Translations of Graphs (DP IB Analysis & Approaches (AA)): Revision Note

Translations of graphs

What are translations of graphs?

• A translation is when:

  • the graph is moved (up or down, left or right) in the plane

    • i.e. its position changes

  • but the shape, size, and orientation remain unchanged

• How far left/right or up/down is specified by a translation vector :

  • is the horizontal displacement

    • Positive moves right

    • Negative moves left

  • is the vertical displacement

    • Positive moves up

    • Negative moves down

How do I find the graph equation after a horizontal translation?

• A horizontal translation of the graph by units to the right, i.e. the vector , is represented by the equation

  • 

• Any vertical asymptotes will also be translated

  • becomes 

  • Horizontal asymptotes stay the same

• To translate a graph units to the left

  • i.e. the vector

    • the equation becomes

How do I find the graph equation after a vertical translation?

• A vertical translation of the graph by units up, i.e. the vector , is represented by

  • 

• Similarly a vertical translation units down

  • i.e. the vector

    • has the equation

• Horizontal asymptotes change

  • becomes 

• Vertical asymptotes stay the same