What you'll learn
This topic covers the four transformations of shapes: translation, rotation, reflection and enlargement. In this guide you will learn how to perform and describe each transformation, the information needed to define them fully, how enlargement changes size with a scale factor, and how to combine transformations. These are core geometry skills at GCSE.
Key terms and definitions
Translation — a slide described by a column vector.
Rotation — a turn about a fixed centre by an angle and direction.
Reflection — a flip in a mirror line.
Enlargement — a resize by a scale factor from a centre of enlargement.
Object and image — the original shape and its transformed result.
Core concepts
Translation
A translation slides a shape without turning or resizing it, described by a column vector (x on top for horizontal, y below for vertical). For example, the vector (3, −2) means 3 right and 2 down. Every point moves by the same vector.
Reflection
A reflection flips a shape in a mirror line, with the image the same distance behind the line as the object is in front. To describe one fully, give the equation of the mirror line (e.g. x = 2, y = −1, or y = x). Each point and its image are equidistant from the line.
Rotation
A rotation turns a shape about a centre of rotation by an angle and direction (clockwise or anticlockwise). To describe it fully, give the centre, angle and direction. Tracing paper helps you carry out the turn accurately.
Enlargement
An enlargement changes size by a scale factor from a centre of enlargement. A factor greater than 1 makes the shape bigger; between 0 and 1 makes it smaller; a negative factor puts the image on the other side of the centre and inverts it. Distances from the centre multiply by the scale factor.
Describing transformations fully
Each transformation needs specific information: a translation needs a vector; a reflection needs the mirror line; a rotation needs centre, angle and direction; an enlargement needs scale factor and centre. Giving all the detail is essential for full marks, and you should name only one transformation when asked to describe a single mapping.
Worked examples
Example 1: Translation
A point at (1, 4) is translated by vector (3, −2). Find its image.
(1 + 3, 4 − 2) = (4, 2).
Example 2: Reflection
Reflect the point (5, 2) in the line x = 3.
It is 2 right of the line, so the image is 2 left: (1, 2).
Example 3: Enlargement
A length of 4 cm is enlarged by scale factor 3. Find the new length.
4 × 3 = 12 cm.
Common mistakes and how to avoid them
Vector the wrong way round. Top number is horizontal, bottom is vertical.
Incomplete rotation description. Give centre, angle and direction.
Forgetting the mirror line equation. Name the exact line for a reflection.
Wrong enlargement centre. Measure distances from the centre, not the origin, unless told.
Naming two transformations. Describe a single mapping with one transformation.
Exam technique for Transformations
Identify the type — translation, reflection, rotation or enlargement.
Give the full description with all required information.
Use tracing paper for rotations and reflections.
Multiply distances from the centre for enlargements.
Check the image has the right size and orientation.
Quick revision summary
There are four transformations. A translation slides a shape by a column vector (horizontal on top, vertical below). A reflection flips it in a mirror line, with object and image equidistant from the line — describe it by the line's equation. A rotation turns a shape about a centre by an angle and direction — describe all three. An enlargement resizes by a scale factor from a centre of enlargement: a factor above 1 enlarges, between 0 and 1 shrinks, and a negative factor inverts the shape on the other side of the centre. To describe a transformation fully, give the specific information each one needs, and name only one transformation for a single mapping. The common errors are writing the vector the wrong way round, giving an incomplete rotation, forgetting the mirror-line equation, using the wrong enlargement centre, and naming two transformations at once. Identify the type, give every required detail, use tracing paper, and check the image's size and orientation.