Describing a transformation means specifying its type and every defining feature — a common source of lost marks. A rotation needs a centre, angle, and direction. A reflection needs the equation of the mirror line. A translation needs a vector. An enlargement needs a centre and scale factor.
Why does describing differ from performing a transformation?
When a question says "transform shape A," you draw the image. When it says "describe the single transformation that maps A onto B," you write a sentence (or sentences) specifying everything about it.
Examiners mark descriptions against a checklist: every missing piece of information costs a mark. A common student error is saying "rotation" and giving the angle but forgetting the centre or direction — that loses marks even though the rotation type was correct.
The golden rule: name the transformation first, then give every defining feature for that type.
How do you fully describe a translation?
A translation slides a shape without rotating or reflecting it. Every point moves the same distance in the same direction.
What to write: name the transformation and give the column vector.
A column vector is written $\binom{a}{b}$ where:
- The top number (a) is the horizontal movement (positive = right, negative = left)
- The bottom number (b) is the vertical movement (positive = up, negative = down)
Example: Shape A has vertices at (1, 2), (3, 2), and (2, 5). Shape B has vertices at (4, −1), (6, −1), and (5, 2). Describe the transformation.
Each point has moved 3 to the right and 3 down.
Answer: Translation by the vector $\binom{3}{-3}$.
Do not say "3 right and 3 down" — give the column vector. Do not describe the direction in words unless a column vector is not asked for.
How do you fully describe a reflection?
A reflection flips a shape across a mirror line.
What to write: name the transformation and state the equation of the mirror line.
Common mirror lines and their equations:
| Mirror line | Equation |
|---|---|
| Horizontal line through y = k | y = k (e.g. y = 0 is the x-axis) |
| Vertical line through x = k | x = k (e.g. x = 0 is the y-axis) |
| Diagonal through origin, slope 1 | y = x |
| Diagonal through origin, slope −1 | y = −x |
Finding the mirror line: the mirror line is the perpendicular bisector of the line segment joining any original point to its image. Find the midpoint of one such segment — it lies on the mirror line.
Example: A is at (2, 5), its image B is at (5, 2). Midpoint = (3.5, 3.5), which lies on y = x. The gradient of AB = (2−5)/(5−2) = −1, which is perpendicular to y = x (gradient 1). ✓
Answer: Reflection in the line y = x.
How do you fully describe a rotation?
A rotation turns a shape about a fixed point called the centre of rotation.
What to write: name the transformation and give the centre, angle, and direction (clockwise or anticlockwise).
Finding the centre: the centre is equidistant from each original point and its image. Practically, draw the perpendicular bisectors of the segments joining two original points to their images — where they cross is the centre.
Example: Triangle A with vertices (1, 1), (3, 1), (1, 4) maps onto triangle B with vertices (−1, 1), (−1, 3), (−4, 1). Describe the transformation.
Checking one point: (1, 1) → (−1, 1). The perpendicular bisector of this horizontal segment is the vertical line x = 0. For (3, 1) → (−1, 3): midpoint is (1, 2); gradient of segment is (3−1)/(−1−3) = −½; perpendicular gradient = 2; bisector: y − 2 = 2(x − 1), i.e. y = 2x. Where do x = 0 and y = 2x meet? At (0, 0). Centre is the origin.
Angle and direction: (1, 1) is at 45° from origin; (−1, 1) is at 135°. The point moved 90° anticlockwise.
Answer: Rotation of 90° anticlockwise about the origin.
How do you fully describe an enlargement?
An enlargement changes the size of a shape, keeping all angles the same.
What to write: name the transformation and give the centre of enlargement and scale factor.
- Scale factor > 1: shape gets bigger
- 0 < scale factor < 1: shape gets smaller (but still called an enlargement)
- Scale factor negative: shape is enlarged AND rotated 180° about the centre
Finding the centre: draw lines through each original vertex and its image, extending in both directions. Where all these lines meet is the centre.
Finding the scale factor: scale factor = image length / original length (for any pair of corresponding sides).
Example: Triangle A has a side of 3 cm. Its image triangle B has a corresponding side of 7.5 cm.
Scale factor = 7.5 / 3 = 2.5. If the lines through corresponding vertices all meet at (2, 0):
Answer: Enlargement, scale factor 2.5, centre (2, 0).
What if two transformations have been applied?
If the question says "describe the single transformation," make sure you identify it correctly. Two reflections can combine to a rotation; two rotations (about the same centre) combine to one rotation. Work with the object and image and identify which single transformation maps one to the other directly.
Frequently asked questions
Do I need to give the direction for a 180° rotation?
No. A 180° rotation is the same whether you go clockwise or anticlockwise, so direction is not needed. Simply state: "Rotation of 180° about (a, b)."
What if the mirror line is not horizontal, vertical, or y = ±x?
You need to find its equation using standard line methods: find the gradient (the mirror line is perpendicular to the line joining each point to its image) and a point it passes through (the midpoint of any object-image pair). Then use y − y₁ = m(x − x₁) to write the equation.
Can the scale factor be a fraction?
Yes. A scale factor of ½ maps a shape to one that is half the size. This is still called an enlargement in GCSE terminology, even though the image is smaller. Do not confuse "enlargement" (the name of the transformation type) with "making bigger" (just one possibility).
How many marks does a full description get?
Typically two to three marks. One mark is usually for naming the correct transformation type. The remaining marks are for the defining features. Every missing feature loses a mark, so always work through your checklist: translation → vector; reflection → equation of line; rotation → centre + angle + direction; enlargement → centre + scale factor.
Work through transformation questions with Professor Pi's guided hints at aitutors.me.