Problem

A projectile's height is modeled by \(h(t)~=~-16(t~-~2)^2~+~64\) for \(0~\le~t~\le~4\), where \(t\) is time in seconds after launch. Interpret when the height is increasing and when it is decreasing during the flight.

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What this problem is really about

Context trims a parabola’s algebraic behavior to the part that represents the actual event. We’ll locate and classify the vertex, describe how height changes on either side of it, and intersect those behaviors with the stated flight interval before attaching time units.

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The height changes direction at 64 seconds because 64 is the important value in the equation, so it increases before 64 and decreases after 64.

The height decreases from 0 to 2 seconds because the coefficient is negative, and it increases from 2 to 4 seconds on the other side of the vertex.

The height increases from 0 to 2 seconds because the graph rises to its vertex at t = 2, and it decreases from 2 to 4 seconds as the graph falls after that maximum.

The height increases until 2 seconds and then decreases forever after that, since a downward-opening parabola keeps falling to the right of the vertex.

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A projectile's height is modeled by \(h(t)~=~-16(t~-~2)^2~+~64\) for \(0~\le~t~\le~4\), where \(t\) is time in seconds after launch. Interpret when the height is increasing and when it is decreasing during the flight.

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