Translations of Graphs (OCR GCSE Maths)

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 −a 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 , e.g. "by "