Combined transformations and invariant points

Doing one transformation after another often gives a result that a single transformation could do. Points that do not move are called invariant.

Practise this lesson free (6 questions)

Describing fully

Rotation: angle, direction and centre. Anticlockwise is positive, clockwise is negative, so −90° means 90° clockwise.
Reflection: the mirror line, such as y = x.
Translation: a column vector.
Enlargement: scale factor and centre.
Rotations, reflections and translations keep lengths and angles, so the image is congruent.

One after another

Reflect (2, 5) in the x-axis: (2, −5). Then in the y-axis: (−2, −5).
The single transformation is a rotation of 180° about (0, 0).
Two translations add: (3 over −1) then (−5 over 4) is (−2 over 3).
Reflecting in x = 1 and then in x = 4 is a translation of 6 units to the right: twice the gap between the lines.

Invariant points

An invariant point stays in the same place.
Reflection: every point on the mirror line is invariant.
Rotation: only the centre is invariant.
Translation: no points are invariant.
Worked example

Reflect the point (2, 5) in the x-axis and then in the y-axis. What single transformation does the same?

  1. (2, 5) → (2, −5) → (−2, −5)
  2. Both coordinates changed sign.

Answer: Rotation of 180° about (0, 0)

Key idea

Describe every transformation fully. Combine them by following one point (or the whole shape). Invariant points: the mirror line for a reflection, the centre for a rotation, none for a translation.

Key words
Reflection
A transformation that flips a shape in a mirror line.
Rotation
A transformation that turns a shape through an angle, in a direction, about a centre of rotation.
Translation
A transformation that slides a shape without turning it, described by a column vector.
Vector
A quantity with size and direction, written as a column vector or in bold, such as a.
Enlargement
A transformation that changes size by a scale factor from a centre of enlargement. Angles stay the same.
Scale factor
The number every length is multiplied by in an enlargement or between similar shapes.

All GCSE and International GCSE Mathematics key terms and definitions →

Quick check

Try two quick questions

See if it has sunk in. Answers are marked straight away.

Extra practice

2 more questions

Try each one, then tap to check.

  1. A shape is rotated 180° about (0, 0). Which pair of reflections gives the same result?

    1. Reflect in the x-axis, then in y = x
    2. Reflect in the x-axis twice
    3. Reflect in the x-axis, then in the y-axis
    4. Reflect in the y-axis, then in y = x
    Show the answer

    Answer: Reflect in the x-axis, then in the y-axis

    Reflecting in both axes sends (x, y) to (−x, −y), a 180° rotation.

  2. What does a rotation of +90° mean?

    1. 90° clockwise
    2. 270° anticlockwise
    3. A reflection
    4. 90° anticlockwise
    Show the answer

    Answer: 90° anticlockwise

    Positive angles are anticlockwise.

Check you have got it

Answer 6 quick questions with instant marking. If you get one wrong, GCSE-ready shows you why and gives you another go. It is free, and you do not need an account.

Practise this lesson free (6 questions)All GCSE and International GCSE Mathematics lessons

Video for this lesson coming soon. Subscribe on YouTube

Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

Related lessons