1. **State the problem:** We need to understand and graph the function $f(x) = -4^x$. 2. **Recall the function form:** The function is an exponential function with base 4 and a negative sign in front, so $f(x) = -(4^x)$. 3. **Important rules:** - Exponential functions of the form $a^x$ where $a > 1$ grow rapidly as $x$ increases. - The negative sign in front reflects the graph across the x-axis. - The domain of $f(x)$ is all real numbers. - The range will be negative values because of the negative sign. 4. **Evaluate some points:** - At $x=0$, $f(0) = -4^0 = -1$. - At $x=1$, $f(1) = -4^1 = -4$. - At $x=-1$, $f(-1) = -4^{-1} = -\frac{1}{4} = -0.25$. 5. **Behavior:** - As $x \to \infty$, $4^x \to \infty$, so $f(x) = -4^x \to -\infty$. - As $x \to -\infty$, $4^x \to 0^+$, so $f(x) \to 0^-$. 6. **Graph features:** - The graph passes through $(0,-1)$. - It approaches 0 from below as $x$ goes to negative infinity (horizontal asymptote at $y=0$). - It decreases without bound as $x$ increases. 7. **Summary:** The graph is an exponential decay reflected below the x-axis with horizontal asymptote $y=0$. Final answer: The function $f(x) = -4^x$ is an exponential decay reflected over the x-axis with domain $(-\infty, \infty)$ and range $(-\infty, 0)$, passing through $(0,-1)$ and approaching 0 from below as $x \to -\infty$.