Problem

A path's signed elevation relative to a reference level is \(E(x)=(x-3)^{2}-16~\text{meters}\), where \(x\) is position in meters on \(-2\le~x\le~8\). Find and interpret its zeros.

Big Picture

What this problem is really about

A zero of this signed-elevation model is a position where the output equals the reference level. We’ll set the model equal to zero, keep both square-root branches, check each candidate against the stated interval—including allowed negative positions—and convert the surviving inputs into graph points and contextual statements.

Answer choices

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Preview only

zero equation=(x−3)²=0; candidate=x=3; domain check=3 is in[-2,8]; zero=(3,0); interpretation=path is at the reference elevation at position3m

zero equation=(x−3)²=16; candidate=x=7; domain check=7 is in[-2,8]; zero=(7,0); interpretation=path is at the reference elevation only at position7m

zero equation=(x−3)²=16; candidates=x=-1,7; domain check=reject -1 because it is negative and keep7; zero=(7,0); interpretation=path is at the reference elevation only at position7m

zero equation=(x−3)²=16; candidates=x=3±4=-1,7; domain check=both in[-2,8]; zeros=(-1,0),(7,0); interpretation=path is at the reference elevation at positions-1m and7m

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A path's signed elevation relative to a reference level is \(E(x)=(x-3)^{2}-16~\text{meters}\), where \(x\) is position in meters on \(-2\le~x\le~8\). Find and interpret its zeros.

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