Home   >  Maths quizzes   >  Grade 6 Quizzes   >  Combined transformations

MathsLinks Logo Combined transformations Worksheets

Combined transformations Quiz

Grade 6 • 6N • 10 Questions (33 Marks)

Grade 1Grade 2Grade 3Grade 4Grade 5Grade 6Grade 7Grade 8/9
Score: 0 / 10
Combined transformations — Notes & Worked Examples Show / Hide

Explanation:

For combined transformations, apply the first transformation to the original shape to draw an intermediate image, then apply the second transformation to that intermediate image.

When asked to describe the single transformation that maps the starting shape directly onto the final shape, check whether the orientation has flipped (reflection) or stayed the same (rotation/translation). Two reflections in perpendicular axes (such as the x-axis and y-axis) are equivalent to a rotation of 180° about their intersection.

Example 1: Shape A has corner (2, 3). It is reflected in the x-axis to give B, then B is reflected in the y-axis to give C. Describe the single transformation from A to C.

A(2,3) B(2,−3) C(−2,−3)

Solution:

(2, 3) → B(2, −3) → C(−2, −3), so (x, y) → (−x, −y)
Answer: Rotation of 180° about (0, 0)

Example 2: Shape P is translated by vector (4, −1) to give Q, then Q is translated by (−6, 5) to give R. Describe the single transformation from P to R.

Solution:

Add the two translation vectors together:
Horizontal: 4 + (−6) = −2
Vertical: −1 + 5 = 4
Answer: Translation by column vector (−2, 4)
Question 1
[2 marks]
xy–6–5–4–3–2–1123456–6–5–4–3–2–1123456OABCD
Square ABCD is transformed by a combined transformation of a reflection in the line x = -1 followed by a rotation.
Under the combined transformation, two vertices of the square ABCD are invariant.
Describe fully one possible rotation.
Square ABCD is transformed by a combined transformation of a reflection in the line x = -1 followed by a rotation.
Under the combined transformation, two vertices of the square ABCD are invariant.
Describe fully one possible rotation.
Worked Solution & Answer
Reflecting ABCD in x = –1 gives A'(–3, 4), B'(–5, 4), C'(–5, 2), D'(–3, 2) M1
Rotating 90° clockwise about (–1, 0) maps B'(–5, 4) → B(3, 4) and D'(–3, 2) → D(1, 2), leaving B and D invariant. C1
Answer: Rotation 90° clockwise about (–1, 0) [or 90° anticlockwise about (–1, 6), or 180° about (–1, 2), or 180° about (–1, 4)]
Question Paper (PDF) • Mark Scheme (PDF)
Question 2
[2 marks]
xyA–4–224–4–224O
Triangle A has vertices (–4, 1), (–3, 1), (–3.5, 4). It is rotated 180° about (–2, 0) followed by translation by vector (–3, 2). Find the coordinates of the invariant point on triangle A.
Triangle **A** is transformed by the combined transformation of a rotation of 180° about the point (–2, 0) followed by a translation with vector (-32)
One point on triangle **A** is invariant under the combined transformation.
Find the coordinates of this point.
Worked Solution & Answer
Rotation of 180° about (–2, 0) maps (x, y) to (–4 – x, –y) M1
Translation by (–3, 2) maps this to (–7 – x, 2 – y)
For invariance: x = –7 – x ⇒ x = –3.5, and y = 2 – y ⇒ y = 1. Thus (–3.5, 1). A1
Answer: –3.5, 1
Question Paper (PDF) • Mark Scheme (PDF)
Question 3
[3 marks]
xyA–6–5–4–3–2–1123456–6–5–4–3–2–1123456O
Triangle A is translated by the vector (6-4) to give triangle B.
Triangle B is rotated 90° clockwise about the point (1, 2) to give triangle C.
Describe fully the single transformation that maps triangle A onto triangle C.
Triangle A is translated by the vector (6-4) to give triangle B.
Triangle B is rotated 90° clockwise about the point (1, 2) to give triangle C.
Describe fully the single transformation that maps triangle A onto triangle C.
Worked Solution & Answer
A has vertices (–3, 2), (–2, 2), (–2, 4) M1
B has vertices (3, –2), (4, –2), (4, 0) M1
Rotating B 90° clockwise about (1, 2) gives C at (–3, 0), (–3, –1), (–1, –1)
Single transformation from A to C is a rotation 90° clockwise about centre (–4, 1). C1
xyABC–6–5–4–3–2–1123456–6–5–4–3–2–1123456O
Answer: Rotation, 90° clockwise, centre (-4, 1)
Question Paper (PDF) • Mark Scheme (PDF)
Question 4
[3 marks]
xyAB–8–7–6–5–4–3–2–112345678–8–7–6–5–4–3–2–112345678O
Shape A can be transformed to shape B by a reflection in the x-axis followed by a translation (cd)
Find the value of c and the value of d.
Shape **A** can be transformed to shape **B** by a reflection in the x-axis followed by a translation (cd)
Find the value of c and the value of d.
Worked Solution & Answer
Reflecting A in the x-axis gives vertices (3, –2), (5, –2), (3, –5) M2
Translating to B (–3, –3), (–1, –3), (–3, –6) gives c = –6, d = –1 A1
xyABA'–8–7–6–5–4–3–2–112345678–8–7–6–5–4–3–2–112345678O
Answer: c = –6, d = –1
Question Paper (PDF) • Mark Scheme (PDF)
Question 5
[3 marks]
xyP–6–5–4–3–2–1123456–6–5–4–3–2–1123456O
Triangle P is reflected in y = –x to give Q, and Q is reflected in x = –1 to give R. Describe fully the single transformation that maps triangle R to triangle P.
Triangle **P** is reflected in the line y = -x to give triangle **Q**.
Triangle **Q** is reflected in the line x = -1 to give triangle **R**.
Describe fully the single transformation that maps triangle **R** to triangle **P**.
Worked Solution & Answer
Vertices of P: (–1, 3), (3, 3), (3, 5) M1
Reflected in y = –x gives Q: (–3, 1), (–3, –3), (–5, –3) M1
Reflected in x = –1 gives R: (1, 1), (1, –3), (3, –3)
Mapping R to P is a rotation of 90° anticlockwise (or 270° clockwise) about (–1, 1). C1
Answer: Rotation of 90° anticlockwise about (–1, 1)
Question Paper (PDF) • Mark Scheme (PDF)
Question 6
[3 marks]
–8–6–4–22468–10–8–6–4–2246OyxP
(a) Enlarge triangle P by scale factor –1.5 with centre (–2, –1). Label your image Q. (2) (b) Triangle P is rotated 90° anticlockwise about O then translated so 1 vertex is invariant. Find one possible column vector. (1)
(a) Enlarge triangle **P** by scale factor -112 with centre of enlargement (-2, -1)
Label your image **Q**. [2 marks]
(b) Find one possible column vector for the translation. [1 mark]
Worked Solution & Answer
(a)
Vectors from (–2, –1) to (–8, 1), (–6, 1), (–8, 5) are (–6, 2), (–4, 2), (–6, 6) M1
multiply by –1.5 to get (9, –3), (6, –3), (9, –9) ⇒ (7, –4), (4, –4), (7, –10) A1
Answer (a): Triangle Q with vertices (7, –4), (4, –4), (7, –10)
(b)
90° anticlockwise sends (x, y) → (–y, x): (–8, 1) → (–1, –8), translation = (–7, 9)
(–6, 1) → (–1, –6), translation = (–5, 7)
(–8, 5) → (–5, –8), translation = (–3, 13) B1
Answer (b): –7, 9
Question Paper (PDF) • Mark Scheme (PDF)
Question 7 AQA Nov 2022 1H Q24
[4 marks]
Triangle ABC is drawn on a grid.
xyO–10–9–8–7–6–5–4–3–2–112345678910–10–9–8–7–6–5–4–3–2–112345678910ABC
ABC is transformed to A'B'C' by a reflection in the line x = 1
A'B'C' is transformed to A''B''C'' by a rotation 90° anticlockwise about (1, -4)
Which **one** point on ABC is invariant under the combined transformation?
You **must** show the result of each transformation on the grid.
Worked Solution & Answer
Reflecting A(–4, 3), B(–4, 1), C(–1, 1) in x = 1 gives A'(6, 3), B'(6, 1), C'(3, 1) M2
Rotating 90° anticlockwise about (1, –4) gives A''(–6, 1), B''(–4, 1), C''(–4, –2) M1
Point B (–4, 1) is invariant. A1
xyO–10–9–8–7–6–5–4–3–2–112345678910–10–9–8–7–6–5–4–3–2–112345678910ABCA'B'C'A''C''
Answer: Answer (-4, 1)
Question Paper (PDF) • Mark Scheme (PDF)
Question 8 OCR Jun 2022 5H Q10
[4 marks]
(b) Triangle **A** and triangle **B** are shown on the coordinate grid below. [4 marks]
–8–8–7–7–6–6–5–5–4–4–3–3–2–2–1–11122334455667788xy0AB
Triangle **A** is mapped onto triangle **B** using a combination of two transformations:
• a transformation T, followed by
• a reflection in the x-axis.
Describe fully transformation T.
Worked Solution & Answer
(b)
Reflecting triangle B in the x-axis gives the intermediate triangle with vertices (2, 2), (4, 2), (4, 6) M3
Mapping A((-2,0),(-2,4),(0,4)) onto ((4,6),(4,2),(2,2)) is a rotation of 180° about (1, 3). C1
Answer (b): Rotation of 180° about centre (1, 3) (or enlargement by scale factor -1 with centre (1, 3)).
Question Paper (PDF) • Mark Scheme (PDF)
Question 9
[4 marks]
The diagram shows a triangle **P** on a grid.
xyP–7–6–5–4–3–2–11234567–7–6–5–4–3–2–1123456O
(a) Describe fully the single transformation that maps triangle **P** onto triangle **R**. [3 marks]
(b) Write down the coordinates of point A. [1 mark]
Worked Solution & Answer
(a)
P vertices: (1,2), (1,5), (2,5) → Q: (–1,–2), (–1,–5), (–2,–5) → R: (4,–4), (4,–7), (3,–7) M2 C1
Answer (a): Rotation of 180° about (2.5, –1)
(b)
Midpoint of (1,2) and (4,–4) is (2.5, –1). B1
Answer (b): (2.5, –1)
Question Paper (PDF) • Mark Scheme (PDF)
Question 10 AQA Jun 2019 3H Q25
[5 marks]
ABCD is a square.
A is (–2, 1), B is (0, –1), C is (2, 1), D is (0, 3)
xyO–2–1123–1123ABCD
(a) A single transformation of ABCD is such that
B is mapped to D
D is mapped to B
A and C are invariant points.
Describe fully the transformation. [2 marks]
(b) A different single transformation of ABCD is such that
B is mapped to D
D is mapped to B
the only invariant point is (0, 1)
Describe fully the transformation. [3 marks]
Worked Solution & Answer
(a)
The invariant line passing through A(–2, 1) and C(2, 1) is the horizontal line y = 1 M1
Reflecting in y = 1 swaps B(0, –1) and D(0, 3). A1
Answer (a): Answer Reflection in the line y = 1
(b)
The centre of the square is (0, 1) M1
A rotation of 180° about (0, 1) maps B(0, –1) to D(0, 3) and D(0, 3) to B(0, –1) M1
The only fixed point is the centre of rotation (0, 1). A1
Answer (b): Answer Rotation of 180° about (0, 1)
Question Paper (PDF) • Mark Scheme (PDF)
Score: 0 / 10 (0 / 33 marks)
Combined transformations Worksheets Next: Capture recapture →

Grade 6 Quizzes

14 Topics
6A - Recurring decimals6B - Harder indices
6C - The product rule6D - Repeated change
6E - Triple brackets6F - Parallel and perpendicular
6G - Inequalities on graphs6H - Similar shapes (area and vol)
6J - Negative scale factors6K - Circle theorems
6L - Cumulative frequency6M - Box plots
6N - Combined transformations6P - Capture recapture