Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \( f(x)=\frac{x^{2}+10 x+24}{x^{2}+8 x+16} \) Find the domain of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain is all real \( x \), except \( x= \) 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 \( x= \) (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. A. The \( y \)-intercept(s) is/are at \( y=\square \). (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no \( y \)-intercepts.

**Step 1. Factor the Function** Given \[ f(x)=\frac{x^{2}+10x+24}{x^{2}+8x+16}, \] we factor both the numerator and denominator. For the numerator: \[ x^2+10x+24=(x+4)(x+6). \] For the denominator: \[ x^2+8x+16=(x+4)^2. \] Thus, for \( x\neq -4 \) (since the factor \( x+4 \) causes the denominator to be 0), \[ f(x)=\frac{(x+4)(x+6)}{(x+4)^2}=\frac{x+6}{x+4}. \] --- **Step 2. Determine the Domain** The denominator of the original function, \[ (x+4)^2, \] is zero when \[ x+4=0 \quad \Longrightarrow \quad x=-4. \] Thus, the domain is all real numbers except \( x=-4 \). *Answer:* A. The domain is all real \( x \), except \( x= -4 \). --- **Step 3. Find the \( x \)-Intercepts** The \( x \)-intercepts occur when \( f(x)=0 \). From the simplified form, \[ \frac{x+6}{x+4}=0, \] the numerator must be zero (and the denominator must not be zero). Setting the numerator equal to zero gives: \[ x+6=0 \quad \Longrightarrow \quad x=-6. \] Since \( -6\neq -4 \), this is valid. *Answer:* A. The \( x \)-intercept is at \( x=-6 \) (or the point \((-6,0)\)). --- **Step 4. Find the \( y \)-Intercepts** The \( y \)-intercept is found by evaluating \( f(0) \). Substitute \( x=0 \) into the simplified form: \[ f(0)=\frac{0+6}{0+4}=\frac{6}{4}=\frac{3}{2}. \] *Answer:* A. The \( y \)-intercept is at \( y=\frac{3}{2} \) (or the point \((0,\frac{3}{2})\)). --- **Additional Graphing Information** 1. **Vertical Asymptote:** Since the denominator \( x+4 \) is zero at \( x=-4 \) and is not canceled, there is a vertical asymptote at \[ x=-4. \] 2. **Horizontal Asymptote:** The degrees of the numerator and denominator (in the simplified form) are both 1. Thus, the horizontal asymptote is given by the ratio of the leading coefficients: \[ y=\frac{1}{1}=1. \] 3. **Summary for Graphing:** - The function simplifies to \( f(x)=\frac{x+6}{x+4} \) for \( x\neq -4 \). - **Domain:** \( x\in\mathbb{R} \setminus \{-4\} \). - **\( x \)-Intercept:** \( x=-6 \) (point \((-6,0)\)). - **\( y \)-Intercept:** \( y=\frac{3}{2} \) (point \((0,\frac{3}{2})\)). - **Vertical Asymptote:** \( x=-4 \). - **Horizontal Asymptote:** \( y=1 \). These details provide a comprehensive view needed to sketch the graph of \( y=f(x) \).