Find all horizontal and vertical asymptotes of \( f(x) \). \( f(x)=\frac{4 x^{2}+11 x-3}{x^{2}+3 x-18} \) Find the horizontal asymptotes. Select the correct choice and, if necessary, fill in the answer box(es) to complete your choi A. The function has one horizontal asymptote, (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is \( \square \) and the bottom asymptote is (Type equations.) The function has no horizontal asymptotes.

To determine the horizontal asymptotes of the function \( f(x) = \frac{4x^{2} + 11x - 3}{x^{2} + 3x - 18} \), we analyze the degrees of the numerator and the denominator. ### Step 1: Compare Degrees - **Numerator:** \( 4x^{2} + 11x - 3 \) has a degree of **2**. - **Denominator:** \( x^{2} + 3x - 18 \) also has a degree of **2**. ### Step 2: Determine Horizontal Asymptote When the degrees of the numerator and denominator are equal, the horizontal asymptote is found by dividing the leading coefficients of the numerator and denominator. - **Leading Coefficient of Numerator:** \( 4 \) - **Leading Coefficient of Denominator:** \( 1 \) Thus, the horizontal asymptote is: \[ y = \frac{4}{1} = 4 \] ### Conclusion **Choice A** is correct: **A. The function has one horizontal asymptote, \( y = 4 \).** ### Additional Information (Optional) While not required for the question, it's useful to note that vertical asymptotes can be found by setting the denominator equal to zero and ensuring the numerator does not also equal zero at those points. - **Denominator Factors:** \( x^{2} + 3x - 18 = (x + 6)(x - 3) \) - **Vertical Asymptotes:** \( x = -6 \) and \( x = 3 \) There are no common factors between the numerator and denominator, so there are no holes in the graph. **Final Answer:** A. The function has one horizontal asymptote, \( y = 4 \).