write an equation for the function graphed below

Explanation:

Step1: Identify vertical asymptotes

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

Step2: Consider the x - intercept

The x - intercept is at $x = 2$, so the numerator must have a factor of $(x - 2)$.

Step3: Determine the y - intercept

The y - intercept is at $y = 1$. Let the function be $y=\frac{a(x - 2)}{x^{2}-9}$. When $x = 0$, $y=\frac{a(0 - 2)}{0^{2}-9}=\frac{-2a}{-9}=\frac{2a}{9}$. Since $y = 1$ when $x = 0$, we have $\frac{2a}{9}=1$, so $a=\frac{9}{2}$.

Answer:

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