# 4.2: Graphs of Exponential Functions

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Skills to Develop

• 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 $$f(x)=b^x$$ 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 $$1$$.

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

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 $$f(x)=ab^x$$, $$b$$ 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.

Figure $$\PageIndex{1}$$ shows the exponential growth function $$f(x)=2^x$$. 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 $$y=0$$.  Note that, unlike a horizontal asymptote for a rational function (see Section 3.7), the graph approaches $$y=0$$ only at one end.

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 $$g(x)={\left(\dfrac{1}{2}\right)}^x$$. Observe how the output values in Table $$\PageIndex{2}$$ change as the input increases by $$1$$.

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

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

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, $$g(x)={\left(\dfrac{1}{2}\right)}^x$$. 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 $$y=0$$.

CHARACTERISTICS OF THE GRAPH OF THE toolkit FUNCTION $$f(x) = b^x$$

An exponential function with the form $$f(x)=b^x$$, $$b>0$$, $$b≠1$$, has these characteristics:

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

Figure $$\PageIndex{3}$$ compares the graphs of exponential growth and decay functions. Figure $$\PageIndex{3}$$ Given an exponential function of the form $$f(x)=b^x$$, graph the function

1. Plot at least $$3$$ points of the graph by finding 3 input-output pairs, including the $$y$$-intercept $$(0,1)$$.
2. Draw a smooth curve through the points.
3. State the domain, $$(−\infty,\infty)$$, the range, $$(0,\infty)$$, and the horizontal asymptote,  $$y=0$$.

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

Since $$b=0.25$$ is between zero and one, we know the function is decreasing. The end behavior of the graph is as follows: as $$x \rightarrow -\infty$$, $$y \rightarrow \infty$$, and as $$x \rightarrow \infty$$, $$y \rightarrow 0$$, so the graph has an asymptote $$y=0$$.
• Find two other points.
• Plot the $$y$$-intercept, $$(0,1)$$, along with the two other points. We will use $$(−1,4)$$ and $$(1,0.25)$$.

Draw a smooth curve connecting the points as in Figure $$\PageIndex{4}$$. Figure $$\PageIndex{4}$$

The domain is $$(−\infty,\infty)$$; the range is $$(0,\infty)$$; the horizontal asymptote is $$y=0$$. $$\PageIndex{1}$$ Sketch the graph of $$f(x)=4^x$$. State the domain, range, and asymptote.

The domain is $$(−\infty,\infty)$$; the range is $$(0,\infty)$$; the horizontal asymptote is $$y=0$$. ## 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 toolkit 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 toolkit 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 toolkit 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 $$\PageIndex{5}$$. Figure $$\PageIndex{5}$$

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 toolkit 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 toolkit 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, $$h(x)=2^{x−3}$$. $$h(x)=2^{x−3}$$. Both horizontal shifts are shown in Figure $$\PageIndex{6}$$. Figure $$\PageIndex{6}$$

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 $$2^{x+3}=2^32^x=(8)2^x$$, 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 $$\left(0,\dfrac{1}{8}\right)$$. 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 toolkit FUNCTION $$f(x) = b^x$$

For any constants $$c$$ and $$d$$,the function $$f(x)=b^{x+c}+d$$ shifts the toolkit function $$f(x)=b^x$$

• 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 $$(0,b^c+d)$$.
• The horizontal asymptote becomes $$y=d$$.
• The range becomes $$(d,\infty)$$.
• The domain, $$(−\infty,\infty)$$,remains unchanged. Given an exponential function with the form $$f(x)=b^{x+c}+d$$, graph the translation

1. Draw the horizontal asymptote $$y=d$$.
2. Shift the graph of $$f(x)=b^x$$ left $$c$$ units if $$c$$ is positive, and right $$c$$ units if $$c$$ is negative.
3. Shift the graph of $$f(x)=b^x$$ up $$d$$ units if $$d$$ is positive, and down $$d$$ units if $$d$$ is negative.
4. State the domain, $$(−\infty,\infty)$$, the range, $$(d,\infty)$$, and the horizontal asymptote $$y=d$$.

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 $$y=−3$$.

Shift the graph of $$f(x)=b^x$$ left $$1$$ unit and down $$3$$ units. Figure $$\PageIndex{7}$$

The domain is $$(−\infty,\infty)$$; the range is $$(−3,\infty)$$; the horizontal asymptote is $$y=−3$$. $$\PageIndex{2}$$ Graph $$f(x)=2^{x−1}+3$$. State domain, range, and asymptote.

The domain is $$(−\infty,\infty)$$; the range is $$(3,\infty)$$; the horizontal asymptote is $$y=3$$.  Given an equation of the form $$f(x)=b^{x+c}+d$$, use a graphing calculator to approximate the solution for $$x$$

• Press [Y=]. Enter the given exponential equation in the line headed “Y1=”.
• Enter the given value for $$f(x)$$ in the line headed “Y2=”.
• Press [WINDOW]. Adjust the $$y$$-axis so that it includes the value entered for “Y2=”.
• Press [GRAPH] to observe the graph of the exponential function along with the line for the specified value of $$f(x)$$.
• To find the value of $$x$$,we compute the point of intersection. Press [2ND] then [CALC]. Select “intersect” and press [ENTER] three times. The point of intersection gives the value of $$x$$for the indicated value of the function.

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

Solve $$42=1.2{(5)}^x+2.8$$ graphically. Round to the nearest thousandth.

Solution

Press [Y=] and enter $$1.2{(5)}^x+2.8$$ next to Y1=. Then enter $$42$$ next to Y2=. For a window, use the values $$–3$$ to $$3$$ for $$x$$ and $$–5$$ to $$55$$ for $$y$$. Press [GRAPH]. The graphs should intersect somewhere near $$x=2$$.

For a better approximation, press [2ND] then [CALC]. Select [5: intersect] and press [ENTER] three times. The x-coordinate of the point of intersection is displayed as $$2.1661943$$. (Your answer may be different if you use a different window or use a different value for Guess?) To the nearest thousandth, $$x≈2.166$$. $$\PageIndex{3}$$ Solve $$4=7.85{(1.15)}^x−2.27$$ graphically. Round to the nearest thousandth.

$$x≈−1.608$$

### 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 toolkit function $$f(x)=b^x$$ by a constant $$|a|>0$$. For example, if we begin by graphing the toolkit 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 $$\PageIndex{8}$$, and the compression, using $$a=\frac{1}{3}$$, to get $$h(x)=\frac{1}{3}{(2)}^x$$ as shown on the right in Figure $$\PageIndex{8}$$. Figure $$\PageIndex{8}$$: (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 $$\frac{1}{3}$$.

STRETCHES AND COMPRESSIONS OF THE toolkit FUNCTION $$f(x)=b^x$$

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

• 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 $$(0,a)$$.
• has a horizontal asymptote at $$y=0$$, a range of $$(0,\infty)$$, and a domain of $$(−\infty,\infty)$$, which are unchanged from the parent function.

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 graph will increase without bound as $$x \rightarrow -\infty$$, and the graph will approach the $$x$$-axis as $$x \rightarrow \infty$$.
• Since $$a=4$$,the graph of $$f(x)=\left(\frac{1}{2}\right)^x$$ will be stretched by a factor of $$4$$.
• Plot the y-intercept, $$(0,4)$$, along with two other points. We will use $$(−1,8)$$ and $$(1,2)$$.

Draw a smooth curve connecting the points, as shown in Figure $$\PageIndex{9}$$. Figure $$\PageIndex{9}$$

The domain is $$(−\infty,\infty)$$; the range is $$(0,\infty)$$; the horizontal asymptote is $$y=0$$. $$\PageIndex{4}$$ Sketch the graph of $$f(x)=\frac{1}{2}{(4)}^x$$. State the domain, range, and asymptote.

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

In addition to shifting, compressing, and stretching a graph, we can also reflect it across the x-axis or the y-axis. When we multiply the toolkit function $$f(x)=b^x$$ by $$−1$$,we get a reflection across the x-axis. When we multiply the input by $$−1$$,we get a reflection across the y-axis. For example, if we begin by graphing the toolkit function $$f(x)=2^x$$, we can then graph the two reflections alongside it. The reflection across the $$x$$-axis, $$g(x)=−2^x$$, is shown on the left side of Figure $$\PageIndex{10}$$, and the reflection across the $$y$$-axis $$h(x)=2^{−x}$$, is shown on the right side of Figure $$\PageIndex{10}$$. Figure $$\PageIndex{10}$$: (a) $$g(x)=−2^x$$ reflects the graph of $$f(x)=2^x$$ across the x-axis. (b) $$g(x)=2^{−x}$$ reflects the graph of $$f(x)=2^x$$ across the $$y$$-axis.

REFLECTIONS OF THE toolkit FUNCTION $$f(x) = b^x$$

The function $$f(x)=−b^x$$

• reflects the toolkit function $$f(x)=b^x$$ across the x-axis.
• has a y-intercept of $$(0,−1)$$.
• has a range of $$(−\infty,0)$$
• has a horizontal asymptote at $$y=0$$ and domain of $$(−\infty,\infty)$$,which are unchanged from the parent function.

The function $$f(x)=b^{−x}$$

• reflects the toolkit function $$f(x)=b^x$$ across the $$y$$-axis.
• has a $$y$$-intercept of $$(0,1)$$, a horizontal asymptote at $$y=0$$, a range of $$(0,\infty)$$, and a domain of $$(−\infty,\infty)$$, which are unchanged from the toolkit function.

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

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

Solution

Since we want to reflect the parent function $$f(x)={(\dfrac{1}{4})}^x$$ about the x-axis, we multiply $$f(x)$$ by $$−1$$ to get, $$g(x)=−{(\dfrac{1}{4})}^x$$. Plot the y-intercept, $$(0,−1)$$,along with two other points. We will use $$(−1,−4)$$ and $$(1,−0.25)$$.

Draw a smooth curve connecting the points: Figure $$\PageIndex{11}$$

The domain is $$(−\infty,\infty)$$; the range is $$(−\infty,0)$$; the horizontal asymptote is $$y=0$$. $$\PageIndex{5}$$

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

The domain is $$(−\infty,\infty)$$; the range is $$(0,\infty)$$; the horizontal asymptote is $$y=0$$. ### 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 $$\PageIndex{3}$$ to arrive at the general equation for translating exponential functions.

Table $$\PageIndex{3}$$: Translations of the Parent Function $$f(x)=b^x$$
Translation Form
Shift
• Horizontally $$c$$ units to the left
• Vertically $$d$$ units up

$$f(x)=b^{x+c}+d$$

Stretch and Compress
• Stretch if $$|a|>1$$
• Compression if $$0<|a|<1$$

$$f(x)=ab^x$$

Reflect across the x-axis

$$f(x)=−b^x$$

Reflect across the y-axis

$$f(x)=b^{−x}=\left(\frac{1}{b}\right)^x$$

General equation for all translations

$$f(x)=ab^{x+c}+d$$

TRANSLATIONS OF EXPONENTIAL FUNCTIONS

A translation of an exponential function has the form

$$f(x)=ab^{x+c}+d$$

Where the toolkit function, $$y=b^x$$, $$b>1$$,is

• shifted horizontally $$c$$ units to the left.
• stretched vertically by a factor of $$|a|$$ if $$|a|>0$$.
• compressed vertically by a factor of $$|a|$$ if $$0<|a|<1$$.
• shifted vertically $$d$$ units.
• reflected across the x-axis when $$a<0$$.

Note that the order of the shifts, transformations, and reflections follows the order of operations.

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 $$f(x)=ab^{x+c}+d$$.  We use the description provided to find $$a, b, c,$$ and $$d$$.

• We are given the toolkit function $$f(x)=e^x$$, so $$b=e$$. Note: $$e$$ is a number, not a variable. It was defined in Section 4.1. Write down its definition, hand it it to your professor, and make her give you extra credit!
• 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: $$e^{−x}$$.
• There is no horizontal shift, so $$c=0$$.
• The graph is shifted vertically 4 units, so $$d=4$$.

Substituting in the general form we get,

$$f(x)=ab^{x+c}+d$$

$$=2e^{−x+0}+4$$

$$=2e^{−x}+4$$

The domain is $$(−\infty,\infty)$$; the range is $$(4,\infty)$$; the horizontal asymptote is $$y=4$$. $$\PageIndex{6}$$ 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 $$\dfrac{1}{3}$$, reflected across the x-axis and then shifted down $$2$$ units.

$$f(x)=−\frac{1}{3}e^{x}−2$$; the domain is $$(−\infty,\infty)$$; the range is $$(−\infty,2)$$; the horizontal asymptote is $$y=-2$$.

Media

Access this online resource for additional instruction and practice with graphing exponential functions.

## Key Equations

 General form for the translation of the toolkit function $$f(x)=b^x$$: $$f(x)=ab^{x+c}+d$$

## Key Concepts

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