IGCSE Transformations: Reflection, Rotation and Enlargement on Coordinate Grids

Audience and search intent: Cambridge IGCSE Mathematics 0580 learners and parents seeking step-by-step transformation methods, particularly Extended enlargement.

A transformation changes where a shape appears on a grid. Sometimes its size changes too. In Cambridge IGCSE Mathematics, you may be asked to draw an image after a transformation, identify what happened to a shape, or describe a transformation precisely.

Four transformations are included in the syllabus: reflection, rotation, enlargement and translation. This guide concentrates on the first three because they are often confused. A separate vector lesson can explain translation notation in detail.

1. Reflection: imagine a mirror line

A reflection flips a shape across a specified line. Every original point and its image must be the same perpendicular distance from the mirror line, but on opposite sides (unless the point lies on the line).

Suppose triangle ABC has vertices A(0,1), B(2,1) and C(1,3). Reflect it in the vertical line x = 2.

• A is two units to the left of x = 2, so A′ is two units to its right: A′(4,1).

• B lies on x = 2, so B′ stays at (2,1).

• C is one unit left of the mirror, so C′ is at (3,3).

Answer: A′(4,1), B′(2,1), C′(3,3).

Notice that a reflection does not change the shape's size. The image is congruent to the original.

Common error: Moving every point the same number of squares instead of reflecting each point according to its own distance from the mirror line.

2. Rotation: centre, angle and direction all matter

A rotation turns a shape about a fixed point. A full description needs the centre of rotation, the angle and the direction (clockwise or anticlockwise). Rotations preserve lengths and angles.

For a 90° anticlockwise rotation about the origin, the coordinate rule is:

(x, y) → (−y, x).

Take P(2,1), Q(4,1) and R(2,3).

• P(2,1) becomes P′(−1,2).

• Q(4,1) becomes Q′(−1,4).

• R(2,3) becomes R′(−3,2).

The rotated triangle is the same size as the original. Each vertex is also the same distance from the centre before and after the turn.

The rule above works for a rotation about (0,0). If the centre is elsewhere, a safe graphical method is to mark the centre, draw lines from the centre to the vertices, rotate those lines by the required angle and keep their lengths unchanged.

Common error: Writing “rotation 90°” without identifying the centre or direction. That description is incomplete.

3. Enlargement: scale factor and centre

An enlargement changes lengths by the absolute value of the scale factor. A factor of 2 doubles lengths; a factor of 1/2 halves them. Angles remain the same, so the shapes are similar.

When the centre is the origin and the scale factor is 1/2, point (4,2) becomes (2,1). This is an easy way to visualise reduction.

But what if the centre is not the origin? Use this three-step approach:

1. Start at the centre and describe the displacement to the original point.

2. Multiply that displacement by the scale factor.

3. Start at the centre again and apply the new displacement.

Worked example: negative scale factor

Enlarge point A(2,1) from centre C(1,1), using scale factor −2.

The displacement from C to A is (1,0). Multiply by −2 to get (−2,0). Add this to C(1,1):

A′ = (−1,1).

For point B(1,3), the displacement from C is (0,2). After multiplying by −2 it becomes (0,−4), so B′ = (1,−3).

A negative scale factor places each image point on the opposite side of the centre from its original point. The factor's magnitude still controls size: −2 doubles every length. For a two-dimensional shape, this also has the effect of a half-turn about the enlargement centre together with scaling.

Common error: Treating −2 as a reduction because it is negative. The size multiplier is 2, not −2.

4. What is different in Core and Extended?

The Cambridge IGCSE Mathematics 0580 syllabus for examinations in 2025–2027 distinguishes the tiers.

Core (C7.1): Reflections are in horizontal or vertical lines. Rotations are through multiples of 90° about specified points such as the origin, vertices or edge midpoints. Enlargements use positive and fractional scale factors. Questions do not combine transformations.

Extended (E7.1): Reflections may be in any straight line, rotations may use any centre through multiples of 90°, enlargements may have positive, fractional or negative scale factors, and questions may combine transformations.

This is why an Extended learner should practise negative scale factors and two-step transformations, while a Core learner should first be secure with the specified single transformations.

5. Two transformations: the order can change the result

Start with point P(2,−1).

First reflect it in the y-axis: P becomes (−2,−1). Then move it three units right: the final point is (1,−1).

Reverse the order. First move P three units right to (5,−1), then reflect in the y-axis: the final point is (−5,−1).

These results differ. In a combined-transformation question, read the order carefully and draw the intermediate image before continuing.

6. Original practice questions

Question 1: Reflect point A(3,−2) in x = −1. Find A′.

Question 2: Rotate B(−2,4) through 90° clockwise about the origin. Find B′.

Question 3 (Extended): Enlarge P(6,3) from centre C(2,−1) by scale factor −1/2. Find P′.

Question 4: A student says, “An enlargement by scale factor −3 makes a triangle three times smaller.” Explain the mistake.

Answers and reasoning

1. A is four units right of x = −1, so its image is four units left: A′(−5,−2).

2. A 90° clockwise rotation about the origin follows (x,y) → (y,−x): B′(4,2).

3. C to P is (4,4). Multiply by −1/2 to get (−2,−2). Add to C: P′(0,−3).

4. The negative sign changes which side of the centre the image lies on. The absolute scale factor is 3, so each length is three times larger, not smaller.

A simple revision routine

Practise one transformation at a time, always naming its essential information. For reflections, mark the mirror line. For rotations, mark the centre and direction. For enlargements, mark the centre and measure the displacement to each vertex. Then try mixed questions and explain the reasoning aloud.

A good diagnostic assessment does more than count correct answers: it identifies whether a learner struggles with coordinates, geometry vocabulary, plotting accuracy or the transformation itself.

Official source: Cambridge IGCSE Mathematics 0580 syllabus, examinations 2025–2027, C7.1 and E7.1. Content and tier distinctions verified on 24 October 2026.

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