1. **State the problem:** We need to graph the exponential function $$f(x) = \left(\frac{5}{3}\right)^x$$ and plot five points on its graph, as well as draw the horizontal asymptote. 2. **Formula and rules:** The general form of an exponential function is $$f(x) = a^x$$ where $a > 0$ and $a \neq 1$. - The function passes through the point $(0,1)$ because any number to the power 0 is 1. - The horizontal asymptote is the line $y=0$ because as $x \to -\infty$, $f(x) \to 0$. 3. **Calculate five points:** - At $x=0$: $$f(0) = \left(\frac{5}{3}\right)^0 = 1$$ - At $x=1$: $$f(1) = \frac{5}{3} \approx 1.6667$$ - At $x=2$: $$f(2) = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \approx 2.7778$$ - At $x=3$: $$f(3) = \left(\frac{5}{3}\right)^3 = \frac{125}{27} \approx 4.6296$$ - At $x=4$: $$f(4) = \left(\frac{5}{3}\right)^4 = \frac{625}{81} \approx 7.7160$$ 4. **Asymptote:** The horizontal asymptote is the line $$y=0$$. 5. **Summary:** The graph passes through points $(0,1)$, $(1,1.6667)$, $(2,2.7778)$, $(3,4.6296)$, $(4,7.7160)$ and approaches $y=0$ as $x$ decreases. This completes the graphing of the exponential function.