Let's graph the original triangle first.
Since we're given the translation 7 units to the left, and 6 units down, we are going to subtract 7 units to the x coordinates and 6 units to the y-coordinates.
A = (-2,3)
A' = (-2 - 7, 3 - 6) = (-9, -3)
B = (-4, 7)
B' = (-4 - 7, 7 - 6) = (-11, 1)
C = (6, 8)
C' = (6 - 7, 8 - 6) = (-1, 2)
The graph of the two triangles is this: