Translations, reflections & rotations
A translation slides every point by the same amount: (x, y) → (x + a, y + b). A reflection over the x-axis maps (x, y) → (x, −y), and over the y-axis (x, y) → (−x, y). A 90° counterclockwise rotation about the origin maps (x, y) → (−y, x).
Translations, reflections & rotations calculator
Move a shape while keeping track of every vertex. Change a value and the diagram, result, and working update together.
Your measurements
The original triangle is A(−2, −1), B(2, −1), C(0, 2). Rotations and dilations are centered at the origin.
Why it works
The idea
Translation adds the same displacement to every point. Reflection reverses one coordinate across a coordinate axis: (x, −y) across the x-axis and (−x, y) across the y-axis. A 90° counterclockwise rotation about the origin sends (x, y) to (−y, x). These transformations preserve distances and angles.
Transformation rules on the coordinate plane
Every rule below acts on the original point P(3, 1). Rotations turn about the origin; positive angles turn counterclockwise. Translations, reflections, and rotations are rigid transformations (isometries): they keep lengths, angles, and area. A dilation keeps angles but scales lengths.
| Transformation | Coordinate rule | P(3, 1) → |
|---|---|---|
| TranslationTranslate by (a, b) (a = −4, b = 2) | (x, y) → (x + a, y + b) | (−1, 3) |
| ReflectionReflect over the x-axis | (x, y) → (x, −y) | (3, −1) |
| ReflectionReflect over the y-axis | (x, y) → (−x, y) | (−3, 1) |
| ReflectionReflect over y = x | (x, y) → (y, x) | (1, 3) |
| ReflectionReflect over y = −x | (x, y) → (−y, −x) | (−1, −3) |
| ReflectionReflect over the line x = h (h = 2) | (x, y) → (2h − x, y) | (1, 1) |
| ReflectionReflect over the line y = k (k = 2) | (x, y) → (x, 2k − y) | (3, 3) |
| RotationRotate 90° counterclockwise | (x, y) → (−y, x) | (−1, 3) |
| RotationRotate 180° | (x, y) → (−x, −y) | (−3, −1) |
| RotationRotate 270° counterclockwise (90° clockwise) | (x, y) → (y, −x) | (1, −3) |
| DilationDilate by k from the origin (k = 2) | (x, y) → (kx, ky) | (6, 2) |
Worked examples
1Reflect a triangle over the x-axis
- Triangle A(1, 2), B(4, 2), C(1, 5).
- Keep each x-coordinate and change the sign of y.
Answer: A′(1, −2), B′(4, −2), C′(1, −5).
2Rotate a point about the origin
- Start with P(2, 5).
- 90° counterclockwise: (x, y) → (−y, x) gives (−5, 2).
- 180°: (x, y) → (−x, −y) gives (−2, −5).
- 270° counterclockwise (90° clockwise): (x, y) → (y, −x) gives (5, −2).
Answer: (−5, 2), (−2, −5), and (5, −2).
3Order matters
- Translate P(2, 5) by (−3, 1), then reflect over the y-axis: (−1, 6) → (1, 6).
- Reflect first, then translate: (−2, 5) → (−5, 6).
Answer: Different orders give (1, 6) and (−5, 6).
Common mistakes
- Rotating the wrong way. Positive angles turn counterclockwise.
- Forgetting the center of rotation. These rules rotate about the origin.
- Mixing up reflections: over y = x the coordinates swap; over the x-axis only y changes sign.
- Calling a dilation rigid. Only translations, rotations, and reflections preserve size.
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Keep exploring
Translations, reflections & rotations: common questions
Wondering about the why? Start here.
What are the rules for reflecting over the x-axis and y-axis?
Reflecting over the x-axis keeps x and changes the sign of y: (x, y) → (x, −y). Reflecting over the y-axis changes the sign of x: (x, y) → (−x, y). For example, (3, 1) becomes (3, −1) or (−3, 1).
Try a reflectionWhat is the rule for a 90° rotation?
A 90° counterclockwise rotation about the origin maps (x, y) → (−y, x). A 90° clockwise rotation, which equals 270° counterclockwise, maps (x, y) → (y, −x). A 180° rotation maps (x, y) → (−x, −y).
What are translations, reflections, and rotations called?
They are rigid transformations, also called isometries or congruence transformations. Each one keeps lengths, angles, and area, so the image is congruent to the original. A dilation is not rigid because it changes size.
Compare with dilationsDoes the order of transformations matter?
Often, yes. Translating (2, 5) by (−3, 1) and then reflecting over the y-axis gives (1, 6), but reflecting first and then translating gives (−5, 6). Apply the steps in the order they are written.