Question

consider the following equation: f(x) = (x² + 4)/(4x² - 4x - 8) name the vertical asymptote(s). ✔ x = -1 and x = 2 ✔ complete because m < n m = n a_m < b_n a_m = b_n ✔ this is where the function is undefined complete name the horizontal asymptote(s). dropdown with options: x = -1 and x = -2, y = -1 and y = -2, x = 1/4, y = 1/4, x = 0, y = 0

Explanation:

Step1: Analyze the degree of numerator and denominator

For the function \( f(x)=\frac{x^{2}+4}{4x^{2}-4x - 8} \), the degree of the numerator \( n = 2 \) (since the highest - power term is \( x^{2} \)) and the degree of the denominator \( m=2 \).

Step2: Use the rule for horizontal asymptotes

When \( m = n \), the horizontal asymptote \( y=\frac{a_{m}}{b_{n}} \), where \( a_{m} \) is the leading coefficient of the numerator and \( b_{n} \) is the leading coefficient of the denominator. Here, \( a_{m}=1 \) (from \( x^{2} \) in the numerator) and \( b_{n}=4 \) (from \( 4x^{2} \) in the denominator). So, \( y = \frac{1}{4} \).

Answer:

\( y=\frac{1}{4} \)