# Translation: Function

How to find the image under the translation of a function: formula, 3 examples, and their solutions.

## Formula

The image of a function y = f(x)

under the translation (x, y) → (x + a, y + b) is

y - b = f(x - a).

Unlike the the translation of a point,

change x to x - a

and change y to y - b.

## Exampley = 2x + 4, Translation (x, y) → (x + 5, y + 3)

The image of [y = 2x + 4] is

under the translation (x, y) → (x + 5, y + 3).

Then the image function is,

change x to x - 5

and change y to y - 3,

y - 3 = 2(x - 5) + 4.

2(x - 5) = 2x - 10

-10 + 4 = -6

y - 3 = 2x - 6

Move -3 to the right side.

-6 + 3 = -3

So y = 2x - 3.

So

y = 2x - 3

is the answer.

This is the graph of [y = 2x + 4]

and its image

under the translation (x, y) → (x + 5, y + 3):

y - 3 = 2(x - 5) + 4.

## Exampley = -x + 1, Translation (x, y) → (x - 2, y + 6)

The image of [y = -x + 1] is

under the translation (x, y) → (x - 2, y + 6).

Then the image function is,

change x to x + 2

and change y to y - 6,

y - 6 = -(x + 2) + 1.

-(x + 2) = -x - 2

-2 + 1 = -1

y - 6 = -x - 1

Move -6 to the right side.

-1 + 6 = +5

So y = -x + 5.

So

y = -x + 5

is the answer.

This is the graph of [y = -x + 1]

and its image

under the translation (x, y) → (x - 2, y + 6):

y - 6 = -(x + 2) + 1.

## Exampley = x^{2}, Translation (x, y) → (x + 5, y + 2)

The image of [y = x^{2}] is

under the translation (x, y) → (x + 5, y + 2).

Then the image function is,

change x to x - 5

and change y to y - 2,

y - 2 = (x - 5)^{2}.

Move -2 to the right side.

So

y = (x - 5)^{2} + 2

is the answer.

This is the graph of [y = x^{2}]

and its image

under the translation (x, y) → (x + 5, y + 2):

y - 2 = (x - 5)^{2}.