Question

question 4 (5 points)
what is the vertical asymptote? show all work for credit.
$y = \frac{x^2 - 25}{x^2 - x - 20}$

Explanation:

Step1: Factor numerator and denominator

First, factor the numerator \(x^{2}-25\) using the difference of squares formula \(a^{2}-b^{2}=(a + b)(a - b)\). So, \(x^{2}-25=(x + 5)(x - 5)\).

Next, factor the denominator \(x^{2}-x - 20\). We need two numbers that multiply to \(-20\) and add to \(-1\). Those numbers are \(-5\) and \(4\). So, \(x^{2}-x - 20=(x - 5)(x + 4)\).

The function becomes \(y=\frac{(x + 5)(x - 5)}{(x - 5)(x + 4)}\).

Step2: Cancel common factors

We can cancel the common factor \((x - 5)\) (for \(x
eq5\)), so the simplified function is \(y=\frac{x + 5}{x + 4}\) (with a hole at \(x = 5\)).

Step3: Find vertical asymptote

To find the vertical asymptote, we set the denominator of the simplified function equal to zero (excluding the hole). So, set \(x+4 = 0\).

Solving for \(x\), we get \(x=-4\).

Answer:

The vertical asymptote is \(x = - 4\)