Applications of Quadratic Equations and Functions

Question

A soccer ball is kicked into the air so that its height, h, in feet, after t seconds is given by the

function:

h(t) = −16t²! + 96

What is the maximum height the ball reaches?

How long is the ball in the air?

Answer 1

When kicked in the air, the ball’s movement creates a parabola shape for its path. We can find the vertex of the formula to determine the highest time and height.

Since the parabola is in vertex form, let’s use transformations to see where the vertex is.

Vertex form is: y= a(x-h)²+k

to find the new vertex, we start with the vertex of the parent function (y=x²), which is (0,0). From our function, we can use the h and k values to see how much we move that point. K is used for moving the y value, in our case, we add 96, shifting it up. For the h value, there is no h value written, so we don’t shift the parabola horizontally.

This gives us a vertex of (0,96), x relating to time and y relating to height. The highest point the soccer ball reached was 96 feet and got to that point in 0 seconds. In a real life example, it would mean the ball was dropped from 96 feet in the air