End Behavior Patterns

Algebra 2 · Axiom Academy

Zoom out on any polynomial and only one thing survives: the leading term. It alone decides which way the two tails run. 1. What End Behavior Really Asks Take a specific polynomial, say f(x) = x^3 - 2x . Near the origin it dips below the axis and climbs back — interesting, but not what end behavior is about. End behavior looks past the middle and asks only about the two far ends of the graph. As (far right), where does f(x) go? As (far left), where does f(x) go? 2. Why Only the Leading Term Survives Now take f(x) = 2x^3 - 5x^2 + 3x - 1 . Its highest-power piece is the leading term 2x^3 . Watch what happens when we factor it out of every term. Factor the leading term out of all four terms As |x| grows, each fraction inside the parentheses — , , — collapses toward 0 , so the whole bracket collapses toward 1 . What is left is f(x) behaving exactly like 2x^3 . 3. The Four End-Behavior Patterns Since only the leading term ax^n decides the ends, just two things settle everything: is the degree n even or odd, and is the leading coefficient a positive or negative? That gives exactly four cases. Read any polynomial's ends in one glance Ignore everything but the leading term. For g(x) = -3x^4 + 9x^3 - x + 6 the leading term is -3x^4 : even degree, negative coefficient, so both tails fall . For h(x) = 5x^7 - 100x^2 the leading term is 5x^7 : odd degree, positive coefficient, so it falls on the left and rises on the right — no graphing required.

This is the written version of the interactive lesson above. See the full Algebra 2 course.