Graphing Exponential Functions

Exponential functions have the form $f(x) = a \\cdot b^{x}$, where:

• $a$ is the initial value (the y-intercept),
• $b$ is the base (growth factor if $b > 1$, decay factor if $0 < b < 1$).

To graph an exponential function, follow these steps:

1. Identify the values of $a$ and $b$.
2. Plot the y-intercept at $(0, a)$.
3. Choose several values for $x$ to find corresponding $f(x)$ values.
4. Plot the points and draw a smooth curve through them.
5. Note the horizontal asymptote, usually at $y = 0$.

Example: Graph $f(x) = 2 \\cdot 3^{x}$

1. Identify $a = 2$ and $b = 3$.
2. Plot the y-intercept: at $x = 0$, $f(0) = 2 \\cdot 3^{0} = 2$. So, plot the point $(0, 2)$.
3. Choose other values:
   • For $x = 1$: $f(1) = 2 \\cdot 3^{1} = 6$. Plot $(1, 6)$.
   • For $x = -1$: $f(-1) = 2 \\cdot 3^{-1} = \frac{2}{3}$. Plot $(-1, \frac{2}{3})$.
4. Draw a smooth curve through the points $(0, 2)$, $(1, 6)$, and $(-1, \frac{2}{3})$.
5. The horizontal asymptote is $y = 0$.

Key Questions:

• What is the effect of changing the value of $a$ on the graph?
• How does the base $b$ influence the growth or decay of the function?
• What point always exists on the graph of an exponential function?
• How do you determine the horizontal asymptote of an exponential function?
• Can you identify exponential growth and decay in real-world scenarios?

Keywords:

• Exponential Function
• Graphing
• Asymptote
• Growth Factor
• Decay Factor