Translation: Function

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

Formula

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.

Example 1

Example

Solution

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.

Graph

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.

Example 2

Example

Solution

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.

Graph

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.

Example 3

Example

Solution

The image of [y = x2] 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.

Graph

This is the graph of [y = x2]
and its image
under the translation (x, y) → (x + 5, y + 2):
y - 2 = (x - 5)2.