which graph represents the function $f(x) = \\frac{3}{2}(2)^x$?

Explanation:

Step1: Identify base function form

The function is $f(x) = \frac{3}{2}(2)^x$, an exponential growth function of the form $f(x)=ab^x$ where $a=\frac{3}{2}$, $b=2>1$.

Step2: Calculate key points

• For $x=0$: $f(0)=\frac{3}{2}(2)^0=\frac{3}{2}=1.5$
• For $x=1$: $f(1)=\frac{3}{2}(2)^1=3$
• For $x=2$: $f(2)=\frac{3}{2}(2)^2=6$
• For $x=-1$: $f(-1)=\frac{3}{2}(2)^{-1}=\frac{3}{4}=0.75$

Step3: Match points to graphs

The top graph has points: $(0, 1.5)$, $(1, 3)$, $(2, 6)$, $(-1, 0.75)$ which align with calculated values.

Answer:

The top graph (first graph shown) represents the function $f(x) = \frac{3}{2}(2)^x$.