Chapter 6 Exponential and Logarithmic Functions
6.2 Graphs of Exponential Functions
Learning Objectives
In this section, you will:
- Graph exponential functions.
- Graph exponential functions using transformations.
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]
| [latex]x[/latex] | [latex]-3[/latex] | [latex]-2[/latex] | [latex]-1[/latex] | [latex]0[/latex] | [latex]1[/latex] | [latex]2[/latex] | [latex]3[/latex] |
| f(x) = 2x | [latex]\frac{1}{8}[/latex] | [latex]\frac{1}{4}[/latex] | [latex]\frac{1}{2}[/latex] | [latex]1[/latex] | 2 | [latex]4[/latex] | [latex]8[/latex] |
Table 1
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][latex]\,b\,[/latex]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[latex]\,a.[/latex]
Notice from the table that
- the output values are positive for all values of [latex]x;[/latex]
- as[latex]\,x\,[/latex]increases, the output values increase without bound; and
- as[latex]\,x\,[/latex]decreases, the output values grow smaller, approaching zero.
Figure 1 shows the exponential growth function [latex]\,f\left(x\right)={2}^{x}.[/latex]
| [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(x)=\left(\frac{1}{2}\right)^{x}[/latex] | [latex]8[/latex] | [latex]4[/latex] | [latex]2[/latex] | [latex]1[/latex] | [latex]\frac{1}{2}[/latex] | [latex]\frac{1}{4}[/latex] | [latex]\frac{1}{8}[/latex] |
Table 2
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[latex]\,x;[/latex]
- as[latex]\,x\,[/latex]increases, the output values grow smaller, approaching zero; and
- as[latex]\,x\,[/latex]decreases, the output values grow without bound.
Figure 2 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]
Based on Table A, answer the following below:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| f(x) | 27 | 9 | 3 | 1 | [latex]\frac