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transformations?

Given the translation T(x,y) → (x – 4, y + 2), find:
a. the image of (2, -1)
b. the preimage of (3, 5)
 
Given the translation T(x,y) → (x – 1, y + 2), find:
a. the image of (-2, 1)
b. the preimage of (3, 2)
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1 Answer

I) Under the transformation T(x,y) → (x-4,y+2), we have that:
 
a) T(2,-1)=(2-4,-1+2)=(-2,1)
 
If a point (x,y) has already been through the transformation, then the previous x-coordinate decreased by 4, and the y-coordinate increased by 2 - hence we perform the opposite operations; i.e.:
 
b) (3,5)=(x-4,y+2); equating x-coordinates: 3=x-4, or x=7
                                equating y-coordinates: 5=y+2 or y=3
   Hence the pre-image of the point (3,5), was (7,3)
 
II) Under the transformation T(x,y)→ (x-1,y+2), we have that:
 
a) T(-2,1)=(-2-1,1+2)=(-3,3)
 
If a point (x,y) has already been through the transformation, then its x-coordinate has decreased by one, and its y-coordinate has increased by two - so we perform the opposite operations to find the pre-image:
 
b) (3,2)=(x-1,y+2); equating x-coordinates: 3=x-1, or x=4
                               equating y-coordinates: 2=y+2, or y=0
   Hence the pre-image of (3,2), is the point (4,0)