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. 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 [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. 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 [latex]1[/latex], 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.

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].

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 [latex]1[/latex].

Again, because the input is increasing by [latex]1[/latex], 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; and
• as x decreases, the output values grow without bound.

The graph 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]

Characteristics of the Graph of  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
• 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]

Compare the graphs of exponential growth and decay functions.

Example

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

Answer:

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].

Summary