Question

graph all vertical and horizontal asymptotes of the rat

$f(x)=\frac{-1}{2 x^{2}-6 x-8}$

Explanation:

Step1: Find vertical asymptotes

Set denominator equal to zero: \(2x^{2}-6x - 8=0\).
Factor: \(2(x^{2}-3x - 4)=0\), then \(2(x - 4)(x+1)=0\).
Solutions \(x = 4\) and \(x=-1\) are vertical asymptotes.

Step2: Find horizontal asymptotes

Degree of numerator \(n = 0\), degree of denominator \(m = 2\).
Since \(n\lt m\), horizontal asymptote is \(y = 0\).

Answer:

Vertical asymptotes: \(x=-1\) and \(x = 4\); Horizontal asymptote: \(y = 0\)