IGCSE Maths transformations: reflect, rotate, enlarge, translate and describe

What a full description needs

Column vectors are printed vertically on exam papers; here (3, -2) means 3 right and 2 down.

What each tier expects

The key ideas

Reflections, rotations and translations keep a shape the same size and shape, so the image is congruent to the original. Enlargements change lengths by the scale factor but keep angles the same, so the image is similar. A scale factor between 0 and 1 makes the shape smaller; a negative scale factor puts the image on the opposite side of the centre, upside down.

Useful coordinate rules about the origin: reflection in the x-axis sends (x, y) to (x, -y); in the y-axis to (-x, y); in y = x to (y, x); rotation 90° anticlockwise sends (x, y) to (-y, x); rotation 180° sends (x, y) to (-x, -y). For other centres, tracing paper is the reliable method, and it is allowed in the exam.

To find a centre of enlargement, draw lines through pairs of corresponding points and extend them; they meet at the centre. To find a centre of rotation, try candidate points with tracing paper, or construct the perpendicular bisectors of two lines joining corresponding points.

Worked example 1: rotate and reflect a triangle

1. Triangle A has vertices (1, 1), (3, 1) and (1, 2).
2. Rotate A 90° anticlockwise about (0, 0) using (x, y) → (-y, x): the image has vertices (-1, 1), (-1, 3) and (-2, 1).
3. Check one point with a sketch: (3, 1) is to the right of the origin, and a quarter turn anticlockwise moves it above, to (-1, 3). Correct.
4. Reflect A in y = x using (x, y) → (y, x): the image has vertices (1, 1), (1, 3) and (2, 1). The point (1, 1) is on the mirror line, so it stays where it is.

Worked example 2: describe an enlargement fully

1. Triangle A has vertices (2, 2), (4, 2), (2, 3). Triangle B has vertices (3, 1), (7, 1), (3, 3). Describe the single transformation from A to B.
2. Compare lengths: the base of A is 2 units and of B is 4 units, so the scale factor is 2.
3. Find the centre C: for scale factor 2, image = C + 2(object - C), so C = 2 × object - image. Using (2, 2) and (3, 1): C = (4 - 3, 4 - 1) = (1, 3).
4. Check with another pair: 2 × (4, 2) - (7, 1) = (1, 3). Same point.
5. Answer: enlargement, scale factor 2, centre (1, 3).

Worked example 3 (Cambridge Extended): negative scale factor and combinations

1. Enlarge triangle A from example 1 by scale factor -2, centre (0, 0). Multiply each coordinate by -2: (-2, -2), (-6, -2), (-2, -4).
2. The image is on the opposite side of the origin, upside down, with lengths doubled and area multiplied by 4.
3. Combination: reflect a shape in the x-axis, then reflect the image in the y-axis. (x, y) → (x, -y) → (-x, -y).
4. (x, y) → (-x, -y) is a rotation of 180° about (0, 0), so the single equivalent transformation is that rotation. (A 180° rotation needs no direction.)

Common mistakes that cost marks

• Naming two transformations when the question asks for a single transformation.
• Missing a detail: a rotation without a direction or centre, or an enlargement without a centre.
• Writing a reflection line as 'the y line' instead of an equation, such as x = 2.
• Writing a translation as coordinates or in words instead of a column vector.
• Confusing x = k (vertical) with y = k (horizontal) as mirror lines.
• Counting the scale factor from the image to the object instead of from the object to the image, giving 1/2 instead of 2.

Exam technique and how a tutor helps

Ask for tracing paper at the start of the exam; both boards allow it. For 'describe fully' questions, write the type first, then every required detail, and check against the table above. When drawing images, label vertices and double-check one vertex by a second method. Use a ruler for all straight edges, which Cambridge requires.

On Cambridge Core papers, questions do not combine transformations, and rotations are about the origin or a point on the shape. Extended questions can ask for the single transformation equivalent to two in sequence.

Transformations are visual, and a tutor in a one-to-one online lesson can rotate or reflect shapes live on the shared whiteboard while the student predicts where they will land. That quickly fixes the commonest confusions, such as clockwise against anticlockwise or which way a negative scale factor goes, and lessons then practise the exact wording that full descriptions need.

Self-check: can you do these?

• Translate (2, -3) by the vector (-4, 5). (Answer: (-2, 2))
• Rotate (3, 1) 90° clockwise about the origin. (Answer: (1, -3))
• Reflect (4, 1) in the line x = 2. (Answer: (0, 1))
• Enlarge (2, 4) by scale factor 1/2, centre (0, 0). (Answer: (1, 2))
• Rotate (3, 2) 180° about (1, 1). (Answer: (-1, 0))