Translations (OCR GCSE Maths)

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Naomi C

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Naomi C

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Translations

What are transformations in maths?

  • There are 4 transformations in GCSE Maths – rotation, reflection, translation and enlargement
  • All 4 change a shape in some way, useful in things like computer graphics.
  • There is some language and notation often used in this topic – the original shape is called the object and the transformed shape is called the image
  • Vertices are labelled to show corresponding points
    • Vertices on the object are labelled A, B, C, etc.
    • Vertices on the image are labelled A’, B’, C’ etc.
    • If there is a second transformation then they will become A”, B”, C” etc.

What is a translation?

  • A translation is the movement of a shape
  • The size, shape and orientation (which way up it is) of the shape stays the same

 

How do I translate a shape?

  • The movement of a translation is described by a vector
  • You need to know how to write a translation using a vector (rather than words)
  • Vectors are written as column vectors in the form open parentheses table row x row y end table close parentheses where:
    • x is the distance moved horizontally
      • Negative means move to the left
      • Positive means move to the right
    • y is the distance moved vertically
      • Negative means move down
      • Positive means move up
  • STEP 1:

    Select a vertex on the original object, move it according to the information given in the column vector

  • STEP 2:

    Repeat STEP 1 for each of the vertices on the original object

  • STEP 3:
    Connect the translated vertices and label the translated image

  • In some cases, where the vectors are small enough, the image can overlap the object
  • The vector is how the shape moves not the size of the gap between the object and the image, watch out for this common error!

 

How do I describe a translation?

  • It is important to fully describe a transformation in order to get full marks
  • For a translation, you must:
    • State that the transformation is a translation
    • Give the column vector that describes the movement

  

Which points are invariant with a translation?

  • Invariant points are points that do not change position when a transformation has been performed
    • Invariant points don't move!
  • With a translation, there are no invariant points, as all points are translated with the object

Examiner Tip

  • Translate one vertex of the shape at a time, put your pencil on the starting position and move across and/up/down the stated number of places and mark the new position of that vertex
  • Be careful not to muddle up which points you are translating, it can be surprisingly easy to count the wrong distance, especially if the original object and the translated image overlap each other!

Worked example

(a)table row blank row blank end table
On the grid below translate shape P using the vector open parentheses table row cell negative 4 end cell row 5 end table close parentheses.
Label your translate shape P'.
 

Translations-Q1, IGCSE & GCSE Maths revision notes

The vector means "4 to the left" and "5 up".
You don't have to draw in any arrows but it is a good idea to mark your paper after counting across and up a couple of times to check that you are in the correct place.

Translations-Q1-working, IGCSE & GCSE Maths revision notes

Translating one vertex and then following around the shape one vertex at a time makes it easier to get the shape in exactly the right position!

Q1-Translations-Solution, IGCSE & GCSE Maths revision notes

 

(b)
Describe fully the single transformation that creates shape B from shape A.

Translations-Q2, IGCSE & GCSE Maths revision notes

This is a case where the image overlaps the object.
You should still see that the shape is the same size and the same way up so it is a translation.

Start at a vertex on the original object that is well away from any overlap area to avoid confusion and count the number of position left/right and up/down that you need to move to reach the corresponding vertex on the translated image.
Take care when counting around the axes!

Translations-Q2-working, IGCSE & GCSE Maths revision notes

Shape A has been translated using the vector stretchy left parenthesis table row 2 row cell negative 3 end cell end table stretchy right parenthesis

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Naomi C

Author: Naomi C

Expertise: Maths

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.