End Behavior

Pre-Calculus · Axiom Academy

LESSON End Behavior of Polynomials Far out on the graph, only the leading term matters — and it dictates which way both arms point. 1. Only the Leading Term Survives Take p(x) = 2x^4 - 3x^2 + 1 . Near the origin the lower terms matter, but push x outward and the 2x^4 piece grows so much faster that the rest becomes rounding error. Watch p(x) (blue) and its leading term 2x^4 (orange) lock together as |x| climbs. the ratio collapses to 1 — the leading term wins 2. Parity Sets the Arms, Sign Sets the Side The exponent's parity decides whether the two arms agree or disagree, and the sign of a_n fixes which way the right arm goes. Below, an even-degree curve x^4 - 3x^2 (both arms agree) is traced beside an odd-degree curve x^3 - 3x (arms split). Both arms point the SAME way — like x^2 : a U (both up) or an upside-down U (both down). The arms point OPPOSITE ways — like x^3 : one arm up, the other down. The right arm goes DOWN: as , . Raising a negative x to an even power gives a positive result, so the left arm copies the right ( on both sides). An odd power keeps the sign of x , so the left arm flips — that single fact is the whole even/odd rule. Two choices — degree even or odd, leading coefficient positive or negative — give exactly four end-behavior pictures. Here each is drawn from a real polynomial; memorize the four corner shapes and you can sketch the tails of any polynomial at a glance.

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