Question

write an equation for the function graphed below
$y =$
question help: video read written example

Explanation:

Step1: Identify vertical asymptotes

The vertical asymptotes are at $x=-3$ and $x=4$, so the denominator is $(x+3)(x-4) = x^2 - x - 12$.

Step2: Set rational function form

The function has the form $y = \frac{a(x - 2)}{(x+3)(x-4)}$, since it has a root at $x=2$ (crosses x-axis here).

Step3: Solve for a using y-intercept

The y-intercept is $(0,2)$. Substitute $x=0, y=2$:
$2 = \frac{a(0 - 2)}{(0+3)(0-4)}$
$2 = \frac{-2a}{-12}$
$2 = \frac{a}{6}$
$a = 12$

Step4: Write final equation

Substitute $a=12$ into the function:
$y = \frac{12(x - 2)}{(x+3)(x-4)} = \frac{12x - 24}{x^2 - x - 12}$

Answer:

$y = \frac{12(x-2)}{(x+3)(x-4)}$ (or expanded form $y = \frac{12x - 24}{x^2 - x - 12}$)