What is important to know about Transformations?

Reflection: needs the mirror line equation (e.g. y = x, x = 2).

What is important to know about Transformations?

Rotation: needs the centre of rotation, angle, and direction (clockwise or anticlockwise).

What is important to know about Transformations?

Translation: described by a column vector (x/y) — positive x moves right, positive y moves up.

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Geometry & Measures · Higher

Transformations

The four transformations are: reflection, rotation, translation and enlargement. At Higher tier you must describe them fully, apply them, and work with negative scale factors for enlargements. Combined transformations may also be examined.

Revising transformations the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Transformations GCSE” on the site for the complete guide.

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Key facts to remember

1Reflection: needs the mirror line equation (e.g. y = x, x = 2).
2Rotation: needs the centre of rotation, angle, and direction (clockwise or anticlockwise).
3Translation: described by a column vector (x/y) — positive x moves right, positive y moves up.
4Enlargement: needs the centre of enlargement and scale factor. Scale factor = image length ÷ original length.
5Negative scale factor: the image is on the opposite side of the centre and inverted.
6Transformations that preserve shape and size (isometries): reflection, rotation, translation.

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Worked examples

Example 1

Describe fully the single transformation that maps triangle A onto triangle B, where A has vertices (1,1),(3,1),(3,4) and B has vertices (−1,1),(−3,1),(−3,4).

Working

Compare the x-coordinates: each x has been negated (1→−1, 3→−3). y-values unchanged.
This is a reflection in the y-axis (the line x = 0).
Check: reflecting (1,1) in x = 0 gives (−1,1) ✓

AnswerReflection in the y-axis (x = 0).

Example 2

Enlarge triangle T with vertices (2,1),(4,1),(4,3) by scale factor −2 about centre (1,1).

Working

Find vector from centre to each vertex, multiply by −2.
(2,1)→(1,0) vector, ×(−2) = (−2,0), image point: (1−2, 1+0) = (−1,1)
(4,1)→(3,0) vector, ×(−2) = (−6,0), image point: (1−6, 1+0) = (−5,1)
(4,3)→(3,2) vector, ×(−2) = (−6,−4), image point: (1−6, 1−4) = (−5,−3)
Image is on opposite side of centre, inverted.

AnswerImage vertices: (−1,1), (−5,1), (−5,−3).

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Common mistakes

✗Giving an incomplete description — always state all required information.

✗Confusing clockwise and anticlockwise for rotations.

✗For enlargements, measuring from the wrong centre.

✗Not inverting the image when using a negative scale factor.

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Exam tips

✓For "describe fully": reflection needs line, rotation needs centre + angle + direction, enlargement needs centre + scale factor.

✓Use tracing paper in exams for rotations and reflections.

✓For negative scale factors, draw lines through the centre of enlargement and go to the other side.

✓A scale factor between −1 and 0 gives a smaller inverted image.

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