Problem

For \(y=2(x+2)(x-6)\), separate the role of the scale factor from the factors that produce zeros. Give both zero equations, \(x\text{-}\text{intercepts}\), and multiplicities, then predict the graph’s behavior at each intercept.

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The zeros are −2 and 6, with multiplicity 1 each, so the graph crosses at both. The required x-intercepts are not written as ordered pairs.

The factor equations give zeros 2 and −6, so the intercepts are (2, 0) and (−6, 0). Solving gives −2 and 6 instead.

The nonzero scale factor 2 does not change the zeros. The zeros are −2 and 6, so the x-intercepts are (−2, 0) and (6, 0); each has multiplicity 1, so the graph crosses at both.

Using 2 = 0 as a zero equation gives x = 2 as an intercept. A nonzero constant factor never equals zero.

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For \(y=2(x+2)(x-6)\), separate the role of the scale factor from the factors that produce zeros. Give both zero equations, \(x\text{-}\text{intercepts}\), and multiplicities, then predict the graph’s behavior at each intercept.

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