1. **State the problem:** Graph the function $f(x) = -2 \left(\frac{1}{5}\right)^x$ and identify the asymptote. 2. **Formula and rules:** The function is an exponential function of the form $f(x) = a b^x$ where $a = -2$ and $b = \frac{1}{5}$. - Since $0 < b < 1$, this is exponential decay. - The negative sign reflects the graph below the x-axis. - The horizontal asymptote for $f(x) = a b^x$ when $b > 0$ is the line $y=0$ (the x-axis). 3. **Plot points:** Calculate $f(0)$ and $f(1)$: $$f(0) = -2 \left(\frac{1}{5}\right)^0 = -2 \times 1 = -2$$ $$f(1) = -2 \left(\frac{1}{5}\right)^1 = -2 \times \frac{1}{5} = -\frac{2}{5}$$ 4. **Asymptote:** The asymptote is the x-axis, which is a horizontal line $y=0$. 5. **Summary:** - The graph approaches $y=0$ but never crosses it. - The points $(0, -2)$ and $(1, -\frac{2}{5})$ lie on the curve. - The graph is an exponential decay reflected below the x-axis. **Final answer:** The asymptote is horizontal, $y=0$.