in a translation, how can you use the relationship between some points and their images to find the images of other points?
Let's see if I understand your question. When you translate (slide) a preimage, you want to know the coordinates of the image? A specific example would help, but in general:
In the above generalization, the coordinates in the first matrix (preimage) are shifted (without rotation or dilation) left or right by x units and up or down by y units. The coordinates of the resulting image is indicated by the sum. Of course, if x is negative the shift is to the left, and if y is negative, the shift is down.
Let's look at a specific example.
TRANSLATE TRIANGLE ABC WITH A(-3, 4), B(-1, 2), C(-3,2) TO TRIANGLE A'B'C' BY SLIDING RIGHT 4 UNITS AND DOWN 6 UNITS.
TRIANGLE A'B'C' HAS COORDINATES A(1, -2), B(3, -4), C(1, -4)
I hope this helped a little.