Question

(consider the vertex you found in the previous problem)
a rocket is launched in the air. its height in feet is given by ( h(t)=-16 t^{2}+96 t ) where ( t ) represents the time in seconds after launch. what does the ( h(t) )-coordinate of the vertex mean?
( \bigcirc ) how long the rocket has been in the air at its peak
( \bigcirc ) how long the rocket has been in the air when it hits the ground
( \bigcirc ) the rockets height at its peak
( \bigcirc ) the rockets height when it hits the ground

Explanation:

The function \( h(t)=-16t^{2}+96t \) is a quadratic function in the form \( y = ax^{2}+bx + c \) (here \( y = h(t)\), \(x=t\), \(a=-16\), \(b = 96\), \(c = 0\)). For a quadratic function \(y=ax^{2}+bx + c\), the vertex \((t_{0},h(t_{0}))\) gives the maximum (since \(a=-16<0\)) value of the function. The \(t\) - coordinate of the vertex gives the time at which the maximum occurs, and the \(h(t)\) - coordinate gives the maximum value of the function. In the context of the rocket's height, the maximum value of \(h(t)\) represents the rocket's maximum height (its height at the peak).

Answer:

the rocket's height at it's peak