Polynomial End Behavior

Far from the origin, one term in a polynomial always wins. Plug in a huge x, like x = 1000, and a term like x³ outgrows a term like 500x² so fast that the smaller-degree term becomes irrelevant by comparison. At x = 1000, x³ is a billion while 500x² is only 500 million — already losing, and the gap only widens as x grows. That's why end behavior, the long-run direction of the graph's arms, depends only on the leading term: the term with the highest degree.

Two facts about that leading term tell you everything: its degree (odd or even) and the sign of its coefficient. Even degree means both ends point the same way — up for positive leading coefficient, down for negative. Odd degree means the ends point opposite ways — up on the right and down on the left for positive leading coefficient, flipped for negative. This mirrors what a single power function xⁿ does, because far from the origin a polynomial behaves almost exactly like its leading term alone.

This sets up a contrast with what happens near the origin, where lower-degree terms, roots, and local turning points shape the graph's details. End behavior is the zoomed-out picture; the rest of Unit 1 — zeros, multiplicity, local extrema — fills in the zoomed-in picture. Knowing both lets you sketch a full polynomial graph from its equation without plotting dozens of points.