Transformations and invariance

Edexcel tests transformations on every paper at both tiers. The defining feature at Higher is the combination of transformations and recognising what is invariant (unchanged).

The four single transformations

• Translation by vector (a; b): every point shifts (a, b). Lengths, angles and orientation are invariant.
• Reflection in a line (x = a, y = a, y = x, y = −x): orientation reverses. Lengths and angles invariant.
• Rotation by angle (90°, 180°, 270°) about a centre: orientation preserved (anti-clockwise default unless stated). Lengths and angles invariant.
• Enlargement by scale factor k about a centre: lengths multiplied by k, areas by k², angles invariant. If k < 0 the image is on the opposite side of the centre and inverted.

Invariance under combinations

When two transformations are applied in sequence:

• Two reflections in parallel lines = a translation perpendicular to the lines, distance = 2 × gap between lines.
• Two reflections in intersecting lines = a rotation about the intersection, angle = 2 × angle between lines.
• A reflection followed by translation parallel to the mirror line = a glide reflection.
• An enlargement followed by an enlargement = an enlargement with scale factor = product of the two factors.

Describing a transformation — full Edexcel marks

Edexcel demands a complete description:

• Translation: state vector as (a; b) column vector.
• Reflection: state the equation of the mirror line.
• Rotation: state angle, direction (clockwise/anti-clockwise) AND centre.
• Enlargement: state scale factor AND centre.

A partial answer (e.g. "rotation 90°") loses at least one mark.

Negative enlargements (Higher only)

If k = −2 about centre C, every point P maps to P' such that P', C and P are collinear and CP' = 2 × CP on the opposite side of C. The image is inverted.

Common Edexcel exam tip

Most "describe fully" questions are 2 or 3 marks. Allocate one mark for the type, one for the parameter (vector/line/angle), one for the centre. If you write only "translation", you score one mark out of three.