Question

How do you find the asymptotes for `y = (8 x^2 + x - 2)/(x^2 + x - 72)`?

Answer 1

Vertical asymptotes: `x=-8,x=9`
Horizontal asymptote: `y=8`

Explanation:

To find the vertical asymptotes of a rational equation, find when the equation's denominator equals `0`:

`x^2-x-72=0`

`(x-9)(x+8)=0`

`x=9,x=-8`

These are where vertical asymptotes will occur.

As for horizontal asymptotes, examine the degree of the numerator and denominator. They are both `2`. When the numerator and denominator of a rational function have the same degree, the horizontal asymptote can be found by dividing the two terms with the largest degree:

`(8x^2)/x^2=8`

Thus there is a horizontal asymptote at `y=8`.

We can check a graph:

graph{(8 x^2 + x - 2)/(x^2 + x - 72) [-13, 12, -50, 50]}