Let P be the transformation representing a reflection over the line y=b and let Q be the transformation representing a reflection over the line y=a, where a < b. Describe a single transformation that would give the same results as P∘Q, if the preimage isn't on neither line y = a nor the line y = b.
P: (x,y) → (x,b-(y-b)) = (x,2b-y)
Q: (x,y) → (x,2a-y)
P∘Q: (x,y) → (x,2a-(2b-y)) = (x,2a-2b+y)
Since b > a, it should be (x, 2b-2a+y)
To describe a single transformation that would give the same results as the composition P∘Q, we can break it down into two steps. First, we would need to reflect the preimage over the line y = a, and then reflect the result over the line y = b.
Let's call the preimage point P(x, y). To reflect P over the line y = a, we can use the transformation R1:
R1: Reflect over y = a
To perform the reflection, we need to find the distance between the preimage point P and the line y = a. Let's call this distance d. Then, the reflection of P would have the y-coordinate y' = a - d.
Next, we take the result of the first reflection and reflect it over the line y = b. We can use another transformation, R2, to do this:
R2: Reflect over y = b
Again, we find the distance between the image point of R1 (let's call it P') and the line y = b. Let's call this distance d'. Then, the reflection of P' would have the y-coordinate y'' = b - d'.
So, the transformation that would give the same results as the composition P∘Q is a reflection over the line y = b'':
R'': Reflect over y = b''
To find the equation of the line y = b'', we need to find the distance between y = b and y = a, and subtract this distance from b. Let's call this distance d'':
d'' = b - a
Now we can define the transformation R'':
R'': Reflect over y = b - (b - a)
So, the single transformation R'' would give the same results as the composition P∘Q if the preimage is not on either line y = a or y = b.