Graphs of Exponential Functions

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 change as the input increases by $1$.

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.

The graph below shows the exponential growth function $f(x)=2^x$.

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

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

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 $g(x)=\left(\tfrac{1}{2}\right)^x$.

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

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

Example. 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.

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

• 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 below.

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 parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent 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 parent 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 parent 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 below.

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 parent 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 parent 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 below.

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 $\left(0,\tfrac{1}{8}\right)$. Again, see that $2^{x-3}=\left(\tfrac{1}{8}\right)2^x$, so the initial value of the function is $\tfrac{1}{8}$.

Example. 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 unit and down 3 units.

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

Example. 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 a different value for Guess?.) To the nearest thousandth, $x\approx2.166$.

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 $f(x)=b^x$ by a constant $|a|>0$. For example, if we begin by graphing the parent function $f(x)=2^x$, we can then graph the stretch, using $a=3$, to get $g(x)=3(2)^x$, and the compression, using $a=\tfrac{1}{3}$, to get $h(x)=\tfrac{1}{3}(2)^x$.

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

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

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

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

• Create a table of points.

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

• 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 below.

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 about the $x$-axis or the $y$-axis. When we multiply the parent 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 parent function $f(x)=2^x$, we can then graph the two reflections alongside it: the reflection about the $x$-axis, $g(x)=-2^x$, and the reflection about the $y$-axis, $h(x)=2^{-x}$.

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

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

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

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:

The domain is $(-\infty,\infty)$; the range is $(-\infty,0)$; 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 below to arrive at the general equation for translating exponential functions.

Example. 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 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,

The domain is $(-\infty,\infty)$; the range is $(4,\infty)$; the horizontal asymptote is $y=4$.

Key equations

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 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 $f(x)=b^x+d$ represents a vertical shift of the parent function $f(x)=b^x$.
• The equation $f(x)=b^{x+c}$ represents a horizontal shift of the parent 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 parent function $f(x)=b^x$.
• When the parent function $f(x)=b^x$ is multiplied by $-1$, the result, $f(x)=-b^x$, is a reflection about the $x$-axis. When the input is multiplied by $-1$, the result, $f(x)=b^{-x}$, is a reflection about 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.

Practice

Graph exponential functions

Graph exponential functions using transformations

Each graph below is a transformation of $y=2^x$. Write an equation describing the transformation.

This section is adapted from Precalculus 2e, Section 4.2: Graphs of Exponential Functions by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated every graph as an accessible inline SVG generated from its exact equation — the parent growth curve $f(x)=2^x$ and its seven tabulated points; the parent decay curve $g(x)=\left(\tfrac{1}{2}\right)^x$ and its seven tabulated points; the generic growth/decay comparison panels $f(x)=b^x$ for $b>1$ and $0<b<1$; the decaying parent $f(x)=0.25^x$ of Example 1; the vertical-shift panel $g(x)=2^x+3$, $f(x)=2^x$, $h(x)=2^x-3$; the horizontal-shift panel $g(x)=2^{x+3}$, $f(x)=2^x$, $h(x)=2^{x-3}$; the shifted curve $f(x)=2^{x+1}-3$ of Example 2; the stretch/compression panels $g(x)=3(2)^x$ and $h(x)=\tfrac{1}{3}(2)^x$ against $f(x)=2^x$; the stretched curve $f(x)=4\left(\tfrac{1}{2}\right)^x$ of Example 4; the reflection-about-$x$-axis and reflection-about-$y$-axis panels $g(x)=-2^x$ and $h(x)=2^{-x}$ against $f(x)=2^x$; the reflected curve $g(x)=-\left(\tfrac{1}{4}\right)^x$ of Example 5; and the unlabeled Practice graph of $y=-2^x+3$. Omitted the section’s Media link to an external graphing-calculator resource, which carries no additional mathematics. Converted every “Try It” into interactive components with instant feedback; the shifted-curve Try It $f(x)=2^{x-1}+3$ is graded as a plot-the-points graph exercise at the five named inputs $x=0$ through $4$ (its outputs land on a half-unit grid), while the other Try Its that ask to “sketch” or “graph” an exponential — whose graphs offer fewer than five grid-reachable points — instead ask for the domain, range, asymptote, or a derived equation that the source’s own solution states in words, with the curve itself not graded. After the first full domain/range/asymptote drill (Try It 1), later Try Its omit repeating the domain, which is $(-\infty, \infty)$ for every function in this section and so carries no new information once established; each still preserves every quantity whose value actually changes with the transformation shown. Adapted thirteen selected end-of-section exercises — the horizontal-asymptote conceptual question, two numeric evaluations, two algebraic reflect-and-stretch/reflect-and-shift compositions with their $y$-intercepts, a vertical-shift asymptote-and-range pair, an end-behavior description split into its two one-sided parts, two single-transformation equation-writing prompts, one graph-reading equation-writing prompt, and two graphing-calculator approximations — into sixteen interactive components in a closing Practice block, one group per objective.