# Reading: Graphs of Exponential Functions

## Graphical Features of Exponential Functions

Graphically, in the function*f*(

*x*) =

*ab*

^{x}.

*a*is the vertical intercept of the graph.*b*determines the rate at which the graph grows:- the function will increase if
*b*> 1,

- the function will decrease if 0 <
*b*< 1.

- the function will increase if
- The graph will have a horizontal asymptote at
*y*= 0. The graph will be concave up if a>0; concave down if*a*< 0. - The domain of the function is all real numbers.
- The range of the function is (0,∞) if
*a*> 0, and (−∞,0) if*a*< 0.

*a*) and (1,

*ab*). The value

*b*will determine the function's long run behavior:

- If
*b*> 1, as*x*→ ∞,*f*(*x*) → ∞, and as*x*→ −∞,*f*(*x*) → 0. - If 0 <
*b*< 1, as*x*→ ∞,*f*(*x*) → 0, and as*x*→ –∞,*f*(*x*) → ∞.

#### Example 6

Sketch a graph of [latex-display]\displaystyle{f{{({x})}}}={4}{(\frac{{1}}{{3}})}^{{x}}[/latex-display] This graph will have a vertical intercept at (0,4), and pass through the point [latex]\displaystyle{({1},frac{{4}}{{3}})}[/latex]. Since*b*< 1, the graph will be decreasing towards zero. Since

*a*> 0, the graph will be concave up. We can also see from the graph the long run behavior: as

*x*→ ∞ ,

*f*(

*x*) → 0, and as

*x*→ –∞,

*f*(

*x*) → ∞. To get a better feeling for the effect of

*a*and

*b*on the graph, examine the sets of graphs below. The first set shows various graphs, where

*a*remains the same and we only change the value for

*b*. Notice that the closer the value of is to 1, the less steep the graph will be. Changing the value of

*b*. In the next set of graphs,

*a*is altered and our value for

*b*remains the same. Changing the value of

*a*. Notice that changing the value for a changes the vertical intercept. Since

*a*is multiplying the

*b*term,

^{x}*a*acts as a vertical stretch factor, not as a shift. Notice also that the long run behavior for all of these functions is the same because the growth factor did not change and none of these values introduced a vertical flip. Try it for yourself using this applet.

### Example 7

Match each equation with its graph.- [latex]\displaystyle{f{{({x})}}}={2}{({1.3})}^{{x}}[/latex]
- [latex]\displaystyle{g{{({x})}}}={2}{({1.8})}^{{x}}[/latex]
- [latex]\displaystyle{h}{({x})}={4}{({1.3})}^{{x}}[/latex]
- [latex]\displaystyle{k}{({x})}={4}{({0.7})}^{{x}}[/latex]

*k*(

*x*) is the easiest to identify, since it is the only equation with a growth factor less than one, which will produce a decreasing graph. The graph of

*h*(

*x*) can be identified as the only growing exponential function with a vertical intercept at (0,4). The graphs of

*f*(

*x*) and

*g*(

*x*) both have a vertical intercept at (0,2), but since

*g*(

*x*) has a larger growth factor, we can identify it as the graph increasing faster.

Shana Calaway, Dale Hoffman, and David Lippman, Business Calculus, " 1.7: Exponential Functions," licensed under a CC-BY license.