What are the four transformations in GCSE Maths?

The four main transformations are **translation, reflection, rotation and enlargement**. Translation moves a shape, reflection creates a mirror image, rotation turns it around a centre, and enlargement changes distances from a centre using a scale factor.

How do I describe a transformation fully?

For a translation, give the column vector. For a reflection, state the mirror line. For a rotation, give the centre, angle and direction. For an enlargement, state the centre and scale factor.

Are negative scale factors on Foundation tier?

Negative scale factors are generally **Higher-tier content** in the major GCSE Mathematics specifications. Foundation students should still understand positive integer and fractional scale factors.

Can an enlargement make a shape smaller?

Yes. If the scale factor satisfies $0<k<1$, the image is smaller than the original. It is still officially called an enlargement.

Does the order of combined transformations matter?

Yes. Applying a reflection and then a translation will usually produce a different image from applying the translation first. Complete each transformation separately and label the intermediate image.

Transformations GCSE Maths: Foundation and Higher

Transformations GCSE Maths questions test whether you can move, turn, reflect or resize a shape accurately. The four main transformations are translation, reflection, rotation and enlargement. You may need to perform a transformation, describe one fully or combine two transformations.

This guide explains each method step by step, with transformations worked examples, common mistakes and exam-style practice. The core ideas appear on both Foundation and Higher tier. Fractional scale factors can appear at Foundation, while negative scale factors are Higher-only content in major GCSE specifications.

The four GCSE transformations

A transformation maps every point on an original shape to a corresponding point on an image. If the original point is called $A$ , its image is usually written $A^{'}$ .

Transformation	Information needed for a full description	Does size change?
Translation	Translation vector	No
Reflection	Equation or clear description of the mirror line	No
Rotation	Centre, angle and direction	No
Enlargement	Centre and scale factor	Usually

Translations, reflections and rotations preserve lengths and angles, so the image is congruent to the original. Enlargements preserve angles but multiply every length by the scale factor, producing a similar shape.

A question that says describe fully the single transformation requires all the relevant information in the table. Naming only the transformation is rarely enough.

Translation

A translation moves every point by the same distance in the same direction. It does not turn, reflect or resize the shape.

Translations are described using a column vector:

(ab)\begin{pmatrix} a\\ b \end{pmatrix}

The top number gives the horizontal movement and the bottom number gives the vertical movement:

positive $a$ : move right
negative $a$ : move left
positive $b$ : move up
negative $b$ : move down

Translation worked example

Translate the triangle with vertices $A (- 2, 1)$ , $B (1, 1)$ and $C (0, 3)$ by

(4−2).\begin{pmatrix} 4\\ -2 \end{pmatrix}.

Add $4$ to every $x$ -coordinate and subtract $2$ from every $y$ -coordinate:

A′=(2,−1),B′=(5,−1),C′=(4,1).A'=(2,-1),\qquad B'=(5,-1),\qquad C'=(4,1).

When drawing on a grid, transform each vertex separately and then join the image vertices in the original order.

Describing a translation

Choose one vertex and its matching image vertex. Count the horizontal and vertical movement from the original to the image. Check another pair of corresponding points before writing the vector.

A frequent mistake is reversing the direction. The vector must describe movement from the original shape to the image.

Reflection

A reflection produces a mirror image in a line. The mirror line is the perpendicular bisector of the segment joining any point to its image. This means the original point and image point must be equal distances from the line.

Common reflection lines include:

the $x$ -axis, with equation $y = 0$
the $y$ -axis, with equation $x = 0$
a vertical line such as $x = 2$
a horizontal line such as $y = - 1$
the diagonal lines $y = x$ and $y = - x$

Reflection worked example

Reflect $P (- 3, 2)$ in the line $x = 1$ .

The point is $4$ units to the left of $x = 1$ , so its image must be $4$ units to the right. The $y$ -coordinate does not change:

P^{'} = (5, 2) .

For a reflection in a vertical line $x = a$ , the coordinate rule is

(x,y)⟼(2a−x,y).(x,y)\longmapsto(2a-x,y).

For the example, $2 (1) - (- 3) = 5$ .

Useful reflection rules

Some coordinate rules can make checking easier:

y=−x:(x,y)↦(−y,−x).\begin{aligned} \text{in the }x\text{-axis: }&(x,y)\mapsto(x,-y),\\ \text{in the }y\text{-axis: }&(x,y)\mapsto(-x,y),\\ \text{in }y=x: &(x,y)\mapsto(y,x),\\ \text{in }y=-x: &(x,y)\mapsto(-y,-x). \end{aligned}

Do not describe a reflection as being in an axis unless that axis is genuinely the mirror line. For example, $x = 2$ is parallel to the $y$ -axis, but it is not the $y$ -axis.

Rotation

A rotation turns a shape around a fixed point called the centre of rotation. A full description states:

the centre of rotation
the angle
the direction, clockwise or anticlockwise

For a $180∘180^\circ$ rotation, the direction does not matter. For $90∘90^\circ$ and $270∘270^\circ$ , it must be stated.

Rotation worked example

Rotate the triangle with vertices $A (1, 1)$ , $B (3, 1)$ and $C (1, 2)$ by $90∘90^\circ$ anticlockwise about the origin.

The coordinate rule is

(x,y)⟼(−y,x).(x,y)\longmapsto(-y,x).

Therefore,

A′=(−1,1),B′=(−1,3),C′=(−2,1).A'=(-1,1),\qquad B'=(-1,3),\qquad C'=(-2,1).

The image remains the same size and every vertex stays the same distance from the origin.

Finding the centre of rotation

If you are given the original and image shapes, join a point to its image and construct the perpendicular bisector of that segment. Repeat this with another pair of corresponding points. The perpendicular bisectors meet at the centre of rotation.

On a coordinate grid, careful counting may be quicker. Test a possible centre by checking that corresponding vertices are the same distance from it and turn through the same angle.

Enlargement

An enlargement changes distances from a fixed centre of enlargement using a scale factor $k$ . For every original point $P$ and image point $P^{'}$ :

CP′=k×CP,CP'=k\times CP,

where $C$ is the centre. Corresponding vertices lie on the same straight line through the centre.

If $k > 1$ , the image is larger. If $0 < k < 1$ , it is smaller. The word enlargement is still used when the image becomes smaller.

Positive enlargement worked example

Enlarge the triangle $A (2, 1)$ , $B (2, 3)$ and $C (1, 3)$ by scale factor $2$ about the centre $(1, 1)$ .

Consider each displacement from the centre:

$A$ is $(1, 0)$ from the centre, so $A^{'}$ is $(2, 0)$ from it, giving $A^{'} = (3, 1)$ .
$B$ is $(1, 2)$ from the centre, so $B^{'}$ is $(2, 4)$ from it, giving $B^{'} = (3, 5)$ .
$C$ is $(0, 2)$ from the centre, so $C^{'}$ is $(0, 4)$ from it, giving $C^{'} = (1, 5)$ .

The side lengths double, but the angles remain unchanged.

Fractional scale factors

A scale factor of $12\frac12$ places each image vertex halfway from the centre to the corresponding original vertex. Do not simply halve the coordinates unless the centre is the origin.

For example, with centre $(2, 2)$ , a point at $(6, 4)$ has displacement $(4, 2)$ . After enlargement by scale factor $12\frac12$ , the displacement is $(2, 1)$ , so the image point is $(4, 3)$ .

Negative scale factors for Higher tier

A negative scale factor places the image on the opposite side of the centre. Its magnitude controls the change in length.

Suppose $C = (1, 1)$ and $P = (3, 2)$ . The displacement from $C$ to $P$ is $(2, 1)$ . With scale factor $- 2$ :

−2(21)=(−4−2).-2\begin{pmatrix}2\\1\end{pmatrix} = \begin{pmatrix}-4\\-2\end{pmatrix}.

Starting at $(1, 1)$ gives

P^{'} = (- 3, - 1) .

A negative enlargement is not the same as a reflection. Every point is scaled through one centre, and both the centre and scale factor are required in the description.

Combined transformations

A combined transformation applies two or more transformations in a stated order. Order matters because changing the order will usually change the final image.

Label an intermediate image after the first transformation. For example, use $A^{'}$ after the first step and $A^{''}$ after the second. This reduces the chance of accidentally applying both instructions to the original shape.

Some questions ask for a single transformation equivalent to a combination. Two reflections in parallel lines, for example, are equivalent to a translation. At GCSE, you should still judge the diagram carefully rather than assuming a rule without checking corresponding points.

Common mistakes in transformations exam questions

Giving an incomplete description

A rotation needs its centre, angle and direction. An enlargement needs its centre and scale factor. A reflection needs its mirror line, and a translation needs its vector.

Transforming only part of the shape

Every vertex must follow exactly the same transformation. Plot each image vertex before joining them, then check that no side has been missed.

Counting from the image to the original

Descriptions normally map the stated original shape onto the image. Reversing that direction changes a translation vector, reverses a rotation and replaces an enlargement scale factor $k$ with $1k\frac1k$ .

Using the origin automatically

The centre of rotation or enlargement is not necessarily $(0, 0)$ . Read the instruction or determine the centre from the diagram before using coordinate rules.

Multiplying coordinates during an enlargement

Multiplying each coordinate by $k$ works only when the centre is the origin. Otherwise, scale the displacement from the given centre.

Embedded exam-style question

Triangle $A B C$ has vertices

A(−3,1),B(−1,1),C(−2,3).A(-3,1),\qquad B(-1,1),\qquad C(-2,3).

The triangle is translated by

(4−2)\begin{pmatrix}4\\-2\end{pmatrix}

and then reflected in the line $y = x$ . Find the coordinates of the final image. [4 marks]

Answer

After the translation:

A′=(1,−1),B′=(3,−1),C′=(2,1).A'=(1,-1),\qquad B'=(3,-1),\qquad C'=(2,1).

Reflection in $y = x$ swaps each point's coordinates:

A′′=(−1,1),B′′=(−1,3),C′′=(1,2).A''=(-1,1),\qquad B''=(-1,3),\qquad C''=(1,2).

Try 4 more transformations exam questions and check the worked solutions.

Four more exam-style practice questions

Point $P (- 4, 2)$ is reflected in the $y$ -axis. Find $P^{'}$ .
Point $Q (2, - 1)$ is rotated $90∘90^\circ$ anticlockwise about the origin. Find $Q^{'}$ .
Point $R (5, 3)$ is enlarged by scale factor $12\frac12$ about the centre $(1, 1)$ . Find $R^{'}$ .
Describe fully the transformation that maps $(x, y)$ to $(x - 3, y + 5)$ .

Answers

$P^{'} = (4, 2)$ .
Using $(x,y)↦(−y,x)(x,y)\mapsto(-y,x)$ gives $Q^{'} = (1, 2)$ .
The displacement from $(1, 1)$ is $(4, 2)$ . Halving it gives $(2, 1)$ , so $R^{'} = (3, 2)$ .
Translation by the vector $(−35)\begin{pmatrix}-3\\5\end{pmatrix}$ .

How to revise transformations effectively

Begin by learning the information required to describe each transformation fully. Then practise plotting transformed vertices, followed by questions in which you must identify the transformation from two shapes.

Use tracing paper when an exam question permits it, especially for rotations. Even when you can visualise the answer, mark the centre and check the distance from the centre to each corresponding vertex.

The MathsGenie transformations revision page brings together lessons, questions and worked solutions. You can also use the transformations exam booklet for longer practice. AQA students can use the AQA transformations resources, while OCR students can use the OCR transformations resources.

Requirements vary slightly between specifications and tiers. Check the GCSE Maths topics by exam board and tier if you are unsure which transformation skills belong to your course.

Conclusion

For transformations GCSE Maths questions, identify the transformation first and write down all the information it needs. Work with corresponding vertices, measure from the correct centre or line, and check that lengths and angles behave as expected.

Translation, reflection and rotation preserve size, while enlargement uses a centre and scale factor. Once the methods are secure, use MathsGenie's topic booklet, practice questions and worked mark schemes to build accuracy under exam conditions.

Sources and referenced URLs

The four GCSE transformations

A transformation maps every point on an original shape to a corresponding point on an image. If the original point is called $A$ , its image is usually written $A^{'}$ .

Transformation	Information needed for a full description	Does size change?
Translation	Translation vector	No
Reflection	Equation or clear description of the mirror line	No
Rotation	Centre, angle and direction	No
Enlargement	Centre and scale factor	Usually

A question that says describe fully the single transformation requires all the relevant information in the table. Naming only the transformation is rarely enough.