In the following graph we observe line segment ab being mapped by translation to line segment a'b'. a. in two or more sentences, explain what test you could perform to check the validity of this translation. b. prove mathematically that the translation is valid using the test you mentioned above.

1. Looking at the graph, the shape of the line is not changing. The line can be described as relatively vertical with point A on the top left of point b which mean there is no rotation or mirroring applied. 
The line also has the same size as the original line so there is no stretch. In this case, the translation should be only shifting.
The shifting can be determined by using this equation to any of the points:
X'a-Xa=  3- (-4)= 7
Y'a-Ya= 6-3= 3
The shift is 7 to right and 3 to up

2. You can prove that it is right by applying the translation into point B. The calculation would be:
X'b= Xb + translation
X'b= -2+ 7 = 5

Y'b= Yb + translation
Y'b= 3+ 3 = 6
The point B coordinate is (5,6) so the translation is true.