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GCSEMathsAQARevision Notes2. AlgebraTransformations of GraphsTranslations of Graphs

Translations of Graphs (AQA 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?

Given a transformation in function notation, then

  • y = f(x + a), represents a translation −a units in the x axis
    • a is inside the bracket 
    • Note the change of sign highlighted!
  • y = f(x) + b, represents a translation + b units in the y direction
    • a is outside the bracket
  • For example, y = f(x + 3) + 2 means
    • a translation of y = f(x) by −3 in x the direction and +2 in the y direction
    • Note the change of sign in the x direction as pointed out above!

How do we describe translations of graphs?

  • Some questions give a transformed function in the form y = f(x + a) or y = f(x) + a and ask you to describe the transformation
  • 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"
  • Remember that
    • if the a is inside the bracket, its a translation by −a in the x-axis
    • if the a is outside the bracket, its a translation by a in the y-axis

Examiner Tip

REMEMBER that;

  • y = f(x + a);  "a" next to x, translates in x-axis by −a
  • y = f(x) + a;  "a" not next to x, translates in y-axis

Worked example

The graph of y equals f open parentheses x close parentheses is shown on the graph below.
On the same graph sketch y equals f open parentheses x plus 3 close parentheses plus 2.

Translation-of-Graph-(before), IGCSE & GCSE Maths revision notes

This is a translation by −3 in the x direction (i.e. 3 to the left, note the change of sign again) and +2 in the y direction (i.e. 2 up)

So we copy the given graph in its new position. Translate key points- endpoints and vertices like the vertex shown below- first, then join the points with straight lines

Translation-of-Graph-(after), IGCSE & GCSE Maths revision notes

Worked example

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