Carry out two transformations in order and describe the single transformation that has the same effect. It is Higher only, aimed at grade 7. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Objectives
Carry out two transformations in the right order.
Track a point to find the single equivalent transformation.
Describe a single transformation with every detail.
Key points
Do the first transformation in the sentence first and apply the second to the image.
Track a point through both transformations using coordinate rules.
Know that two reflections in parallel lines make a translation, and two reflections in lines that cross make a rotation.
Describe a single transformation fully: mirror line, centre with angle and direction, vector, or centre with scale factor.
Worked examples
Example 1
The point (4, 1) is reflected in the y-axis and the image is translated by (2, 3). Where does it end up?
2 marks
Reflect in the y-axis: (4, 1) goes to (-4, 1)
Add the vector to the image: (-4 + 2, 1 + 3)
The final point is (-2, 4)
Answer: (-2, 4)
Common slips
Slip: Translate first: (6, 4) The reflection is done first.
Slip: Add the vector to the original: (4 + 2, 1 + 3) The second step acts on the image.
Slip: The final point is (6, 4) Use the image, not the original point.
Example 2
The point (3, 2) is rotated 90 degrees anticlockwise about the origin and then reflected in the x-axis. Where does it end up?
2 marks
Rotation 90 degrees anticlockwise: (x, y) goes to (-y, x), so (3, 2) goes to (-2, 3)
Reflection in the x-axis changes the sign of y: (-2, 3) goes to (-2, -3)
The final point is (-2, -3)
Answer: (-2, -3)
Common slips
Slip: Rotation gives (2, -3) That is a clockwise quarter turn.
Slip: Reflection in the x-axis changes the sign of x The x-axis reflection flips y.
Slip: The final point is (2, 3) Apply the reflection to the image of the rotation.
Example 3
Shape A is reflected in the line x = 1 and then in the line x = 4. The two reflections are equivalent to a translation. How many units does it move the shape?
3 marks
The mirror lines are parallel and 4 - 1 = 3 units apart
The translation is twice the distance between the lines
2 x 3 = 6 units
Answer: 6
Common slips
Slip: The lines cross, so it is a rotation Lines x = 1 and x = 4 are both vertical, so they are parallel.
Slip: The translation is the same as the distance between the lines Each reflection throws the shape across its own line.
Slip: 3 units Double the distance between the lines.
Questions and answers
GCSE Maths combined transformations questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
The point (2, 5) is reflected in the x-axis and then translated by (3, 1). Where does it end up?
Show answer
Answer: (5, -4)
Reflect in the x-axis: (2, -5).
Add the vector: (5, -4).
M1 — Reflect in the x-axis: (2, -5).
A1 — Answer (5, -4)
Question 2 Easy2 marks
The point (-3, 4) is reflected in the y-axis and then translated by (0, -2). Where does it end up?
Show answer
Answer: (3, 2)
Reflect in the y-axis: (3, 4).
Subtract 2 from y: (3, 2).
M1 — Reflect in the y-axis: (3, 4).
A1 — Answer (3, 2)
Question 3 Medium2 marks
The point (2, 3) is rotated 90 degrees anticlockwise about the origin and then translated by (1, 1). Where does it end up?
Show answer
Answer: (-2, 3)
Rotation gives (-3, 2).
Add (1, 1): (-2, 3).
M1 — Rotation gives (-3, 2).
A1 — Answer (-2, 3)
Question 4 Medium2 marks
The point (4, 0) is reflected in the line y = x and then in the x-axis. Where does it end up?
Show answer
Answer: (0, -4)
Reflect in y = x: (0, 4).
Reflect in the x-axis: (0, -4).
M1 — Reflect in y = x: (0, 4).
A1 — Answer (0, -4)
Question 5 Hard2 marks
The point (3, 1) is rotated 180 degrees about the origin and then translated by (5, 4). Where does it end up?
Show answer
Answer: (2, 3)
Half turn: (-3, -1).
Add (5, 4): (2, 3).
M1 — Half turn: (-3, -1).
A1 — Answer (2, 3)
Question 6 Hard2 marks
Two reflections are made in the parallel lines y = 1 and y = 6. How many units is the single translation?
Show answer
Answer: 10
The lines are 5 units apart.
The translation is twice that.
M1 — The lines are 5 units apart.
A1 — Answer 10
Question 7 Exam-style4 marks
Shape S is rotated 180 degrees about the origin. The image is then translated by the vector (4, -2). Describe fully the single transformation that is equivalent to this combination.
Show answer
Answer: rotation 180 degrees about (2, -1)
A half turn sends (x, y) to (-x, -y).
The translation then gives (4 - x, -2 - y).
A half turn about (p, q) gives (2p - x, 2q - y), so 2p = 4 and 2q = -2.
The centre is (2, -1).
M1 — Rotation by 180 degrees sends (x, y) to (-x, -y)
M1 — Adds the translation to get (4 - x, -2 - y)
M1 — Compares with (2p - x, 2q - y) or finds the fixed point
A1 — Rotation 180 degrees about (2, -1)
Common mistakes
Most lost marks on this topic are one slip.
Applying the second transformation to the original shape
Apply it to the image of the first transformation
Listing two transformations when one is asked for
Track a point and name one transformation with every detail
Moving the shape by the gap between parallel mirror lines
Use twice the gap between the lines
Spot the slip
A shape is reflected in the x-axis and then rotated, but a student starts with the rotation. Why is that a problem?
The first transformation named is done first, so the order changes the image
Rotations must always be done last
Reflections cannot be combined with rotations
Show the slip
The first transformation named is done first, so the order changes the image The second transformation acts on the image of the first, so swapping them usually changes the result.
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.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.