1. **State the problem:** We are given the function $f(x) = -\left(\frac{5}{2}\right)^x$ and want to understand its behavior and graph. 2. **Formula and rules:** This is an exponential function of the form $f(x) = -a^x$ where $a = \frac{5}{2} > 1$. - Since $a > 1$, $a^x$ grows exponentially as $x$ increases. - The negative sign in front reflects the graph across the x-axis. - The horizontal asymptote of $a^x$ is $y=0$, so for $f(x)$ it remains $y=0$. 3. **Behavior analysis:** - As $x \to \infty$, $\left(\frac{5}{2}\right)^x \to \infty$, so $f(x) = -\left(\frac{5}{2}\right)^x \to -\infty$. - As $x \to -\infty$, $\left(\frac{5}{2}\right)^x \to 0^+$, so $f(x) \to -0^-$ (approaches zero from below). 4. **Intercepts:** - At $x=0$, $f(0) = -\left(\frac{5}{2}\right)^0 = -1$. - There is no x-intercept because $f(x)$ is always negative. 5. **Summary:** The graph starts just below zero on the left, decreases slowly, passes through $-1$ at $x=0$, and then decreases rapidly to negative infinity as $x$ increases. **Final answer:** The function $f(x) = -\left(\frac{5}{2}\right)^x$ is a reflected exponential growth function with horizontal asymptote $y=0$ and passes through $(0,-1)$, decreasing without bound as $x$ increases.