What is G8: Combined transformations?

Higher tier only. In a combination, apply transformations in the stated order because the image from one step becomes the object for the next.

How many marks are G8 questions worth?

In verified past-paper appearances, G8 questions on Edexcel GCSE Maths 1MA1 have been worth 2–3 marks.

Which past exam papers has G8 appeared in?

G8 has appeared in 4 verified past-paper questions indexed on this page, with direct links to the official question paper and mark scheme for each.

Describe the changes and invariance achieved by combinations of rotations, reflections and translations [Higher only]

Higher only

Combined transformations

Worked answers, methods and verified real exam appearances for G8 on Edexcel GCSE Maths 1MA1.

Explanation

Higher tier only. In a combination, apply transformations in the stated order because the image from one step becomes the object for the next.
Two translations combine by adding their vectors. Two reflections in parallel lines produce a translation; two reflections in intersecting lines produce a rotation, with angle twice the angle between the mirror lines.
Rotations, reflections and translations preserve lengths, angles, area and parallelism. One reflection reverses orientation; two reflections restore it.
Coordinate rules are a reliable way to identify a single equivalent transformation.
Reversing the order can change the result, so transformations do not generally commute.

Successive reflections in parallel lines give one translation.

Worked example

A point is reflected in the $x$ -axis and then in the $y$ -axis. Describe the single equivalent transformation.

1.Reflection in the $x$ -axis maps $(x,y)$ to $(x,-y)$ .
2.Reflection in the $y$ -axis then maps $(x,-y)$ to $(-x,-y)$ .
3.The rule $(x,y)\mapsto(-x,-y)$ is a rotation of $180^\circ$ about the origin.

Answer: A rotation of $180^\circ$ about the origin.

Common mistakes

Don't apply the second transformation to the original shape instead of the first image.
Don't say that two reflections always make a translation, ignoring whether the mirror lines intersect.

Exam tip

To identify a combined transformation, track one general point and then state every parameter of the single result.

Worked practice

Tier 1 · Easy

A shape is rotated and then translated. State two properties of the shape that must remain invariant.

(2)

(Total for Question 1 is 2 marks)

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Mark scheme
1	Any two of: side lengths, angle sizes, area, parallelism, or orientation remain unchanged.	2	Both a rotation and a translation are rigid transformations. Each preserves lengths and angles, so it also preserves area and parallel lines; neither reverses orientation.

Tier 2 · Standard

A shape is translated first by

\begin{pmatrix}3\\-2\end{pmatrix}

and then by

\begin{pmatrix}-5\\7\end{pmatrix}

. Describe the single equivalent transformation.

(2)

(Total for Question 2 is 2 marks)

Mark scheme

Mark scheme for question 2
Question	Answer	Mark	Mark scheme
2	A translation by $\begin{pmatrix}-2\\5\end{pmatrix}$ .	2	Add the translation vectors component by component: $\begin{pmatrix}3\\-2\end{pmatrix}+\begin{pmatrix}-5\\7\end{pmatrix}=\begin{pmatrix}-2\\5\end{pmatrix}$ .

Tier 3 · Hard

A shape is reflected in the

x

-axis and then reflected in the line

y=x

. Describe the single equivalent transformation and state whether orientation is preserved.

(4)

(Total for Question 3 is 4 marks)

Mark scheme

Mark scheme for question 3
Question	Answer	Mark	Mark scheme
3	A rotation of $90^\circ$ anticlockwise about the origin; orientation is preserved.	4	The first reflection maps $(x,y)$ to $(x,-y)$ . Reflection in $y=x$ then swaps the coordinates, giving $(-y,x)$ . This is the coordinate rule for a $90^\circ$ anticlockwise rotation about the origin. A rotation preserves orientation.

Tier 1 · Easy

Horizontal lines

m

and

n

are parallel and

3

cm apart, with

n

above

m

. A shape is reflected first in line

m

, then in line

n

. Describe the single equivalent transformation.

(2)

(Total for Question 4 is 2 marks)

Mark scheme

Mark scheme for question 4
Question	Answer	Mark	Mark scheme
4	A vertical translation of $6$ cm upwards	2	Two reflections in parallel lines give a translation perpendicular to the lines. Its distance is twice the separation, so it is $2\times3=6$ cm from $m$ towards $n$ , which is upwards.

Tier 2 · Standard

Point

P(2,-5)

is reflected in the

y

-axis and then rotated

90^\circ

clockwise about the origin. Find the final image. Write down whether orientation is preserved by the combination.

(3)

(Total for Question 5 is 3 marks)

Mark scheme

Mark scheme for question 5
Question	Answer	Mark	Mark scheme
5	$(-5,2)$ ; orientation is reversed	3	Reflection in the $y$ -axis sends $(2,-5)$ to $(-2,-5)$ . A $90^\circ$ clockwise rotation sends $(x,y)$ to $(y,-x)$ , giving $(-5,2)$ . One reflection reverses orientation and the rotation preserves it, so the combination reverses orientation.

Tier 3 · Hard

A shape is reflected first in line

l

and then in line

m

. The lines meet at

O

, and the clockwise angle from

l

m

38^\circ

. Describe the single equivalent transformation. Write down two properties that remain invariant.

(4)

(Total for Question 6 is 4 marks)

Mark scheme

Mark scheme for question 6
Question	Answer	Mark	Mark scheme
6	A rotation of $76^\circ$ clockwise about $O$ ; any two of lengths, angle sizes, area, parallelism and orientation are invariant.	4	Two reflections in intersecting lines give a rotation about their intersection. The rotation angle is twice the directed angle from the first line to the second, so it is $2\times38^\circ=76^\circ$ clockwise about $O$ . A rotation preserves lengths, angles, area, parallelism and orientation.

Tier 2 · Standard

A shape of area

34

^2

undergoes two rotations and three reflections. Write down whether its final image is congruent to the original, whether its orientation is preserved, and its final area.

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
Question	Answer	Mark	Mark scheme
7	It is congruent to the original; orientation is reversed; area $=34$ cm $^2$	3	Rotations and reflections preserve all lengths and area, so the final image is congruent and still has area $34$ cm $^2$ . Rotations preserve orientation. Each reflection reverses it, and three is an odd number of reversals, so the final orientation is reversed.

Tier 3 · Hard

A shape is rotated

90^\circ

clockwise about the origin and then rotated

90^\circ

clockwise about

(4,0)

. Describe the single equivalent transformation and state whether orientation is preserved.

(4)

(Total for Question 8 is 4 marks)

Mark scheme

Mark scheme for question 8
Question	Answer	Mark	Mark scheme
8	A rotation of $180^\circ$ about $(2,2)$ ; orientation is preserved	4	The first rotation maps $(x,y)$ to $(y,-x)$ . Rotating this image $90^\circ$ clockwise about $(4,0)$ gives $(4-x,4-y)$ . This is the rule for a $180^\circ$ rotation about $(2,2)$ . A rotation preserves orientation.

Tier 3 · Hard

A line segment has endpoints

A(-1,2)

and

B(3,5)

. In route 1, reflect the segment in the line

x=2

and then translate it by

\begin{pmatrix}3\\-4\end{pmatrix}

. In route 2, apply the same translation first and then reflect in

x=2

. Work out the final endpoints for both routes and explain how your answers show that the order matters.

(5)

(Total for Question 9 is 5 marks)

Mark scheme

Mark scheme for question 9
Question	Answer	Mark	Mark scheme
9	Route 1: $A_1=(8,-2)$ and $B_1=(4,1)$ ; route 2: $A_2=(2,-2)$ and $B_2=(-2,1)$ ; the different endpoint pairs show that these transformations do not commute	5	Reflection in $x=2$ maps $(x,y)$ to $(4-x,y)$ . For route 1 this sends $A$ and $B$ to $(5,2)$ and $(1,5)$ , then the translation gives $A_1=(8,-2)$ and $B_1=(4,1)$ . For route 2 the translation first gives $(2,-2)$ and $(6,1)$ ; reflecting these gives $A_2=(2,-2)$ and $B_2=(-2,1)$ . Since corresponding final endpoints are different, reversing the order changes the result.

Q10

Tier 3 · Hard

A shape is rotated

180^\circ

about

(3,-2)

and then reflected in the line

x=3

. Describe the single equivalent transformation, state whether orientation is preserved, and describe all points that remain fixed.

(4)

(Total for Question 10 is 4 marks)

Mark scheme

Mark scheme for question 10
Question	Answer	Mark	Mark scheme
10	Reflection in the line $y=-2$ ; orientation is reversed; every point on $y=-2$ is fixed	4	The half-turn maps $(x,y)$ to $(6-x,-4-y)$ . Reflection in $x=3$ then maps this to $(x,-4-y)$ . This leaves the $x$ -coordinate unchanged and places the final $y$ -coordinate equally far across $y=-2$ , so it is reflection in $y=-2$ . A reflection reverses orientation, and precisely the points on its mirror line are fixed.

Verified exam appearances

Series	Paper	Question	Marks	Calculator	Tier	Links
2024-06	1H	Q11	3	Non-calculator	Higher	QP MS
2019-11	2H	Q24	2	Allowed	Higher	QP MS
2022-11	3H	Q18	2	Allowed	Higher	QP MS
2019-06	1H	Q5	3	Non-calculator	Higher	QP MS

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