You try to do a trick shot and make a hole in one. The ball is at (2,2) and the hole is located at (8,3). Youbank it off the side wall at (5,8).

Question

Write an equation for the path of the ball and determine whether you will make the shot.

Answer 1

This type of problem most likely uses an absolute value equation due to the "bounce" off the wall. These equations are almost exactly the same as a typical linear function except for the fact that the x components are within absolute value brackets.

 

First, we have to find the vertex or the spot where the ball changes direction. Because the ball hits the wall and changes direction at (5,8), this is our vertex:

 

f(x)  = a|x-5|+8

 

Now, we need to find the slope and direction of the function. Does it open up or down? How much does it stretch/shrink? We can find this by using the slope formula. We'll use the vertex and starting point: (2,2) and (8,3)

 

(y1 - y2)/(x1 - x2) = (2-3)/(2-8) = 1/6

 

The graph moves from left to right, so we know that the first half of the graph has a positive slope at 1/6. This means that when the graph changes direction, it makes an upside down "V", so the graph is negative:

 

f(x) = -(1/6)|x-5|+8

 

Now, all we have to do is plug in the hole's coordinates (8,3) and see if the function is still true!

 

3 = -(1/6)|3-5|+8

3 = -(1/6)*2+8

3 = 8-(1/3)

 

3 is not equal to 8-(1/3), so we know that the ball never makes it to the hole.

 

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