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

These notes recap section 7.1 (Transformations) of the Cambridge O Level Mathematics (Syllabus D) 4024 syllabus for 2025, 2026 and 2027. There is one set of outcomes for every candidate. The study guide has the longer explanations. To check what you remember, try the practice questions. See also the course hub, the printable checklist, coordinate geometry practice and the free diagnostics.

Key vocabulary

• Object: the original shape. Image: the shape after the transformation. A point maps to A′.
• Congruent: same size and shape. Reflection, rotation and translation give congruent images.
• Similar: same shape, different size. Enlargement gives a similar image.
• Invariant point: a point that does not move.
• Column vector (x over y): x units right, y units up. Negative means left or down.

Describe fully: what to write

Transformation	Include
Reflection	the mirror line written as an equation, such as x = 3, y = −1, y = x or y = −x
Rotation	the size of the turn, its direction, and the point it turns about
Enlargement	k, and the point the shape grows from
Translation	the column vector

Give one transformation name only when the question asks for a single transformation.

Coordinate rules

Move	Image of the point (x, y)
Mirror in the x-axis	(x, −y)
Mirror in the y-axis	(−x, y)
Mirror in the line x = a	(2a − x, y)
Mirror in the line y = b	(x, 2b − y)
Mirror in the line y = x	(y, x)
Mirror in the line y = −x	(−y, −x)
Quarter turn anticlockwise, origin	(−y, x)
Quarter turn clockwise, origin	(y, −x)
Half turn, origin	(−x, −y)
Enlarge from the origin by k	(kx, ky)
Translate by (a over b)	(x + a, y + b)

For a centre (p, q) that is not the origin: subtract the centre from the point, apply the rule, add the centre back. For enlargement the image is (p + k(x − p), q + k(y − q)).

Reflection in steps

1. Draw the mirror line with a ruler.
2. For each vertex, measure the perpendicular distance to the line.
3. Mark the image the same distance on the other side.
4. Join the images with a ruler.

To describe: points with the same x value (or y value) before and after tell you the line is horizontal (or vertical). The line is halfway between the two values. For a sloping line, find the midpoint of a point and its image and draw the perpendicular through it.

Worked reminder: reflect (7, −2) in x = 4. The new x is 2 × 4 − 7 = 1, so the image is (1, −2).

Rotation in steps

1. Locate the centre. Work out each point relative to the centre.
2. Turn it through the angle in the stated direction.
3. Add the centre back, or use tracing paper and a pencil point on the centre.
4. Join the images with a ruler.

To find a centre: draw the perpendicular bisector of a point and its image, do the same for a second pair, and the centre is where they cross. For 180° the centre is the midpoint of every point and its image.

To find the angle: look at the lines from the centre to a point and to its image. Decide the direction with a sketch.

Worked reminder: rotate (6, 2) through 90° clockwise about (1, 4). Relative position (5, −2). Clockwise (u, v) → (v, −u) gives (−2, −5). Add the centre: (−1, −1).

Enlargement in steps

1. Draw lines from the centre through each vertex.
2. Multiply each distance from the centre by the scale factor. Mark the image on that line.
3. If the scale factor is negative, mark the image on the opposite side of the centre.
4. Join the images with a ruler.

To describe: the scale factor is image length ÷ object length for matching sides, with a negative sign if the image is on the opposite side of the centre. The centre is where the lines through matching points meet.

Scale factor k	Image
k > 1	bigger, same side as the object
0 < k < 1	smaller, same side as the object
−1 < k < 0	smaller, opposite side, inverted
k < −1	bigger, opposite side, inverted
k = −1	same size, same as a 180° rotation about the centre

Lengths are multiplied by |k|. Areas are multiplied by k². Angles are unchanged.

Worked reminder: area of the object 20 cm², scale factor 3/2. Image area = 20 × 9/4 = 45 cm².

Worked reminder: a triangle with vertices (2, 2), (2, 4), (4, 2) maps to one with vertices (5, 5), (5, 11), (11, 5). The side from (2, 2) to (2, 4) has length 2 and its image has length 6, so the scale factor is 3. The image has not been turned over, so k = +3. The centre C satisfies image − C = 3 × (object − C), so C = (3 × object − image) ÷ 2 = ((6 − 5) ÷ 2, (6 − 5) ÷ 2) = (0.5, 0.5).

Translation in steps

1. Count the units right or left, then up or down, for one point.
2. Apply the same move to every vertex.
3. To describe: image minus object, one coordinate at a time.

Worked reminder: P(3, −1) maps to P′(−2, 5). Vector: (−2 − 3, 5 − (−1)) = (−5 over 6).

Combinations

• Do each step in the order given and label every stage.
• Two reflections in parallel lines make a translation, perpendicular to the lines, of twice the distance between them.
• Two reflections in lines that cross at a point make a rotation about that point. For the axes, the result is a rotation of 180° about the origin.
• Two rotations about the same centre add their angles.
• A translation followed by a translation is a translation: add the vectors.
• To describe a combination as one transformation, follow one or two points through, then compare the start and end positions.

Worked reminder: triangle M has a vertex at (1, 2). It is rotated 90° clockwise about the origin, then reflected in the y-axis. The vertex goes (1, 2) → (2, −1) → (−2, −1). Now compare with the rule for y = −x, which sends (x, y) to (−y, −x): (1, 2) → (−2, −1). The two agree, so the combination is a reflection in the line y = −x.

Must-know distinctions

• Rotation direction is stated. Clockwise 90° is anticlockwise 270°.

• Scale factor 2 and scale factor −2 give the same size, but opposite sides of the centre.

• The centre of enlargement is not always inside the shape.

• Reflection reverses orientation. Rotation and translation do not.

• “Enlargement by a scale factor of 1/2” makes the shape smaller. It is still an enlargement.

• A shape and its image under reflection are mirror images, so the order of the vertices runs the opposite way round.

• Under a negative enlargement, every line on the image is parallel to the matching line on the object.

Quick self-test

1. Find the image of (3, −2) in the line y = x.
2. Find the image of (1, 5) in the line x = −2.
3. Rotate (−4, 3) through 90° anticlockwise about the origin.
4. Rotate (5, 3) through 90° clockwise about (1, 1).
5. Enlarge (2, −1) by scale factor −3, centre the origin.
6. Enlarge (8, 5) by scale factor 1/3, centre (2, 2).
7. A shape has area 10 cm². It is enlarged by scale factor 4. State the new area. Then state the area after scale factor 1/2.
8. Point (3, −2) maps to (−1, 4) by a translation. Write the column vector.
9. Reflect (2, 3) in y = 1, then reflect the image in y = 6. State the final point and the single transformation.
10. Rotate (−2, 7) through 180° about (1, 3).

Answers

1. (−2, 3)
2. (−5, 5)
3. (−3, −4)
4. (3, −3)
5. (−6, 3)
6. (4, 3)
7. 160 cm², then 2.5 cm²
8. (−4 over 6)
9. (2, 13). The single transformation is a translation by (0 over 10).
10. (4, −1)

Where marks are usually lost

• Giving only part of a description, such as “rotation 90°” with the centre and direction missing.
• Saying “enlargement of 2” instead of “scale factor 2, centre (a, b)”.
• Writing the centre of an enlargement from a rough look at the diagram, rather than from lines drawn through matching points.
• Drawing freehand lines. The syllabus requires a ruler for all straight edges.
• Writing a mirror line as a word, such as “vertical”, with no equation.
• Reading a negative scale factor as a reduction. Check |k| for size and the sign for side.
• Rotating about the origin when the centre is elsewhere.
• Mixing up the order in a combination.
• Giving a translation as coordinates in a bracket with a comma, when the question asks for a column vector.

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.