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Transformations

Transformations

Reflection and translation

Reflection- On Grid Paper 1 + 2 8 min read

On this page
  1. In this note
  2. Formula bank
  3. Describing a transformation
  4. Reflection
  5. Translation
  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: reflections and translations
  • 7.1Draw and describe a reflection in a vertical or horizontal line, and in \(y=x\) or \(y=-x\) .
  • 7.1Draw and describe a translation by a column vector.
  • 7.1Know how to describe a transformation FULLY for the marks.

Before you start: coordinates and the lines \(x=a\), \(y=b\) (Coordinate geometry). Next: Rotation.

Formula bank

Reflect in x = aLearn it\((x,y)\to(2a-x,\ y)\)the y-axis is x = 0.Use it when a vertical mirror.
Reflect in y = bLearn it\((x,y)\to(x,\ 2b-y)\)the x-axis is y = 0.Use it when a horizontal mirror.
Reflect in y = xLearn it\((x,y)\to(y,\ x)\)swap.Use it when a diagonal mirror.
Reflect in y = −xLearn it\((x,y)\to(-y,\ -x)\)swap and change both signs.Use it when the other diagonal.
TranslationLearn it\(\begin{pmatrix}a\\b\end{pmatrix}\): a right (left if negative), b up (down if negative)Use it when a slide.

Describing a transformation

A transformation moves a shape (the object) to a new position (the image). In a reflection, rotation or translation the image is the same size and shape (congruent); an enlargement changes the size.

TransformationTo describe it FULLY, give…
Reflectionthe word reflection + the equation of the mirror line
Rotationrotation + angle + direction + centre
Translationtranslation + the column vector
Enlargementenlargement + scale factor + centre

Name ONE transformation only. Two ("reflection and translation") scores zero.

Reflection

Triangle A reflected in the y-axis gives B-5-4-3-2-112345-11234xyAB
Each corner of B is the same distance from the mirror as A, on the other side.

The mirror line is like a fold. Every point and its image are the same distance from the mirror, and the line joining them crosses the mirror at right angles.

General methodHow to draw a reflection

  1. Draw the mirror line (x = a is vertical, y = b is horizontal).
  2. For each corner: count the squares to the mirror at right angles, then the same number on the other side.
  3. Join the new corners. Check: the image is 'flipped'.

General methodHow to describe a reflection

  1. Join a corner to its image and find the midpoint. Do this for two corners.
  2. The mirror line passes through the midpoints. Write its equation (x = …, y = …, y = x or y = −x).
  3. Answer: 'Reflection in the line …'.
Triangle E reflected in y = x gives F12345671234567xyy = xEF
Mirror y = x: the coordinates swap.

In \(y=x\) every point swaps its coordinates: \((6,1)\to(1,6)\). In \(y=-x\) they swap AND both change sign: \((3,-1)\to(1,-3)\) .

Example Reflect the triangle \((1,1), (4,1), (1,3)\) in the line \(y=-1\).

Horizontal mirror: x stays, y goes to \(2(-1)-y\).

  1. \((1,1)\to(1,-3)\)
  2. \((4,1)\to(4,-3)\)
  3. \((1,3)\to(1,-5)\)

Answer \((1,-3), (4,-3), (1,-5)\)

Try it yourself

1
Reflect the point \((2,5)\) in the line \(x=3\).
Show answer

Answer (4, 5)

2
Reflect \((3,-1)\) in the line \(y=x\).
Show answer

Answer (−1, 3)

3
A corner at \((-2,3)\) maps to \((5,3)\). What is the mirror line? Type x = …
Show answer

Answer x = 1.5

Translation

Triangle A translated by (4, 1) gives D12345678912345xyAD
Every corner moves 4 right and 1 up: vector (4, 1).

A translation slides the shape without turning it. The column vector \(\begin{pmatrix}a\\b\end{pmatrix}\) says how far: top number = across (+ right, − left), bottom number = up/down (+ up, − down).

General methodHow to answer any translation question

  1. Drawing: move every corner by the vector, then join.
  2. Describing: pick one corner and its image. Vector = (image x − object x, image y − object y).
  3. Answer: 'Translation by the vector …'.

Example Describe the transformation that maps \(P(-2,3)\) to \(Q(1,-1)\).

Image minus object, for each coordinate.

  1. across: \(1-(-2)=3\)
  2. up: \(-1-3=-4\)

Answer Translation by \(\begin{pmatrix}3\\-4\end{pmatrix}\)

Try it yourself

1
Translate \((4,-2)\) by \(\begin{pmatrix}-5\\3\end{pmatrix}\).
Show answer

Answer (−1, 1)

2
Vector that maps \((2,2)\) to \((6,-1)\)? Type it as top, bottom.
Show answer

Answer 4, −3

Ways they test it

Type 1 Draw a reflection

"Reflect shape A in the line x = −1." (2 marks)

Spot it a grid, a named line.

  1. Draw the line first.
  2. Count squares at right angles.

Type 2 Describe a reflection

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

Spot it the image is flipped, same size.

  1. Midpoints of corner-to-image joins.
  2. 'Reflection, line …'.

For example \((-2,3)\to(5,3)\)

  1. midpoint \(x=1.5\)

Answer Reflection in x = 1.5

Type 3 Draw a translation

"Translate shape A by \(\begin{pmatrix}-4\\2\end{pmatrix}\)." (2 marks)

Spot it a column vector.

  1. Move each corner.
  2. Check it has not turned.

Type 4 Describe a translation

"Describe fully…" (2 marks)

Spot it same way up, just moved.

  1. Image − object.
  2. 'Translation, vector …'.

Mistakes to avoid

  • ✗ x = 2 drawn as a horizontal line✓ x = 2 is VERTICALEvery point on it has x = 2.
  • ✗ Writing a translation as a coordinate (3, −4)✓ use a column vectorCoordinates are positions, vectors are moves.
  • ✗ Leaving out the mirror line✓ 'reflection in the line y = x'The description needs it for the second mark.
  • ✗ Naming two transformations✓ one single transformation

Summary

  • Reflection: same distance either side of the mirror; describe with the line's equation.
  • In \(y=x\) swap; in \(y=-x\) swap and change signs.
  • Translation: column vector = image − object.
  • Describe ONE transformation with all its details.

If you see …

  • shape flipped, same sizereflection.
  • same way up, same sizetranslation.

Practice

Practice: easy, medium, hard

Easy

1
Reflect \((3,2)\) in the y-axis.
Show answer

Answer (−3, 2)

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

Answer (3, −2)

3
Translate \((1,1)\) by \(\begin{pmatrix}3\\-2\end{pmatrix}\).
Show answer

Answer (4, −1)

Medium

1
Reflect \((-1,5)\) in \(y=2\).
Guide me

What is it asking? a point

What do we know? horizontal mirror, 3 above it

Plan 3 below: y = −1

Show working
  1. (−1, −1)

Answer (−1, −1)

2
Reflect \((3,-1)\) in \(y=-x\).
Guide me

What is it asking? a point

What do we know? swap and change signs

Plan \((1,-3)\)

Show working
  1. (1, −3)

Answer (1, −3)

3
Vector mapping \((-2,3)\) to \((1,-1)\)?
Guide me

What is it asking? a vector

What do we know? image − object

Plan 3 and −4

Show working
  1. \(\begin{pmatrix}3\\-4\end{pmatrix}\)

Answer 3, −4

Hard

1
A maps to B. Corner \((-2,3)\) goes to \((5,3)\) and \((-1,1)\) goes to \((4,1)\). Mirror line? Type x = …
Guide me

What is it asking? a line

What do we know? midpoints (1.5, 3) and (1.5, 1)

Plan vertical line through them

Show working
  1. x = 1.5

Answer x = 1.5

PartAnswer / workingMarksGuidance
ReflectionB1
x = 1.5B1
Total: 2 marks
2
Reflect the triangle \((1,1),(4,1),(1,3)\) in \(y=x\). Image of \((4,1)\)?
Guide me

What is it asking? a point

What do we know? y = x: swap

Plan \((1,4)\)

Show working
  1. (1, 4)

Answer (1, 4)

PartAnswer / workingMarksGuidance
(1, 4)B1
Total: 1 mark
3
Translate \((4,-2)\) by \(\begin{pmatrix}-5\\3\end{pmatrix}\).
Guide me

What is it asking? a point

What do we know? 4 − 5 across

Plan −2 + 3 up

Show working
  1. (−1, 1)

Answer (−1, 1)

PartAnswer / workingMarksGuidance
(−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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