Question

1. what is the vertical asymptote(s) for $y = \frac{x - 5}{x^2 - 4x - 12}$?

• no vertical asymptotes
• $x = 5$
• $x = -6, x = 2$
• $x = -2, x = 6$

Explanation:

Step1: Factor denominator

$x^2 - 4x - 12 = (x + 2)(x - 6)$

Step2: Find undefined points

Set denominator to 0: $(x + 2)(x - 6) = 0$ → $x = -2$ or $x = 6$

Step3: Check numerator at these x

At $x=-2$: $-2 -5 = -7 ≠0$; at $x=6$: $6 -5 =1≠0$ → no cancellation

Step4: Confirm vertical asymptotes

Vertical asymptotes at $x=-2$ and $x=6$

Answer:

x = -2, x = 6