Question

Which of the following is the graph of the rational function? y=x^2+2x-8/x^2-9

Answer 1

Notice that we can simplify both numerator and denominator of our rational function. In the numerator we have a quadratic expression of the form
`ax^2+bx+c`. 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.

`x^2+2x-8=(x+4)(x-2)`
In the denominator we have a difference of squares:

`x^2-9=x^2-3^2=(x+3)(x-3)`

Now we can rewrite our function:

`y= (x^2+2x-8)/(x^2-9) = ((x+4)(x-2))/((x+3)(x-3))`

From the simplified form of our rational function we can infer that its graph has two vertical asymptotes at
`x=3` and
`x-3`

We can conclude that the graphic of our rational function is: