asic Concepts 5.2 Quiz \[ \begin{array}{l}f(x)=\frac{x+4}{x+7} \\ \text { Match the rational function with the appropriate description. The vertical asymptote is } x=-4 \text {. } \\ \text { B. The vertical asymptote is } x=-7 \text {. } \\ \text { C. The } x \text {-intercept is }(7,0) \text {. } \\ \text { D. The } y \text {-intercept is }(0,4) \text {. }\end{array} \]

To analyze the rational function \( f(x) = \frac{x+4}{x+7} \) and match it with the appropriate descriptions, we will evaluate the vertical asymptote, x-intercept, and y-intercept step by step. ### Step 1: Find the vertical asymptote The vertical asymptote occurs where the denominator is equal to zero. Set the denominator equal to zero: \[ x + 7 = 0 \] Solving for \( x \): \[ x = -7 \] Thus, the vertical asymptote is at \( x = -7 \). ### Step 2: Find the x-intercept The x-intercept occurs where the function equals zero. This happens when the numerator is equal to zero. Set the numerator equal to zero: \[ x + 4 = 0 \] Solving for \( x \): \[ x = -4 \] Thus, the x-intercept is at \( (-4, 0) \). ### Step 3: Find the y-intercept The y-intercept occurs where \( x = 0 \). We substitute \( x = 0 \) into the function: \[ f(0) = \frac{0 + 4}{0 + 7} = \frac{4}{7} \] Thus, the y-intercept is at \( (0, \frac{4}{7}) \). ### Summary of Results - The vertical asymptote is at \( x = -7 \) (Description B). - The x-intercept is at \( (-4, 0) \) (not listed in the descriptions). - The y-intercept is at \( (0, \frac{4}{7}) \) (not listed in the descriptions). ### Matching Descriptions Now we can match the results with the descriptions provided: - **A. The vertical asymptote is \( x = -4 \)**: **Incorrect**. - **B. The vertical asymptote is \( x = -7 \)**: **Correct**. - **C. The x-intercept is \( (7, 0) \)**: **Incorrect**. - **D. The y-intercept is \( (0, 4) \)**: **Incorrect**. ### Final Matches - **B**: The vertical asymptote is \( x = -7 \). The other descriptions do not match the function \( f(x) = \frac{x+4}{x+7} \).