Translations of Graphs (OCR GCSE Maths)

Translations of Graphs

What are translations of graphs?

• The equation of a graph can be changed in certain ways

  • This has an effect on the graph

    • How a graph changes is called a graph transformation

• A translation is a type of graph transformation that shifts (moves) a graph (up or down, left or right) in the xy plane

  • The shape, size, and orientation of the graph remain unchanged

• A particular translation is specified by a translation vector

How do I translate graphs?

Vertical translations: y=f(x) + a

• Given an original equation in x, the graph of that equation will be translated +a units in the y direction by adding a outside the bracket

  • The graph moves up for positive values of  

  • The graph moves down for negative values of 

  • The x-coordinates stay the same

• 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 a

Horizontal translations: y=f(x + a)

• 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

  • The graph moves left for positive values of 

    • This is often the opposite direction to which people expect

  • The graph moves right for negative values of 

  • The y-coordinates stay the same

• 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)!

How does a translation affect the equation of the graph?

• For a horizontal translation of a units to the right

  • is subtracted from throughout the equation

  • Every instance of in the equation is replaced with

• E.g. the graph undergoes a translation of 6 units to the right

  • is replaced throughout the equation by

    • is the new equation

  • The equation can be left in this form or expanded and simplified

    • 

    • 

• For a vertical translation of a units up

  • is added to the equation as a whole

• E.g. the graph undergoes a translation of 5 units down

  • 5 is subtracted from the equation as a whole

    • 

  • The equation can be left in this form or simplified

    • 

How do I apply a combined translation?

• For a horizontal translation of units and vertical translation of units combined

• E.g. the graph undergoes a translation of 2 units up and 1 unit to the left

  • is replaced throughout the equation by

  • 2 is added to the equation as a whole

    • 

• Note that when the equation is in the form

  • the vertex is

  • the value of does not affect the vertex coordinates