Notice that we can simplify both numerator and denominator of our rational function. In the numerator we have a quadratic expression of the form [tex]ax^2+bx+c[/tex]. To to simplify it, we are going to find tow numbers that add to 2 and multiply to -8; those numbers are 4 and -2.
[tex]x^2+2x-8=(x+4)(x-2)[/tex]
In the denominator we have a difference of squares:
[tex]x^2-9=x^2-3^2=(x+3)(x-3)[/tex]
Now we can rewrite our function:
[tex]y= \frac{x^2+2x-8}{x^2-9} = \frac{(x+4)(x-2)}{(x+3)(x-3)} [/tex]
From the simplified form of our rational function we can infer that its graph has two vertical asymptotes at [tex]x=3[/tex] and [tex]x-3[/tex]
We can conclude that the graphic of our rational function is:
