Precalculus Functions and Graphs Word Problem

Question

A ball is thrown from 5 inches above the ground. After 2 seconds, the ball reaches a maximum height of 9 inches, and then lands back on the ground 5 seconds after it was thrown. Write a function that models the path of the ball in vertex form.

Answer 1

Remember, vertex form is f(x) = a(x -h)² + k where (h,k) is the vertex.  Remember, x can be replaced with t if the function is tracking time.

 

If you read this problem closely, it describes 3 different times and the corresponding locations of the ball.  In essence, it is describing three different ordered pairs for your function.

 

The initial position of the ball (t = 0) was 5 inches.  Therefore, a coordinate pair is (0, 5).

 

At two seconds (t = 2), ball hits a maximum height of 9 inches.  Therefore, a coordinate pair is (2,9).  This coordinate pair happens to be the vertex, since the vertex is always the maximum or minimum of a function.

 

Finally, the ball hits the ground (h=0) at 5 seconds (t=5).  Therefore, a coordinate pair is (5, 0).

 

If you put your vertex into the function, you get f(t) = a(t - 2)² + 9.

We need to solve for a, and you can do that by putting either coordinate pair into the function so you only have to solve for a.  (It doesn't matter which, both will give you the same answer for a.)

 

Let's use the first point.

 

5 = a(0-2)² + 9

5 = 4a + 9

-4 = 4a

-1 = a

 

Therefore the function is f(t) = -1(t - 2)² + 9