Question

How do you graph `f(x)=(x^3-16x)/(-3x^2+3x+18)` using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

Factoring Steps:

`f(x)=(x^3-16x)/(-3x^2+3x+18)`

`f(x)=((x)(x^2-16))/(-3(x-3)(x+2))`

`f(x)=((x)(x-4)(x-4))/(-3(x-3)(x+2))`

Analysis of the rational equation:

There are no holes, because none of the terms cancel each other out in the numerator/denominator.

There are x intercepts at `x=0,4,-4.`

There are vertical asymptotes at `x=-2, and 3.`

because there is an x-intercept at `x=0`, that means that the t-intercept is also 0.

Therefore, the graph would look like this:

graph{(x^3-16x)/(-3x^2+3x+18) [-10, 10, -5, 5]}