Translation of a Function

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).

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

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

The given function is
y = 2x + 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) = 2x - 10

-10 + 4 = -6

y - 3 = 2x - 6
Move -3 to the right side.

Then y = 2x - 6 + 3.

-6 + 3 = 3

So y = 2x - 3.
This is the image.

Let's see the graphs of
the given function y = 2x + 4 (white)
and its image y = 2x - 3 (brown).

As you can see,
the image is under the translation
(x, y) → (x [+ 5], y [+ 3]).

Example 2

Find the image of y = -x + 1 whose image is under the translation (x, y) -> (x - 2, y + 6).

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

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

The given function is
y = x2.

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 = x2 - 2⋅5⋅x + 52

Square of a difference (a - b)2

-2⋅5⋅x = -10x
52 = 25

25 + 2 = 27

So y = x2 - 10x + 27.

This is the image.

Let's see the graphs of
the given function y = x2 (white)
and its image y = x2 - 10x + 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 = x2, (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