1. **State the problem:** We are given the function $$y = - (0.8)^x + 5$$ and want to understand its behavior. 2. **Identify the function type:** This is an exponential function of the form $$y = -a^x + c$$ where $$a = 0.8$$ and $$c = 5$$. 3. **Key properties:** - Since $$0 < a < 1$$, $$a^x$$ is a decreasing exponential function. - The negative sign in front flips the graph vertically. - The $$+5$$ shifts the graph upward by 5 units. 4. **Intercepts:** - To find the y-intercept, set $$x=0$$: $$y = - (0.8)^0 + 5 = -1 + 5 = 4$$ - To find the x-intercept, set $$y=0$$: $$0 = - (0.8)^x + 5$$ $$ (0.8)^x = 5$$ Taking natural logarithm on both sides: $$x \ln(0.8) = \ln(5)$$ $$x = \frac{\ln(5)}{\ln(0.8)}$$ Since $$\ln(0.8) < 0$$, $$x$$ will be negative. 5. **Extrema:** - Exponential functions of this form have no local maxima or minima except horizontal asymptotes. 6. **Horizontal asymptote:** - As $$x \to \infty$$, $$ (0.8)^x \to 0$$, so $$y \to 5$$. - As $$x \to -\infty$$, $$ (0.8)^x \to \infty$$, so $$y \to -\infty$$. **Final summary:** The graph is a decreasing exponential flipped vertically, shifted up by 5, with y-intercept at 4 and x-intercept at $$x = \frac{\ln(5)}{\ln(0.8)}$$.