# How do you graph #f(x)=(x^2-12x)/(x^2-2x-3)# using holes, vertical and horizontal asymptotes, x and y intercepts?

##### 1 Answer

By considering asymptotes, yields;

graph{y = (x^2-12x)/(x^2-2x-3) [-19.82, 19.83, -9.9, 9.92]}

#### Explanation:

First we can factorise to give

Now we can consider verticle asymptotes, hence at

Now we can cosnider Horizontal asymptotes:

Hence

as

Hence Horizontal asymptote ;

We now can consider roots;

Hence

Hence has roots,

Now can cosnider the nature of function as they approach the verticle asymptotes from possitive and negative direction:

We can compute the follwoing limits, via letting

We this is efficiant information to be able to sketch this function,

So we note that as

We can repeat this for the other side of the asymptote, noting

Hence;

graph{y = (x^2-12x)/(x^2-2x-3) [-19.82, 19.83, -9.9, 9.92]}