Final answer:
To find a translation T such that T⁶=S, we can assume that T has the form T:(x,y) → (x+a, y+b) and solve for the values of 'a' and 'b', which turns out to be a=2 and b=-1. Therefore, the translation T is given by T:(x,y) → (x+2, y-1), where T⁶=S.
Step-by-step explanation:
To find a translation T such that T⁶=S, we need to determine the translation rule for T. Let's assume that T has the form T:(x,y) → (x+a, y+b), where 'a' and 'b' are constants. We can now substitute this into T⁶ and equate it to S:
T⁶:(x,y)→(x+6a, y+6b) = S:(x+12, y-3)
Now, equating the x-coordinate and y-coordinate of both sides, we get two equations:
x+6a=x+12
y+6b=y-3
Simplifying these equations, we find that a=2 and b=-1. Therefore, the translation T is given by T:(x,y) → (x+2, y-1), where T⁶=S.