write an equation for the function graphed below y = question help: video

Explanation:

Step1: Identify vertical asymptotes

The vertical asymptotes are at $x = - 3$ and $x=3$. So the denominator of a rational - function will have factors of $(x + 3)$ and $(x - 3)$, and the denominator is $q(x)=(x + 3)(x - 3)=x^{2}-9$.

Step2: Consider the form of the rational function

Let the rational function be $y=\frac{p(x)}{q(x)}$. Since the function has a zero at $x = 2$, the numerator $p(x)$ has a factor of $(x - 2)$. Let the function be $y=\frac{a(x - 2)}{x^{2}-9}$.

Step3: Find the value of $a$

We know that the function passes through the point $(0,-2)$. Substitute $x = 0$ and $y=-2$ into $y=\frac{a(x - 2)}{x^{2}-9}$: [ \begin{align*} -2&=\frac{a(0 - 2)}{0^{2}-9}\ -2&=\frac{-2a}{-9}\ -2&=\frac{2a}{9}\ a&=-9 \end{align*} ]

Answer:

$y=\frac{-9(x - 2)}{x^{2}-9}=\frac{-9x + 18}{x^{2}-9}$