You have to translate the triangle ABC using the translation rule:
(x,y) → (x-1,y+2)
This rule indicates that the triangle will be translated 1 unit left and 2 units up. To determine the coordinates of each vertex after the translation, you have to subtract 1 to the x-coordinate of each point and add 2 to the y-coordinate.
Original → Translation
A(6,1) → A'(6-1,1+2) = A'(5,3)
B(1,7) → B'(1-1,7+2) = B'(0,9)
C(-3,-2) → C'(-3-1,-2+2) = C'(-4,0)
The coordinates of the vertices of triangle A'B'C' will be
A'(5,3)
B'(0,9)
C'(-4,0)
The correct option is C.