End Behavior Analysis

Calculus Readiness · Axiom Academy

LESSON End Behavior of Polynomials Zoom far enough out and only the leading term is left — degree parity and its sign decide where the two tails go. 1. Far Out, Only the Leading Term Survives Take f(x) = x^3 - 6x^2 + 4x + 12 . Near the origin the -6x^2 and +4x terms bend the curve around. But end behavior asks what happens as — and out there the highest power grows so much faster than the rest that the lower terms become a rounding error. The leading term — what survives far out Its share of the value , so f and x^3 merge 2. Two Questions Decide the Arms Since only the leading term a_n x^n matters, end behavior comes down to two yes/no questions about it. Watch the arms swing as we cycle the four cases — the degree's parity sets whether the arms match or oppose, and the sign of a_n flips them up or down. Both arms up: on each side. A wide bowl, like x^2 . Both arms down: on each side. An upside-down bowl. Down-left, up-right: as , as . Like x^3 . Up-left, down-right: the mirror of the case above. Reading a polynomial off the page For g(x) = -x^5 + 3x^2 - 4 : the leading term is -x^5 , degree 5 (odd) with a negative coefficient. Odd opposite ends; negative right end down. So as and as . Not in standard form? Just find the highest power first. p(x) = 5 + x - 3x^2 + 2x^3 reorders to 2x^3 - 3x^2 + x + 5 — leading term 2x^3 , odd and positive, so on the left and on the right.

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