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Transformations

Transformations

Combined transformations

Introduciton Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. One after the other
  4. The order matters
  5. Undoing a transformation
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 7.1: combining transformations Extended
  • 7.1Carry out one transformation and then another, in the right order.
  • 7.1Describe the single transformation that does the same job as two.
  • 7.1Find the inverse that undoes a transformation.

Before you start: Reflection and translation, Rotation, Enlargement.

Formula bank

Two reflections, axesLearn itin the y-axis then the x-axis = rotation 180° about OUse it when mirrors at right angles.
Two reflections, parallel linesLearn it= a translation, twice the gap between the linesUse it when parallel mirrors.
InverseLearn itreflection: the same; rotation: same angle, opposite way; translation: minus the vector; enlargement: 1/k, same centreUse it when 'undo', 'maps B back onto A'.

One after the other

A reflected in the y-axis gives B; B reflected in the x-axis gives C-5-4-3-2-112345-4-3-2-11234xyABC
Do the first transformation, then do the second one to the IMAGE.

'Reflect A in the y-axis to give B, then reflect B in the x-axis to give C.' Always apply the second transformation to the new shape, not to the original.

General methodHow to answer a combined-transformation question

  1. Do the first transformation. Label the image.
  2. Do the second transformation to that image.
  3. Asked for a single transformation? Compare the FIRST and LAST shapes only: same way up (translation)? flipped (reflection)? turned (rotation)? resized (enlargement)?
  4. Describe it fully, and check two corners with the rule.

Example Reflect A \((1,1),(4,1),(1,3)\) in the y-axis, then in the x-axis. Describe the single transformation from A to the final image C.

Compare A and C directly.

  1. C is \((-1,-1),(-4,-1),(-1,-3)\)
  2. every point \((x,y)\to(-x,-y)\)

Answer Rotation, 180°, centre (0, 0)

Example Reflect in x = 1, then in x = 4. What single transformation is this?

Follow the origin: (0, 0) → (2, 0) → (6, 0). Same way up, so a translation.

  1. move \(=2\times\) the gap \(=2\times3\)

Answer Translation by \(\begin{pmatrix}6\\0\end{pmatrix}\)

Try it yourself

1
Reflect \((3,2)\) in \(x=1\), then translate by \(\begin{pmatrix}2\\3\end{pmatrix}\).
Show answer

Answer (1, 5)

2
Reflect in \(y=-1\), then in \(y=2\). Where does the origin end up?
Show answer

Answer (0, 6)

The order matters

Swapping the order can give a different answer. Point \((2,1)\): reflect in \(y=x\) then rotate 90° clockwise about O gives \((2,-1)\), a reflection in the x-axis overall. Rotate first, then reflect, and you get \((-2,1)\): a reflection in the y-axis.

Undoing a transformation

TransformationInverse (undoes it)
Reflection in a linethe same reflection
Rotation 90° clockwise, centre Crotation 90° anticlockwise, centre C
Translation by \(\begin{pmatrix}a\\b\end{pmatrix}\)translation by \(\begin{pmatrix}-a\\-b\end{pmatrix}\)
Enlargement, scale factor k, centre Cenlargement, scale factor \(\tfrac1k\), centre C

Try it yourself

1
An enlargement has scale factor 3, centre (1, 2). Scale factor of its inverse? (Type a fraction.)
Show answer

Answer 1/3

Ways they test it

Type 1 Two steps, draw both

"Reflect A in y = 1 to give B. Rotate B 90° clockwise about O to give C." (4 marks)

Spot it 'then', two shapes to draw.

  1. First image.
  2. Second transformation on THAT image.

Type 2 Single equivalent

"Describe fully the single transformation that maps A onto C." (3 marks)

Spot it after two steps.

  1. Compare A and C only.
  2. Name and detail it.

For example Reflect in y-axis, then x-axis

  1. \((x,y)\to(-x,-y)\)

Answer Rotation 180° about O

Type 3 Inverse

"Describe the transformation that maps B back onto A." (2 marks)

Spot it 'back onto', 'undo'.

  1. Use the inverse table.

Mistakes to avoid

  • ✗ Doing the second step to the ORIGINAL shape✓ do it to the first image
  • ✗ Describing both steps when asked for a single one✓ one transformation, A to C
  • ✗ Inverse of a rotation in the same direction✓ turn back the other way

Summary

  • Second transformation acts on the first image.
  • Single equivalent: compare the first and last shapes.
  • Order can change the answer.
  • Inverses: same reflection, opposite rotation, minus vector, 1/k.

If you see …

  • 'then'two steps in order.
  • 'single transformation'first vs last only.

Practice

Practice: easy, medium, hard

Easy

1
Inverse of a translation by the vector (4, −1): top number?
Show answer

Answer −4

2
Inverse of a reflection in y = x: which line? Type y = …
Show answer

Answer y = x

3
Reflect \((3,2)\) in the y-axis, then in the x-axis.
Show answer

Answer (−3, −2)

Medium

1
Reflect \((3,2)\) in \(x=1\), then translate by \(\begin{pmatrix}2\\3\end{pmatrix}\).
Guide me

What is it asking? a point

What do we know? \(x=1\): (−1, 2)

Plan then + (2, 3)

Show working
  1. (1, 5)

Answer (1, 5)

2
Reflect \((2,1)\) in \(y=x\), then rotate 90° clockwise about O.
Guide me

What is it asking? a point

What do we know? swap: (1, 2)

Plan clockwise \((x,y)\to(y,-x)\)

Show working
  1. (2, −1)

Answer (2, −1)

3
Reflect in x = 1, then x = 4. Translation vector top number?
Guide me

What is it asking? a vector

What do we know? gap 3

Plan twice the gap

Show working
  1. 6

Answer 6

Hard

1
Rotate \((2,1)\) 90° clockwise about O, then reflect in \(y=x\).
Guide me

What is it asking? a point

What do we know? clockwise: (1, −2)

Plan swap: (−2, 1)

Show working
  1. (−2, 1)

Answer (−2, 1)

PartAnswer / workingMarksGuidance
(1, −2)M1
(−2, 1)A1
Total: 2 marks
2
Reflect in \(y=-1\), then in \(y=2\). The single transformation is a translation. Its vector's bottom number?
Guide me

What is it asking? a vector

What do we know? gap 3

Plan twice the gap, upwards

Show working
  1. \(\begin{pmatrix}0\\6\end{pmatrix}\)

Answer 6

PartAnswer / workingMarksGuidance
translationB1
\(\begin{pmatrix}0\\6\end{pmatrix}\)B1
Total: 2 marks
3
A rotation is 90° clockwise about (1, 1). Its inverse is 90° ___ about (1, 1). Type the direction.
Guide me

What is it asking? a direction

What do we know? undo the turn

Plan turn back

Show working
  1. anticlockwise

Answer anticlockwise

PartAnswer / workingMarksGuidance
anticlockwise, (1, 1)B1
Total: 1 mark

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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