Transformations: Translation, Rotation, Reflection and Enlargement — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the four transformations that move or resize a shape on a grid. By the end of this guide you should be able to carry out each one accurately and describe one fully when given an object and its image.

You should also be able to work with negative and fractional scale factors, find a centre of rotation or enlargement, combine two transformations, and say which properties of a shape each transformation preserves.

The organising idea is that describing a transformation fully means naming the type and then supplying exactly the right extra information — no more and no less. A translation needs a vector. A reflection needs the equation of the mirror line. A rotation needs three things: angle, direction and centre. An enlargement needs two: scale factor and centre. Marks are awarded for each item, so an answer that names the type but omits the centre scores a fraction of what it could. Learning which details belong to which transformation is most of this topic.

Key terms and definitions

Object — the original shape. Image — the shape after the transformation.

Congruent — identical in shape and size. Three of the four transformations produce congruent images.

Translation — a slide, described by a column vector.

Reflection — a flip in a mirror line.

Rotation — a turn about a fixed point.

Enlargement — a resize by a scale factor from a centre.

Centre of rotation — the fixed point a shape turns about.

Centre of enlargement — the fixed point from which a shape is scaled.

Invariant point — a point that does not move under the transformation.

Core concepts

Translation

A translation slides every point of the shape by the same amount, with no turning and no resizing.

It is described by a column vector: the top number is the movement right, the bottom number the movement up. Negative entries mean left and down. So a vector of 3 over −2 means 3 right and 2 down.

To describe one fully: the word translation and the vector. Nothing else is needed, and writing "moved right and down" without the vector loses marks.

The image is congruent to the object, and the shape keeps its orientation.

Reflection

A reflection flips the shape in a mirror line, with each image point the same perpendicular distance from the line as its object point, on the opposite side.

To describe one fully: the word reflection and the equation of the mirror line — such as x = 2, y = −1, y = x or y = −x.

Giving the line as "the vertical line" or "the diagonal" is not enough; it needs its equation.

Points on the mirror line do not move, making them invariant.

The image is congruent but its orientation is reversed, so a reflected shape appears "flipped".

To reflect in y = x, swap each point's coordinates: (3, 5) becomes (5, 3). To reflect in y = −x, swap and negate both: (3, 5) becomes (−5, −3).

Rotation

A rotation turns the shape about a fixed centre of rotation.

To describe one fully you need three things: the angle, the direction (clockwise or anticlockwise) and the centre, given as coordinates.

The direction can be omitted only for a half turn, since 180° clockwise and 180° anticlockwise give the same image.

Tracing paper is the reliable method: trace the object, put a pencil point on the centre, turn the paper through the angle, and mark the new position.

The image is congruent, and the centre of rotation is invariant.

Enlargement

An enlargement resizes the shape by a scale factor from a centre of enlargement.

To describe one fully: the word enlargement, the scale factor, and the centre as coordinates.

Each point moves along the line from the centre through it, ending up scale-factor times as far from the centre as it began.

Unlike the other three, an enlargement does not produce a congruent image unless the scale factor is 1 — the image is similar to the object, with equal angles and proportional sides.

Scale factors between 0 and 1

A scale factor between 0 and 1 makes the shape smaller, even though the word "enlargement" is still used. There is no separate term at GCSE.

A scale factor of ½ halves every distance from the centre, so the image sits between the object and the centre.

Negative scale factors

A negative scale factor sends the image to the opposite side of the centre, and turns it upside down.

With a scale factor of −2, each point moves to twice its distance from the centre but on the other side, so the image is inverted as well as enlarged.

The effect is identical to enlarging by the positive factor and then rotating 180° about the centre, which is a useful way to picture it.

Finding the centre

For an enlargement, join each vertex of the image to the corresponding vertex of the object and extend the lines. They all meet at the centre.

For a rotation, join two corresponding points, construct the perpendicular bisector of that line, then repeat with another pair. The two bisectors cross at the centre. Trial and error with tracing paper also works and is accepted.

Invariant points

An invariant point is one that does not move under the transformation, and questions ask for them directly.

Under a reflection, every point on the mirror line is invariant. Under a rotation, only the centre is. Under an enlargement, only the centre is. Under a translation, no point is invariant, since everything shifts by the same vector.

A question asking "which points on the shape are invariant" is therefore asking where the shape meets the mirror line, or whether it passes through the centre.

The effect on coordinates

Some transformations have a rule that can be applied directly to coordinates, which is quicker than drawing.

A translation by a vector adds that vector to every coordinate pair.

Reflection in the x-axis negates the y-coordinate, so (3, 4) becomes (3, −4). Reflection in the y-axis negates the x-coordinate, giving (−3, 4).

A rotation of 180° about the origin negates both, so (3, 4) becomes (−3, −4).

A rotation of 90° anticlockwise about the origin sends (x, y) to (−y, x), and 90° clockwise sends it to (y, −x).

These rules are worth knowing as a check on a drawn answer rather than as a replacement for the drawing, since exam questions usually want the image on the grid.

Combining transformations

Two transformations performed one after another can often be described as a single equivalent one.

A reflection in one line followed by a reflection in a parallel line gives a translation. A reflection in one line followed by reflection in a perpendicular line gives a rotation of 180° about their intersection.

When asked for "a single transformation equivalent to" two others, carry both out, then describe the object-to-final-image relationship from scratch rather than trying to combine the descriptions.

Order matters: doing transformation A then B usually gives a different result from B then A.

What each transformation preserves

Translation, reflection and rotation all produce congruent images: lengths and angles are unchanged.

Enlargement preserves angles but multiplies lengths by the scale factor, producing a similar image.

All four preserve the shape's basic form — a triangle stays a triangle.

Only reflection reverses orientation.

Worked examples

Example 1: Describing a rotation fully

Triangle A has vertices at (1, 1), (3, 1) and (1, 4). Triangle B has vertices at (−1, 1), (−3, 1) and (−1, 4). Describe the single transformation mapping A to B.

The image is the same size, so it is not an enlargement. It is a mirror image across the y-axis.

The transformation is a reflection in the line x = 0, which is the y-axis.

Note that a rotation of 180° about the origin would not work: it would send (1, 4) to (−1, −4), not to (−1, 4). Checking one awkward vertex settles which transformation it is.

Example 2: A negative scale factor

Enlarge the point (4, 2) by scale factor −2 about the centre (1, 1).

Find the movement from the centre to the point: 3 right and 1 up.

Multiply by the scale factor of −2: 6 left and 2 down.

Apply that to the centre: (1 − 6, 1 − 2) = (−5, −1).

The image is on the opposite side of the centre and twice as far away, which is exactly what a negative factor does.

Example 3: Combining two transformations

A shape is reflected in the line x = 1, then reflected in the line x = 4. Describe the single equivalent transformation.

The two mirror lines are parallel, so the result is a translation.

The distance between the lines is 3, and a double reflection in parallel lines translates by twice that distance, in the direction from the first line to the second.

The single transformation is a translation by the vector 6 over 0, so 6 units to the right.

Testing a point confirms it: a point at x = 0 reflects to x = 2, then to x = 6 — a move of 6 right.

Common mistakes and how to avoid them

Describing a rotation without the centre. Angle, direction and centre are all required.

Giving a mirror line in words. Use its equation, such as y = −1.

Omitting the vector from a translation. "Moved left" is not a description.

Giving two transformations when one was asked for. Re-read the question; "a single transformation" means one.

Forgetting that an enlargement is not congruent. Only the other three preserve size.

Mishandling a negative scale factor. The image goes to the opposite side of the centre and is inverted.

Assuming order does not matter when combining. A then B is generally different from B then A.

Exam technique for "Transformations"

Write down which details the transformation needs before describing it — vector, mirror line, or angle-direction-centre, or scale-factor-and-centre. Each is a separate mark.

Use tracing paper for rotations. It is permitted, it is quick, and it is far more reliable than estimating.

Give centres as coordinates in brackets and mirror lines as equations.

Check a rotation or reflection by testing a single awkward vertex, since several transformations can map some points correctly while failing on others.

For combined transformations, carry both out on the grid and then describe the overall result fresh, rather than reasoning about the two descriptions.

Look out for the phrase "a single transformation", which tells you one description is wanted and that a combination will not score.

Quick revision summary

Describing fully means naming the type and supplying exactly the right details, each worth a mark:

• Translation — the word plus a column vector.
• Reflection — the word plus the equation of the mirror line.
• Rotation — the word plus angle, direction and centre.
• Enlargement — the word plus scale factor and centre.

Translation, reflection and rotation give congruent images. Enlargement gives a similar image, with angles preserved and lengths scaled.

A scale factor between 0 and 1 shrinks the shape; a negative scale factor puts the image on the opposite side of the centre, inverted.

Reflecting in y = x swaps the coordinates; reflecting in y = −x swaps and negates them.

Find a centre of enlargement by joining corresponding vertices and extending; find a centre of rotation with perpendicular bisectors or tracing paper.

Two reflections in parallel lines make a translation; in perpendicular lines, a 180° rotation. Order matters when combining.