1. **State the problem:** We need to graph the exponential function $$f(x) = - \left(\frac{3}{5}\right)^{-x}$$ and plot five points on its graph, as well as draw the horizontal asymptote. 2. **Recall the formula and properties:** The function is an exponential function with base $$\frac{3}{5}$$ raised to the power $$-x$$, and multiplied by a negative sign. Important rules: - For any base $$a$$ where $$0 < a < 1$$, $$a^{-x} = \left(\frac{1}{a}\right)^x$$ which is an increasing exponential function. - The negative sign in front reflects the graph across the x-axis. - The horizontal asymptote of $$a^x$$ is $$y=0$$, so here it remains $$y=0$$. 3. **Simplify the function:** $$f(x) = - \left(\frac{3}{5}\right)^{-x} = - \left(\frac{5}{3}\right)^x$$ 4. **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 $$x=-1$$: $$f(-1) = - \left(\frac{5}{3}\right)^{-1} = - \frac{3}{5} = -0.6$$ - At $$x=-2$$: $$f(-2) = - \left(\frac{5}{3}\right)^{-2} = - \left(\frac{3}{5}\right)^2 = - \frac{9}{25} = -0.36$$ 5. **Describe the asymptote:** The horizontal asymptote is $$y=0$$, which the graph approaches but never crosses. 6. **Summary:** The graph is a decreasing negative exponential curve approaching zero from below, passing through the points calculated. **Final answer:** The function is $$f(x) = - \left(\frac{5}{3}\right)^x$$ with horizontal asymptote $$y=0$$ and points at $$(0,-1), (1,-1.6667), (2,-2.7778), (-1,-0.6), (-2,-0.36)$$.