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 the table below 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 [latex]f\left(x\right)=a{b}^{x}[/latex], 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
• as x decreases, the output values grow smaller, approaching zero

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

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

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

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 the table below 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 [latex]\frac{1}{2}[/latex].

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
• as x decreases, the output values grow without bound

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

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

Example: Sketching the Graph of an Exponential Function of the Form [latex]f\left(x\right)={b}^{x}[/latex]

Sketch a graph of [latex]f\left(x\right)={0.25}^{x}[/latex]. 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
  [latex]f\left(x\right)={0.25}^{x}[/latex]	64	16	4	1	0.25	0.0625	0.015625

• 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.

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

Watch the following video for another example of graphing an exponential function. The base of the exponential term is between [latex]0[/latex] and [latex]1[/latex], so this graph will represent decay.

The next example shows how to plot an exponential growth function where the base is greater than [latex]1[/latex].

Example

Sketch a graph of [latex]f(x)={\sqrt{2}(\sqrt{2})}^{x}[/latex]. State the domain and range.

Show Solution

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

• Since b = [latex]\sqrt{2}[/latex], which is greater than one, we know the function is increasing, and we can verify this by creating a table of values. The left tail of the graph will get really close to the x-axis and the right tail will increase without bound.
• Create a table of points.
  

  x	[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)=\sqrt{2}{(\sqrt{2})}^{x}[/latex]	[latex]0.5[/latex]	[latex]0.71[/latex]	[latex]1[/latex]	[latex]1.41[/latex]	[latex]2[/latex]	[latex]2.83[/latex]	[latex]4[/latex]

• Plot the y-intercept, [latex]\left(0,1.41\right)[/latex], along with two other points. We can use [latex]\left(-1,1\right)[/latex] and [latex]\left(1,2\right)[/latex].

Draw a smooth curve connecting the points.

The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(0,\infty \right)[/latex].

The next video example includes graphing an exponential growth function and defining the domain and range of the function.

Try It

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

Show Solution

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