Math basics › Precalculus
Rational functions
Factor both parts first
Everything else follows from the factors. Nothing about a rational function is clear until the top and bottom are factored.
Holes versus asymptotes
A factor that cancels leaves a hole. A factor left on the bottom gives a vertical asymptote. Same-looking algebra, completely different graphs.
What happens far out
Compare the degrees. Bottom bigger: y → 0. Equal: y goes to the ratio of the leading coefficients. Top bigger by one: a slant asymptote, found by dividing.
x-intercepts
Where the top is zero and the bottom is not. A value that zeroes both is a hole, not an intercept.