Question

0/1 pt 3 98 details
write an equation for the function graphed below
$y = $
question help:

Explanation:

Step1: Identify vertical asymptotes

Vertical asymptotes at $x=-2$ and $x=4$, so denominator is $(x+2)(x-4) = x^2 - 2x - 8$.

Step2: Identify x-intercept

x-intercept at $x=2$, so numerator has factor $(x-2)$. Assume numerator is $a(x-2)$ where $a$ is a constant.

Step3: Solve for constant $a$

Use y-intercept $(0,2)$: substitute $x=0, y=2$ into $y=\frac{a(x-2)}{x^2 - 2x - 8}$
$2 = \frac{a(0-2)}{0 - 0 - 8}$
$2 = \frac{-2a}{-8} = \frac{a}{4}$
$a = 8$

Step4: Form final equation

Substitute $a=8$ into the function.

Answer:

$y=\frac{8(x-2)}{x^2 - 2x - 8}$