Determining End Behavior

Pre-Calculus · Axiom Academy

EXAMPLE Determining End Behavior Reading a polynomial's end behavior straight from its leading term. Determine the end behavior of each polynomial function — that is, describe what happens to the output as and as . Nice work — you read both polynomials' end behavior off a single term each. The leading term alone decides how a polynomial "ends". Only the leading term matters: the lower-degree terms ( 3x^2 , -x , 5 , -2x^3 , …) are swamped as |x| grows, so they never change the end behavior. Degree parity sets whether the arms match: even degree both ends point the same way; odd degree the two ends point opposite ways. The leading coefficient's sign sets the direction: a positive lead sends the right arm up; a negative lead sends it down. Results: has even degree and a negative lead, so at both ends ( ); has odd degree and a positive lead, so it falls on the left and rises on the right ( ). Both arms point down: at each end. Falls on the left, rises on the right: opposite ends. You can predict how any polynomial behaves at the edges of its graph just by looking at one term — degree parity tells you whether the ends match, and the leading sign tells you which way they point.

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