponential equation of the form $ f(x) = 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 $ (-c, d) $, so the shift is $ (-1, -3) $.
Shift the graph of $f(x) = b^x$ 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). — **Figure $ \PageIndex{ 6 } $**

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

Checkpoint $ \PageIndex{ 2 } $

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

Example $ \PageIndex{ 3 } $: Approximating the Solution of an Exponential Equation

Solve $42 = 1.2( 5 )^x +2.8$ using graphing technology. Round to the nearest thousandth.

Solution: If you use Desmos, then you would go to the graphing calculator and type in $ f(x) = 1.2 ( 5 )^x +2.8$. On the next line, type $ y = 42 $. For a window, use the values –3 to 3 for $x$ and –5 to 55 for $y$. If you click where the two graphs intersect, Desmos will display the point $ \left( 2.1662, 42 \right) $. Therefore, the solution, which is the value of $ x $ where these graphs intersect, is approximately $ 2.166 $.

Checkpoint $ \PageIndex{ 3 } $

Solve $4=7.85 ( 1.15 )^x −2.27$ graphically. Round to the nearest thousandth.

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 function $f(x) = b^x$ by a constant $|a|>0$. For example, if we begin by graphing the function $ f(x) = 2^x $, we can then graph the stretch, using $a=3$, to get $ g(x) = 3(2)^x $ as shown on the left in Figure $ \PageIndex{ 7 } $, and the compression, using $ a = \frac{1}{3} $, to get $ h(x) = \frac{1}{3} (2)^x $ as shown on the right in Figure $ \PageIndex{ 7 } $.

Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression. — **Figure $ \PageIndex{ 7 } $**

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

Example $ \PageIndex{ 4 } $: Graphing the Stretch of an Exponential Function

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

Solution

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 $f(x)= \left( \frac{1}{2} \right)^x$ will be stretched by a factor of $4$.

Create a table of points as shown in Table $ \PageIndex{ 4 } $.

**Table $ \PageIndex{ 4 } $**
$x$	$−3$	$−2$	$−1$	$0$	$1$	$2$	$3$
$ f(x) =4 \left( \frac{1}{2} \right)^x $	$32$	$16$	$8$	$4$	$2$	$1$	$0.5$

Plot the $ y $-intercept, $ (0,4) $, along with two other points. We can use $( −1,8 )$ and $( 1,2 )$.

Draw a smooth curve connecting the points, as shown in Figure $ \PageIndex{ 8 } $.

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). — **Figure $ \PageIndex{ 8 } $**

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

Checkpoint $ \PageIndex{ 4 } $

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

Graphing Reflections

In addition to shifting, compressing, and stretching a graph, we can also reflect it about the $ x $-axis or the $ y $-axis. When we multiply the function $f(x) = b^x$ by $−1$, we get a reflection about the $ x $-axis. When we multiply the input by $−1$, we get a reflection about the $ y $-axis. For example, if we begin by graphing the function $ f(x) = 2^x $, we can then graph the two reflections alongside it. The reflection about the $ x $-axis, $g(x)= −2^x $, is shown on the left side of Figure $ \PageIndex{ 9 } $, and the reflection about the $ y $-axis $h(x)= 2^{−x} $, is shown on the right side of Figure $ \PageIndex{ 9 } $.

Two graphs where graph a is an example of a reflection about the x-axis and graph b is an example of a reflection about the y-axis. — **Figure $ \PageIndex{ 9 } $**

(a) $g(x)=− 2^x$ reflects the graph of $ f(x) = 2^x $ about the $x$-axis.
(b) $g(x)= 2^{−x}$ reflects the graph of $ f(x) = 2^x $ about the $ y $-axis.

Example $ \PageIndex{ 5 } $: Writing and Graphing the Reflection of an Exponential Function

Find and graph the equation for a function, $g(x)$, that reflects $ f(x) = \left( \frac{1}{4} \right)^x $ about the $ x $-axis. State its domain, range, and asymptote.

Solution

Since we want to reflect the parent function $ f(x) = \left( \frac{1}{4} \right)^x $ about the $ x $-axis, we multiply $f(x)$ by $−1$ to get, $ g(x) = -\left( \frac{1}{4} \right)^x $. Next we create a table of points as in Table $ \PageIndex{ 5 } $.

**Table $ \PageIndex{ 5 } $**
$x$	$−3$	$−2$	$−1$	$0$	$1$	$2$	$3$
$ g(x) = −\left( \frac{1}{4} \right)^x $	$−64$	$−16$	$−4$	$−1$	$−0.25$	$−0.0625$	$−0.0156$

Plot the $ y $-intercept, $( 0,−1 )$, along with two other points. We can use $( −1,−4 )$ and $( 1,−0.25 )$.

Draw a smooth curve connecting the points:

Graph of the function, g(x) = -(0.25)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, -4), (0, -1), and (1, -0.25). — **Figure $ \PageIndex{ 10 } $**

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

Checkpoint $ \PageIndex{ 5 } $

Find and graph the equation for a function, $g(x)$, that reflects $f(x)= 1.25^x$ about the $ y $-axis. State its domain, range, and asymptote.

Summarizing Translations of the Exponential Function

Now that we have worked with each type of translation for the exponential function, we can summarize them in Table $ \PageIndex{ 6 } $ to arrive at the general equation for translating exponential functions.

**Table $ \PageIndex{ 6 } $**

*Transformations of the Parent Function $ f(x) = b^x $*
Transformation	Form
Shift Horizontally $c$ units to the left (if $ c > 0 $) Vertically $d$ units up (if $ d > 0 $)	\[f(x)= b^{x+c} +d \nonumber \]
Stretch and Compress Stretch if $\| a \|>1$ Compression if $0<\| a \|<1$	\[f(x)=a b^x \nonumber \]
Reflect about the $ x $-axis	\[f(x)=− b^x \nonumber \]
Reflect about the $ y $-axis	\[f(x)= b^{−x} = \left( \frac{1}{b} \right)^x \nonumber \]
General equation for all transformations	\[f(x)=a b^{x+c} +d \nonumber \]

Example $ \PageIndex{ 6 } $: Writing a Function from a Description

Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.

$f(x)= e^x$ is vertically stretched by a factor of $2$ , reflected across the $ y $-axis, and then shifted up $4$ units.

Solution

We want to find an equation of the general form $f(x)=a b^{x+c} + d$. We use the description provided to find $a$, $b$, $c$, and $d$.

We are given the parent function $f(x)= e^x $, so $b=e$.
The function is stretched by a factor of $2$ , so $a=2$.
The function is reflected about the $ y $-axis. We replace $x$ with $−x$ to get: $e^{−x} $.
The graph is shifted vertically $ 4 $ units, so $d=4$.

Substituting in the general form we get,\[f(x) =a b^{x+c} +d = 2 e^{−x+0} + 4 = 2 e^{−x} +4. \nonumber \]The domain is $ (-\infty, \infty) $; the range is $( 4, \infty )$; the horizontal asymptote is $y=4$.

Checkpoint $ \PageIndex{ 6 } $

Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.

$f(x)= e^x$ is compressed vertically by a factor of $ \frac{1}{3} $, reflected across the $ x $-axis and then shifted down $2$ units.

\(x\)	\(−3\)	\(−2\)	\(−1\)	\(0\)	\(1\)	\(2\)	\(3\)
\(f(x)= 2^x\)	\( \frac{1}{8} \)	\( \frac{1}{4} \)	\( \frac{1}{2} \)	\(1\)	\(2\)	\(4\)	\(8\)

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

\(x\)	\(−3\)	\(−2\)	\(−1\)	\(0\)	\(1\)	\(2\)	\(3\)
\( f(x) = 0.25^x \)	\(64\)	\(16\)	\(4\)	\(1\)	\(0.25\)	\(0.0625\)	\(0.015625\)

\(x\)	\(−3\)	\(−2\)	\(−1\)	\(0\)	\(1\)	\(2\)	\(3\)
\( f(x) =4 \left( \frac{1}{2} \right)^x \)	\(32\)	\(16\)	\(8\)	\(4\)	\(2\)	\(1\)	\(0.5\)

\(x\)	\(−3\)	\(−2\)	\(−1\)	\(0\)	\(1\)	\(2\)	\(3\)
\( g(x) = −\left( \frac{1}{4} \right)^x \)	\(−64\)	\(−16\)	\(−4\)	\(−1\)	\(−0.25\)	\(−0.0625\)	\(−0.0156\)

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Learning Objectives

Graphing Exponential Functions

Theorem: Properties of the Graphs of Exponential Functions

Example \( \PageIndex{ 1 } \): Sketching the Graph of an Exponential Function of the Form \(f( x ) = b^x \)

Checkpoint \( \PageIndex{ 1 } \)

Graphing Transformations of Exponential Functions

Graphing a Vertical Shift

Graphing a Horizontal Shift

Example \( \PageIndex{ 2 } \): Graphing a Shift of an Exponential Function

Checkpoint \( \PageIndex{ 2 } \)

Example \( \PageIndex{ 3 } \): Approximating the Solution of an Exponential Equation

Checkpoint \( \PageIndex{ 3 } \)

Graphing a Stretch or Compression

Example \( \PageIndex{ 4 } \): Graphing the Stretch of an Exponential Function

Checkpoint \( \PageIndex{ 4 } \)

Graphing Reflections

Example \( \PageIndex{ 5 } \): Writing and Graphing the Reflection of an Exponential Function

Checkpoint \( \PageIndex{ 5 } \)

Summarizing Translations of the Exponential Function

Example \( \PageIndex{ 6 } \): Writing a Function from a Description

Checkpoint \( \PageIndex{ 6 } \)

Transformation	Form
Shift Horizontally \(c\) units to the left (if \( c > 0 \)) Vertically \(d\) units up (if \( d > 0 \))	\[f(x)= b^{x+c} +d \nonumber \]
Stretch and Compress Stretch if \(\| a \|>1\) Compression if \(0<\| a \|<1\)	\[f(x)=a b^x \nonumber \]
Reflect about the \( x \)-axis	\[f(x)=− b^x \nonumber \]
Reflect about the \( y \)-axis	\[f(x)= b^{−x} = \left( \frac{1}{b} \right)^x \nonumber \]
General equation for all transformations	\[f(x)=a b^{x+c} +d \nonumber \]