This is a problem asking you to graph the exponential function:

$f(x) = \frac{3}{2} \cdot (3)^x$

Steps to Solve:

1. Understand the formula:

   • This is an exponential growth function where the base is $3$ and the coefficient is $\frac{3}{2}$.

2. Generate Points:

   • To graph this, choose several $x$-values (e.g., -2, -1, 0, 1, 2) and calculate $f(x)$.

Calculations:

$f(x) = \frac{3}{2} \cdot 3^x$

• For $x = -2$: $f(-2) = \frac{3}{2} \cdot 3^{-2} = \frac{3}{2} \cdot \frac{1}{9} = \frac{1}{6}$
• For $x = -1$: $f(-1) = \frac{3}{2} \cdot 3^{-1} = \frac{3}{2} \cdot \frac{1}{3} = \frac{1}{2}$
• For $x = 0$: $f(0) = \frac{3}{2} \cdot 3^0 = \frac{3}{2} \cdot 1 = \frac{3}{2} = 1.5$
• For $x = 1$: $f(1) = \frac{3}{2} \cdot 3^1 = \frac{3}{2} \cdot 3 = 4.5$
• For $x = 2$: $f(2) = \frac{3}{2} \cdot 3^2 = \frac{3}{2} \cdot 9 = 13.5$

Points to Plot:

$(-2, \frac{1}{6}), (-1, \frac{1}{2}), (0, 1.5), (1, 4.5), (2, 13.5)$

Graph:

• Plot these points on the graph.
• Connect them smoothly as the curve increases exponentially for $x > 0$ and approaches $y = 0$ for $x < 0$.

Would you like help with a visualization or further explanation?

Related Questions:

1. What is the horizontal asymptote of $f(x) = \frac{3}{2} \cdot 3^x$?
2. How does changing the coefficient ($\frac{3}{2}$) affect the graph?
3. How does changing the base ($3$) affect the growth rate of the function?
4. Can you identify the domain and range of $f(x)$?
5. How would the graph change for $f(x) = \frac{3}{2} \cdot (3)^{-x}$?

Tip:

When graphing exponential functions, focus on key points like $x = -2, -1, 0, 1, 2$ for clarity and note the rapid growth or decay!