Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \[ f(x)=\frac{x^{2}+13 x+36}{x^{2}+8 x+16} \] tina the aomain or \( \mathrm{r}(\mathrm{X}) \). seiect the correct cnoice deiow ana, it necessary, till in the answer dox to complete your cnoice. A. The domain is all real \( x \), except \( x= \) \( \square \) (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The domain is all real \( x \). Find the \( x \)-intercepts of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( x \)-intercept(s) is/are at \( x= \) \( \square \) . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts. Find the \( y \)-intercepts of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

To analyze the function \( f(x) = \frac{x^{2} + 13x + 36}{x^{2} + 8x + 16} \), we will follow these steps: ### Step 1: Determine the Domain The domain of a rational function is all real numbers except where the denominator is zero. 1. **Find the denominator**: \[ x^{2} + 8x + 16 = 0 \] This can be factored as: \[ (x + 4)(x + 4) = 0 \quad \Rightarrow \quad (x + 4)^{2} = 0 \] Thus, \( x = -4 \) is the only value that makes the denominator zero. 2. **Conclusion about the domain**: The domain is all real \( x \) except \( x = -4 \). ### Step 2: Find the \( x \)-intercepts The \( x \)-intercepts occur where \( f(x) = 0 \). This happens when the numerator is zero. 1. **Find the numerator**: \[ x^{2} + 13x + 36 = 0 \] This can be factored as: \[ (x + 4)(x + 9) = 0 \] Thus, the solutions are: \[ x = -4 \quad \text{and} \quad x = -9 \] 2. **Check for \( x = -4 \)**: Since \( x = -4 \) is also where the denominator is zero, it is not an \( x \)-intercept. 3. **Conclusion about the \( x \)-intercepts**: The \( x \)-intercept is at \( x = -9 \). ### Step 3: Find the \( y \)-intercept The \( y \)-intercept occurs where \( x = 0 \). 1. **Calculate \( f(0) \)**: \[ f(0) = \frac{0^{2} + 13(0) + 36}{0^{2} + 8(0) + 16} = \frac{36}{16} = \frac{9}{4} \] 2. **Conclusion about the \( y \)-intercept**: The \( y \)-intercept is at \( y = \frac{9}{4} \). ### Summary of Findings - **Domain**: The domain is all real \( x \), except \( x = -4 \). - **\( x \)-intercepts**: The \( x \)-intercept is at \( x = -9 \). - **\( y \)-intercept**: The \( y \)-intercept is at \( y = \frac{9}{4} \). ### Final Answers A. The domain is all real \( x \), except \( x = -4 \). B. The \( x \)-intercept(s) is/are at \( x = -9 \). C. The \( y \)-intercept is at \( y = \frac{9}{4} \). Now, let's summarize the information in the answer boxes: - A. The domain is all real \( x \), except \( x = \) \( -4 \). - A. The \( x \)-intercept(s) is/are at \( x = \) \( -9 \). - The \( y \)-intercept is at \( y = \frac{9}{4} \).