1. **State the problem:** We are comparing the graphs of three functions: $$f(x) = 5^x$$ $$g(x) = 5^x - 8$$ $$h(x) = 5^{x+4}$$ We want to understand how the graphs of $g(x)$ and $h(x)$ differ from $f(x)$ and compare their domain, range, asymptotes, and y-intercepts. 2. **Recall the base function properties:** - The function $f(x) = 5^x$ is an exponential function with base 5. - Its domain is all real numbers: $(-\infty, \infty)$. - Its range is $(0, \infty)$ because $5^x$ is always positive. - It has a horizontal asymptote at $y=0$. - The y-intercept is found by evaluating $f(0) = 5^0 = 1$. 3. **Analyze $g(x) = 5^x - 8$:** - This is a vertical shift of $f(x)$ downward by 8 units. - Domain remains $(-\infty, \infty)$ because shifting vertically does not affect domain. - Range shifts down by 8: original range $(0, \infty)$ becomes $(-8, \infty)$. - Horizontal asymptote shifts from $y=0$ to $y = 0 - 8 = -8$. - Y-intercept: $g(0) = 5^0 - 8 = 1 - 8 = -7$. 4. **Analyze $h(x) = 5^{x+4}$:** - This is a horizontal shift of $f(x)$ to the left by 4 units. - Domain remains $(-\infty, \infty)$ because exponential functions are defined for all real $x$. - Range remains $(0, \infty)$ because horizontal shifts do not affect range. - Horizontal asymptote remains $y=0$. - Y-intercept: $h(0) = 5^{0+4} = 5^4 = 625$. 5. **Summary table:** | Function | Domain | Range | Asymptote | Y-intercept | |---|---|---|---|---| | $f(x) = 5^x$ | $(-\infty, \infty)$ | $(0, \infty)$ | $y=0$ | 1 | | $g(x) = 5^x - 8$ | $(-\infty, \infty)$ | $(-8, \infty)$ | $y=-8$ | -7 | | $h(x) = 5^{x+4}$ | $(-\infty, \infty)$ | $(0, \infty)$ | $y=0$ | 625 | 6. **Interpretation:** - $g(x)$ shifts the graph of $f(x)$ down by 8 units, lowering the asymptote and y-intercept. - $h(x)$ shifts the graph of $f(x)$ left by 4 units, increasing the y-intercept significantly but keeping the asymptote the same. This completes the comparison of the graphs of $f(x)$, $g(x)$, and $h(x)$.