Question

Sara kicked a football. The height of the ball (in meters above the ground)
\[t\] seconds after Sara kicked it is modeled by
\[h(t)=-5t^2+20t\]
Sara wants to know the height of the ball above the ground at its highest point.
1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation.
\[h(t)=\]
2) At its highest point, how far above the ground was the ball?
meters

Answer 1

1) To rewrite the function \( h(t) = -5t^2 + 20t \) in vertex form, we can complete the square.

First, we can factor out \(-5\) from the \(t^2\) and \(t\) terms:

\[
h(t) = -5(t^2 - 4t)
\]

Next, we need to complete the square for the expression \(t^2 - 4t\). To do this, we take half of the coefficient of \(t\) (which is \(-4\)), square it, and add and subtract this number inside the parentheses. Half of \(-4\) is \(-2\), and squaring it gives us \(4\):

\[
h(t) = -5(t^2 - 4t + 4 - 4)
\]
\[
h(t) = -5((t - 2)^2 - 4)
\]
\[
h(t) = -5(t - 2)^2 + 20
\]

Thus, the rewritten function in vertex form is:

\[
h(t) = -5(t - 2)^2 + 20
\]

2) The highest point of the ball occurs at the vertex of the parabola represented by the height equation. From the vertex form \(h(t) = -5(t - 2)^2 + 20\), we can see that the vertex is at \((2, 20)\).

Therefore, at its highest point, the ball was:

\[
\text{Height} = 20 \text{ meters}
\]

In conclusion:
1) \(h(t) = -5(t - 2)^2 + 20\)
2) The height of the ball at its highest point is \(\boxed{20}\) meters.