Exponential and Logarithmic Functions
5.3 Graphs of Exponential Functions
Core Graph Features
Exponential graphs are smooth, never cross the horizontal asymptote, and stay positive when $a>0$.
Fact: Features of $f(x)=ab^x$ with $a>0$
- Domain: $(-\infty,\infty)$
- Range: $(0,\infty)$
- Horizontal asymptote: $y=0$
- y-intercept: $(0,a)$
Example: Find key graph features
For
$$f(x)=4\left(\frac13\right)^x,$$
find y-intercept, asymptote, and whether it grows or decays.
Show Solution
At $x=0$:
$$f(0)=4,$$
so y-intercept is $(0,4)$.
Base $\frac13$ is between 0 and 1, so decay.
Horizontal asymptote is $y=0$.
MyOpenMath: Practice: exponential graph features (placeholder)
Transforming Exponential Graphs
Use known parent graph behavior and apply shifts or reflections.
Example: Analyze a transformed exponential
Describe the graph of
$$g(x)=2^{x-1}+3.$$
Show Solution
Start with $2^x$.
- $x-1$ shifts right 1.
- $+3$ shifts up 3.
New horizontal asymptote is
$$y=3.$$
