A translation is a type of rigid transformation, which means it changes the position or orientation of an image without changing its size or shape.

Just like how we have up/down and left/right buttons to control the movement of a video game character, we can use translations to control the movement of an image:

Not quite!

Enter the translation that takes us from our starting point to the blue, ending point:

Translation:

$($

$0$

$,$

$0$

$)$

A translation is a set of directions in the form of a point

$($$xchange$$,$$ychange$$)$

or a description ($x$ units right/left and $y$ units up/down) that tells us how to move an image or shape.

Adding the translation to each original point of the image gives us the new points of the image.

$original points+translation=new points$

Check out our step-by-step translations calculator to practice translating images or keep scrolling to learn more about how to do horizontal and/or vertical translations on a graph.

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Translations Calculator

Step 1. Identify the translation.

How is the translation described?

Step 2. Turn the description of the translation into a point.

Description

→

Value

$0$ right

→

$0$

$0$ up

→

$0$

$0$ right and $0$ up → $($$0$$,$$0$$)$

Step 3. Identify the original points.

How many points does your shape have?

Step 4. Add the translation to the original points to get the new points.

Original Point

$+$

Translation

$=$

New Point

Step 5. Plot the new points and connect the dots.

Key Steps 🗝 How to Do Translations

Step 1. Identify the translation.

How is the translation described?

Step 2. Turn the description of the translation into a point.

Step 3. Add the translation to the original points to get the new points.

Original Point

$+$

Translation

$=$

New Point

$(x_{1},y_{1})$

$+$

$(x,y)$

$=$

$(x_{1}+x,y_{1}+y)$

$(x_{2},y_{2})$

$+$

$(x,y)$

$=$

$(x_{2}+x,y_{2}+y)$

...

...

...

...

...

$(x_{n},y_{n})$

$+$

$(x,y)$

$=$

$(x_{n}+x,y_{n}+y)$

Step 4. Plot the new points and connect the dots.

Let’s check out some example problems of horizontal and vertical translations!

Translations

A translation works just like the up/down and left/right arrows we use to control the movement of a character of a video game.A translation is a set of directions for how to move an image and can be in the form of a point $(x,y)$ or a description ($x$ units up/down and $y$ units right/left).

If $x$and $y$,the translation is both horizontal and vertical.

Adding the translation to each original point of the shape gives us the new points of the shape.

$original points+translation=new points$

Let’s walk through some examples to see this in action!

Let's start by translating this image $(3,1)$:

Step 1. Identify the translation.

The translation is $($$3$$,$$1$$)$, which means we're translating $3$ right and $1$ up.

$($$,$$)$

Step 2. Turn the description of the translation into a point.

Description

→

Value

$3$ right

→

$3$

$1$ up

→

$1$

$3$ right and $1$ up → $($$3$$,$$1$$)$

Step 2. Identify the original points.

Original

$(x_{original},y_{original})$

Point 1

$($$,$$)$

Point 2

$($$,$$)$

Point 3

$($$,$$)$

From the diagram, we can see that our original points are: $(4,5),(3,−2),(1,2)$.

Step 3. Add the translation to the original points to get the new points.

Original Point

$+$

Translation

$=$

New Point

$(4,5)$

$+$

$(3,1)$

$=$

$(7,6)$

$(3,−2)$

$+$

$(3,1)$

$=$

$(6,−1)$

$(1,2)$

$+$

$(3,1)$

$=$

$(4,3)$

Step 4. Plot the new points and connect the dots.

Nice work! Now let's try translating this image $2$ to the left and $3$ down:

Step 1. Identify the translation.

The translation is $2$ left and $3$ down.

↔️units↕️units

Step 2. Turn the description of the translation into a point.

Description

→

Value

$2$ left

→

$−2$

$3$ down

→

$−3$

$2$ left and $3$ down → $($$−2$$,$$−3$$)$

Step 3. Identify the original points.

Original

$(x_{original},y_{original})$

Point 1

$($$,$$)$

Point 2

$($$,$$)$

Point 3

$($$,$$)$

Point 4

$($$,$$)$

From the diagram, we can see that our original points are: $(2,2),(3,4),(5,4),(6,2)$.

Step 4. Add the translation to the original points to get the new points.

Original Point

$+$

Translation

$=$

New Point

$(2,2)$

$+$

$(2,3)$

$=$

$(4,5)$

$(3,4)$

$+$

$(2,3)$

$=$

$(5,7)$

$(5,4)$

$+$

$(2,3)$

$=$

$(7,7)$

$(6,2)$

$+$

$(2,3)$

$=$

$(8,5)$

Step 5. Plot the new points and connect the dots.

If you’d like additional practice with translations, take a moment to complete the quick practice below before closing out this lesson! ⚡

Practice: Translations

Question 1 of 10: Translate the following figure by $(3,5)$:

Step 1. Identify the translation.

$($$,$$)$

Step 2. Turn the description of the translation into a point.

$3$ right and $5$ up →

$($$,$$)$

Step 2. Identify the original points.

Enter the original points of the shape:

Original Points

$(x_{original},y_{original})$

Point 1

$($$,$$)$

Point 2

$($$,$$)$

Point 3

$($$,$$)$

Point 4

$($$,$$)$

Step 3. Add the translation to the original points to get the new points.

Original Points

+

Translation

=

New Point

$(1,1)$

+

$(3,5)$

=

$($$,$$)$

$(−1,1)$

+

$(3,5)$

=

$($$,$$)$

$(−1,−1)$

+

$(3,5)$

=

$($$,$$)$

$(1,−1)$

+

$(3,5)$

=

$($$,$$)$

Step 4. Plot the new points and connect the dots.

Awesome job! Our new translated shape is the blue shape:

When you feel like you've mastered this lesson, click for a celebration ⬇️!

Nice work, look at you go! Thanks for checking out this lesson ☺️🙏. Where to next?