Translations

Q: How do I tell left from right in a rule?

Look at the sign attached to $x$. In $(x, y) \to (x + 4, y)$ the figure moves $4$ units right; in $(x, y) \to (x - 4, y)$ it moves $4$ units left. The same logic applies to $y$: plus is up, minus is down.

A translation slides a figure — every point moves the same distance in the same direction. Nothing flips, nothing turns, nothing changes size. If you slide a triangle $4$ units right, all three vertices move $4$ units right and the image is congruent to the original.

On the coordinate plane, a translation is written as a rule like $(x, y) \to (x + 3, y - 5)$ . Once you can read that rule — and build one from a description or from a pair of points — translation problems become pure arithmetic.

Reading a translation rule

Every translation rule has the form $(x, y) \to (x + a, y + b)$ . The number added to $x$ controls horizontal movement: positive means right, negative means left. The number added to $y$ controls vertical movement: positive means up, negative means down.

So $(x, y) \to (x + 3, y - 5)$ says: slide $3$ units right and $5$ units down. To apply it to a specific point, substitute the coordinates and do the arithmetic — the point $(1, 2)$ becomes $(1 + 3, 2 - 5) = (4, -3)$ .

Watch the signs when the starting coordinates are negative. Applying $(x, y) \to (x + 3, y - 5)$ to $(-7, -1)$ gives $(-7 + 3, -1 - 5) = (-4, -6)$ .

Finding the rule from two points

Sometimes you're given a point and its image and asked for the rule. Subtract, coordinate by coordinate: image minus original. The horizontal change is $a$ and the vertical change is $b$ .

The graph below shows a point at $(-3, -2)$ and its image at $(1, 1)$ . Horizontal change: $1 - (-3) = 4$ . Vertical change: $1 - (-2) = 3$ . The rule is $(x, y) \to (x + 4, y + 3)$ — a slide of $4$ units right and $3$ units up, shown by the dashed segment.

Translating a whole figure

To translate a polygon, apply the rule to each vertex and reconnect them in the same order. Because every vertex moves the same distance and direction, the image has the same side lengths and the same angles as the original — a translation is a rigid motion.

This also gives you a quick check: after translating, pick any two matching vertices and confirm they differ by the same amounts as every other pair. If one vertex moved differently, recheck its arithmetic.

The triangle below is slid by the rule $(x, y) \to (x + 5, y + 2)$ . Each image vertex ( $A'$ , $B'$ , $C'$ ) is the same distance and direction from its original, as the dashed segment from $A$ to $A'$ shows.

Worked examples

Example 1: apply a rule to a point

The translation $(x, y) \to (x + 2, y - 6)$ is applied to the point $(-1, 4)$ . Find the image.

Apply the rule to the

x

-coordinate

x = -1 + 2 = 1

Apply the rule to the

y

-coordinate

y = 4 - 6 = -2

Write the image

(1, -2)

Answer: $(1, -2)$

Example 2: write a rule from a description

Write the rule for a translation of $5$ units left and $2$ units up.

Left

5

subtracts from the

x

-coordinate

x - 5

2

adds to the

y

-coordinate

y + 2

Combine into one rule

(x, y) \to (x - 5, y + 2)

Answer: $(x, y) \to (x - 5, y + 2)$

Example 3: find the rule from two points

A translation maps $(2, -3)$ to $(-1, 4)$ . Write the rule.

Horizontal change: image

x

minus original

x

-1 - 2 = -3

Vertical change: image

y

minus original

y

4 - (-3) = 7

Write the rule

(x, y) \to (x - 3, y + 7)

Answer: $(x, y) \to (x - 3, y + 7)$

Try one yourself

Common questions

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

No. A translation is a rigid motion — every point moves the same distance in the same direction, so all lengths and angles are preserved. The image is congruent to the original and even faces the same way.

How do I tell left from right in a rule?

Look at the sign attached to $x$ . In $(x, y) \to (x + 4, y)$ the figure moves $4$ units right; in $(x, y) \to (x - 4, y)$ it moves $4$ units left. The same logic applies to $y$ : plus is up, minus is down.

What if only one coordinate changes?

Then the slide is purely horizontal or purely vertical. The rule $(x, y) \to (x, y - 3)$ moves every point straight down $3$ units, and $(x, y) \to (x + 6, y)$ moves every point straight right $6$ units.

How do I find the rule when I'm given a point and its image?

Subtract the original coordinates from the image coordinates, one at a time. If $(1, 5)$ maps to $(4, 2)$ , then $a = 4 - 1 = 3$ and $b = 2 - 5 = -3$ , so the rule is $(x, y) \to (x + 3, y - 3)$ .

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Translations

Reading a translation rule

Finding the rule from two points

Translating a whole figure

Worked examples

Example 1: apply a rule to a point

Example 2: write a rule from a description

Example 3: find the rule from two points

Try one yourself

Common questions

Keep learning

Reflections

Compositions of Transformations

Symmetry: Line & Rotational

Rotations