Translations

Q: How do I know if the rule moves the point left or right?

Look at the sign on the number added to $x$. In $(x, y) \to (x + a, y + b)$, a positive $a$ moves right and a negative $a$ moves left. The same idea works for $b$: positive is up, negative is down.

A translation slides every point of a figure the same distance in the same direction. Nothing flips, nothing turns, nothing grows — the figure just moves. The image (the new figure) is the exact same size and shape as the original, which is why the original and its image are congruent.

On the coordinate plane, a translation is just arithmetic. One rule tells you what to add to or subtract from each coordinate, and you apply that same rule to every point. Once you can read the rule, translating a whole triangle is no harder than translating one point.

The translation rule

Every translation can be written as $(x, y) \to (x + a, y + b)$ . The number $a$ moves the point right or left, and the number $b$ moves it up or down.

The signs tell you the direction. If $a$ is positive, the point moves right; if $a$ is negative, it moves left. If $b$ is positive, the point moves up; if $b$ is negative, it moves down. So the rule $(x, y) \to (x + 3, y - 5)$ means every point slides $3$ right and $5$ down.

To apply a rule, handle one coordinate at a time: do the $x$ arithmetic, then the $y$ arithmetic. Most mistakes come from rushing both at once and dropping a sign.

Reading a translation from a graph

When you are given the original figure and its image and asked for the translation, do not stare at the whole shape. Pick one vertex of the original and find its matching vertex on the image — the corner in the same position on the shape.

Then count: how far did that vertex move horizontally, and how far did it move vertically? Those two numbers are the whole translation. In the graph below, the black triangle's left corner is at $(-4, 1)$ and its image's matching corner is at $(1, -2)$ — that is $5$ right and $3$ down, so the rule is $(x, y) \to (x + 5, y - 3)$ .

What stays the same

A translation changes only the position of a figure. Side lengths, angle measures, and the way the figure faces all stay exactly the same. That makes translations easy to check: if your image looks stretched, flipped, or turned, something went wrong.

Image points are named with a prime mark: point $A$ translates to $A'$ , read "A prime." The prime just means "the image of $A$ " — same point of the figure, new location.

Worked examples

Example 1: apply a rule to a point

A figure is translated by the rule $(x, y) \to (x + 5, y - 3)$ . Find the image of $(2, 4)$ .

Write the rule

(x, y) \to (x + 5, y - 3)

Add

5

to the

x

-coordinate

2 + 5 = 7

Subtract

3

from the

y

-coordinate

4 - 3 = 1

Answer: $(7, 1)$

Example 2: translate a whole triangle

Translate triangle $ABC$ with $A(-3, 1)$ , $B(-1, 4)$ , $C(0, 1)$ four units right and two units down.

Right

4

, down

2

as a rule

(x, y) \to (x + 4, y - 2)

Apply the rule to

A(-3, 1)

A' = (1, -1)

Apply the rule to

B(-1, 4)

B' = (3, 2)

Apply the rule to

C(0, 1)

C' = (4, -1)

Answer: $A'(1, -1)$ , $B'(3, 2)$ , $C'(4, -1)$

Example 3: find the rule, then use it

A translation moves $(1, -2)$ to $(4, 3)$ . Where does the same translation move $(0, 0)$ ?

Compare the

x

-coordinates

4 - 1 = 3

Compare the

y

-coordinates

3 - (-2) = 5

Write the rule

(x, y) \to (x + 3, y + 5)

Apply it to

(0, 0)

(0 + 3, 0 + 5) = (3, 5)

Answer: $(3, 5)$

Try one yourself

Common questions

What does the prime symbol mean, like

A'

It marks the image of a point. If a transformation moves point $A$ , its new location is named $A'$ , read "A prime." The whole image triangle is $A'B'C'$ .

Does a translation change the size or shape of a figure?

No. A translation only changes position. The image is congruent to the original — same side lengths, same angles, facing the same way.

How do I know if the rule moves the point left or right?

Look at the sign on the number added to $x$ . In $(x, y) \to (x + a, y + b)$ , a positive $a$ moves right and a negative $a$ moves left. The same idea works for $b$ : positive is up, negative is down.

Do I have to translate every point of the figure?

Only the vertices. Apply the rule to each corner, plot the images, and connect them — the sides come along automatically because every point of the figure moves the same way.

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Translations

The translation rule

Reading a translation from a graph

What stays the same

Worked examples

Example 1: apply a rule to a point

Example 2: translate a whole triangle

Example 3: find the rule, then use it

Try one yourself

Common questions

Keep learning

Reflections

Rotations

Dilations