Finding Asymptotes

If you factor both the numerator and denominator in the function y = (x2 - x - 6)/(x2 - 9), you will change the function from standard form to factored form. In the factored form, the above function will reveal two main locations:

1) Location of any vertical asymptotes.

2) Location of any x-axis intercepts.

Here what the above function looks like in factored form:

y = (x + 2)/(x + 3)

Once the original function has been factored, the denominator roots will equal our vertical asymptotes and the numerator roots will equal our x-axis intercepts. This means that when the denominator equals zero we have found a vertical asymptote. When the denominator equals zero the whole function equals zero.

NOTE: In terms of the factored form of the original function or y = (x + 2)/(x + 3), there is removable singularity for the numerator at x = -2 and at x = -3 for the denominator.

Here's what y =(x+2)/(x+3) looks like with removable singularity at x = -2 and at x = -3:

As you can see, there is a vertical asymptote at x = -3. This is because the denominator has gone to zero. That makes the whole function go to positive or negative infinity.

Hopefully you can see that an asymptote can often be found by factoring a function to create a simple expression in the denominator. That denominator will reveal your asymptotes.

By Mr. Feliz
(c) 2005