Question

Two students are bouncing-passing a ball between them. The first student bounces the ball from 6 feet high and it bounces 5 feet away from her. The second student is 4 feet away from where the ball bounced.Create an absolute value function to represent the situation. How high did the ball bounce for the second student to catch it?

Answer 1

We need to find a function of the form:

`a\mleft|x-b\mright|+c`

Such that it equals 6 when x=0 and 0 when x=5.

Substitute x=5 and assume that the expression is equal to 0:

`a\mleft|5-b\mright|+c=0`

Since the vertex of the function is at the point (5,0), we can assume that c=0.

Therefore:

`a\mleft|5-b\mright|=0`

Divide both sides by a:

`|5-b|=0`

Using that for any number k, |k| = 0 if and only if k=0, then:

`5-b=0`
`\text{Therefore, b=5.}`

Next, substitute x=0 and assume that the expression is equal to 6 to find a.

`a\mleft|0-5\mright|=6`

Since |0-5|=5:

`5a=6`

Dividing both sides by 5, we get the value of a:

`a=(6)/(5)`

Substitute a=6/5, b=5 and c=0 in the original equation:

`(6)/(5)|x-5|`

You can check that this expression is equal to 6 when x=0 and it is equal to 0 when x=5.

To find the height at which the second studend should catch the ball, notice that the second student will be placed at x=5+4=9.

Substitute x=9 to find how high will the ball bounce for the second student to catch it:

`(6)/(5)|9-5|=(6)/(5)|4|=(6\cdot4)/(5)=(24)/(5)=4.8\text{ ft}`

To summarize:

`\begin{gathered} \text{The function which represents this situation is:} \\ f(x)=(6)/(5)|x-5| \\ \\ \text{The ball bounces up to }4.8\text{ feet for the second student to catch it.} \end{gathered}`