Multiplicity Effects

Algebra 2 · Axiom Academy

How a zero's multiplicity decides whether the graph crosses the x-axis or just touches it — and how flat it looks doing so. 1. A Single Factor: Crossing Straight Through When a factor appears exactly once, its zero has multiplicity 1 . Near that point the graph behaves like a straight line with a nonzero slope: it comes from one side of the x-axis and passes straight to the other. The value of f changes sign, so the graph crosses . One factor of (x − 2) → a zero of multiplicity 1 at x = 2 2. A Squared Factor: Touch and Turn Back When a factor is squared , its zero has multiplicity 2 . Near that point the graph flattens onto the x-axis, touches it, and turns back the way it came — like a ball bouncing off the ground. The value of f never changes sign. A repeated factor (x − 2)² → multiplicity 2, tangent to the axis 3. A Cubed Factor: Cross, but Flattened When a factor is cubed , its zero has multiplicity 3 . The multiplicity is odd, so the graph still crosses — but it flattens as it passes through, hugging the axis before continuing (an inflection). Compare it with the straight, steady crossing of multiplicity 1. A tripled factor (x − 2)³ → multiplicity 3, a flattened crossing 4. Turning the Multiplicity Dial

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