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Maths · Transformations and constructions

Translations and combinations of different transformations

Translate shapes using column vectors, describe translations, and combine transformations to find a single equivalent transformation.

  • 6 key terms
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Student handouts

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Warm-up

Answer each one, then check.

  1. 1

    What is the reflection of \((3, 1)\) in the y-axis?

    Show answerHide answer

    \((-3, 1)\)

  2. 2

    What is the image of \((2, 3)\) under a \(180^\circ\) rotation about the origin?

    Show answerHide answer

    \((-2, -3)\)

  3. 3

    What does congruent mean?

    Show answerHide answer

    Same shape and size

  4. 4

    Work out \((5, 2) + (3, -4)\) as coordinates.

    Show answerHide answer

    \((8, -2)\)

  5. 5

    What is a scale factor?

    Show answerHide answer

    The number lengths are multiplied by

Learning Objectives

  1. 1Translate a shape using a column vector.
  2. 2Describe a translation with a vector.
  3. 3Carry out and describe combinations of transformations.
  4. 4Find 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).

Translating a Shape

Triangle A has vertices \((1, 1)\), \((1, 3)\) and \((3, 1)\). Translate it by the vector \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\).

Show the solutionHide the solution
  1. 1 Add 4 to each x-coordinate (move right) \(1 + 4 = 5\)
  2. 2 Subtract 3 from each y-coordinate (move down) \(1 - 3 = -2\)
  3. 3 Apply it to every vertex \((1, 1) \to (5, -2)\), \((1, 3) \to (5, 0)\), \((3, 1) \to (7, -2)\)

AnswerThe 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.

Show the solutionHide the solution
  1. 1 Change in x \(-1 - 2 = -3\): 3 to the left
  2. 2 Change in y \(2 - 5 = -3\): 3 down
  3. 3 Write as a vector \(\begin{pmatrix} -3 \\ -3 \end{pmatrix}\)

AnswerA translation by the vector \(\begin{pmatrix} -3 \\ -3 \end{pmatrix}\).

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.

Show the solutionHide the solution
  1. 1 B has vertices \((-1, 1)\), \((-1, 3)\), \((-3, 1)\)
  2. 2 C is B reflected in the x-axis \((-1, -1)\), \((-1, -3)\), \((-3, -1)\)
  3. 3 Compare A and C Every coordinate has changed sign
  4. 4 Identify A half turn about the origin

AnswerA rotation of \(180^\circ\) about \((0, 0)\).

Slide and Flip

Shape A has vertices \((1, 1)\), \((3, 1)\), \((1, 2)\). Translate A by \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\) 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 \(x = 0\)).

Can I...?

  1. 1Translate using a column vector.
  2. 2Describe a translation with a vector.
  3. 3Add vectors for two translations.
  4. 4Do a combination in the right order.
  5. 5Label images clearly.
  6. 6Find a single equivalent transformation.
  7. 7Identify invariant points.
  8. 8Describe every transformation fully.

Summary & Exam Focus

  • 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) (3 marks)

Find the coordinates of A and C and compare them. Every sign changing points to a half turn about the origin.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

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.

Questions and answers

12 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Non-calculator 2 marks Easier

Triangle A has vertices \((1, 1)\), \((1, 3)\) and \((3, 1)\). It is translated by the vector \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\). Write down the coordinates of the vertices of the image.

Mark scheme — 2 marks available

  • Two vertices correct — B1
  • All three correct — B1

Model answer

\((5, -2)\), \((5, 0)\) and \((7, -2)\).

2. Exam question Non-calculator 2 marks Easier

The point \(P(2, 5)\) is mapped to \(P'(-1, 2)\) by a translation. Write the translation as a column vector.

Mark scheme — 2 marks available

  • One number correct — M1
  • Both correct — A1

Model answer

\(\begin{pmatrix} -3 \\ -3 \end{pmatrix}\).

3. Exam question Non-calculator 4 marks Easier

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.

Three triangles A, B and C on a grid, in three of the four quadrants.

Mark scheme — 4 marks available

  • (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 \(y = 0\)).

4. Exam question Non-calculator 3 marks Easier

Using the diagram in the last question, describe fully the single transformation that maps triangle A onto triangle C.

Mark scheme — 3 marks available

  • Rotation — B1
  • \(180^\circ\) — B1
  • Centre \((0, 0)\) — B1

Model answer

A rotation of \(180^\circ\) about the origin \((0, 0)\).

5. Exam question Non-calculator 2 marks Easier

A shape is translated by \(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\) and then by \(\begin{pmatrix} -5 \\ 4 \end{pmatrix}\). Write down the single translation that does the same job.

Mark scheme — 2 marks available

  • Adding the vectors — M1
  • \(\begin{pmatrix} -2 \\ 6 \end{pmatrix}\) — A1

Model answer

Add the vectors: \(\begin{pmatrix} 3 + (-5) \\ 2 + 4 \end{pmatrix} = \begin{pmatrix} -2 \\ 6 \end{pmatrix}\).

6. Exam question Non-calculator 3 marks Easier

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.

Mark scheme — 3 marks available

  • Translation — B1
  • Distance 6 (twice the gap between the lines) — M1
  • \(\begin{pmatrix} 6 \\ 0 \end{pmatrix}\) — A1

Model answer

A translation by \(\begin{pmatrix} 6 \\ 0 \end{pmatrix}\): twice the distance between the mirror lines (\(2 \times 3 = 6\)) to the right.

7. Multiple choice 1 mark Easier

A translation by \(\begin{pmatrix} 3 \\ -2 \end{pmatrix}\) moves a point...

  1. A 3 up and 2 left
  2. B 3 right and 2 down Correct
  3. C 3 left and 2 up
  4. D 2 right and 3 down

Why: 3 to the right and 2 down.

8. Multiple choice 1 mark Core

What is the image of \((1, 4)\) under the translation \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\)?

  1. A \((2, 3)\)
  2. B \((3, 4)\)
  3. C \((3, 5)\) Correct
  4. D \((2, 5)\)

Why: \((1 + 2, 4 + 1) = (3, 5)\).

9. Multiple choice 1 mark Core

Which transformation changes the size of a shape?

  1. A Translation
  2. B Reflection
  3. C Rotation
  4. D Enlargement Correct

Why: Only an enlargement changes size.

10. Multiple choice 1 mark Core

Two translations \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\) and \(\begin{pmatrix} 3 \\ -4 \end{pmatrix}\) equal one translation of...

  1. A \(\begin{pmatrix} 4 \\ -2 \end{pmatrix}\) Correct
  2. B \(\begin{pmatrix} 2 \\ 6 \end{pmatrix}\)
  3. C \(\begin{pmatrix} -2 \\ 6 \end{pmatrix}\)
  4. D \(\begin{pmatrix} 3 \\ -8 \end{pmatrix}\)

Why: Add the vectors: \(\begin{pmatrix} 4 \\ -2 \end{pmatrix}\).

11. Multiple choice 1 mark Core

An invariant point under a transformation is a point that...

  1. A Moves the furthest
  2. B Does not move Correct
  3. C Doubles in distance
  4. D Is the centre of the grid

Why: It stays in the same place.

12. Multiple choice 1 mark Stretch

Reflecting in the y-axis and then the x-axis is equivalent to...

  1. A A translation
  2. B A reflection in \(y = x\)
  3. C A rotation of \(180^\circ\) about the origin Correct
  4. D An enlargement

Why: A half turn about the origin.