The problem provides the exponential function $f(x) = 2 \cdot 3^x$. We are given a graph and are expected to plot the function $f(x)$.

1. Problem Description

The problem provides the exponential function $f(x) = 2 \cdot 3^x$. We are given a graph and are expected to plot the function $f(x)$.

2. Solution Steps

To sketch the graph, we can calculate the value of the function for some $x$ values:

If $x = 0$, then $f(0) = 2 \cdot 3^0 = 2 \cdot 1 = 2$. Thus, the point $(0, 2)$ is on the graph.

If $x = 1$, then $f(1) = 2 \cdot 3^1 = 2 \cdot 3 = 6$. Thus, the point $(1, 6)$ is on the graph.

If $x = 2$, then $f(2) = 2 \cdot 3^2 = 2 \cdot 9 = 18$. Thus, the point $(2, 18)$ is on the graph.

If $x = -1$, then $f(-1) = 2 \cdot 3^{-1} = 2 \cdot \frac{1}{3} = \frac{2}{3} \approx 0.67$. Thus, the point $(-1, \frac{2}{3})$ is on the graph.

If $x = -2$, then $f(-2) = 2 \cdot 3^{-2} = 2 \cdot \frac{1}{9} = \frac{2}{9} \approx 0.22$. Thus, the point $(-2, \frac{2}{9})$ is on the graph.

The function is an exponential growth function. As $x$ gets larger, $f(x)$ grows rapidly. As $x$ approaches $-\infty$, $f(x)$ approaches

0. Plotting the points $(0, 2)$, $(1, 6)$, $(2, 18)$, $(-1, \frac{2}{3})$ on the graph and sketching the exponential curve should give a good representation of the graph of $f(x) = 2 \cdot 3^x$.

3. Final Answer

The graph of $f(x) = 2 \cdot 3^x$ is an exponential curve passing through $(0, 2)$, $(1, 6)$ and $(2, 18)$. As x approaches negative infinity, the curve gets closer to the x-axis, which is the asymptote for this function.