Transformations: reflection, rotation, translation, enlargement (incl. negative/fractional) – GCSE Maths exam questions with mark scheme

1Grade 1–3FoundationNon-calculator2 marks

Triangle A is drawn on the grid.

Reflect triangle A in the $y$-axis.
Write down the coordinates of the vertices of the image. [2]

at least two vertices correct, or a correct reflection in a different vertical line (correct size and orientation)B1
fully correct: $(-1, 1)$, $(-3, 1)$, $(-1, 4)$B1
Answer: $(-1, 1)$, $(-3, 1)$, $(-1, 4)$
Examiner tip: Each point moves to the same distance on the OTHER side of the mirror line — count squares to the line, then the same number past it.

2Grade 4–5Non-calculator3 marks

Triangle A and triangle C are drawn on the grid.

Describe fully the single transformation that maps triangle A onto triangle C. [3]

rotation (do not accept 'turn')B1
$90°$ clockwise (accept $270°$ anticlockwise, or $-90°$)B1
centre $(0, 0)$ (accept 'about the origin')B1
Answer: Rotation $90°$ clockwise about $(0, 0)$
Examiner tip: 'Describe fully' a rotation needs three things: the word rotation, the angle AND direction, and the centre. Giving two transformations scores 0.

3Grade 4–5Non-calculator2 marks

Triangle P is drawn on the grid.

Enlarge triangle P by scale factor 2, centre $(1, 1)$.
Give the coordinates of the vertices of the image. [2]

a triangle of the correct size and orientation in the wrong position, or two vertices correctB1
fully correct: $(3, 3)$, $(5, 3)$, $(5, 7)$B1
Answer: $(3, 3)$, $(5, 3)$, $(5, 7)$
Examiner tip: Measure from the centre of enlargement to each vertex and double that distance — don't just double the lengths of the sides anywhere on the grid.

4Grade 6–7HigherNon-calculator3 marks

Triangle A and triangle B are drawn on the grid.

Describe fully the single transformation that maps triangle A onto triangle B. [3]

enlargementB1
scale factor $\tfrac{1}{2}$ (accept 0.5)B1
centre $(2, 8)$B1
Answer: Enlargement, scale factor $\tfrac{1}{2}$, centre $(2, 8)$
Examiner tip: Find the centre by drawing lines through corresponding vertices — where they meet is the centre. A shape getting smaller is still an 'enlargement'.

5Grade 8–9HigherNon-calculator2 marks

Triangle T is drawn on the grid.

Enlarge triangle T by scale factor $-2$, centre $(1, 1)$.
Give the coordinates of the vertices of the image. [2]

a triangle with correct size and orientation (rotated $180°$) in the wrong position, or two vertices correct, or correct enlargement with scale factor $+2$ centre $(1,1)$B1
fully correct: $(-1, -1)$, $(-5, -1)$, $(-1, -3)$B1
Answer: $(-1, -1)$, $(-5, -1)$, $(-1, -3)$
Examiner tip: A negative scale factor puts the image on the OPPOSITE side of the centre and upside down — go through the centre, then twice as far again.

6Grade 6–7HigherNon-calculator3 marks

Triangle A is drawn on the grid.

Triangle A is reflected in the line $y = x$ to give triangle B.
Triangle B is then rotated $180°$ about $(0, 0)$ to give triangle C.

Describe fully the single transformation that maps triangle A onto triangle C. [3]

triangle B correct, vertices $(1, 1)$, $(1, 3)$, $(4, 1)$, or triangle C correct, vertices $(-1, -1)$, $(-1, -3)$, $(-4, -1)$ (may be seen on a diagram). A fully correct description with no working scores all 3 marks.B1
reflectionB1
in the line $y = -x$ (accept $x + y = 0$; dependent on 'reflection')B1
Answer: Reflection in the line $y = -x$
Examiner tip: Draw B and C before you describe anything — guessing the single transformation from the instructions alone usually goes wrong.

7Grade 1–3FoundationNon-calculator2 marks

Shape A and shape B are drawn on the grid.

Describe fully the single transformation that maps shape A onto shape B. [2]

translation (do not accept 'move', 'slide' or 'transformation')B1
by vector $\begin{pmatrix} 7 \\ -6 \end{pmatrix}$ (accept an equivalent description in words, e.g. '7 right and 6 down'; do not accept coordinates such as $(7, -6)$)B1
Answer: Translation by $\begin{pmatrix} 7 \\ -6 \end{pmatrix}$
Examiner tip: Write the movement as a column vector, across on top and up/down underneath (down is negative); writing it as coordinates $(7, -6)$ loses the second mark.

8Grade 1–3FoundationNon-calculator3 marks

Shape A is drawn on the grid.

(a) Translate shape A by the vector $\begin{pmatrix} -5 \\ 4 \end{pmatrix}$.
Write down the coordinates of the four vertices of the image. [2]

(b) Write down the column vector of the translation that maps the image back onto shape A. [1]

(a) shape of the correct size and orientation with one component of the translation correct (e.g. 5 left but wrong vertical move), or at least 3 vertices correctB1
(a) fully correct: $(-4, 1)$, $(-1, 1)$, $(-1, 3)$, $(-3, 3)$ (vertices in any order; accept correct image drawn on the grid)B1
(b) $\begin{pmatrix} 5 \\ -4 \end{pmatrix}$ (do not accept coordinates $(5, -4)$; accept '5 right, 4 down')B1
Answer: (a) $(-4, 1)$, $(-1, 1)$, $(-1, 3)$, $(-3, 3)$ (b) $\begin{pmatrix} 5 \\ -4 \end{pmatrix}$
Examiner tip: The top number moves left/right and the bottom number moves up/down — swapping them, or reversing only one sign for the way back, loses the mark.

9Grade 4–5Non-calculator4 marks

Shape A and shape B are drawn on the grid.

(a) Reflect shape A in the line $x = -1$.
Write down the coordinates of the four vertices of the image. [2]

(b) Describe fully the single transformation that maps shape A onto shape B. [2]

(a) shape of the correct size reflected in any vertical line (correct orientation, wrong position), or at least 3 vertices correctB1
(a) fully correct: $(-3, 1)$, $(-5, 1)$, $(-5, 2)$, $(-3, 4)$ (any order)B1
(b) reflection (do not accept 'mirror' or 'flip'; award no marks in (b) if more than one transformation is given)B1
(b) in the line $y = -1$ (dependent on 'reflection'; do not accept 'in $x = -1$' or 'in the $y$-axis')B1
Answer: (a) $(-3, 1)$, $(-5, 1)$, $(-5, 2)$, $(-3, 4)$ (b) Reflection in the line $y = -1$
Examiner tip: $x = -1$ is a vertical line and $y = -1$ is a horizontal line — students who mix them up reflect the wrong way and lose both marks.

10Grade 4–5Non-calculator5 marks

Triangle T and triangle U are drawn on the grid.

(a) Rotate triangle T $90°$ anticlockwise about the point $(1, -1)$.
Write down the coordinates of the vertices of the image. [2]

(b) Describe fully the single transformation that maps triangle T onto triangle U. [3]

(a) triangle of the correct size and orientation in the wrong position, or rotation $90°$ clockwise about $(1, -1)$, i.e. $(3, -2)$, $(3, -4)$, $(4, -4)$, or 2 vertices correctB1
(a) fully correct: $(-1, 0)$, $(-1, 2)$, $(-2, 2)$ (any order)B1
(b) rotation (do not accept 'turn'; award no marks in (b) if more than one transformation is given)B1
(b) $180°$ (direction not required)B1
(b) centre $(0, -1)$. Higher tier: 'enlargement, scale factor $-1$, centre $(0, -1)$' scores full marks.B1
Answer: (a) $(-1, 0)$, $(-1, 2)$, $(-2, 2)$ (b) Rotation $180°$ about $(0, -1)$
Examiner tip: For a centre that is not the origin, use tracing paper pinned at the centre — rotating about $(0, 0)$ by habit gives an image in the wrong place.

11Grade 8–9HigherNon-calculator3 marks

Triangle A and triangle B are drawn on the grid.

Describe fully the single transformation that maps triangle A onto triangle B. [3]

enlargement (award no marks if more than one transformation is given)B1
scale factor $-\tfrac{1}{2}$ (accept $-0.5$; scale factor $\tfrac{1}{2}$ or $-2$ scores B0)B1
centre $(-1, 3)$B1
Answer: Enlargement, scale factor $-\tfrac{1}{2}$, centre $(-1, 3)$
Examiner tip: B is half the size AND upside down relative to A, so the scale factor must be negative — find the centre where the lines joining matching vertices cross.

12Grade 6–7HigherNon-calculator4 marks

Shape A is drawn on the grid. The lines $x = 1$ and $x = 4$ are also drawn.

Shape A is reflected in the line $x = 1$ to give shape B.
Shape B is then reflected in the line $x = 4$ to give shape C.

(a) Describe fully the single transformation that maps shape A onto shape C. [3]

(b) Ella reflects shape A in the line $x = 4$ first, and then reflects the image in the line $x = 1$.
She says she will also get shape C.
Is Ella correct? Give a reason for your answer. [1]

(a) shape B correct, vertices $(5, 1)$, $(3, 1)$, $(3, 4)$, $(5, 2)$, or shape C correct, vertices $(3, 1)$, $(5, 1)$, $(5, 4)$, $(3, 2)$ (may be seen on the diagram). A fully correct answer with no working scores 3 marks.B1
(a) translation (award no further marks if more than one transformation is given)B1
(a) by $\begin{pmatrix} 6 \\ 0 \end{pmatrix}$ (accept '6 to the right'; dependent on 'translation')B1
(b) No, with a reason, e.g. the image would be translated by $\begin{pmatrix} -6 \\ 0 \end{pmatrix}$ / 6 to the left, so it is not C (accept correct vertices $(-9, 1)$, $(-7, 1)$, $(-7, 4)$, $(-9, 2)$ shown)C1
Answer: (a) Translation by $\begin{pmatrix} 6 \\ 0 \end{pmatrix}$ (b) No — it gives a translation by $\begin{pmatrix} -6 \\ 0 \end{pmatrix}$
Examiner tip: Two reflections in parallel lines always give a translation of twice the gap between the lines — and the direction depends on the order.

13Grade 8–9HigherNon-calculator3 marks

Triangle A is drawn on the grid.

Triangle A is rotated $180°$ about the point $(2, 3)$ to give triangle B.
Triangle B is then translated by the vector $\begin{pmatrix} 2 \\ -4 \end{pmatrix}$ to give triangle C.

Describe fully the single transformation that maps triangle A onto triangle C. [3]

triangle B correct, vertices $(3, 1)$, $(1, 1)$, $(1, -1)$, or triangle C correct, vertices $(5, -3)$, $(3, -3)$, $(3, -5)$ (may be seen on the diagram). A fully correct answer with no working scores 3 marks.B1
rotation $180°$ (accept 'enlargement, scale factor $-1$'; award no further marks if more than one transformation is given)B1
centre $(3, 1)$B1
Answer: Rotation $180°$ about $(3, 1)$
Examiner tip: Draw B and C first, then join each vertex of A to its partner on C — the midpoints all meet at the new centre $(3, 1)$, not the original $(2, 3)$.

14Grade 4–5Non-calculator4 marks

Triangle A is drawn on the grid.

(a) Enlarge triangle A by scale factor $\tfrac{1}{3}$, centre $(-4, -4)$.
Write down the coordinates of the vertices of the image. [2]

(b) Kai says, “An enlargement always makes a shape bigger.”
Use your answer to part (a) to explain why Kai is wrong. [1]

(a) triangle with sides $\tfrac{1}{3}$ of the length (1 by 2 right-angled triangle, correct orientation) in the wrong position, or 2 vertices correctB1
(a) fully correct: $(-2, -3)$, $(-1, -3)$, $(-1, -1)$ (any order)B1
(b) a scale factor between 0 and 1 (e.g. $\tfrac{1}{3}$) makes the shape smaller — accept 'the image in (a) is smaller than A' or 'the sides are a third of the length'C1
(c) $k = 9$ (accept 'area of image is $\tfrac{1}{9}$ of the area of A'; do not accept 3)B1
Answer: (a) $(-2, -3)$, $(-1, -3)$, $(-1, -1)$ (b) a fractional scale factor (between 0 and 1) makes it smaller (c) $k = 9$
Examiner tip: Measure from the centre to each vertex and take a third of that distance — measuring from the origin, or dividing the coordinates by 3, puts the image in the wrong place.

15Grade 6–7HigherNon-calculator4 marks

Triangle A is drawn on the grid. The line $y = x$ is also drawn.

(a) Triangle A is reflected in the line $y = x$ to give triangle B.
Write down the coordinates of the vertices of triangle B. [2]

(b) Describe the points of triangle A that are invariant under this reflection. [1]

(c) Triangle A is translated by the vector $\begin{pmatrix} 2 \\ -1 \end{pmatrix}$.
Explain why no point of triangle A is invariant under this translation. [1]

(a) vertex $(1, 5)$ correct (the image of $(5, 1)$), or triangle A correctly reflected in $y = -x$ (copying A's own vertices $(1, 1)$ and $(3, 3)$ alone scores B0)B1
(a) fully correct: $(1, 1)$, $(1, 5)$, $(3, 3)$ (any order)B1
(b) all the points on the side joining $(1, 1)$ and $(3, 3)$ — accept 'the side that lies on $y = x$' (just the two vertices $(1, 1)$ and $(3, 3)$ scores B0)B1
(c) every point moves 2 right and 1 down, so no point stays in the same place (accept 'all points move by the same vector, which is not zero')C1
Answer: (a) $(1, 1)$, $(1, 5)$, $(3, 3)$ (b) every point on the side from $(1, 1)$ to $(3, 3)$ (c) a translation moves every point by the same non-zero vector
Examiner tip: Invariant means 'does not move' — for a reflection that is every point ON the mirror line, so a whole side can be invariant, not just two corners.

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