G7, G8, G24Transformations

Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Geometry and measures

Practise Transformations. 1 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.

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Revision notes

Translations, reflections, rotations and enlargements move or resize a shape. You carry them out on a coordinate grid, describe them fully, and deal with combinations of transformations; Higher also includes enlargements with negative scale factors.

Grade by grade

What you need to be able to do, from the first marks up to the top grade.

3
Reflect a shape in a mirror lineReflect in lines such as x = 2, y = −1, y = x and y = −x; each image point is the same distance from the line.
3
Translate a shape using a column vectorThe top number moves right (+) or left (−); the bottom number moves up (+) or down (−).
4
Rotate a shape about a given centreUse tracing paper, the angle and the direction (clockwise or anticlockwise).
4
Enlarge a shape by a positive scale factorMultiply the distance from the centre of enlargement to each vertex by the scale factor.
5
Describe a single transformation fullyGive the type and every detail, e.g. rotation 90° clockwise about (1, 2).
5
Enlarge by a fractional scale factorA scale factor between 0 and 1 makes the shape smaller, still measured from the centre.
5
Carry out and describe combined transformationsDo one transformation then the next, then describe the single transformation that has the same effect.

Notes

Translation and reflection

A translation slides a shape. It is described by a column vector \(\begin{pmatrix} x \\ y \end{pmatrix}\): x across (right is positive) and y up (down is negative).
A reflection flips a shape in a mirror line. Each image point is the same perpendicular distance from the line as the original point, on the other side.
x = a is a vertical line and y = b is a horizontal line. Also know the diagonal lines y = x and y = −x.

Rotation

A rotation turns a shape through an angle, clockwise or anticlockwise, about a centre of rotation.
Use tracing paper: trace the shape, hold the centre still with your pencil point and turn the paper.
To find the centre, try points with tracing paper until the shape lands on its image.
A rotation of 180° needs no direction, as clockwise and anticlockwise give the same image.

Enlargement

An enlargement changes the size of a shape by a scale factor from a centre of enlargement. Angles stay the same, so the image is similar to the original.
Multiply the distance (or the vector) from the centre to each vertex by the scale factor.
A scale factor between 0 and 1 makes the shape smaller; ½ halves every length.
To find the centre, draw straight lines through corresponding vertices of the shape and image; they meet at the centre.

Describing and combining

Describe fully: translation + vector; reflection + equation of the mirror line; rotation + angle, direction and centre; enlargement + scale factor and centre.
For a combination, do the first transformation, then the second on the image. The result can often be described as one single transformation.
Invariant points stay in the same place, e.g. points on the mirror line in a reflection, or the centre of a rotation.
Translations, reflections and rotations give congruent images; enlargements give similar images.

Cheatsheet

Translation: column vector (top: right +/left −; bottom: up +/down −)
Reflection: equation of the mirror line
Rotation: angle, direction and centre
Enlargement: scale factor and centre
x = a is vertical; y = b is horizontal
Image distance from the centre = scale factor × original distance
Invariant point: a point that does not move

How to answer each type of question

Describe fully the single transformation

2 to 3 marks5

Decide the type: same size and same way round (translation), flipped (reflection), turned (rotation) or resized (enlargement).
Find every detail: vector, mirror line, angle, direction, centre or scale factor.
Write one transformation only, with all its details.

Example. Triangle A has vertices (1, 1), (3, 1) and (1, 2). Triangle B has vertices (−1, 1), (−1, 3) and (−2, 1).
Describe fully the single transformation that maps triangle A onto triangle B.

Show the model answer

Rotation (B1)
90° anticlockwise (B1)
about (0, 0) (B1)

Enlarge a shape by a fractional scale factor

2 marks5

Write the vector from the centre to each vertex.
Multiply each vector by the scale factor.
Add each new vector to the centre to plot the image.

Example. Triangle P has vertices (4, 2), (8, 2) and (8, 6).
Enlarge triangle P by scale factor ½ with centre (2, 0).

Show the model answer

Vectors from (2, 0) to the vertices: (2, 2), (6, 2), (6, 6); halved: (1, 1), (3, 1), (3, 3)
Image vertices (3, 1), (5, 1) and (5, 3) (B2)
(B1 for a triangle of the correct size and orientation in the wrong position)

Combined transformations

2 to 3 marks5

Carry out the first transformation, then apply the second to its image.
Compare the final image with the original shape.
Describe the single transformation that maps one onto the other.

Example. Shape S is reflected in the y-axis to give shape T. Shape T is then reflected in the x-axis to give shape U.
Describe fully the single transformation that maps shape S onto shape U.

Show the model answer

e.g. the point (2, 1) goes to (−2, 1) and then to (−2, −1), so every (x, y) goes to (−x, −y)
Rotation (B1) 180° about (0, 0) (B1)

Shortcuts and memory tricks

Rotations about (0, 0): 90° clockwise takes (x, y) to (y, −x); 90° anticlockwise takes (x, y) to (−y, x); 180° takes (x, y) to (−x, −y).
Reflection in y = x swaps the coordinates: (x, y) goes to (y, x). In y = −x: (x, y) goes to (−y, −x).
Ask for tracing paper for rotations and reflections.
Check an enlargement with straight lines from the centre through each vertex: the image vertices must lie on them.

Where marks are lost

Describing two transformations when the question asks for a single transformation; this scores no marks.
Mixing up x = 3 (a vertical line) and y = 3 (a horizontal line).
Leaving out a detail, such as the direction or centre of a rotation, or the centre of an enlargement.
Enlarging from the origin when the centre is somewhere else.
Writing 'turn', 'flip' or 'move' instead of rotation, reflection or translation.
Writing a translation as a coordinate such as (3, −2) instead of a column vector.

Exam technique

'Describe fully' needs the name of the transformation and every detail; use the number of marks as a check.
Only describe one transformation when the question says 'single transformation'.
Draw images accurately and check two or three vertices using their coordinates.

Quick recall

Cover the answers and test yourself. The app has these as flashcards that come back just before you'd forget them.

Shape A is rotated to give shape B. Jo describes the transformation as 'a rotation of 90°'. Jo's description is not complete. Write down the two other pieces of information that are needed to describe the rotation fully.: The direction (clockwise or anticlockwise) and the centre of rotation.

Sample questions

Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.

Question 1Easy2 marks

Shape A is rotated to give shape B.
Jo describes the transformation as 'a rotation of 90°'.
Jo's description is not complete. Write down the two other pieces of information that are needed to describe the rotation fully.[2]

Show the answer and mark scheme

Answer: The direction (clockwise or anticlockwise) and the centre of rotation.

B1 for the direction of the rotation (clockwise or anticlockwise)
B1 for the centre of rotation

Worked solution: A rotation is described fully by its angle, its direction and its centre. Jo has given only the angle.

Question 2Medium4 marks

The grid shows triangle A.

[object Object]

(a) On the grid, draw the image of triangle A after a rotation of 180° about the point (1, −1).
Label the image B.[2]

(b) On the grid, draw the image of triangle A after a reflection in the line \(y = -x\).
Label the image C.[2]

Show the answer and mark scheme

(a) Answer: Triangle B with vertices (−1, −4), (−1, −6), (−4, −6).

B2 for triangle B drawn with vertices (−1, −4), (−1, −6), (−4, −6) (B1 for a rotation of 180° about a different centre, or a rotation about (1, −1) by the wrong angle or in the wrong direction)

Worked solution: Apply the transformation to each vertex of A: (3, 2) \(\to\) (−1, −4), (3, 4) \(\to\) (−1, −6), (6, 4) \(\to\) (−4, −6).

(b) Answer: Triangle C with vertices (−2, −3), (−4, −3), (−4, −6).

B2 for triangle C drawn with vertices (−2, −3), (−4, −3), (−4, −6) (B1 for a reflection of A in the line \(y = x\) (the wrong diagonal), or for triangle C with at least two vertices correct)

Worked solution: Reflecting in \(y = -x\) maps \((x, y) \to (-y, -x)\): (3, 2) \(\to\) (−2, −3), (3, 4) \(\to\) (−4, −3), (6, 4) \(\to\) (−4, −6).

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