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10.6 Combining Transformations

GCSE Maths Foundation · Edexcel · Chapter 10: Transformations

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: describing any transformation fully, doing two transformations in a row, and finding a single equivalent transformation. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Know what information is needed to describe each type of transformation
  • Identify the transformation that maps one shape onto another
  • Carry out two transformations, one after the other
  • Find a single transformation equivalent to two transformations

Key words

  • Combined transformation: Two (or more) transformations done one after the other.
  • Single transformation: One transformation that has the same overall effect as a combination.
  • Equivalent: Having the same effect.
  • Congruent: Same shape and size. Translations, reflections and rotations give congruent images.

Describing a transformation fully

In an exam, “describe fully” means give the name of the transformation and all of its details.

  • Translation; Details you must give column vector; Size of image same (congruent)
  • Reflection; Details you must give equation of the mirror line; Size of image same (congruent)
  • Rotation; Details you must give angle, direction and centre; Size of image same (congruent)
  • Enlargement; Details you must give scale factor and centre; Size of image different (similar)

Tip: Ask yourself: Has the size changed? (enlargement) Is it flipped? (reflection) Is it turned? (rotation) Has it only slid? (translation)

One transformation after another

Do the first transformation, label the image, then apply the second transformation to that image. Then compare the final shape with the original.

Worked example: two reflections in the axes

Vertex (4, 1) of A goes to (−4, 1) on B, then to (−4, −1) on C. A and C are the same way round, but upside down.
Every point (x, y) ends up at (−x, −y): this is a half turn about the origin.

Answer: single transformation: rotation 180° about (0, 0)

Combining translations and reflections

Worked example: two translations

A shape is translated by and then by .
Add the vectors: = .

Answer: translation by

Worked example: reflections in parallel lines

A shape is reflected in x = 1 and the image is reflected in x = 4.
Follow (0, 1): it goes to (2, 1), then to (6, 1). Every point moves 6 to the right (twice the gap of 3).

Answer: translation by

The order matters

Doing two transformations in the opposite order can give a different final image. Always do them in the order given.

Worked example: changing the order

Start with the point (1, 2). Rotate 90° clockwise about (0, 0), then translate by : (1, 2) → (2, −1) → (4, −1).
Translate by first, then rotate: (1, 2) → (3, 2) → (2, −3).

Answer: different images, so the order matters

Common mistakes

  • Describing two transformations when the question asks for a single transformation.
  • Leaving out details: the centre of a rotation or enlargement, or the vector of a translation.
  • Applying the second transformation to the original shape instead of to the first image.
  • Mixing up a reflection and a rotation of 180°. A reflected shape is flipped; a rotated shape is not.

Revision questions

Try these, then check your answers below each one.

Question 1. Which transformation is described fully by a column vector only? (Choose from: Translation, Reflection, Rotation, Enlargement.)

Answer: Translation

Working: A translation is described by a column vector. The others need a line, a centre, an angle or a scale factor.

Question 2. A shape is reflected in the x-axis, then the image is reflected in the y-axis. Which single transformation has the same effect? (Choose from: Rotation 180° about (0, 0), Reflection in y = x, Rotation 90° clockwise about (0, 0), Reflection in the x-axis.)

Answer: Rotation 180° about (0, 0)

Working: (3, 1) → (3, −1) → (−3, −1). In general (x, y) → (-x, -y), a half turn about the origin.

Question 3. Doing two transformations in the opposite order always gives the same final image.

Answer: False

Working: Rotate then translate: (1, 2) → (2, −1) → (4, −1). Translate then rotate: (1, 2) → (3, 2) → (2, −3). Different, so false.

Related lessons

Continue in the full course

This lesson is section 10.6 (EGCMF1006) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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