Translations of Graphs (OCR GCSE Maths)

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Daniel I

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Daniel I

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Translations of Graphs

What are transformations of graphs?

  • A transformation is simply a change of some sort
    • Reflections (using either the x-axis or y-axis as a mirror line)
    • Translations (moving the whole graph in the x and/or y direction)
  • You should be able to recognise these two different sorts of transformation and apply them to a given graph.

How do we translate the graph of a function?

Translations in the x-axis

  • Given an original equation in x, the graph of that equation will be translated units in the x direction by adding a inside a bracket next to x
    • For example, in relation to y = x2;
      • y = (x 2)2 is a translation 2 units to the right/ a translation by +2 in the x-axis
      • y = (x + 3)2 is a translation 3 units to the left/ a translation by 3 in the x-axis
    • Another example, in relation to y = sin(x) where x is in degrees;
      • y = sin(x + 90) is a translation 90 degrees to the left/ a translation by 90 degrees in the x-axis
      • y = sin(x 180) is a translation 180 degrees to the right/ a translation by +180 degrees in the x-axis
    • Note that for changes in the x direction, the translation is in the opposite direction to the sign of a (as highlighted)!

Translations in the y-axis

  • Given an original equation in x, the graph of that equation will be translated +b units in the y direction by adding b outside the bracket
    • For example, in relation to y = x2;
      • y = x2 + 2 is a translation 2 units up/ a translation by +2 in the y-axis
      • y = x2 − 3 is a translation 3 units down/ a translation by −3 in the y-axis
    • Another example, in relation to y = sin(x) where x is in degrees;
      • y = sin(x) − 2 is a translation 2 units down/ a translation by −2 units in the y-axis
      • y = sin(x) + 1 is a translation 1 unit up/ a translation by +1 unit in the y-axis
    • Note that for translations in the y direction, the direction is the same as the sign of b

How do we describe translations of graphs?

  • Some questions give a transformed a transformed graph of an equation with an original graph of an equation and ask you to describe the transformation
  • As with translations of shapes, to describe a translation fully, you must include;
    1. the transformation: "translation"
    2. the direction in the x-axis and in the y-axis
      • this can be given as a worded description, e.g. "3 left and 2 up" / "−3 in the x-axis and +2 in the y-axis"
      • or it can be given as a vector open parentheses table row x row y end table close parentheses, e.g. "by open parentheses table row cell negative 3 end cell row 2 end table close parentheses"

Examiner Tip

REMEMBER that:

  • number next to x (inside the bracket): the graph translates in the x direction but with the opposite sign

Worked example

The graph of y equals sin open parentheses x degree close parentheses is shown on the graph below.
On the same graph sketch y equals sin open parentheses x degree close parentheses plus 3.

ocr-7-graphs-transformations-translations-1

This is a translation by +3 in the y direction (i.e. 3 up)
So we copy the given graph in its new position. Translate key points first- x-intercepts, maximums and minimums as shown below

ocr-7-graphs-transformations-translations2

Now join your new points with a curved line. The new curve should go through the key points shown in the answer below

ocr-7-graphs-transformations-translations3

Worked example

Describe the transformation that maps the graph of y equals x cubed to the graph of y equals open parentheses x minus 4 close parentheses cubed minus 6.


The number inside the bracket (next to x) is −4 so this is a translation by +4 in the x-axis (note the change in sign again)
The number outside the bracket (not next to x!) is −6 so this is a translation by −6 in the y-axis

Translation by begin bold style stretchy left parenthesis table row 4 row cell negative 6 end cell end table stretchy right parenthesis end style
or Translation 4 right and 6 down
or Translation by +4 in the x-axis and −6 in the y-axis

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Daniel I

Author: Daniel I

Expertise: Maths

Daniel has taught maths for over 10 years in a variety of settings, covering GCSE, IGCSE, A-level and IB. The more he taught maths, the more he appreciated its beauty. He loves breaking tricky topics down into a way they can be easily understood by students, and creating resources that help to do this.