Quadratic Functions, Equations and Inequalities

A ball’s height \(h\) metres, \(t\) seconds after release, is modelled by
\[h=-(t-2)^2+9.\] The model applies from release until the ball first reaches the ground.

1. Find:

   1. the initial height;

   2. the greatest height and when it occurs;

   3. the time when the ball reaches the ground.

   Hence state the practical time interval.

2. Find the times when the ball is \(5\) m above the ground.

3. A horizontal line \(h=H\) is drawn on the height–time graph.

   Considering only the practical time interval, determine the values of \(H\) for which the ball is at height \(H\):

   1. exactly twice;

   2. exactly once;

   3. never.

   Include the boundary values carefully.