What is a transformation?
A transformation takes every point of a shape to a new point. The starting shape is the object. The new shape is the image. We name image points with a dash: A goes to A′.
There are four main kinds: translation (slide), reflection (flip), rotation (turn) and enlargement or dilation (resize).
Translation, reflection and rotation are rigid motions (also called isometries). They keep lengths and angles, so the image is congruent to the object.
Translation: slide by a vector
A translation moves every point the same distance in the same direction. We write it as a vector, for example (3, 2): 3 right, 2 up.
Rule: (x, y) → (x + a, y + b). Negative a means left, negative b means down.
The shape does not turn or flip. It faces the same way.
Reflection: flip in a mirror line
Each point goes to the other side of the mirror line, at the same distance. The line joining a point and its image is at right angles to the mirror.
- In the x-axis: (x, y) → (x, −y)
- In the y-axis: (x, y) → (−x, y)
- In the line y = x: (x, y) → (y, x)
Points on the mirror line do not move. The image is reversed (left and right swap), like writing in a mirror.
Rotation: turn about a centre
You need three things: the centre, the angle and the direction (clockwise or anticlockwise).
- 90° anticlockwise about O: (x, y) → (−y, x)
- 180° about O: (x, y) → (−x, −y)
- 90° clockwise (= 270° anticlockwise) about O: (x, y) → (y, −x)
Every point stays the same distance from the centre. Tracing paper helps: trace, pin the centre with a pencil, turn.
Enlargement (dilation) and similarity
An enlargement has a centre and a scale factor k. From centre O: (x, y) → (kx, ky).
- k > 1: the image is bigger.
- 0 < k < 1: the image is smaller (still called an enlargement).
- Negative k: the image is on the other side of the centre, upside down.
Lengths are multiplied by k, area by k². Angles stay the same, so the image is similar, not congruent (unless k = 1 or −1).
Describing and combining transformations
To describe a transformation fully, give: translation → the vector; reflection → the mirror line; rotation → centre, angle, direction; enlargement → centre and scale factor. Doing one after another is a composition. Order can matter.
Key formulas and definitions
- Translation by (a, b): (x, y) → (x + a, y + b)
- Reflect in x-axis: (x, y) → (x, −y)
- Reflect in y-axis: (x, y) → (−x, y)
- Reflect in y = x: (x, y) → (y, x)
- Rotate 90° anticlockwise about O: (x, y) → (−y, x)
- Rotate 180° about O: (x, y) → (−x, −y)
- Enlarge by k from O: (x, y) → (kx, ky); area × k²
Worked examples
1. Translate P(2, −1) by the vector (−5, 4).
Add: x = 2 + (−5) = −3, y = −1 + 4 = 3. So P′ = (−3, 3).
2. Reflect Q(4, 3) in the x-axis, then in the y-axis.
In the x-axis: (4, −3). Then in the y-axis: (−4, −3). Note: the result is the same as a 180° rotation about O.
3. Rotate R(3, 1) by 90° anticlockwise about the origin.
Rule (x, y) → (−y, x): R′ = (−1, 3). Check distance from O: √(9 + 1) = √10 before and √(1 + 9) = √10 after.
4. A square with side 3 cm is enlarged by scale factor 4. Find the new side and new area.
Side = 3 × 4 = 12 cm. Old area = 9 cm², new area = 9 × 4² = 144 cm².
5. Triangle A(1, 1), B(3, 1), C(1, 3) maps to A′(−1, 1), B′(−1, 3), C′(−3, 1). Describe the transformation.
Test (x, y) → (−y, x): A(1,1) → (−1, 1) ✓, B(3,1) → (−1, 3) ✓, C(1,3) → (−3, 1) ✓. It is a rotation of 90° anticlockwise about the origin.
6. Enlarge D(2, 6) by scale factor ½ from O, then translate by (1, −1).
Enlarge: (1, 3). Translate: (1 + 1, 3 − 1) = (2, 2).
Common mistakes
- Mixing up the reflection rules: in the x-axis it is y that changes sign.
- Forgetting to state the centre and direction of a rotation; '90°' alone is not a full description.
- Thinking an enlargement with k = ½ is not an enlargement; it is, it just makes the shape smaller.
- Multiplying area by k instead of k² after an enlargement.