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

Translations of Graphs

What are translations of graphs?

• When you alter a function in certain ways, the effects on the graph of the function can be described by geometrical transformations

• For a translation:

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

    • Its position changes

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

• A particular translation (how far left/right, how far up/down) is specified by a translation vector :

  • x is the horizontal displacement

    • Positive moves right

    • Negative moves left

  • y is the vertical displacement

    • Positive moves up

    • Negative moves down

What effects do horizontal translations have on the graphs and functions?

• A horizontal translation of the graph by the vector  is represented by 

  • 

• The x-coordinates change

  • The value a is added to them

• The y-coordinates stay the same

• The coordinates become 

• Horizontal asymptotes stay the same

• Vertical asymptotes change

  • becomes 

What effects do vertical translations have on the graphs and functions?

• A vertical translation of the graph  by the vector  is represented by

  • 

  • This is often rearranged to 

• The x-coordinates stay the same

• The y-coordinates change

  • The value b is added to them

• The coordinates become 

• Horizontal asymptotes change

  • becomes 

• Vertical asymptotes stay the same