A translation is a slide: a transformation that moves every point of a figure the same distance in the same direction, without turning or flipping it. The image is congruent to the original. On the coordinate plane, a translation adds the same amounts to every point: (x, y) → (x + a, y + b).
Every vertex slides the same way: 5 units right and 3 units down. The image A′B′C′ has the same size, shape and orientation as ABC.
Formal definition and notation
A translation by the vector⟨ a, b ⟩ is the rigid motion that maps each point (x, y) of the plane to (x + a, y + b). Every segment from a point to its image has the same length, √a2 + b2, and the same direction, so all those segments are parallel. Translations preserve lengths, angle measures, parallel lines and orientation.
Symbol
Read it as
Meaning
(x, y) → (x + a, y + b)
x, y maps to x plus a, y plus b
the coordinate rule: move a right (left if negative) and b up (down if negative)
A′
A prime
the image of point A after the translation
T⟨ 5, −3 ⟩
translation by 5, negative 3
a translation written with its vector, common in high school geometry
Key facts
A translation is a rigid motion: side lengths and angle measures stay the same, so the image is congruent to the original.
Orientation is kept: if the vertices go A, B, C counterclockwise before the slide, they still do after it. A reflection reverses it.
Every point moves the same distance in the same direction. If two points move different amounts, the move is not a translation.
Two translations in a row make one translation: 2 right then 5 right is 7 right, (x, y) → (x + 7, y).
Translating the graph of y = f(x) right h and up k gives y = f(x − h) + k: the minus sign is easy to get backward.
Translation rule
Translation rule
(x, y) → (x + a, y + b)
a
horizontal shift: positive moves right, negative moves left
b
vertical shift: positive moves up, negative moves down
(x, y)
any point of the original figure (the pre-image)
To find the rule from a point and its image, subtract: a = x′ − x and b = y′ − y.
Try it: slide a shape
Interactive
Set how far to move right or left and up or down. Watch every vertex move by the same amounts and read the coordinate rule.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Translation examples, step by step
Example 1: Translate a point
Warm-up
Translate P(2, −6) 4 units left and 7 units up. What are the coordinates of P′?
Left means subtract from x: 2 − 4 = −2.
Up means add to y: −6 + 7 = 1.
Write the image: P′(−2, 1).
Answer:P′ is at (−2, 1).
Example 2: Find the rule from a point and its image
Core
A translation maps M(−3, 5) to M′(4, 1). Write the rule, then find the image of N(0, −2).
Horizontal change: a = 4 − (−3) = 7.
Vertical change: b = 1 − 5 = −4.
Rule: (x, y) → (x + 7, y − 4).
Apply it to N: (0 + 7, −2 − 4) = (7, −6).
Answer: The rule is (x, y) → (x + 7, y − 4), and N′ is (7, −6).
Example 3: How far does each point move?
Test-style
SquareABCD is translated by the rule (x, y) → (x − 6, y + 8). How far does vertex A move, and what is the perimeter of the image if each side of ABCD is 5 units?
Each point moves 6 left and 8 up. Those are the legs of a right triangle.
Distance moved: √62 + 82 = √100, which is 10 units.
A translation keeps side lengths, so each side of the image is still 5.
Perimeter of the image: 4 × 5 = 20 units.
Answer: Vertex A moves 10 units, and the image still has a perimeter of 20 units.
Common mistakes
✗ MistakeMoving (3, 2) “4 down” to (7, 2).
✓ InsteadDown changes y: (3, 2 − 4) = (3, −2).
Left and right change x; up and down change y. Moving down subtracts from the y-coordinate.
✗ MistakeTranslating only one vertex and redrawing the shape by eye, so the image is stretched.
✓ InsteadApply the same rule to every vertex, then connect the image points in the same order.
A translation moves every point by the same amounts. If the sides change length, a mistake has turned the slide into a different transformation.
✗ MistakeGraph of y = (x − 3)2 is the graph of y = x2 moved 3 left.
✓ InsteadIt is moved 3 right: the vertex goes from (0, 0) to (3, 0).
Inside the function, x − 3 reaches the old input 0 when x = 3, so the whole graph shifts to the right.
A translation slides a figure and keeps its orientation. A reflection flips it over a line and reverses orientation: across the y-axis, (2, 5) goes to (−2, 5).
A rotation turns a figure around a fixed point, so points move different distances. In a translation every point moves the same distance in the same direction.
Transformation is the umbrella word for any rule that moves or resizes a figure. A translation is one kind: the slide.
Quick check: 3 questions
Question 1 of 3
Point K(−1, 4) is translated by (x, y) → (x + 3, y − 5). Where is K′?
Show answer and explanation
Answer: A.−1 + 3 = 2 and 4 − 5 = −1, so K′(2, −1). (−4, 9) reverses both moves, (4, −1) adds 3 to the wrong coordinate after a sign slip, and (2, 9) adds 5 instead of subtracting.
Question 2 of 3
A translation maps (5, −2) to (1, 3). Which rule is it?
Show answer and explanation
Answer: C. Subtract image minus original: 1 − 5 = −4 and 3 − (−2) = 5. The other rules have one or both signs backward, which happens when you subtract in the wrong order.
Question 3 of 3
After a translation, which of these can change?
Show answer and explanation
Answer: D. A translation is a rigid motion, so lengths, angles and area stay the same. Only the position changes, which means the coordinates of the vertices change (unless the translation is by 0 and 0).
Translation: frequently asked questions
What is the rule for a translation?
Add the same horizontal amount a to every x-coordinate and the same vertical amount b to every y-coordinate: (x, y) → (x + a, y + b). A positive a moves right and a negative a moves left; a positive b moves up and a negative b moves down. So (x, y) → (x − 2, y + 6) means 2 left and 6 up.
Is a translation a rigid transformation?
Yes. A translation keeps every distance and every angle the same, so the image is congruent to the original figure. It also keeps orientation: the figure is not flipped. Translations, reflections and rotations are the three rigid motions; a dilation is not rigid, because it changes size.
How do you describe a translation in words?
Name the distance and direction for each axis, such as “6 units left and 2 units up”, or give the vector ⟨ −6, 2 ⟩. In high school geometry you may also describe it as sliding a figure along a directed segment from one point to another, where every point moves parallel to that segment by the same length.
TransformationA transformation moves or changes a figure on the coordinate plane. The four common transformations are translations (slides), reflections (flips), rotations (turns) and dilations (resizing). The original figure is the preimage and the result is the image, labeled with primes such as A′. Only a dilation changes the size.Geometry · Grade 7, Grade 8, Algebra 1Congruent FiguresCongruent figures have exactly the same size and shape. One can be moved onto the other by sliding, flipping or turning, so every pair of matching sides and angles is equal. Two squares with side 5 are congruent; a 3-4-5 triangle and a 6-8-10 triangle are not, because their sizes differ. The symbol is ≅.Geometry · Grade 7, Grade 8Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7VectorA vector is a quantity that has both a size, called its magnitude, and a direction. It is drawn as an arrow from an initial point to a terminal point and written in components, such as v = ⟨ 4, 3 ⟩: 4 units right and 3 units up. Velocity and force are vectors; speed and mass are not.Trigonometry · PrecalculusLengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3Parallel LinesParallel lines are lines in the same plane that never intersect. They stay the same distance apart and point in the same direction, like the rails of a straight railroad track. On a coordinate plane, parallel lines have the same slope but different y-intercepts. The symbol ∥ means is parallel to.Geometry · Grade 4, Grade 5, Grade 6SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2CongruentCongruent means exactly the same size and shape. Two segments are congruent when they have the same length, two angles when they have the same measure, and two figures when one can be turned, flipped or slid onto the other. The symbol is ≅: AB ≅ CD reads “segment AB is congruent to segment CD.”Geometry · Grade 8, GeometrySquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3PerimeterPerimeter is the total distance around a two-dimensional shape. To find it, add the lengths of all the sides. A rectangle with sides 5 and 3 has perimeter 5 + 3 + 5 + 3 = 16. Perimeter is a length, so it uses length units such as inches or meters, never square units.Geometry · Grade 3, Grade 4, Grade 5Right TriangleA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the Pythagorean theorem: a2 + b2 = c2.Geometry · Grade 4, Grade 6, Grade 8ReflectionA reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).Geometry · Grade 8, GeometryY-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x-axis at the origin, (0, 0). A point’s y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6, Grade 7RotationA rotation is a transformation that turns a figure around a fixed point, called the center, by a given angle and direction. Every point stays the same distance from the center, so size and shape do not change. About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x).Geometry · Grade 8, GeometryDilationA dilation is a transformation that enlarges or shrinks a figure by a scale factor k from a fixed center point. Each point moves along the ray from the center to k times its distance. With center at the origin, (x, y) → (kx, ky). The image keeps its shape, so it is similar, not congruent.Geometry · Grade 8, GeometryOrdered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6, Grade 7IntegerAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 12 are not integers.Number System · Grade 6, Grade 7, Grade 8Similar FiguresSimilar figures have the same shape but not necessarily the same size. Corresponding angles match, and corresponding side lengths are proportional: each side of one figure is the same multiple, the scale factor, of the matching side of the other. A scale drawing is similar to the real object.Geometry · Grade 7, Grade 8, Algebra 1