1. The problem is to graph the exponential function $$f(x) = -2 \left(\frac{3}{2}\right)^x$$ and plot five points on its graph. 2. The general form of an exponential function is $$f(x) = a \cdot b^x$$ where $a$ is the initial value and $b$ is the base. Here, $a = -2$ and $b = \frac{3}{2} > 1$, so the function is an exponential growth reflected over the x-axis because of the negative $a$. 3. To plot points, substitute values of $x$ into the function and calculate $f(x)$: - For $x=0$: $$f(0) = -2 \left(\frac{3}{2}\right)^0 = -2 \cdot 1 = -2$$ - For $x=1$: $$f(1) = -2 \left(\frac{3}{2}\right)^1 = -2 \cdot \frac{3}{2} = -3$$ - For $x=2$: $$f(2) = -2 \left(\frac{3}{2}\right)^2 = -2 \cdot \frac{9}{4} = -\frac{18}{4} = -\frac{9}{2} = -4.5$$ - For $x=-1$: $$f(-1) = -2 \left(\frac{3}{2}\right)^{-1} = -2 \cdot \frac{2}{3} = -\frac{4}{3} \approx -1.333$$ - For $x=-2$: $$f(-2) = -2 \left(\frac{3}{2}\right)^{-2} = -2 \cdot \left(\frac{2}{3}\right)^2 = -2 \cdot \frac{4}{9} = -\frac{8}{9} \approx -0.889$$ 4. The five points to plot are: $$(-2, -0.889), (-1, -1.333), (0, -2), (1, -3), (2, -4.5)$$ 5. The graph is an exponential curve decreasing from left to right (due to the negative coefficient), passing through these points. 6. Summary: We evaluated the function at five $x$ values to get corresponding $y$ values and identified the shape of the graph. Final answer: The five points on the graph of $$f(x) = -2 \left(\frac{3}{2}\right)^x$$ are $$(-2, -0.889), (-1, -1.333), (0, -2), (1, -3), (2, -4.5)$$.