Question

Ruth hit's a baseball to the right field. The ball is 4 ft above the ground when she hits it. Three seconds later it reaches its maximum height of 148 ft.

Write an equation in vertex form for the quadratic function expressing the relationship between the height of the ball and time. Transform it so that height is expressed as a function of time.

Answer 1

Vertex form:
`h(t)=-16(t-3)^2+148`

Height expressed as a function of time:
`h(t)=-16t^2+96t+4`

Explanation:

We have been given that the ball is 4 ft above the ground when she hits it.

This means that initial value (y-intercept) of ball is 4.

We are also told that three seconds later it reaches its maximum height of 148 ft. This means
`h(3)=148`.

We know that vertex form of a parabola is in format
`y=a(x-h)^2+k`.

Let us solve for a using our given information.

`4=a(0-3)^2+148`

`4=a(-3)^2+148`

`4=a*9+148`

`4-148=9a+148-148`

`-144=9a`

`a=(-144)/(9)`

`a=-16`

Therefore, the vertex form of our given function would be
`y=-16(x-3)^2+148`.

Since we need height as a function of time, so we will get:

`h(t)=-16(t-3)^2+148`

`h(t)=-16(t^2-6t+9)+148`

`h(t)=-16t^2+96t-144+148`

`h(t)=-16t^2+96t+4`

Therefore, the function where height is expressed as a function of time would be
`h(t)=-16t^2+96t+4`.