Graphs of Polynomial and Simple Rational Functions

A polynomial's degree sets a ceiling on how much it can bend. A parabola, degree 2, turns once. A cubic, degree 3, can turn twice, because each degree above 1 buys one more possible change of direction. The factors control where those turns happen relative to the x-axis, and the exponent on each factor controls what the curve does when it gets there. A factor like (x - p) to an odd power, usually just 1, means the graph crosses straight through at x = p. A factor squared, like (x - p)², means the graph touches the axis at x = p and turns back without crossing, because the sign of that factor never flips.

Rational functions add division to the mix, and division breaks the graph wherever the denominator hits zero. If a factor in the denominator cancels with the same factor in the numerator, the function is undefined there but the graph doesn't blow up, it just leaves a hole, a single missing point. If a denominator factor does not cancel, the function's values shoot toward infinity on either side, and that's a vertical asymptote, a vertical line the curve approaches but never touches.

This all flows from the same idea: factored form is a map of a function's zeros, and what happens at each zero, crossing, bouncing, canceling, or blowing up, depends entirely on whether that factor survives in the final expression and how many times it appears. Once you can read factored form fluently, sketching a polynomial or simple rational function becomes a matter of listing intercepts, checking multiplicities, and checking for cancellation, not plotting points.