How to find a translation image
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 xvalues and yvalues 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)
2 = x  2
x = 4
5 = y + 5
y = 10
(x , y) = (4, 10)

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