Rational Functions and End Behavior

A rational function is just one polynomial divided by another. Near the origin, every term in both polynomials matters, and the graph can bend and wiggle. But as x grows huge, in either direction, the highest-degree term in each polynomial dominates all the smaller ones. So the whole function starts behaving like the ratio of just those two leading terms.

To predict the far-away shape, compare the degree of the numerator to the degree of the denominator. If they're equal, the lower terms cancel out in importance and the graph flattens toward a horizontal line at height a/b, the ratio of leading coefficients. If the numerator's degree is exactly one more than the denominator's, the division leaves a leftover linear piece, and the graph runs alongside a slanted line instead of a flat one. If the denominator's degree is bigger, the whole function shrinks toward zero.

This is the leading term idea from earlier in the unit, applied to a quotient instead of a single polynomial. It also sets up rational functions as a whole: the same leading-term comparison that gives you end behavior here will later help you spot holes, vertical asymptotes, and the overall shape of the graph near its breaks.