Question

How do you identify all asymptotes or holes for `f(x)=(x^3-7x^2+12x)/(-4x^2+8x)`?

Answer 1

First step is to factor the cubic and quadratic equation on the numerator and denominator, respectively.

To factor the cubic expression you can either use a graphing calculator, or find a factor and carry on with synthetic or long division to find the other two factors.

1. Holes- Any common factor in the numerator and denominator that will cancel out is a hole.
2. Any factor in the denominator that does not cancel out is going to be your vertical asymptote.
3. For the horizontal asymptote, these are the three rules. Commit these to memory, as it will always work to find your horizontal asymptote:

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = a/b. A and B being the leading coefficients of the numerator and denominator.

If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Thus, in your equation, there will be no horizontal asymptote.