Polynomial Roller Coasters

Pre-Calculus · Axiom Academy

A polynomial's graph is a roller-coaster track: smooth hills and valleys in the middle, two arms shooting off the ends. The shape is decided by just two things — the degree and the leading term. A polynomial is a ride you can see Plot any polynomial and you don't get a jagged mess — you get one smooth, unbroken track that rises and dips like a roller coaster. Before any algebra, watch one play out: where the track turns around, and where its arms head as the ride leaves the screen. Send the cart down a degree-4 coaster. Every time it crests a hill or bottoms out in a valley, the track changes direction — that's a turning point . Count them as you go, and notice both arms ride off the same way at the ends. A degree-4 track turns at most 3 times — and here it uses all 3. The arms both point up, because the leading term is positive. Click through the degrees. A line (degree 1) never turns. A parabola (degree 2) turns once. Each step up buys the coaster room for one more hill or valley — a degree- n track can turn at most n − 1 times. Max turns is always one less than the degree — the turns shown here hits that ceiling for every preset. The arms: where the ride ends up The hills live in the middle; the arms are decided by the very first term — the leading one. Drag its coefficient. Flip it negative and watch both ends swap direction. For this degree-3 track the arms split (one up, one down) and the sign says which way.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.