Translate, reflect, rotate and enlarge shapes on coordinate grids, then describe each transformation fully using the exact detail Edexcel mark schemes require.
Geometry and MeasuresFoundation and HigherGrades 3 to 7Skill
Pearson Edexcel GCSE Maths geometry G7: identify, describe and construct rotation, reflection, translation and enlargement, including fractional and negative scale factors; G8: describe changes and invariance from combinations of transformations.
Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.
Theory
A transformation moves or resizes a shape. The original shape is the object; the new shape is the image.
There are four GCSE transformations: translation, reflection, rotation and enlargement.
When Edexcel says describe fully, the description must include all required details: translation needs a column vector; reflection needs the mirror line; rotation needs centre, angle and direction; enlargement needs scale factor and centre.
If a question asks for a single transformation, give one transformation only. Writing two steps, such as a reflection then a translation, can score zero even if the movement is true.
A translation slides every vertex by the same vector. The top number is horizontal movement, the bottom number is vertical movement. Right and up are positive; left and down are negative.
A reflection flips the shape in a mirror line. Vertical lines are written x = k and horizontal lines are written y = k. The y-axis is x = 0 and the x-axis is y = 0.
A rotation turns a shape around a centre. State the centre, angle and direction. A 180 degree rotation does not need a direction because clockwise and anticlockwise land in the same place.
An enlargement changes size from a centre. State the scale factor and centre. Enlargements make similar shapes; translations, reflections and rotations make congruent shapes.
A fractional enlargement, such as scale factor 1/2, makes the image smaller but it is still called an enlargement.
Higher: a negative scale factor puts the image on the opposite side of the centre and turns it through the centre.
Tracing paper is allowed in Edexcel exams and is especially useful for rotations.
Key ruleDescribe fully: translation by column vector; reflection in a line; rotation with centre, angle and direction; enlargement with scale factor and centre.
Diagram guide
TranslationEvery vertex moves the same way. Use a column vector: top number for sideways movement, bottom number for up or down.ReflectionName the mirror line with its equation. Vertical mirror lines are x = k; horizontal mirror lines are y = k.Diagonal reflectionReflection in y = x swaps the coordinates. Diagonal mirror lines are a common grade 5 trap.RotationState the angle, direction and centre of rotation. Use tracing paper for accuracy.EnlargementMultiply each journey from the centre by the scale factor. The image is similar to the object.Fractional enlargementA scale factor between 0 and 1 makes the image smaller. It is still called an enlargement.Higher: negative scale factorA negative scale factor sends the image through the centre to the opposite side.Combined transformationsTwo reflections in the coordinate axes have the same single effect as a 180 degree rotation about the origin.
Worked examples
Translate a triangle
Triangle A has vertices (1, 1), (3, 1) and (1, 4). Translate A by the column vector [3, -2].
Example: translation by [3, -2]Add 3 to every x-coordinate and subtract 2 from every y-coordinate.
The top number is +3, so move every vertex 3 right.
The bottom number is -2, so move every vertex 2 down.
(1, 1) -> (4, -1).
(3, 1) -> (6, -1).
(1, 4) -> (4, 2).
Answer: Image vertices: (4, -1), (6, -1), (4, 2)
Reflect in a vertical line
Reflect the triangle with vertices (2, 1), (4, 1), (4, 3) in the line x = 1.
Example: reflection in x = 1Each vertex lands the same perpendicular distance on the other side of the mirror line.
The line x = 1 is vertical.
(2, 1) is 1 square right of x = 1, so it maps to (0, 1).
(4, 1) is 3 squares right of x = 1, so it maps to (-2, 1).
(4, 3) maps to (-2, 3).
Answer: Image vertices: (0, 1), (-2, 1), (-2, 3)
Reflect in y = x
Reflect the triangle (1, 2), (4, 2), (4, 3) in the line y = x.
Example: diagonal reflectionReflection in y = x swaps the coordinates.
For reflection in y = x, use (a, b) -> (b, a).
(1, 2) -> (2, 1).
(4, 2) -> (2, 4).
(4, 3) -> (3, 4).
Answer: Image vertices: (2, 1), (2, 4), (3, 4)
Rotate about the origin
Rotate the triangle (1, 1), (4, 1), (1, 3) by 90 degrees clockwise about (0, 0).
Example: 90 degrees clockwiseFor a 90 degree clockwise rotation about the origin, (x, y) maps to (y, -x).
Use the rule (x, y) -> (y, -x).
(1, 1) -> (1, -1).
(4, 1) -> (1, -4).
(1, 3) -> (3, -1).
Answer: Image vertices: (1, -1), (1, -4), (3, -1)
Enlarge from the origin
Enlarge the triangle (1, 1), (2, 1), (1, 3) by scale factor 3, centre (0, 0).
Example: scale factor 3Because the centre is the origin, multiply each coordinate by 3.
(1, 1) -> (3, 3).
(2, 1) -> (6, 3).
(1, 3) -> (3, 9).
Every side is 3 times longer, so the image is similar.
Answer: Image vertices: (3, 3), (6, 3), (3, 9)
Fractional enlargement
Enlarge the triangle (3, 1), (7, 1), (3, 5) by scale factor 1/2, centre (1, 1).
Example: scale factor 1/2Halve each journey from the centre. The image sits between the centre and the object.
From centre (1, 1) to (3, 1) is [2, 0]. Half is [1, 0], so the image point is (2, 1).
From (1, 1) to (7, 1) is [6, 0]. Half is [3, 0], so the image point is (4, 1).
From (1, 1) to (3, 5) is [2, 4]. Half is [1, 2], so the image point is (2, 3).
Answer: Image vertices: (2, 1), (4, 1), (2, 3)
Higher: negative enlargement
Enlarge the triangle (2, 1), (3, 1), (2, 3) by scale factor -2, centre (1, 1).
Example: scale factor -2Multiply each journey from the centre by -2. The negative sends the image to the opposite side.
From (1, 1) to (2, 1) is [1, 0]. Multiply by -2 to get [-2, 0], so the image point is (-1, 1).
From (1, 1) to (3, 1) is [2, 0]. Multiply by -2 to get [-4, 0], so the image point is (-3, 1).
From (1, 1) to (2, 3) is [1, 2]. Multiply by -2 to get [-2, -4], so the image point is (-1, -3).
Triangle A has vertices (1, 1), (3, 1), (1, 2). Triangle B has vertices (5, 4), (7, 4), (5, 5). Describe fully the single transformation from A to B.
Example: same size, same orientationSame size and same orientation means translation. Count from any vertex to its image.
The image is the same size and has the same orientation.
From (1, 1) to (5, 4) is 4 right and 3 up.
Write the movement as a column vector with horizontal movement on top.
Answer: Translation by column vector [4, 3]
Combine two reflections
Triangle A has vertices (1, 2), (3, 2), (1, 5). A is reflected in the y-axis, then the image is reflected in the x-axis. Describe the single transformation from A to the final image.
Example: two reflections as one transformationReflection in both coordinate axes has the same single effect as a 180 degree rotation about the origin.
Reflecting in the y-axis changes (x, y) to (-x, y).
Reflecting in the x-axis then changes (-x, y) to (-x, -y).
Overall, every point (x, y) becomes (-x, -y).
That is a rotation 180 degrees about (0, 0).
Answer: Rotation 180 degrees about (0, 0)
Common mistakes
Using coordinate brackets instead of a column vector for translations.
Putting the vertical movement on top of a translation vector.
Confusing x = k and y = k. The y-axis is x = 0.
Forgetting the direction in a 90 degree or 270 degree rotation.
Forgetting the centre of rotation or centre of enlargement.
Naming two transformations when the question asks for a single transformation.
Calling a scale factor 1/2 enlargement a reduction.
Drawing a negative enlargement on the same side of the centre instead of the opposite side.
Quick exercise
Try these before moving to the exam-style questions.
A shape moves 6 squares left and 1 square up. Type the translation vector as [-6, 1].Quick check: column vectorHorizontal movement goes on top; vertical movement goes underneath.
The point (3, -2) is reflected in the x-axis. Write down the image coordinate.Quick check: x-axis reflectionReflecting in the x-axis changes the sign of the y-coordinate.
The point (2, 5) is rotated 180 degrees about the origin. Write down the image coordinate.Quick check: 180 degree rotationA 180 degree rotation about the origin maps (x, y) to (-x, -y).
A shape is enlarged by scale factor 1/2. Type: halved and similar.Quick check: fractional enlargementA fractional enlargement changes side lengths and keeps angles the same.
Which three transformations always produce a congruent image? Type them separated by commas.Quick check: congruent transformationsConguent means same shape and same size.
Higher: the point (4, 2) is enlarged by scale factor -1, centre (0, 0). Write down the image coordinate.Quick check: negative scale factorScale factor -1 about the origin sends (x, y) to (-x, -y).
Exam-style questions
Practise the same skill at three levels.
These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.
Basic GCSE styleFoundationNon-calculator2 marks
Triangle A has vertices (2, 2), (5, 2) and (2, 4). Translate triangle A by the column vector [-3, 2]. Label the image B.
Question diagram: translationMove every vertex 3 left and 2 up.
Standard exam styleFoundation and HigherNon-calculator2 marks
Triangle A has vertices (1, 1), (3, 1), (3, 4). Triangle B has vertices (-3, 1), (-5, 1), (-5, 4). Describe fully the single transformation that maps A onto B.
Question diagram: find the mirror lineThe shapes are mirror images. The mirror line is halfway between corresponding vertices.
reflectionmirror linedescribe fully
Standard exam styleFoundation and HigherNon-calculator3 marks
Triangle A has vertices (2, 1), (4, 1), (2, 2). Triangle B has vertices (2, -1), (2, -3), (3, -1). Describe fully the single transformation that maps A onto B.
Question diagram: rotationFor rotations, the full description needs type, angle, direction and centre.
rotationcentre of rotationdescribe fully
Standard exam styleFoundation and HigherNon-calculator2 marks
Triangle A has vertices (1, 3), (3, 3) and (1, 4). Enlarge triangle A by scale factor 2, centre (0, 2).
Question diagram: enlargementMeasure each journey from the centre, double it, then plot the image.
enlargementscale factorcentre of enlargement
ChallengeHigherNon-calculator3 marks
Triangle A has vertices (3, 1), (5, 1), (3, 5). Triangle B has vertices (0, 1), (-1, 1), (0, -1). Describe fully the single transformation that maps A onto B.
Question diagram: negative enlargementThe image is smaller and on the opposite side of the centre, which signals a negative fractional scale factor.
ChallengeFoundation and HigherNon-calculator4 marks
Triangle A has vertices (1, 2), (3, 2) and (1, 5). (a) Reflect triangle A in the y-axis to give B. (b) Reflect B in the x-axis to give C. (c) Describe fully the single transformation that maps A onto C.
Question diagram: two reflectionsThe final answer for part (c) must be one transformation, not the two steps already given.
combined transformationsreflectionrotation 180invariant centre
Next step in the app
Use the notes with a real answer.
Reading the method helps, but the progress comes from answering, marking, and fixing one question at a time.