Transformations

Edexcel GCSE Mathematics requires you to perform and describe four types of transformation: reflection, rotation, translation and enlargement. You need to know what information is required to describe each one fully, and how to carry out each transformation on a coordinate grid. Higher tier includes fractional and negative scale factors for enlargement.

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Key Vocabulary

Term	Meaning
Object	The original shape before a transformation
Image	The shape after the transformation
Congruent	Same shape and same size
Similar	Same shape but different size (angles preserved, sides in proportion)
Invariant point	A point that does not move during a transformation

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The Four Transformations at a Glance

Transformation	What changes?	Congruent?	Information needed to describe
Reflection	Position (flipped)	Yes	Line of reflection
Rotation	Position and orientation	Yes	Centre, angle, direction (clockwise/anticlockwise)
Translation	Position only	Yes	Column vector
Enlargement	Size (and position)	No (similar)	Scale factor and centre of enlargement

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Reflection

A reflection flips a shape across a mirror line (line of reflection).

How to Reflect

1. For each vertex, count the perpendicular distance to the mirror line.
2. Plot the image vertex the same distance on the other side.
3. Join the image vertices.

Common mirror lines on a coordinate grid:

• x = 0 (the y-axis)
• y = 0 (the x-axis)
• y = x (diagonal line through origin at 45°)
• y = -x (diagonal line through origin at -45°)
• x = a or y = b (vertical or horizontal lines)

Worked Example 1

Reflect triangle A with vertices (1, 2), (3, 2), (1, 5) in the line x = 4.

For each point, find the horizontal distance to x = 4 and go the same distance beyond: (1, 2): distance to x = 4 is 3, so image = (7, 2) (3, 2): distance to x = 4 is 1, so image = (5, 2) (1, 5): distance to x = 4 is 3, so image = (7, 5)

Describing a Reflection: You MUST state the equation of the mirror line (e.g. "Reflection in the line y = -x"). Just saying "reflection" without the line scores 1 out of 2 marks.

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Rotation

A rotation turns a shape around a fixed point (the centre of rotation).

How to Describe a Rotation (3 things needed)

1. Centre of rotation (a coordinate)
2. Angle of rotation
3. Direction: clockwise or anticlockwise

Worked Example 2

Describe the single transformation that maps triangle A at (1, 1), (3, 1), (3, 2) to triangle B at (1, -1), (1, -3), (2, -3).

By inspection or tracing:

• Centre: (0, 0)
• Angle: 90°
• Direction: clockwise

Answer: A rotation of 90° clockwise about the origin (0, 0).

Exam Tip: If the question says "describe fully", you need all three pieces of information (centre, angle, direction) for a rotation to get full marks.

Finding the Centre of Rotation

Join a point on the object to its corresponding image point. Construct the perpendicular bisector of this line. Repeat for another pair. The centre of rotation is where the perpendicular bisectors meet.

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Translation

A translation slides every point of a shape by the same distance in the same direction. It is described using a column vector.

The column vector (a over b) means move a units right (negative = left) and b units up (negative = down).

Worked Example 3

Translate the triangle with vertices (2, 3), (5, 3), (2, 6) by the vector (3 over -4) [i.e. 3 right and 4 down].

(2, 3) -> (5, -1) (5, 3) -> (8, -1) (2, 6) -> (5, 2)

Describing a Translation: You MUST give the column vector. Saying "3 right and 4 down" without the vector loses a mark.

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Enlargement

An enlargement changes the size of a shape. Every length is multiplied by the scale factor (SF), and all angles stay the same.

What You Need to Describe an Enlargement

1. Scale factor (SF)
2. Centre of enlargement

Worked Example 4

Enlarge the triangle with vertices (2, 1), (4, 1), (2, 3) by scale factor 2, centre of enlargement (0, 0).

Multiply each coordinate by 2: (2, 1) -> (4, 2) (4, 1) -> (8, 2) (2, 3) -> (4, 6)

Finding the Scale Factor

Scale factor = image length / object length

Or: Scale factor = distance from centre to image point / distance from centre to object point

Worked Example 5

A triangle has been enlarged. The original base is 3 cm; the image base is 9 cm. Find the scale factor.

SF = 9 / 3 = 3

Fractional Scale Factors [H]

A scale factor between 0 and 1 makes the image smaller than the object.

Worked Example 6 [H]

Enlarge the rectangle with vertices (6, 3), (12, 3), (12, 9), (6, 9) by scale factor 1/3, centre (0, 0).

(6, 3) -> (2, 1) (12, 3) -> (4, 1) (12, 9) -> (4, 3) (6, 9) -> (2, 3)

Negative Scale Factors [H]

A negative scale factor produces an image on the opposite side of the centre of enlargement, and the image is inverted.

Worked Example 7 [H]

Enlarge point (4, 2) by scale factor -2, centre (1, 1).

Vector from centre to point: (4 - 1, 2 - 1) = (3, 1) Multiply by -2: (-6, -2) Image point: (1 + (-6), 1 + (-2)) = (-5, -1)

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Combining Transformations

Two (or more) transformations applied in sequence can sometimes be described as a single transformation.

Worked Example 8

A shape is reflected in the y-axis, then reflected in the x-axis. The single transformation equivalent is a rotation of 180° about the origin.

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Common Mistakes and Misconceptions