Cambridge O Level Mathematics: Transformations: Reflection, Rotation, Enlargement and Translation (4024)

This guide covers section 7.1 (Transformations) of the Cambridge O Level Mathematics (Syllabus D) 4024 syllabus for 2025, 2026 and 2027. The syllabus is not tiered, so every candidate studies all of it. Sections 7.2 and 7.3 (vectors and magnitude) are separate. Use the revision notes for a short recap and the practice questions to test yourself. The course hub and the printable checklist show where this unit fits. For work on coordinates and lines, try coordinate geometry practice. To find out what to revise first, take a free diagnostic.

What you must be able to do

Outcome (7.1)	What it means
1 Reflection	Draw the image of a shape in a straight line, and describe a reflection by giving the line
2 Rotation	Rotate a shape about a centre through multiples of 90°, and describe it with centre, angle and direction
3 Enlargement	Enlarge from a centre by a positive, fractional or negative scale factor, and describe it with centre and scale factor
4 Translation	Move a shape by a column vector (x over y), and describe a translation by its vector
Notes	Questions may combine transformations. A ruler must be used for all straight edges

Describing a transformation fully means giving every piece of information in the right-hand column of the second table. Leaving out one piece makes the description incomplete.

Transformation	Name it and state
Reflection	the mirror line, as an equation
Rotation	how far it turns, which way (clockwise or anticlockwise), and the point it turns about
Enlargement	the multiplier k and the fixed point it grows from
Translation	the column vector

Use one transformation word only. “Reflection and translation” in one description is not a full answer to a single-transformation question.

Object and image

The original shape is the object. The shape after the transformation is the image. A point on the image is written with a dash: A maps to A′. A reflection, rotation or translation gives an image that is congruent to the object (same size and shape). An enlargement gives an image that is similar to the object. A point that does not move is invariant.

1. Reflection

Every point and its image are the same distance from the mirror line, on opposite sides. The line joining a point to its image is perpendicular to the mirror line.

Rules you can use for lines on a coordinate grid:

Mirror line	Image of a point (x, y)
x-axis (y = 0)	(x, −y)
y-axis (x = 0)	(−x, y)
x = a	(2a − x, y)
y = b	(x, 2b − y)
y = x	(y, x)
y = −x	(−y, −x)

Worked example 1: drawing a reflection

Triangle T has vertices (−4, 1), (−1, 1) and (−1, 5). Reflect T in the line x = 1, then reflect triangle D with vertices (2, −1), (5, −1) and (5, −3) in the line y = −x.

For x = 1, each point keeps its y value and x becomes 2 − x. (−4, 1) → (6, 1); (−1, 1) → (3, 1); (−1, 5) → (3, 5).

For y = −x, (x, y) becomes (−y, −x). (2, −1) → (1, −2); (5, −1) → (1, −5); (5, −3) → (3, −5).

Draw the mirror line with a ruler, plot the image vertices and join them with a ruler. A quick check is that each vertex and its image are the same perpendicular distance from the line.

Worked example 2: describing a reflection

Point (1, −3) maps to (1, 5) and point (6, 2) maps to (6, 0). Both x values are unchanged, so the mirror line is horizontal. The line is halfway between the y values: (−3 + 5) ÷ 2 = 1 and (2 + 0) ÷ 2 = 1.

Reflection in the line y = 1.

For a sloping mirror line, join a point to its image, find the midpoint and draw the perpendicular through it. Always write the equation, such as y = x, not “the diagonal”.

2. Rotation

A rotation needs a centre, an angle and a direction. The centre is invariant. Every point stays the same distance from the centre. Tracing paper is a reliable way to check a rotation in practice, but the rules below let you do it by calculation.

To rotate about a centre (a, b), find the position of each point relative to the centre, rotate that, then add the centre back.

Turn about (0, 0)	Image of a point (x, y)
quarter turn anticlockwise	(−y, x)
quarter turn clockwise	(y, −x)
half turn	(−x, −y)
three quarter turns anticlockwise	(y, −x)

Worked example 3: drawing a rotation

Rotate triangle A with vertices (3, 3), (6, 3) and (6, 5) through 90° clockwise about (2, 1).

Positions relative to (2, 1): (1, 2), (4, 2), (4, 4). Rotate 90° clockwise, (u, v) → (v, −u): (2, −1), (2, −4), (4, −4). Add (2, 1) back: (4, 0), (4, −3), (6, −3).

Worked example 4: finding the centre

A rotation maps (4, 1) to (−1, 2) and (5, 0) to (0, 3). The centre is the same distance from a point and its image, so it lies on the perpendicular bisector of each pair.

Pair 1: midpoint (1.5, 1.5), and the line joining the points has gradient −1/5, so the bisector has gradient 5: y = 5x − 6. Pair 2: midpoint (2.5, 1.5), joining gradient −3/5, so the bisector has gradient 5/3: 3y = 5x − 8. Solve: 3(5x − 6) = 5x − 8, so 10x = 10, x = 1 and y = −1.

The angle is the angle between the line from the centre to a point and the line from the centre to its image. From (1, −1), the point (4, 1) is 3 right and 2 up, and its image (−1, 2) is 2 left and 3 up. That is a quarter turn anticlockwise.

Rotation of 90° anticlockwise about (1, −1).

A turn of 180° has no direction issue, and its centre is the midpoint of any point and its image. Rotating (3, 5) through 180° about (−1, 2) gives (−5, −1).

3. Enlargement

An enlargement needs a centre and a scale factor k. Each image point lies on the line from the centre through the object point, at k times the distance from the centre. Lengths are multiplied by |k|. Angles do not change.

Scale factor	Effect
k > 1	image larger, same side of the centre
0 < k < 1	image smaller (fractional), same side of the centre
k < 0	image on the opposite side of the centre and upside down

Rule from centre (a, b): the image of (x, y) is (a + k(x − a), b + k(y − b)).

Worked example 5: positive and fractional scale factors

Enlarge the triangle with vertices (1, 4), (3, 4) and (3, 5) by scale factor 2, centre (−1, 2). Relative positions: (2, 2), (4, 2), (4, 3). Double them: (4, 4), (8, 4), (8, 6). Add (−1, 2): (3, 6), (7, 6), (7, 8).

Enlarge the triangle with vertices (7, 3), (11, 3) and (11, 7) by scale factor 1/2, centre (3, −1). Relative positions: (4, 4), (8, 4), (8, 8). Halve: (2, 2), (4, 2), (4, 4). Add (3, −1): (5, 1), (7, 1), (7, 3).

Worked example 6: negative scale factor

Enlarge the triangle with vertices (2, 1), (4, 1) and (4, 2) by scale factor −2, centre (1, 0). Relative positions: (1, 1), (3, 1), (3, 2). Multiply by −2: (−2, −2), (−6, −2), (−6, −4). Add (1, 0): (−1, −2), (−5, −2), (−5, −4).

The image is on the opposite side of (1, 0), twice as big, and turned upside down. Check: a line from (2, 1) through (1, 0) continues to (−1, −2).

Worked example 7: describing an enlargement

A(1, 2) maps to A′(5, 10) and B(2, 2) maps to B′(8, 10). The length AB is 1 and A′B′ is 3, so the scale factor has size 3. The image is in the same orientation, so it is positive. For the centre, draw the lines AA′ and BB′ with a ruler. They meet at the centre. By calculation, the centre C satisfies A′ − C = 3(A − C), so C = (3A − A′) ÷ 2 = ((3 − 5) ÷ 2, (6 − 10) ÷ 2).

Enlargement, scale factor 3, centre (−1, −2).

Area

If lengths are multiplied by k, areas are multiplied by k². A triangle of area 6 cm² enlarged by scale factor 1.5 has area 6 × 2.25 = 13.5 cm². Scale factor −2 multiplies area by 4. The negative sign affects position, not area.

4. Translation

A translation moves every point the same distance in the same direction. The column vector (x over y) means x units right (left if negative) and y units up (down if negative). There is no turning and no centre.

Worked example 8

Triangle with vertices (1, 2), (3, 2), (3, 5) is translated by (−4 over 3): subtract 4 from each x and add 3 to each y, giving (−3, 5), (−1, 5), (−1, 8).

To describe a translation, subtract object from image for any one point. A(−3, 4) maps to A′(2, 1): vector = (2 − (−3), 1 − 4) = (5 over −3).

Combinations of transformations

Do the transformations one at a time, in the order given, and plot each stage.

Some combinations are equal to a single transformation.

Worked example 9

Triangle B has vertices (2, 1), (5, 1) and (5, 3). Reflect it in the x-axis, then reflect the image in the y-axis. After the first reflection: (2, −1), (5, −1), (5, −3). After the second: (−2, −1), (−5, −1), (−5, −3). Each point went from (x, y) to (−x, −y), so the single transformation is a rotation of 180° about the origin.

Two reflections in parallel lines give a translation. Reflecting (0, 0) in x = 1 gives (2, 0). Reflecting that in x = 4 gives (6, 0). The shift is 2 × (4 − 1) = 6, at right angles to the lines, so the translation is (6 over 0).

Common errors

• Describing a rotation without a centre, or a reflection as “flipped”.
• Giving the enlargement centre as the middle of the shape.
• Using the distance between object and image corners as the scale factor. Compare corresponding lengths.
• Forgetting that a negative scale factor puts the image on the other side of the centre.
• Rotating the wrong way. Check clockwise or anticlockwise with a quick sketch.
• Mixing up the sign of a translation vector. Moving left or down gives a negative component.
• Drawing sloping edges freehand. The syllabus says a ruler must be used for all straight edges.

Official syllabus

Cambridge O Level Mathematics (Syllabus D) 4024 syllabus for 2025, 2026 and 2027, published by Cambridge University Press & Assessment (Cambridge International), section 7.1 Transformations.