What translation rule can be used to describe the result of the composition of T<-5,-3>(X,Y) and T<-7,-1> (X,Y)?
A. T<12,-4>(X,Y)
B. T<-12,-4>(X,Y)
C. T<2,-2>(X,Y)
D. T<-4,-12>(X,Y)
B
Just add the components for X and Y
To find the result of the composition of T<-5,-3>(X,Y) and T<-7,-1>(X,Y), we need to apply the translation rules in the correct order.
When two transformations are composed, we perform the second transformation first, and then apply the first transformation to the result.
Let's start with T<-7,-1>(X,Y):
This translation rule shifts the coordinates (X,Y) by -7 units horizontally (to the left) and -1 unit vertically (downwards).
Next, let's apply T<-5,-3>(X,Y) to the result of T<-7,-1>(X,Y):
This translation rule shifts the coordinates by -5 units horizontally (to the left) and -3 units vertically (downwards).
Therefore, the result of the composition of T<-5,-3>(X,Y) and T<-7,-1>(X,Y) is T<-5,-3>[T<-7,-1>(X,Y)]:
T<-5,-3>[T<-7,-1>(X,Y)] = T<-5,-3>(X-7, Y-1).
Simplifying this expression, we get T<-5,-3>(X-7, Y-1) = (X-7-5, Y-1-3) = T<-12,-4>(X,Y).
Therefore, the correct translation rule to describe the result of the composition of T<-5,-3>(X,Y) and T<-7,-1>(X,Y) is:
B. T<-12,-4>(X,Y).
To find the result of the composition of two translations, we need to apply one translation followed by the other. Let's first perform the second translation, T<-7,-1>(X,Y), and then apply the first translation, T<-5,-3>(X,Y), to the result.
For the second translation, T<-7,-1>(X,Y), we simply subtract 7 from the X-coordinate and subtract 1 from the Y-coordinate. Therefore, the result of the second translation is (X-7, Y-1).
Next, we apply the first translation, T<-5,-3>(X,Y), to the result obtained from the second translation. This means we subtract 5 from the X-coordinate and subtract 3 from the Y-coordinate of (X-7, Y-1).
(X-7 - 5, Y-1 - 3) = (X-12, Y-4).
Hence, the result of the composition of T<-5,-3>(X,Y) and T<-7,-1>(X,Y) is T<-12,-4>(X,Y).
Therefore, option B, T<-12,-4>(X,Y), is the correct answer.