Translation of a Function

Translation of a Function

How to solve the translation of a function problems: formula, examples, and their solutions.

Formula

The image of a function y = f(x), whose image is under the translation (x, y) → (x + a, y + b), is y - b = f(x - a).

The image of a function y = f(x),
whose image is under the translation
(x, y) → (x + a, y + b),
is y - b = f(x - a).

Be careful of the signs of a and b.

It's different from the translation of a point.

Example 1

Find the image of y = 2x + 4 whose image is under the translation (x, y) → (x + 5, y + 3).

Translation:
(x, y) → (x + 5, y + 3)

Function:
y = f(x) → y - 3 = f(x - 5)

Example 2

Find the image of y = -x + 1 whose image is under the translation (x, y) → (x - 2, y + 7).

Translation:
(x, y) → (x - 2, y + 7)

Function:
y = f(x) → y - 7 = f(x - (-2))

Example 3

Find the image of y = x^2 whose image is under the translation (x, y) → (x + 5, y + 2).

Translation:
(x, y) → (x + 5, y + 2)

Function:
y = f(x) → y - 2 = f(x - 5)

This is the vertex form of the quadratic function.

See the graph of the image below. (red)
The vertex is (5, 2),
which can be found by the vertex form formula.

This is why the vertex form is true.

(x - 5)2 = x2 - 2⋅x⋅5 + 52

Square of a difference