Question

find equations for the vertical asymptotes, if any, for the following rational function.
$f(x) = \frac{2x^2 + 2x - 4}{-x - 7}$
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Explanation:

Step1: Recall vertical asymptote rule

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), vertical asymptotes occur where \( D(x) = 0 \) and \( N(x)
eq0 \) (no common factors).

Step2: Find denominator zero

Denominator \( D(x)=-x - 7 \). Set \( -x - 7 = 0 \). Solve: \( -x=7\Rightarrow x=-7 \).

Step3: Check numerator at \( x = -7 \)

Numerator \( N(x)=2x^{2}+2x - 4 \). Substitute \( x=-7 \):
\( N(-7)=2(-7)^{2}+2(-7)-4=2(49)-14 - 4=98 - 14 - 4 = 80
eq0 \).

Answer:

\( x = -7 \)