1. **State the problem:** We need to graph the function $f(x) = 2^x + 4$, and determine its domain, range, horizontal asymptote, and y-intercept. 2. **Formula and transformations:** The base function is $g(x) = 2^x$. - The function $f(x) = 2^x + 4$ is a vertical shift of $g(x)$ upward by 4 units. - The horizontal asymptote of $g(x)$ is $y=0$, so for $f(x)$ it shifts to $y=4$. 3. **Domain:** The domain of $2^x$ is all real numbers because exponential functions are defined for all real $x$. - Therefore, the domain of $f(x)$ is also all real numbers. - In interval notation: $$(-\infty, \infty)$$ 4. **Range:** The base function $2^x$ has range $(0, \infty)$. - Adding 4 shifts the range up by 4, so the range of $f(x)$ is $(4, \infty)$. 5. **Horizontal asymptote:** The horizontal asymptote of $f(x)$ is the line $y=4$. 6. **Y-intercept:** To find the y-intercept, evaluate $f(0)$: $$f(0) = 2^0 + 4 = 1 + 4 = 5$$ - So the y-intercept is at $(0, 5)$. 7. **Summary:** - Domain: $$(-\infty, \infty)$$ - Range: $$(4, \infty)$$ - Horizontal asymptote: $$y=4$$ - Y-intercept: $$(0, 5)$$ 8. **Graphing:** The graph is the exponential curve $2^x$ shifted up by 4 units, approaching but never touching $y=4$ from above, crossing the y-axis at 5.