Question

A ball is thrown straight up from a height of 3 ft with a speed of 32 ft/s. Its height above the ground after x seconds is given by the quadratic function y = -16x2 + 32x + 3. Explain the steps you would use to determine the path of the ball in terms of a transformation of the graph of y = x2.

Answer 1

You can rewrite the trajectory equation as y = -16(x-1)^2 + 19

Therefore, the transformation from y = x^2 are as follow:

1.  Right shift 1 unit: (x-1)^2

2. Shrink 16 times horizontally: 16(x-1)^2

3. Inverse the whole shape: -16(x-1)^2

4. Upward shift 19 units: -16(x-1)^2 + 19

Answer 2

Complete the square to get the equation in vertex form with 
a = -16, h = 1, and k = 19. The path is a reflection over the x-axis and narrower. It is also translated right 1 unit and up 19 units.