To find a translation image of a shape, you can use the following rule or formula.

Suppose you want to translate or slide point P a units horizontally and b units vertically. Then, change the x-values and y-values of the coordinates of P.

P = (x, y) translates to P’ = (x + a, y + b)

Example #1

Find the image of point P (-2, 1) under the translation (x, y) →(x + 3, y – 5)

Under this translation, P (-2, 1) translates to (-2 + 3, 1 – 5) or P'(1, -4)

Example #2

Find the image of triangle ABC under the translation (x, y) →(x + 6, y – 3)

The points of the triangle of are A(-3, 1), B(-4, 3), and C(-2, 4). Use the rule to find each vertex in the translated image.

A(-3, 1) translates to (-3 + 6, 1 – 3) or A'(3, -2)

A(-4, 3) translates to (-4 + 6, 3 – 3) or A'(2, 0)

A(-2, 4) translates to (-2 + 6, 4 – 3) or A'(4, 1)

Example #3

A translation maps (8, -12) → (0, 14). Find a rule to describe this translation.

(8, -12) → (0, 14)

We are looking for a and b such that the rule is (x + a, y + b). We know x = 8 and y = -12

Therefore, (0, 14) → (8 + a, -12 + b)

0 = 8 + a

a = -8

14 = -12 + b

b = 14 + 12 = 26

The rule is (x, y) → (x – 8, y + 26)

Example #4

The coordinate of the image of a point is (2, -5) under the translation (x , y) → (x – 2, y + 5). Find the coordinate of the preimage.

We are just looking for x and y and we know (2, -5) → (x – 2, y + 5)