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 [latex]f\left(x\right)={b}^{x}[/latex] whose base is greater than one. We’ll use the function [latex]f\left(x\right)={2}^{x}.[/latex] Observe how the output values in Table 1 change as the input increases by [latex]1.[/latex]

Each output value is the product of the previous output and the base, [latex]2.[/latex] We call the base [latex]2[/latex] the constant ratio or growth factor. In fact, for any exponential function with the form [latex]f\left(x\right)=a{b}^{x},[/latex] [latex]b[/latex] is the constant ratio of the function. This means that each time we increase the input by 1, we multiply the output by [latex]b[/latex]. Notice from the table that the output values are positive for all values of [latex]x.[/latex]

End Behavior of [latex]f\left(x\right)=ab^x[/latex]

Often we want to know what happens to the output value of [latex]f\left(x\right)[/latex] as [latex]x[/latex] moves far to the left or far to the right.  This is known as the end behavior or long term behavior of the function.  We will continue to study the function [latex]f\left(x\right)=2^x[/latex] and determine its end behavior.

Begin by looking at two tables of values.  Table 2 shows function values as x moves far to the left.  We choose x values of -10, -100 and -250 and evaluate [latex]f\left(x\right)[/latex] at these values so we can observe what is happening to the output values to the far left.  Note that -250 is not considered long term behavior for most functions but for an exponential function it is about the limit of what our current technology can compute.  In Table 3, we move far toward the right choosing our x values to be 10, 100, and 250 and evaluate the function so we can observe what happens to the output when x gets large.

Recognize that scientific notation is being used in Table 2 and Table 3 for the output values.  In Table 2, we observe that as [latex]x[/latex] decreases or becomes more and more negative, the output values get closer and closer to zero from above.  We capture this idea using arrow notation and write as [latex]x\to-\infty,f\left(x\right)\to0.[/latex] This is read, “As x decreases without bound, [latex]f[/latex] of x goes to zero.” When we are studying end behavior and we observe that the output is getting closer and closer to a value, we say that there is a horizontal asymptote.  In this example, [latex]y=0[/latex] is the horizontal asymptote on the left hand side.

Further, Table 3 shows that as [latex]x[/latex] increases or becomes larger and larger, the output values also become larger and larger or increase without bound.  We write as [latex]x\to\infty,f\left(x\right)\to\infty.[/latex] Since these output values increase without bound, there is not a horizontal asymptote in this direction.

Figure 1 shows the exponential growth function [latex]f\left(x\right)={2}^{x}.[/latex]

Figure 1: Notice that the graph gets close to the x-axis, but never touches it.

We observe in the graph above that as x becomes more negative, the graph is getting closer to the x-axis but never touches it demonstrating the horizontal asymptote of [latex]y=0.[/latex]  Other characteristics of the graph can also be observed.  The domain of [latex]f\left(x\right)={2}^{x}[/latex] is all real numbers, and the range is [latex]\left(0,\infty \right).[/latex]  Notice that the function is also increasing and concave up on [latex]\left(-\infty,\infty \right).[/latex]

Exponential Decay Graphically

To get a sense of the behavior of exponential decay, we can create a table of values for a function of the form [latex]f\left(x\right)={b}^{x}[/latex] whose base is between zero and one. We’ll use the function [latex]g\left(x\right)={\left(\frac{1}{2}\right)}^{x}.[/latex] Observe how the output values in Table 4 change as the input increases by 1.

Again, notice that each time the input is increased by 2, the output is multiplied by the base, or constant ratio [latex]b=\frac{1}{2}.[/latex]

To look at the end behavior of the exponential decay function, we again create tables with input values to the far left and right.

Notice from the tables above that:

• the output values are positive for all values of [latex]x.[/latex]
• as [latex]x[/latex] decreases, the output values grow without bound so as [latex]x\to-\infty,g\left(x\right)\to\infty.[/latex]
• as [latex]x[/latex] increases without bound, the output values approach zero from above so as [latex]x\to\infty,g\left(x\right)\to0.[/latex] The horizontal asymptote is [latex]y=0[/latex] on the right hand side.

Figure 2 shows the exponential decay function, [latex]g\left(x\right)={\left(\frac{1}{2}\right)}^{x}.[/latex]

Figure 2 The graph shows that as x gets larger, the output gets close to zero.

Again we observe the end behavior and see that as x increases, the graph approaches the x-axis and there is a horizontal asymptote of [latex]y=0.[/latex]  Other characteristics can also be observed from the graph.  The domain of [latex]g\left(x\right)={\left(\frac{1}{2}\right)}^{x}[/latex] is all real numbers, and the range is [latex]\left(0,\infty \right).[/latex]  Notice that the function is decreasing and concave up on [latex]\left(-\infty,\infty \right).[/latex]

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

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

• one-to-one function
• horizontal asymptote: [latex]y=0[/latex] on one side
• domain: [latex]\left(–\infty , \infty \right)[/latex]
• range: [latex]\left(0,\infty \right)[/latex]
• x-intercept: none
• y-intercept: [latex]\left(0,1\right)[/latex]
• increasing if [latex]b>1[/latex]
• decreasing if [latex]b<1[/latex]
• concave up

Figure 3 compares the graphs of exponential growth and decay functions.

Figure 3

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

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

Show Solution

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

• Since [latex]b=0.25[/latex] is between zero and one, we know the function is decreasing. The left end behavior of the graph will increase without bound, and the right end behavior will approach the horizontal asymptote [latex]y=0.[/latex]
• Create a table of points as in Table 7.
  

  Table 7
  [latex]x[/latex]	[latex]-3[/latex]	[latex]-2[/latex]	[latex]-1[/latex]	[latex]0[/latex]	[latex]1[/latex]	[latex]2[/latex]	[latex]3[/latex]
  [latex]f\left(x\right)={0.25}^{x}[/latex]	[latex]64[/latex]	[latex]16[/latex]	[latex]4[/latex]	[latex]1[/latex]	[latex]0.25[/latex]	[latex]0.0625[/latex]	[latex]0.015625[/latex]

• Plot the y-intercept, [latex]\left(0,1\right),[/latex] along with two other points. We can use [latex]\left(-1,4\right)[/latex] and [latex]\left(1,0.25\right).[/latex]

Draw a smooth curve connecting the points as in Figure 4.

Figure 4

The domain is [latex]\left(-\infty ,\infty \right);[/latex] the range is [latex]\left(0,\infty \right);[/latex] the horizontal asymptote is [latex]y=0[/latex] on the right side.

Graphing Transformations of Exponential Functions

Transformations of exponential graphs behave similarly to those of other functions. Just as with our toolkit functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the exponential function [latex]f\left(x\right)={b}^{x}[/latex] 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.

NOTE: In this section we will be using different notation for horizontal and vertical shifts. Recall that in section 1.6, we considered functions in the form [latex]g\left(x\right)=af\left(b\left(x-h\right)\right)+k.[/latex] In this notation, [latex]k[/latex] indicated the vertical shift, [latex]h[/latex] indicated the horizontal shift, [latex]a[/latex] indicated the vertical stretch/compression, and [latex]b[/latex] indicated the horizontal stretch/compression.  When we study exponential functions, we have already designated [latex]k[/latex] to indicate continuous growth.  Therefore, we will modify our notation and use [latex]c[/latex] to represent horizontal shifts and [latex]d[/latex] to represent vertical shifts. Vertical stretches/compressions will still be represented by [latex]a.[/latex] The variable [latex]b[/latex] will represent the base of the exponential function and not represent a horizontal stretch or compression.

Graphing a Vertical Shift

The first transformation occurs when we add a constant [latex]d[/latex] to the exponential function [latex]f\left(x\right)={b}^{x},[/latex] giving us a vertical shift [latex]d[/latex] units in the same direction as the sign. For example, if we begin by graphing the function, [latex]f\left(x\right)={2}^{x},[/latex] we can then graph two vertical shifts alongside it, using [latex]d=3[/latex] and [latex]d=-3:[/latex] the upward shift, [latex]g\left(x\right)=f\left(x\right)+3={2}^{x}+3[/latex] and the downward shift, [latex]h\left(x\right)=f\left(x\right)-3={2}^{x}-3.[/latex] Both vertical shifts are shown in Figure 5.

Figure 5

Observe the results of shifting [latex]f\left(x\right)={2}^{x}[/latex] vertically:

• The domain, [latex]\left(-\infty ,\infty \right)[/latex] remains unchanged.
• When the function is shifted up [latex]3[/latex] units to [latex]g\left(x\right)={2}^{x}+3:[/latex]
  • The y-intercept shifts up [latex]3[/latex] units to [latex]\left(0,4\right).[/latex]
  • The horizontal asymptote shifts up [latex]3[/latex] units to [latex]y=3[/latex] on the left side.
  • The range becomes [latex]\left(3,\infty \right).[/latex]
• When the function is shifted down [latex]3[/latex] units to [latex]h\left(x\right)={2}^{x}-3:[/latex]
  • The y-intercept shifts down [latex]3[/latex] units to [latex]\left(0,-2\right).[/latex]
  • The horizontal asymptote also shifts down [latex]3[/latex] units to [latex]y=-3[/latex] on the left side.
  • The range becomes [latex]\left(-3,\infty \right).[/latex]

Graphing a Horizontal Shift

The next transformation occurs when we subtract a constant [latex]c[/latex] from the input of the exponential function [latex]f\left(x\right)={b}^{x},[/latex] giving us a horizontal shift [latex]c[/latex] units in the direction of the sign of [latex]c[/latex]. The equation is given by [latex]f\left(x-c\right)=b^{x-c}.[/latex] For example, if we begin by graphing the function [latex]f\left(x\right)={2}^{x},[/latex] we can then graph two horizontal shifts alongside it, using

[latex]c=-3:[/latex] the shift left, [latex]g\left(x\right)=f\left(x+3\right)={2}^{x+3},[/latex] and using

[latex]c=3:[/latex] the shift right, [latex]h\left(x\right)=f\left(x-3\right)={2}^{x-3}.[/latex]

Both horizontal shifts are shown in Figure 6.

Figure 6

Observe the results of shifting [latex]f\left(x\right)={2}^{x}[/latex] horizontally:

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

Shifts of the Function y = b^x

For any constants [latex]c[/latex] and [latex]d,[/latex] the function [latex]f\left(x\right)={b}^{x-c}+d[/latex] shifts the exponential function [latex]y={b}^{x}[/latex]

• vertically [latex]d[/latex] units, in the direction of the sign of [latex]d,[/latex] and
• horizontally [latex]c[/latex] units, in the direction of the sign of [latex]c.[/latex]
• The vertical intercept becomes [latex]\left(0,{b}^{-c}+d\right).[/latex]
• The horizontal asymptote becomes [latex]y=d[/latex] on the same side.
• The range becomes [latex]\left(d,\infty \right).[/latex]
• The domain, [latex]\left(-\infty ,\infty \right),[/latex] remains unchanged.

How To

Given an exponential function with the form [latex]f\left(x\right)={b}^{x-c}+d,[/latex] graph the translation.

1. Draw the horizontal asymptote [latex]y=d.[/latex]
2. Identify [latex]c[/latex] and [latex]d[/latex].  Shift the graph of [latex]y={b}^{x}[/latex] right [latex]c[/latex] units if [latex]c[/latex] is positive, and left [latex]c[/latex] units if [latex]c[/latex] is negative.
3. Shift the graph of [latex]y={b}^{x}[/latex] up [latex]d[/latex] units if [latex]d[/latex] is positive, and down [latex]d[/latex] units if [latex]d[/latex] is negative.
4. State the domain, [latex]\left(-\infty ,\infty \right),[/latex] the range, [latex]\left(d,\infty \right),[/latex] and the horizontal asymptote [latex]y=d.[/latex]

Example 2:  Graphing a Shift of an Exponential Function

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

Show Solution

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

Draw the horizontal asymptote [latex]y=d[/latex], so draw [latex]y=-3.[/latex]  Shift the graph of [latex]y={2}^{x}[/latex] left 1 unit and down 3 units.

Figure 7

The domain is [latex]\left(-\infty ,\infty \right);[/latex] the range is [latex]\left(-3,\infty \right);[/latex] the horizontal asymptote is [latex]y=-3[/latex] on the left side.

Try it #2

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

Show Solution

The domain is [latex]\left(-\infty ,\infty \right);[/latex] the range is [latex]\left(3,\infty \right);[/latex] the horizontal asymptote is [latex]y=3[/latex] on the left side.

Graphing a Vertical 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 exponential function [latex]f\left(x\right)={b}^{x}[/latex] by a constant [latex]|a|>0.[/latex] For example, if we begin by graphing the function [latex]f\left(x\right)={2}^{x},[/latex] we can then graph the vertical stretch, using [latex]a=3,[/latex] to get [latex]g\left(x\right)=3f\left(x\right)=3{\left(2\right)}^{x}[/latex] as shown on the left in Figure 8a, and the vertical compression, using [latex]a=\frac{1}{3},[/latex] to get [latex]h\left(x\right)=\frac{1}{3}f\left(x\right)=\frac{1}{3}{\left(2\right)}^{x}[/latex] as shown on the right in Figure 8b.

Figure 8. (a) [latex]g\left(x\right)=3{\left(2\right)}^{x}[/latex] stretches the graph of [latex]f\left(x\right)={2}^{x}[/latex] vertically by a factor of [latex]3.[/latex] (b) [latex]h\left(x\right)=\frac{1}{3}{\left(2\right)}^{x}[/latex] compresses the graph of [latex]f\left(x\right)={2}^{x}[/latex] vertically by a factor of [latex]\frac{1}{3}.[/latex]

Stretches and Compressions of the Function y = b^x

For any factor [latex]a>0,[/latex] the function [latex]f\left(x\right)=a{\left(b\right)}^{x}[/latex]

• stretches [latex]y={b}^{x}[/latex] vertically by a factor of [latex]a[/latex] if [latex]|a|>1.[/latex]
• compresses [latex]y={b}^{x}[/latex] vertically by a factor of [latex]a[/latex] if [latex]|a|<1.[/latex]
• has a vertical intercept of [latex]\left(0,a\right).[/latex]
• has a horizontal asymptote at [latex]y=0,[/latex] a range of [latex]\left(0,\infty \right),[/latex] and a domain of [latex]\left(-\infty ,\infty \right),[/latex] which are unchanged from [latex]y={b}^{x}.[/latex]

Example 3:  Graphing the Stretch of an Exponential Function

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

Show Solution

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

• Since [latex]b=\frac{1}{2}[/latex] is between zero and one, the left end behavior of the graph will increase without bound as [latex]x[/latex] decreases without bound, and the right end behavior will approach the x-axis as [latex]x[/latex] increases without bound.
• Since [latex]a=4,[/latex] the graph of [latex]y={\left(\frac{1}{2}\right)}^{x}[/latex] will be stretched by a factor of [latex]4.[/latex]
• Create a table of points as shown in Table 8.
  

  Table 8

  [latex]x[/latex]	[latex]-3[/latex]	[latex]-2[/latex]	[latex]-1[/latex]	[latex]0[/latex]	[latex]1[/latex]	[latex]2[/latex]	[latex]3[/latex]
  [latex]f\left(x\right)=4\left(\frac{1}{2}\right)^{x}[/latex]	[latex]32[/latex]	[latex]16[/latex]	[latex]8[/latex]	[latex]4[/latex]	[latex]2[/latex]	[latex]1[/latex]	[latex]0.5[/latex]

• Plot the vertical intercept, [latex]\left(0,4\right),[/latex] along with two other points. We can use [latex]\left(-1,8\right)[/latex] and [latex]\left(1,2\right).[/latex]

Draw a smooth curve connecting the points, as shown in Figure 9.

Figure 9

The domain is [latex]\left(-\infty ,\infty \right);[/latex] the range is [latex]\left(0,\infty \right);[/latex] the horizontal asymptote is [latex]y=0[/latex] on the right side.

Try it #3

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

Show Solution

The domain is [latex]\left(-\infty ,\infty \right);[/latex] the range is [latex]\left(0,\infty \right);[/latex] the horizontal asymptote is [latex]y=0[/latex] on the left side.

Graphing Reflections

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

Figure 10 (a) [latex]g\left(x\right)=-{2}^{x}[/latex] reflects the graph of [latex]f\left(x\right)={2}^{x}[/latex] about the x-axis. (b) [latex]g\left(x\right)={2}^{-x}[/latex] reflects the graph of [latex]f\left(x\right)={2}^{x}[/latex] about the y-axis.

Reflections of the  Function y = f(x) = b^x

The function [latex]g\left(x\right)=-{b}^{x}[/latex]

• reflects the function [latex]y={b}^{x}[/latex] over the x-axis.
• has a vertical intercept of [latex]\left(0,-1\right).[/latex]
• has a range of [latex]\left(-\infty ,0\right).[/latex]
• has a horizontal asymptote at [latex]y=0[/latex] and domain of [latex]\left(-\infty ,\infty \right),[/latex] which are unchanged from the function [latex]y={b}^{x}.[/latex]

The function [latex]h\left(x\right)={b}^{-x}[/latex]

• reflects the  function [latex]y={b}^{x}[/latex] over the y-axis.
• has a vertical intercept of [latex]\left(0,1\right),[/latex] a horizontal asymptote at [latex]y=0,[/latex] a range of [latex]\left(0,\infty \right),[/latex] and a domain of [latex]\left(-\infty ,\infty \right),[/latex] which are unchanged from the function [latex]y={b}^{x}.[/latex]

Example 4:  Writing and Graphing the Reflection of an Exponential Function

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

Show Solution

Since we want to reflect the function [latex]f\left(x\right)={\left(\frac{1}{4}\right)}^{x}[/latex] about the x-axis, we multiply [latex]f\left(x\right)[/latex] by [latex]-1[/latex] to get, [latex]g\left(x\right)=-{\left(\frac{1}{4}\right)}^{x}.[/latex] Next we create a table of points as in Table 9.

Table 9

[latex]x[/latex]	[latex]-3[/latex]	[latex]-2[/latex]	[latex]-1[/latex]	[latex]0[/latex]	[latex]1[/latex]	[latex]2[/latex]	[latex]3[/latex]
[latex]g\left(x\right)=-\left(\frac{1}{4}\right)^{x}[/latex]	[latex]-64[/latex]	[latex]-16[/latex]	[latex]-4[/latex]	[latex]-1[/latex]	[latex]-0.25[/latex]	[latex]-0.0625[/latex]	[latex]-0.0156[/latex]

Plot the y-intercept, [latex]\left(0,-1\right),[/latex] along with two other points. We can use [latex]\left(-1,-4\right)[/latex] and [latex]\left(1,-0.25\right).[/latex]

Draw a smooth curve connecting the points:

Figure 11.

The domain is [latex]\left(-\infty ,\infty \right);[/latex] the range is [latex]\left(-\infty ,0\right);[/latex] the horizontal asymptote is [latex]y=0[/latex] on the right side.

Try it #4

Find and graph the equation for a function, [latex]g\left(x\right),[/latex] that reflects [latex]f\left(x\right)={1.25}^{x}[/latex] over the y-axis. State its domain, range, and horizontal asymptote.

Show Solution

The function is [latex]g\left(x\right)={1.25}^{-x}={0.8}^{x}.[/latex] The domain is [latex]\left(-\infty ,\infty \right);[/latex] the range is [latex]\left(0,\infty \right);[/latex] the horizontal asymptote is [latex]y=0[/latex] on the right side.