Combined transformations - Edexcel GCSE Maths (Calculator) Exam Questions

Practise 28 exam-style questions on Combined transformations for Edexcel GCSE Maths (Calculator). Write your answer as you would in the exam and get it marked in seconds against the mark scheme, point by point, with an AI tutor on hand if you get stuck.

28 questions
1–5 marks each
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20 questions, marked as you go

Sample questions with model answers

Here are 4 of the questions in this topic, each with a model answer, an explanation of why it earns the marks and the mark scheme an examiner would use.

Question 1

2 marks

The diagram shows triangle T, with vertices (1, 1), (4, 1) and (3, 3), and the line y = 1.
Triangle T is reflected in the line y = 1.

Which points of triangle T are invariant under this reflection?

A coordinate grid with x from −1 to 6 and y from −2 to 4; each grid square is 1 unit across, and the axes cross at the origin O. Shape T is a shaded triangle with vertices (1, 1), (4, 1) and (3, 3). The line labelled "y = 1" is a horizontal straight line crossing the y-axis at (0, 1).

Model answer

Points on the mirror line do not move when reflected.
The side from (1, 1) to (4, 1) lies on the line y = 1, so the vertices (1, 1) and (4, 1), and every point on the side between them, are invariant.
The vertex (3, 3) moves to (3, −1), so it is not invariant.

Explanation

In a reflection, the invariant points are exactly the points on the mirror line. Here one side of the triangle lies along the mirror line, so that whole side stays where it is.

Mark scheme

1 mark
B1 for the vertices (1, 1) and (4, 1)
correct answer with no working scores full marks
1 mark
B1 for stating that every point on the side joining (1, 1) to (4, 1) is invariant (because it lies on the mirror line)
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Question 2

3 marks

The diagram shows triangle A and the lines x = 1 and x = 4.
Triangle A is reflected in the line x = 1 to give triangle B.
Triangle B is reflected in the line x = 4 to give triangle C.

Describe fully the single transformation that maps triangle A onto triangle C.

A coordinate grid with x from −4 to 8 and y from −1 to 5; each grid square is 1 unit across, and the axes cross at the origin O. Shape A is a shaded triangle with vertices (−3, 1), (−1, 1) and (−3, 4). The line labelled "x = 1" is a vertical straight line crossing the x-axis at (1, 0). The line labelled "x = 4" is a vertical straight line crossing the x-axis at (4, 0).

Model answer

A has vertices (−3, 1), (−1, 1), (−3, 4).
Reflect in x = 1: B has vertices (5, 1), (3, 1), (5, 4).
Reflect in x = 4: C has vertices (3, 1), (5, 1), (3, 4).
Each vertex of C is 6 to the right of the matching vertex of A.
Translation by \(\begin{pmatrix}6\\0\end{pmatrix}\)

Explanation

Two reflections in parallel lines give a translation perpendicular to the lines, by twice the distance between them (2 × 3 = 6). The second reflection turns the shape back the right way round, which is why the result is not a reflection.

Mark scheme

1 mark
M1 for the correct position of B, (5, 1), (3, 1), (5, 4), or of C, (3, 1), (5, 1), (3, 4)
1 mark
B1 for translation
award no marks for the description if more than one transformation is given
1 mark
B1 for \(\begin{pmatrix}6\\0\end{pmatrix}\)

Question 3

3 marks

The diagram shows triangle A on a grid.
Triangle A is reflected in the y-axis to give triangle B.
Triangle B is rotated 90° clockwise about the origin to give triangle C.

Describe fully the single transformation that maps triangle A onto triangle C.

A coordinate grid with x from −5 to 6 and y from −5 to 6; each grid square is 1 unit across, and the axes cross at the origin O. Shape A is a shaded triangle with vertices (2, 1), (5, 1) and (2, 3).

Model answer

A: (2, 1), (5, 1), (2, 3)
B (reflect in the y-axis): (−2, 1), (−5, 1), (−2, 3)
C (rotate 90° clockwise about O): (1, 2), (1, 5), (3, 2)
Each vertex (x, y) of A has gone to (y, x).
Reflection in the line y = x.

Explanation

One reflection and one rotation reverse the orientation once, so the single transformation must be a reflection. Swapping the coordinates is the reflection in y = x. Plotting B and C on the grid is a good check.

Mark scheme

1 mark
M1 for the vertices of B, (−2, 1), (−5, 1), (−2, 3), or of C, (1, 2), (1, 5), (3, 2)
1 mark
B1 for reflection
award no marks for the description if more than one transformation is given
1 mark
B1 for in the line y = x

Question 4

5 marks

A shape is rotated 90° clockwise about the origin.
The image is then translated by the vector \(\begin{pmatrix}2\\0\end{pmatrix}\).

Find the coordinates of the point that is invariant under this combination.
Hence describe fully the single transformation that is equivalent to the combination.

Model answer

(x, y) → (y, −x) after the rotation → (y + 2, −x) after the translation.
For an invariant point: x = y + 2 and y = −x.
So x = −x + 2, x = 1 and y = −1. The invariant point is (1, −1).
(Check: (1, −1) → (−1, −1) → (1, −1).)
The shape is turned through 90° clockwise and the point (1, −1) does not move, so the combination is a rotation of 90° clockwise about (1, −1).

Explanation

A rotation followed by a translation turns the shape through the same angle, so it is a rotation through that angle about some new centre. The centre is the only point that does not move, so solve for the invariant point.

Mark scheme

1 mark
M1 for the image of a general point, (x, y) → (y, −x) → (y + 2, −x)
1 mark
M1 for setting up equations for an invariant point, x = y + 2 and y = −x
1 mark
A1 for (1, −1)
cao
1 mark
B1 for rotation of 90° clockwise
the translation does not turn the shape, so the angle is still 90° clockwise
1 mark
B1 for centre (1, −1)
ft their invariant point

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