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

Stretches of graphs

What are stretches of graphs?

• A stretch is when

  • the graph is stretched vertically or horizontally

  • by a scale factor,

    • The size of the graph changes

    • The orientation of the graph remains unchanged

• Stretches act parallel to a coordinate axis

  • e.g. for a vertical stretch of scale factor

    • points above the -axis move vertically upwards

    • points below the -axis move vertically downwards

    • i.e. a vertical stretch acts parallel to the -axis

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

• A vertical stretch of the graph  by a scale factor parallel to the -axis has the equation

  • 

  • Any horizontal asymptotes change

    • becomes 

    • Vertical asymptotes stay the same

• If the scale factor is

  • , then points

    • above the -axis stretch upwards

    • below the -axis stretch downwards

  • , then points

    • above the -axis "compress / squash" downwards

    • below the -axis "compress / squash" upwards

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

• A horizontal stretch of the graph by a scale factor parallel to the -axis has the equation

  • 

    • Any horizontal asymptotes stay the same

    • but vertical asymptotes change

    • e.g. becomes

• This means the equation where represents

  • a horizontal stretch of scale factor

    • i.e. a horizontal "compression / squash" of scale factor