Before we can figure out how to correct the path of the ball so that it will hit the brick, let's look at the uninterrupted path it will take.
If we move our paddle out of the way rather than using it to help us hit the brick, we will miss the target by 2 steps to the right. However, because the ball will always deflect at a right angle, we know that the slope will always change in the same way.
Before deflection:& m=-1
After deflection:& m=1
How can we use this to figure out where can we place our paddle so that the ball is deflected in just the right way to hit the brick? Let's trace our steps backwards from the brick and see where it intersects with the original path.
If we place our paddle at (7,1) the ball will change direction in time to accurately hit the brick.
b
Let's compare the graph of the ball's path to the parent absolute value function. First, we need to graph them on the same coordinate plane.
How can we write a transformation of the parent function such that the new function will be the same as the ball's path? The slopes are the same for both functions, so we know that it will only be translations. So, how can we get from the parent vertex to the transformed vertex?
The graph is shifted by 1 unit up, a vertical translation, and 7 units to the right, a horizontal translation. Let's write the general form for a vertical and a horizontal translation.
Vertical translation:& g(x)=f(x)+k
Horizontal translation:& g(x)=f(x-h)
In this form, k is the number of units the graph is shifted up (for positive numbers) or down (for negative numbers), and h is the number of units the graph is shifted right (for positive numbers) or left (for negative numbers). The graph of the function is shifted 1 unit up and 7 units to the right. Let's write the equation for the transformation.
y=|x-7|+1
