Graphing Behavior at Zeros

Pre-Calculus · Axiom Academy

LESSON Graphing Behavior at Zeros Sketch a polynomial straight from its factors — each zero's local shape plus the end behavior gives you the whole curve. 1. Read the Zeros Right Off the Factors A zero is an x where f(x)=0 . Set each factor to 0 and the zero falls out; the exponent on that factor is its multiplicity — how many times the factor appears. Watch each factor drop a marker onto the x -axis at its zero. Three factors → three zeros, with multiplicities 2, 1, and 3 2. Multiplicity Sets the Local Shape Zoom in on a single zero and the multiplicity tells you exactly one of three things. The reason is the sign: an odd power lets the factor change sign as x passes through (the curve crosses); an even power keeps it one sign (the curve bounces); and a higher power presses the curve flat against the axis first. Odd power. The curve cuts straight through like a line. The curve touches and turns back , like a parabola at its vertex. The curve flattens , hugs the axis, then crosses — an S -bend. Odd multiplicity the graph crosses . Even multiplicity the graph bounces . The larger the multiplicity, the flatter the curve sits against the axis as it arrives. 3. End Behavior Frames the Two Arms Far from the zeros, only the leading term matters. Multiply the factors' degrees: 2+1+3=6 , an even degree with a positive leading coefficient. So both arms shoot the same way — up . Even degree means the ends match; odd degree would send them opposite ways.

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