Translations of Graphs (Cambridge (CIE) IGCSE International 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?

• Let be the equation of the original graph

Vertical translations: y=f(x) + a

• is a vertical translation by the vector 

  • The graph moves up for positive values of  

  • The graph moves down for negative values of 

  • The x-coordinates stay the same

Horizontal translations: y=f(x + a)

•   is a horizontal translation by the vector 

  • 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

What happens to asymptotes when a graph is translated?

• Any asymptotes of f(x) are also translated

  • An asymptote parallel to the direction of translation will not be affected

How does a translation affect the equation of the graph?

• For a horizontal translation of the graph

  • 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

  • becomes

  • 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 the graph

  • is added to the equation as a whole

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

  • becomes

  • 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

  • becomes

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

  • will become

  • 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