Transformation in Math: Definition, Types & Rules

Definitionnoun

What is a transformation in math?

A 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.

In Spanish: transformación
Also called: move on a plane, geometric transformation, mapping
Grade level: 6-8

Grade 7Grade 8Algebra 1CCSS 8.G.A.1CCSS 8.G.A.3CCSS HSG-CO.A.2SATACT

Transformation in a picture

Slide, flip, turn, resize (CCW means counterclockwise). In the first three the green image is the same size as the blue original; only the dilation makes it bigger.

Formal definition and notation

A transformation of the plane is a function that takes each point P as input and gives exactly one point P′ as output. Translations, reflections and rotations are rigid motions: they preserve distance and angle measure, so the image is congruent to the preimage. A dilation with center O and scale factor k > 0 sends P to the point P′ on ray OP with OP′ = k · OP, producing a similar figure.

Symbol	Read it as	Meaning
A → A′	A maps to A prime	point A of the preimage goes to point A′ of the image
(x, y) → (x + 4, y + 1)	x, y maps to x plus 4, y plus 1	coordinate rule for a translation 4 right and 1 up
A′′	A double prime	the image after a second transformation

Key facts

A dilation changes size; translations, reflections, and rotations keep size the same.
Rigid motions (translation, reflection, rotation) produce congruent figures. A dilation produces similar figures.
A dilation with scale factor k > 1 enlarges; 0 < k < 1 shrinks. All lengths multiply by k and areas by k².
A reflection reverses orientation (clockwise order of the vertices becomes counterclockwise); translations and rotations do not.
Order can matter: a reflection followed by a translation does not always give the same image as the translation followed by the reflection.

Coordinate rules

translation by (a, b): (x, y) → (x + a, y + b)

(x, y) → (x, −y): reflection across the x-axis
(x, y) → (−x, y): reflection across the y-axis
(x, y) → (−y, x): rotation 90° counterclockwise about the origin
(x, y) → (−x, −y): rotation 180° about the origin
(x, y) → (kx, ky): dilation by scale factor k centered at the origin

Apply the rule to every vertex, then connect the new points in the same order.

Try it: transform a shape

Interactive

Pick a transformation and change its settings. The readout gives the coordinate rule and where each vertex lands.

This interactive needs JavaScript. The worked examples below show the same ideas step by step.

Transformation examples, step by step

Example 1: Translate a point

Warm-up

Point A(2, −3) is translated by the rule (x, y) → (x − 5, y + 4). Where is A′?

New x: 2 − 5 = −3 (5 units left).
New y: −3 + 4 = 1 (4 units up).
Write the image point: A′(−3, 1).

Answer: A′ is at (−3, 1).

Example 2: Reflect a triangle

Core

Triangle PQR has vertices P(1, 2), Q(4, 2) and R(4, 5). Reflect it across the x-axis.

Reflection across the x-axis: (x, y) → (x, −y). Keep x, change the sign of y.
P′(1, −2), Q′(4, −2), R′(4, −5).
Check one side: QR and Q′R′ both have length 3, so size is kept.

Answer: P′(1, −2), Q′(4, −2) and R′(4, −5).

Example 3: Dilation and area

Test-style

A rectangle has vertices (0, 0), (6, 0), (6, 4) and (0, 4). It is dilated by a scale factor of 1.5 centered at the origin. What is the area of the image?

Multiply every coordinate by 1.5: (0, 0), (9, 0), (9, 6), (0, 6).
The image is 9 by 6, so its area is 9 × 6 = 54.
Check with the area rule: the original area is 24, and areas scale by 1.5² = 2.25. 24 × 2.25 = 54.
Trap to avoid: the area does not grow by only 1.5 (that would give 36).

Answer: The image has an area of 54 square units.

Common mistakes

✗ MistakeA dilation is a rigid motion, so the image is congruent.

✓ InsteadA dilation changes size; translations, reflections, and rotations keep size the same.

A dilation multiplies every length by the scale factor, so the image is similar, not congruent (unless the factor is 1).

✗ MistakeReflecting (3, 4) across the x-axis gives (−3, 4).

✓ InsteadIt gives (3, −4).

Reflecting across the x-axis flips up and down, so the y-coordinate changes sign. (−3, 4) is the reflection across the y-axis.

✗ MistakeRotating (2, 5) 90° counterclockwise about the origin gives (5, −2).

✓ Instead(x, y) → (−y, x) gives (−5, 2).

(5, −2) is the 90° clockwise rotation. Check with a sketch: a point in Quadrant I turns counterclockwise into Quadrant II.

✗ Mistake(x, y) → (x + 2, y − 3) moves a figure 2 left and 3 up.

✓ InsteadIt moves the figure 2 right and 3 down.

Adding to x moves right; subtracting from y moves down.

Not to be confused with

Transformationvs.Congruent Figures

Congruence describes two figures with the same size and shape. A transformation is the motion; a rigid motion is what shows that two figures are congruent.

Transformationvs.Similar Figures

Similar figures have the same shape but maybe a different size. A dilation (possibly with rigid motions) is the transformation that turns one into the other.

Quick check: 3 questions

Question 1 of 3

Which transformation turns a figure around a fixed point?

Show answer and explanation

Answer: A. A rotation turns a figure through an angle about a fixed center. A translation slides, a reflection flips over a line, and a dilation resizes.

Question 2 of 3

What is the image of (−2, 5) after a reflection across the y-axis?

Show answer and explanation

Answer: B. Across the y-axis, (x, y) → (−x, y): (2, 5). Choice A reflects across the x-axis, C rotates 180°, and D swaps the coordinates.

Question 3 of 3

Which transformation does the rule (x, y) → (x − 3, y) describe?

Show answer and explanation

Answer: D. Subtracting 3 from every x-coordinate and keeping y slides each point 3 units left. A translation 3 units down would change y instead.

Transformation: frequently asked questions

What are the four types of transformations?

Translation (slide every point the same distance in the same direction), reflection (flip over a line), rotation (turn around a point) and dilation (enlarge or shrink from a center). The first three keep the size and shape, so they are called rigid motions. A dilation keeps the shape but changes the size.

What is a rigid transformation?

A rigid transformation, or rigid motion, moves a figure without changing its size or shape: distances and angle measures stay the same. Translations, reflections and rotations are rigid; any combination of them is rigid too. If one figure can be moved onto another by rigid motions, the two figures are congruent.

What does the prime symbol mean in transformations?

The prime mark names the image. If triangle ABC is reflected, the image is A′B′C′, read A prime, B prime, C prime. After a second transformation the points become A′′, B′′, C′′ (double prime). Keeping the same letters shows which image point came from which original point.

Practice and tools

Distance and Midpoint CalculatorCheck that a rigid motion kept every length Function GrapherSee graphs shift, flip and stretch

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.G.A.1, CCSS 8.G.A.3, CCSS HSG-CO.A.2

Cite this entry

Effortless Math Team. “Transformation.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/transformation/

Flashcard

TermTransformationslide, flip, turn, resize

DefinitionA move or change of a figure: translation, reflection, rotation or dilation. (x, y) → (x + 2, y) slides 2 right.

Study the full deck

Browse all terms in the Math Dictionary · More geometry terms · Random term

TranslationA 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).Geometry · Grade 8, GeometryReflectionA 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, GeometryRotationA 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, GeometryCongruentCongruent 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, 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, GeometryScale FactorA scale factor is the number used to multiply side lengths when making a similar figure larger or smaller. A scale factor of 2 doubles every length, and a scale factor of 12 makes lengths half as large. Find it by dividing a new length by the matching original length. Areas change by the square of the scale factor.Ratios and Proportions · 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 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 1LengthLength 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 3TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7SideA 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 2OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8SquareA 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 3Coordinate 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 7Ordered 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 7

Transformation

What is a transformation in math?

Transformation in a picture

Formal definition and notation

Key facts

Coordinate rules

Try it: transform a shape

Transformation examples, step by step

Example 1: Translate a point

Example 2: Reflect a triangle

Example 3: Dilation and area

Common mistakes

Not to be confused with

Quick check: 3 questions

Transformation: frequently asked questions

What are the four types of transformations?

What is a rigid transformation?

What does the prime symbol mean in transformations?

Related terms

Practice and tools

Sources

Flashcard

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