Question

AB has a vértices at (-5,1) and (-3,7). After a transformation, the image had endpoints at (3,-1) and (5,5). Describe the transformation

Answer 1

If the image is A'B' we have, for the point A x=-5 goes to x = 3 ( a translation)

Its a translation of 8 to the right.

The same goes for point B the x coordinate goes 8 to the right.

The y coordinate for A translates 1 to -1 That is 2 units down

Same goes for B coordinate of y

Translation is 8 units to right and 2 down.

Answer 2

That looks like a translation; let's check.  We have

A(-5,1), B(-3,7), A'(3,-1), B'(5,5)

If it's a translation by T(x,y) we'd have

A' = A + T

B' = B + T

so 

T = A' - A = (3,-1) - (-5,1) = (8,-2)

and also 

T = B' - B = (5, 5) - (-3, 7) = (8,-2)

They're the same so we've verified this transformation is a translation by (8,-2), eight right, two down.