This problem is really asking about the coordinates of the vertex. The x-coordinate of the vertex represents time (t) and the y-coordinate represents height (h).
To find the x-coordinate of the vertex of a quadratic of the form , use:
Once you've found the x-coordinate, that represents the time at which the object reaches its highest point. To find the height AT THAT TIME, simply plug in the x-coordinate to your equation (for t) and you'll get out the height.