# How do you graph #f(x)=(x^3-x)/(x^3+2x^2-3x)# using holes, vertical and horizontal asymptotes, x and y intercepts?

##### 1 Answer

You need to first factor to see if you can eliminate anything (this is when holes will occur).

There will be two holes: at

The exact coordinates of the holes can be obtained by substituting

Hence, the holes will be at

For this function, there will be a horizontal asymptote at the ratio between the coefficents of the terms with highest degree in the numerator and denominator.

The horizontal asymptote is given by

As for intercepts, set the function to

**y intercept:**

there are none, because both are eliminated when factoring (even though it does appear that there is a y-intercept on the graph, this is in fact a hole.

**x-intercept:**

You will find there is an x-intercept at

The last thing that is required to graph a rational function like this is end behavior. This can be found by picking a few numbers close to the asymptotes and checking their trend. For example, you can pick

Doing this for all the vertical and horizontal asymptotes, you should get the following graph.

graph{(x^3 - x)/(x^3 + 2x^2 - 3x) [-10, 10, -5, 5]}

Hopefully this helps!