EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Translations and combinations of different transformations
Transformations and constructions · Lesson 4 of 8
Teacher copy - includes the notes for whoever is teaching from it.
Warm-up
Answer each one, then check.
1. What is the reflection of (3, 1) in the y-axis?
(−3, 1)
2. What is the image of (2, 3) under a 180° rotation about the origin?
(−2, −3)
3. What does congruent mean?
Same shape and size
4. Work out (5, 2) + (3, −4) as coordinates.
(8, −2)
5. What is a scale factor?
The number lengths are multiplied by
Learning Objectives
1. Translate a shape using a column vector.
2. Describe a translation with a vector.
3. Carry out and describe combinations of transformations.
4. Find the single transformation equivalent to a combination.
The Key Idea
A translation slides every point of a shape the same distance in the same direction.
It is described by a column vector: the top number is the move right (negative means left), the bottom number is the move up (negative means down).
A Translation
|
Every point moves the same way. |
A triangle translated 4 units right and 3 units down with arrows from each vertex to its image.
Translating a Shape
|
Triangle A has vertices (1, 1), (1, 3) and (3, 1). Translate it by the vector beginpmatrix 4 \ −3 endpmatrix. |
1. Add 4 to each x-coordinate (move right)
1 + 4 = 5
2. Subtract 3 from each y-coordinate (move down)
1 − 3 = −2
3. Apply it to every vertex
(1, 1) → (5, −2), (1, 3) → (5, 0), (3, 1) → (7, −2)
Answer: The image has vertices (5, −2), (5, 0) and (7, −2).
Describing a Translation
|
Point P is at (2, 5) and its image is at (−1, 2). Describe the translation. |
1. Change in x
−1 − 2 = −3: 3 to the left
2. Change in y
2 − 5 = −3: 3 down
3. Write as a vector
beginpmatrix −3 \ −3 endpmatrix
Answer: A translation by the vector beginpmatrix −3 \ −3 endpmatrix.
PART TWO
Combining Transformations
Do them one after the other.
A Combination of Two Transformations
Apply the first, then the second to the image.
▸ Do them in order. The order can change the answer, so work through the steps carefully.
▸ Label each image. Name the shapes A, B and C to keep track.
▸ Find the single equivalent. Look for the one transformation that takes the original straight to the final image.
Useful Combinations
|
Two reflections in parallel lines Equivalent to a translation. |
Two reflections in lines that cross Equivalent to a rotation about the crossing point. |
|
Two rotations about the same centre Equivalent to one rotation about that centre. |
Two translations Equivalent to one translation: add the vectors. |
Describing a Combination
|
Triangle A has vertices (1, 1), (1, 3), (3, 1). Triangle B is A reflected in the y-axis. Triangle C is B reflected in the x-axis. Describe the single transformation that maps A onto C. |
1. B has vertices
(−1, 1), (−1, 3), (−3, 1)
2. C is B reflected in the x-axis
(−1, −1), (−1, −3), (−3, −1)
3. Compare A and C
Every coordinate has changed sign
4. Identify
A half turn about the origin
Answer: A rotation of 180° about (0, 0).
Key Terms
|
Translation A slide of every point by the same vector. |
Column vector A pair of numbers, one above the other, giving a move right and up. |
|
Combination Two or more transformations done one after the other. |
Single transformation One transformation that does the same job as a combination. |
|
Invariant point A point that does not move under a transformation. |
Image The shape after a transformation. |
Your Task: Slide and Flip
12 minutes
|
Shape A has vertices (1, 1), (3, 1), (1, 2). Translate A by beginpmatrix 2 \ 3 endpmatrix to get B, then reflect B in the y-axis to get C. Write the vertices of B and C, and describe fully the single transformation that maps B onto C. 1. Do the translation first. 2. Reflect the image. 3. Compare with the original. |
A good answer shows: B has vertices (3, 4), (5, 4), (3, 5). C has vertices (−3, 4), (−5, 4), (−3, 5). B maps onto C by a reflection in the y-axis (the line x = 0).
Note: Ask what the combination would be if the order were swapped.
Can I...?
☐ Translate using a column vector.
☐ Describe a translation with a vector.
☐ Add vectors for two translations.
☐ Do a combination in the right order.
☐ Label images clearly.
☐ Find a single equivalent transformation.
☐ Identify invariant points.
☐ Describe every transformation fully.
Summary
✓ Translation vector: right/left on top, up/down underneath.
✓ Two translations add.
✓ Work through combinations one step at a time.
✓ Always describe fully: name plus details.
|
EXAM FOCUS Triangle B is the reflection of triangle A in the y-axis. Triangle C is the reflection of triangle B in the x-axis. Describe fully the single transformation that maps triangle A onto triangle C. (3 marks) Find the coordinates of A and C and compare them. Every sign changing points to a half turn about the origin. |
Exam Practice: Translations and Combinations
Answer all questions. Show your working. · 30 minutes
▸ Question 1 · 2 marks · Non-calculator. Triangle A has vertices (1, 1), (1, 3) and (3, 1). It is translated by the vector beginpmatrix 4 \ −3 endpmatrix. Write down the…
▸ Question 2 · 2 marks · Non-calculator. The point P(2, 5) is mapped to P'(−1, 2) by a translation. Write the translation as a column vector.
▸ Question 3 · 4 marks · Non-calculator. The diagram shows triangles A, B and C. (a) Describe fully the single transformation that maps A onto B. (b) Describe fully the single…
▸ Question 4 · 3 marks · Non-calculator. Using the diagram in the last question, describe fully the single transformation that maps triangle A onto triangle C.
▸ Question 5 · 2 marks · Non-calculator. A shape is translated by beginpmatrix 3 \ 2 endpmatrix and then by beginpmatrix −5 \ 4 endpmatrix. Write down the single translation that…
▸ Question 6 · 3 marks · Non-calculator. The shape P is reflected in the line x = 1 and then the image is reflected in the line x = 4. Describe fully the single transformation that…
Question 1 · 2 marks · Non-calculator
|
“Triangle A has vertices (1, 1), (1, 3) and (3, 1). It is translated by the vector beginpmatrix 4 \ −3 endpmatrix. Write down the coordinates of the vertices of the image.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 1 · mark scheme
2 marks available. Award a mark for each point made.
▸ Two vertices correct. B1
▸ All three correct. B1
▸ Model answer. (5, −2), (5, 0) and (7, −2).
Question 2 · 2 marks · Non-calculator
|
“The point P(2, 5) is mapped to P'(−1, 2) by a translation. Write the translation as a column vector.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 2 · mark scheme
2 marks available. Award a mark for each point made.
▸ One number correct. M1
▸ Both correct. A1
▸ Model answer. beginpmatrix −3 \ −3 endpmatrix.
Question 3 · 4 marks · Non-calculator
|
The diagram shows triangles A, B and C. (a) Describe fully the single transformation that maps A onto B. (b) Describe fully the single transformation that maps B onto C. (4 marks) |
|
Question 3 · mark scheme
4 marks available. Award a mark for each point made.
▸ (a) Reflection. B1
▸ (a) The y-axis, or x = 0. B1
▸ (b) Reflection. B1
▸ (b) The x-axis, or y = 0. B1
▸ Model answer. (a) A reflection in the y-axis (the line x = 0). (b) A reflection in the x-axis (the line y = 0).
Question 4 · 3 marks · Non-calculator
|
“Using the diagram in the last question, describe fully the single transformation that maps triangle A onto triangle C.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 4 · mark scheme
3 marks available. Award a mark for each point made.
▸ Rotation. B1
▸ 180°. B1
▸ Centre (0, 0). B1
▸ Model answer. A rotation of 180° about the origin (0, 0).
Question 5 · 2 marks · Non-calculator
|
“A shape is translated by beginpmatrix 3 \ 2 endpmatrix and then by beginpmatrix −5 \ 4 endpmatrix. Write down the single translation that does the same job.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 5 · mark scheme
2 marks available. Award a mark for each point made.
▸ Adding the vectors. M1
▸ beginpmatrix −2 \ 6 endpmatrix. A1
▸ Model answer. Add the vectors: beginpmatrix 3 + (−5) \ 2 + 4 endpmatrix = beginpmatrix −2 \ 6 endpmatrix.
Question 6 · 3 marks · Non-calculator
|
“The shape P is reflected in the line x = 1 and then the image is reflected in the line x = 4. Describe fully the single transformation that is equivalent to these two reflections.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 6 · mark scheme
3 marks available. Award a mark for each point made.
▸ Translation. B1
▸ Distance 6 (twice the gap between the lines). M1
▸ beginpmatrix 6 \ 0 endpmatrix. A1
▸ Model answer. A translation by beginpmatrix 6 \ 0 endpmatrix: twice the distance between the mirror lines (2 × 3 = 6) to the right.