Does anyone know any good quick tutorials or guides on how to sketch rational functions? Not just x/x^2+1, I get that, but how to do things like (x^2-1x+2)^7(x-4)^9/x^11(4x-1)^4 ? Might not of been paying full attention in my Pre-Calculus class due to being tired and not having my glasses.
Input some values into the function, get some results, connect the dots? All you have to worry about is near the poles of the function, and the poles are easy to figure out, because every pole is a root of the denominator (although not vice versa). For the poles, you have to figure out the left and right limit at those poles, and draw the corresponding vertical asymptotes in the correct direction.
Example: (x³-1)/(x³+x²-x-1) = (x-1)(x²+x+1)/(x+1)²(x-1), so you have a pole at x=-1 of order 2 and a reducible singularity (no pole) at x=1. Since the order of the pole is even, both limits go in the same direction, and you can always figure out the direction of the right asymptote from the sign of lim(x->-1, f(x)*(x+1)²), which is incidentally 1.
So (x³-1)/(x³+x²-x-1) has no zeros (the pole and zero at x=1 cancel each other out) and one pole of order 2 at x=-1 with positive divergence from both sides.
EDIT: The horizontal asymptote is 0 if the degree of the denominator is greater than the degree of the numerator, undefined if the degree of the denominator is smaller, and equal to a/b when the degrees are both equal to n, where a and b are the coefficients of x^n for the numerator and denominator respectively.