# Translation of a Function

How to find the function under the given translation: formula, 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 translation of a point,

write -*a* and -*b*.

Translation of a point

## Example 1

The given function is*y* = 2*x* + 4.

Its image is under the translation

(*x*, *y*) → (*x* [+ 5], *y* [+ 3]).

Then the image is,

change the signs of the changes,*y* [- 3] = 2(*x* [- 5]) + 4.

2(*x* - 5) = 2*x* - 10

-10 + 4 = -6

*y* - 3 = 2*x* - 6

Move -3 to the right side.

Then *y* = 2*x* - 6 + 3.

-6 + 3 = 3

So *y* = 2*x* - 3.

This is the image.

Let's see the graphs of

the given function *y* = 2*x* + 4 (white)

and its image *y* = 2*x* - 3 (brown).

As you can see,

the image is under the translation

(*x*, *y*) → (*x* [+ 5], *y* [+ 3]).

## Example 2

The given function is*y* = -*x* + 1.

Its image is under the translation

(*x*, *y*) → (*x* [- 2], *y* [+ 6]).

Then the image is,

change the signs of the changes,*y* [- 6] = -(*x* [+ 2]) + 1.

-(*x* + 2) = -*x* - 2

-2 + 1 = -1

*y* - 6 = -*x* - 1

Move -6 to the right side.

Then *y* = -*x* - 1 + 6.

-1 + 6 = 5

So *y* = -*x* + 5.

This is the image.

Let's see the graphs of

the given function *y* = -*x* + 1 (white)

and its image *y* = -*x* + 5 (brown).

As you can see,

the image is under the translation

(*x*, *y*) → (*x* [- 2], *y* [+ 6]).

## Example 3

The given function is*y* = *x*^{2}.

Its image is under the translation

(*x*, *y*) → (*x* [+ 5], *y* [+ 2]).

Then the image is,

change the signs of the changes,*y* [- 2] = (*x* [- 5])^{2}.

Move -2 to the right side.

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

(*x* - 5)^{2} = *x*^{2} - 2⋅5⋅*x* + 5^{2}

Square of a difference (*a* - *b*)^{2}

-2⋅5⋅*x* = -10*x*

5^{2} = 25

25 + 2 = 27

So *y* = *x*^{2} - 10*x* + 27.

This is the image.

Let's see the graphs of

the given function *y* = *x*^{2} (white)

and its image *y* = *x*^{2} - 10*x* + 27 (brown).

As you can see,

the image is under the translation

(*x*, *y*) → (*x* [+ 5], *y* [+ 2]).

See the change of the vertices.

The vertex of *y* = *x*^{2}, (0, 0),

is moved to (5, 2).

And the quadratic function of the image is*y* = (*x* - 5)^{2} + 2,

which is in vertex form.

This is why

when the vertex of the quadratic function is (*h*, *k*),

the quadratic function in vertex form is*y* = (*x* - *h*)^{2} + *k*.

Quadratic function - Vertex form