Problem

For \(y=-(x+3)^{2}\), solve the zero equation and state the \(x\text{-}\text{intercept}\) and its multiplicity. Explain how the even multiplicity and negative leading factor determine the graph’s behavior at and around the intercept.

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Set x + 3 = 0, giving zero −3 and x-intercept (−3, 0). Its multiplicity is 2, so the graph touches and turns at the x-axis and remains at or below it.

x + 3 = 0 gives zero 3 and intercept (3, 0). Solving gives −3, not 3.

The zeros are −1 and −3 because the negative sign is also set equal to zero. The factor −1 never equals zero.

The squared factor gives two distinct intercepts at (−3, 0), each with multiplicity 1, so the graph crosses twice. A repeated zero has one intercept and multiplicity 2.

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For \(y=-(x+3)^{2}\), solve the zero equation and state the \(x\text{-}\text{intercept}\) and its multiplicity. Explain how the even multiplicity and negative leading factor determine the graph’s behavior at and around the intercept.

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