Reflect, rotate, translate and enlarge shapes on a grid, and describe a transformation fully. It is on Foundation and Higher, aimed at grade 3. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Four transformations move a shape: translation (slide by a vector (xy)), reflection (flip in a given line), rotation (turn about a point through an angle, clockwise or anticlockwise), and enlargement (scale from a centre). Each needs a full description: a rotation is not '90 degrees' on its own — you need the centre and the direction too.
Count squares. A translation (3−2) is 3 right and 2 down for every vertex. Reflection in x=2 keeps the same y and mirrors the x distance the other side of the line. Enlargement at this grade is positive scale factor; negative scale factors wait until a later topic.
Reflection: the mirror is the perpendicular bisector of the join. Rotation: 90° clockwise about O is (x,y)→(y,−x). Translation: add the vector. Enlargement at this grade is a positive integer scale factor from a centre — count rays.
Objectives
Every point moves by the same vector.
Every point is the same distance the other side of the mirror line.
A rotation needs a centre, an angle and a direction.
Key points
Translate a shape using a column vector.
Reflect a shape in a line such as the x-axis, y-axis, y=x or x=2.
Rotate a shape about a centre by 90° or 180°, clockwise or anticlockwise.
Enlarge a shape by a scale factor from a centre of enlargement.
Worked examples
Example 1
The point (−2, −3) is translated by the vector (3, −2). What are the image coordinates?
2 marks
The vector (3, −2) moves x by 3 and y by −2.
New x: −2 + 3 = 1. New y: −3 + (−2) = −5.
The image is (1, −5).
Answer: (1, -5)
Common slips
Slip: The vector (3, −2) moves x by −2 and y by 3. The top number is the x change and the bottom number is the y change.
Slip: New x: −2 + (−2) = −4. New y: −3 + 3 = 0. These sums use the two parts of the vector the wrong way round.
Slip: The image is (−4, 0). The image is wrong because the vector was swapped.
Example 2
The point (−1, −1) is rotated 90° clockwise about the origin. What are the image coordinates?
2 marks
A 90° clockwise turn about the origin sends (x, y) to (y, −x).
(−1, −1) goes to (−1, 1).
The image is (−1, 1).
Answer: (-1, 1)
Common slips
Slip: A 90° clockwise turn about the origin sends (x, y) to (−y, x). That rule is for an anticlockwise turn.
Slip: (−1, −1) goes to (1, −1). This uses the anticlockwise rule.
Slip: The image is (1, −1). The image is on the wrong side of the origin for a clockwise turn.
Example 3
The point (5, 1) is reflected in the x-axis. What are the image coordinates?
3 marks
Reflection in the x-axis keeps x the same and changes the sign of y.
(5, 1) goes to (5, −1).
The image is (5, −1).
Answer: (5, -1)
Common slips
Slip: Reflection in the x-axis keeps y the same and changes the sign of x. That is a reflection in the y-axis. A reflection in the x-axis changes y.
Slip: (5, 1) goes to (−5, 1). The wrong coordinate has been changed.
Slip: The image is (−5, 1). The point has moved sideways instead of up or down.
Questions and answers
GCSE Maths transformations questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
The point (1, 3) is translated by the vector (−2, 4). What are the image coordinates?
Show answer
Answer: (-1, 7)
The vector (−2, 4) moves x by −2 and y by 4.
New x: 1 + (−2) = −1. New y: 3 + 4 = 7.
M1 — The vector (−2, 4) moves x by −2 and y by 4.
A1 — Answer (-1, 7)
Question 2 Easy2 marks
The point (2, 5) is rotated 90° clockwise about the origin. What are the image coordinates?
Show answer
Answer: (5, -2)
A 90° clockwise turn about the origin sends (x, y) to (y, −x).
(2, 5) goes to (5, −2).
M1 — A 90° clockwise turn about the origin sends (x, y) to (y, −x).
A1 — Answer (5, -2)
Question 3 Medium2 marks
The point (−2, 8) is rotated 90° clockwise about the origin. What are the image coordinates?
Show answer
Answer: (8, 2)
A 90° clockwise turn about the origin sends (x, y) to (y, −x).
(−2, 8) goes to (8, 2).
M1 — A 90° clockwise turn about the origin sends (x, y) to (y, −x).
A1 — Answer (8, 2)
Question 4 Medium2 marks
The point (−1, 2) is reflected in the x-axis. What are the image coordinates?
Show answer
Answer: (-1, -2)
Reflection in the x-axis keeps x the same and changes the sign of y.
(−1, 2) goes to (−1, −2).
M1 — Reflection in the x-axis keeps x the same and changes the sign of y.
A1 — Answer (-1, -2)
Question 5 Hard2 marks
The point (2, −3) is reflected in the x-axis. What are the image coordinates?
Show answer
Answer: (2, 3)
Reflection in the x-axis keeps x the same and changes the sign of y.
(2, −3) goes to (2, 3).
M1 — Reflection in the x-axis keeps x the same and changes the sign of y.
A1 — Answer (2, 3)
Question 6 Hard2 marks
The point (3, −1) is translated by the vector (3, −2). What are the image coordinates?
Show answer
Answer: (6, -3)
The vector (3, −2) moves x by 3 and y by −2.
New x: 3 + 3 = 6. New y: −1 + (−2) = −3.
M1 — The vector (3, −2) moves x by 3 and y by −2.
A1 — Answer (6, -3)
Question 7 Exam-style3 marks
The point (−1, 5) is translated by the vector (−2, 4). What are the image coordinates?
Show answer
Answer: (-3, 9)
The vector (−2, 4) moves x by −2 and y by 4.
New x: −1 + (−2) = −3. New y: 5 + 4 = 9.
The image is (−3, 9).
M1 — The vector (−2, 4) moves x by −2 and y by 4.
M1 — New x: −1 + (−2) = −3. New y: 5 + 4 = 9.
A1 — Answer (-3, 9)
Common mistakes
Most lost marks on this topic are one slip.
Writing a translation as a reflection.
If the shape has not flipped, it is not a reflection.
Giving a rotation without a centre.
The centre is part of the description.
Adding the vector to only one vertex.
Every vertex moves by the same vector.
Spot the slip
A translation needs
a vector
a mirror line
a centre and an angle
Show the slip
a vector A vector says how far across and how far up.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
A full description has all its parts: reflection (line), rotation (angle, direction, centre), translation (vector), enlargement (scale factor, centre).
Use tracing paper for rotations — ask for it if you do not have it.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.