1. **State the problem:** We need to graph the exponential function $$f(x) = \left(\frac{5}{2}\right)^x$$ and plot five points on its graph, as well as draw the horizontal asymptote. 2. **Formula and properties:** The general form of an exponential function is $$f(x) = a^x$$ where $$a > 0$$ and $$a \neq 1$$. - For $$a > 1$$, the function represents exponential growth. - The horizontal asymptote is the line $$y = 0$$ because exponential functions never reach zero but approach it as $$x \to -\infty$$. 3. **Calculate five points:** Choose $$x = -2, -1, 0, 1, 2$$. - $$f(-2) = \left(\frac{5}{2}\right)^{-2} = \left(\frac{2}{5}\right)^2 = \frac{4}{25} = 0.16$$ - $$f(-1) = \left(\frac{5}{2}\right)^{-1} = \frac{2}{5} = 0.4$$ - $$f(0) = \left(\frac{5}{2}\right)^0 = 1$$ - $$f(1) = \frac{5}{2} = 2.5$$ - $$f(2) = \left(\frac{5}{2}\right)^2 = \frac{25}{4} = 6.25$$ 4. **Interpretation:** The function grows rapidly as $$x$$ increases and approaches zero as $$x$$ decreases. 5. **Asymptote:** The horizontal asymptote is $$y = 0$$. **Final answer:** The graph of $$f(x) = \left(\frac{5}{2}\right)^x$$ passes through points $$( -2, 0.16 ), ( -1, 0.4 ), ( 0, 1 ), ( 1, 2.5 ), ( 2, 6.25 )$$ and has a horizontal asymptote at $$y = 0$$.