IGCSE Maths · Transformations and Vectors · Core & Extended

Transformations: Reflection, Rotation, Translation, Enlargement

Four ways to move or resize a shape, each with its own precise language for describing exactly what happened. Reflection, rotation, translation, and enlargement with a positive scale factor are shared by both tiers. Negative/fractional scale factors and combined transformations are Extended-only, tagged throughout.

1. Reflection

A reflection flips a shape over a mirror line — every point ends up the same distance from the line, on the opposite side. To describe a reflection fully, state the equation of the mirror line.

Worked example x=4 (1,1) (7,1) Triangle A (blue) is reflected in the line x = 4 to give triangle B (gold). Describe the transformation.
Answer: a reflection in the line x = 4. Every point’s x-coordinate reflects across 4 (e.g. (1,1) → (7,1), since 4 is exactly halfway between 1 and 7), while y-coordinates stay the same.
Try it — Reflection
  1. A point (2,5) is reflected in the line y = 3. Find its image.
  2. A point (6,2) is reflected in the line x = 1. Find its image.
  3. A point (3,4) is reflected in the line y = x. Find its image (hint: this mirror line swaps coordinates).
Show answers
  1. y=3 is the mirror; distance from 5 to 3 is 2, so image is at y=1: (2,1)
  2. x=1 is the mirror; distance from 6 to 1 is 5, so image is at x=−4: (−4,2)
  3. (4,3) — reflection in y=x always swaps the x and y coordinates

2. Rotation

A rotation turns a shape about a fixed point. To describe a rotation fully, state the centre, the angle, and the direction (clockwise or anticlockwise).

Worked example centre (3,3) Triangle A (blue) is rotated to give triangle B (gold), about the marked centre. Describe the transformation.
Answer: a rotation of 90° anticlockwise about the point (3, 3). Check with one vertex: (4,4) is 1 right, 1 up from the centre; after a 90° anticlockwise turn it becomes 1 up, 1 left from the centre — landing at (2,4), matching the image.
Try it — Rotation
  1. A point (5,2) is rotated 90° anticlockwise about the origin (0,0). Find its image. (Rule: (x,y) → (−y,x))
  2. A point (3,1) is rotated 180° about the origin. Find its image. (Rule: (x,y) → (−x,−y))
  3. A point (2,6) is rotated 90° clockwise about the origin. Find its image. (Rule: (x,y) → (y,−x))
Show answers
  1. (−2,5)
  2. (−3,−1)
  3. (6,−2)

3. Translation

A translation slides a shape without turning or resizing it. Describe it fully with a column vector.

Vector (x,y): x = right/left movement, y = up/down movement
Worked example vector (5,3) Triangle A (blue) is translated to give triangle B (gold). Describe the transformation.
Answer: a translation by the vector (5, 3) — every point moves 5 right and 3 up. Check: (1,1) → (6,4) ✓
Try it — Translation
  1. Point (2,3) is translated by the vector (4,−2). Find its image.
  2. Point (−1,5) is translated by the vector (−3,2). Find its image.
  3. A shape is translated by (6,1) and its image includes the point (9,4). Find the corresponding point on the original shape.
Show answers
  1. (6,1)
  2. (−4,7)
  3. (9−6, 4−1) = (3,3)

4. Enlargement

An enlargement resizes a shape from a fixed centre. Describe it fully with a centre and a scale factor.

Image point = Centre + k × (Original point − Centre)
Worked example — positive scale factor centre (2,2) Triangle A (blue) is enlarged to give triangle B (gold), from the marked centre. Describe the transformation.
Answer: an enlargement, scale factor 2, centre (2, 2). Check: point (3,2) is 1 right, 0 up from centre; doubled is 2 right, 0 up, landing at (4,2) — matching the image.
Try it — Enlargement (Positive Scale Factor)
  1. Point (5,3) is enlarged by scale factor 3, centre (1,1). Find its image.
  2. Point (6,4) is enlarged by scale factor ½, centre (2,2). Find its image.
Show answers
  1. Centre(1,1)+3×((5,3)−(1,1)) = (1,1)+3×(4,2) = (1,1)+(12,6) = (13,7)
  2. (2,2)+0.5×((6,4)−(2,2)) = (2,2)+0.5×(4,2) = (2,2)+(2,1) = (4,3)
Worked example — negative scale factor Extended only A negative scale factor places the image on the opposite side of the centre from the original.
Enlarge the triangle from Section 4 (vertices (3,2),(5,2),(3,4)) by scale factor −1, centre (2,2).
(3,2) → (2,2) + (−1)×(1,0) = (1,2)
(5,2) → (2,2) + (−1)×(3,0) = (−1,2)
(3,4) → (2,2) + (−1)×(1,2) = (1,0)
The image is the same size (|scale factor|=1) but flipped to the opposite side of the centre — it looks like the original has been rotated 180° about the centre.
Try it — Negative and Fractional Scale Factors Extended
  1. Point (4,3) is enlarged by scale factor −2, centre (1,1). Find its image.
  2. Point (0,6) is enlarged by scale factor −1, centre (2,2). Find its image.
  3. Explain why a scale factor of −1 makes a shape end up the same size, just repositioned, rather than “shrinking to nothing.”
Show answers
  1. (1,1)+(−2)×(3,2) = (1,1)+(−6,−4) = (−5,−3)
  2. (2,2)+(−1)×(−2,4) = (2,2)+(2,−4) = (4,−2)
  3. The magnitude of the scale factor (1, ignoring the sign) controls size — a magnitude of 1 keeps the size unchanged. The negative sign only affects position, flipping the image to the opposite side of the centre.

5. Combined Transformations Extended only

Sometimes a question asks you to apply one transformation, then another, to the result — or to describe the single overall effect of two transformations combined.

Worked example A shape is first reflected in the line x = 0 (the y-axis), then translated by the vector (2, −3). A point starts at (3, 5). Find its final position.
Step 1 — reflect in x=0: (3,5) → (−3,5) (x-coordinate negates; y stays the same)
Step 2 — translate by (2,−3): (−3,5) → (−3+2, 5−3) = (−1, 2)
Try it — Combined Transformations Extended
  1. A point (4,2) is rotated 90° anticlockwise about the origin, then translated by (1,1). Find its final position.
  2. A point (5,5) is reflected in the line y=0 (the x-axis), then enlarged by scale factor 2 from the origin. Find its final position.
Show answers
  1. Rotate: (4,2)→(−2,4). Translate: (−2+1,4+1)=(−1,5)
  2. Reflect: (5,5)→(5,−5). Enlarge: (5×2,−5×2)=(10,−10)
Common mistakes to watch for
  • Describing a transformation incompletely — a rotation needs centre, angle, and direction; an enlargement needs centre and scale factor; a reflection needs the mirror line’s equation.
  • Forgetting the direction of rotation (clockwise vs anticlockwise) — the angle alone isn’t a full description.
  • Mixing up which coordinate changes in a reflection — reflecting in a vertical line (x=k) changes the x-coordinate; reflecting in a horizontal line (y=k) changes the y-coordinate.
  • Assuming a negative scale factor means the shape shrinks — the size depends only on the size (magnitude) of the scale factor, not its sign.
  • In combined transformations, applying the two steps in the wrong order — always follow the sequence exactly as given.

6. Quick-Fire Challenge Round

True or False?
  1. A full description of a rotation needs a centre, an angle, and a direction.
  2. A translation changes the size of a shape.
  3. An enlargement with scale factor −1 produces an image the same size as the original.
  4. Reflecting a point in the line y=x swaps its coordinates.
Show answers
  1. True.
  2. False — a translation only slides a shape; size and orientation stay the same.
  3. True. Ext.
  4. True.
Spot the Error
  1. “The transformation is a rotation of 90°.” (as a complete description)
  2. “Reflecting (3,7) in the line y=2: the image is (3,−3).”
  3. “An enlargement of scale factor −3 makes the shape three times bigger and flips it upside down… so it’s now 1/3 the size.”
Show answers
  1. Incomplete — a rotation description needs the centre and direction too, not just the angle.
  2. Distance from 7 to the mirror line (y=2) is 5, so the image should be 5 below the line: y=2−5=−3… this is actually correct! (3,−3) is right. Not every “spot the error” has an error — always check rather than assume.
  3. A scale factor of −3 makes the shape three times bigger (magnitude 3) AND flips it to the opposite side of the centre — it doesn’t shrink to 1/3 the size at any point in that reasoning.

7. Mixed Practice — Bringing It All Together

Core & Extended
  1. Point (4,6) is reflected in the line x=2. Find its image.
  2. Point (1,4) is rotated 180° about the origin. Find its image.
  3. Point (3,3) is translated by the vector (−2,5). Find its image.
  4. Point (4,3) is enlarged by scale factor 2, centre (1,1). Find its image.
Show answers
  1. (0,6)
  2. (−1,−4)
  3. (1,8)
  4. (1,1)+2×(3,2) = (7,5)
Extended-level mixed set Extended only
  1. Point (2,5) is enlarged by scale factor −2, centre (0,0). Find its image.
  2. Point (6,1) is reflected in x=0, then rotated 90° anticlockwise about the origin. Find its final position.
Show answers
  1. (−4,−10)
  2. Reflect: (6,1)→(−6,1). Rotate: (−1,−6)

Practice at Home

A large set to open the Transformations and Vectors unit — take it steadily.

Part A — Reflection
  1. Point (7,2) is reflected in the line x=3. Find its image.
  2. Point (2,8) is reflected in the line y=5. Find its image.
  3. Point (5,1) is reflected in the line y=x. Find its image.
Part B — Rotation
  1. Point (3,6) is rotated 90° anticlockwise about the origin. Find its image.
  2. Point (4,2) is rotated 90° clockwise about the origin. Find its image.
  3. Point (5,3) is rotated 180° about the origin. Find its image.
Part C — Translation and enlargement
  1. Point (3,4) is translated by the vector (−5,2). Find its image.
  2. Point (7,5) is enlarged by scale factor 3, centre (2,1). Find its image.
  3. Point (8,6) is enlarged by scale factor ¼, centre (0,2). Find its image.
Part D — Negative scale factors and combinations Extended
  1. Point (5,4) is enlarged by scale factor −3, centre (1,1). Find its image.
  2. Point (2,3) is rotated 90° anticlockwise about the origin, then reflected in x=0. Find its final position.
Part E — Applied / challenge
  1. A shape is enlarged by scale factor 2, centre (0,0), then translated by (3,−1). A point on the original shape is at (2,4). Find its final position after both transformations.
  2. Describe a single transformation that has exactly the same effect as “rotate 90° clockwise about the origin, then rotate 90° clockwise about the origin again.”
  3. A point P is enlarged by scale factor k from centre C, and ends up back at its own original position. What must be true about k, or about P’s relationship to C?

Key Vocabulary Recap

Reflection — a flip across a mirror line
Rotation — a turn about a fixed centre, through an angle, in a direction
Translation — a slide, described by a column vector
Enlargement — a resizing from a fixed centre, by a scale factor
Scale factor — the multiplier connecting original and image lengths

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) section C7.1 (Core subject content) and section E7.1 (Extended subject content) — Transformations. Reflection, rotation, translation, and enlargement with a positive scale factor are shared by both tiers. Negative and fractional scale factors, and combined transformations, are Extended-only, marked above.

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