What is the vertex form of the equation that represents the height of the water balloon, h, after t seconds?

Question

Justin launches a water balloon vertically from a platform on his roof. The height of the balloon (in feet) is represented by the equation h=-16t^2+32t+48  where  is the time (in seconds) after he launches the water balloon.

What is the vertex form of the equation that represents the height of the water balloon,  after  seconds?

Answer 1

Hello, Angela,

I'm not clear on this type of problem, so please read cautiously. I believe the vertex form of a function is represented by the equation f (x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. For this equation, I get h = -16(t-1)^2 + 64. The vertex is therefore (1, 64). The h is the shift to the right and k is the vertical shift.

When I plot the graph, it is an inverted parabola, with a maximum at around t = I. I initially had difficulty explaining it, but it appears the height at t = 0 is 48 feet, which I assume is the roof height. Due to the fact that the balloon goes up upon release, I must assume it is literally launched upwards until gravity slows it down to zero at the vertex and it starts falling at 32ft/sec², the acceleration due to gravity. It appears that it will hit ground at 3 seconds.

Good luck. I hope this is correct, but please check carefully.

Bob