Stretches of Graphs (DP IB Analysis & Approaches (AA): SL): 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, $a$

    • 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 $a>1$

    • points above the $x$-axis move vertically upwards

    • points below the $x$-axis move vertically downwards

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

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

• A vertical stretch of the graph $y=f\left(x\right)$ by a scale factor $a$ parallel to the $y$-axis has the equation

  • $y=af\left(x\right)$

  • Any horizontal asymptotes change

    • $y=k$ becomes $y=ak$

    • Vertical asymptotes stay the same

• If the scale factor is

  • $a>1$, then points

    • above the $x$-axis stretch upwards

    • below the $x$-axis stretch downwards

  • $0<a<1$, then points

    • above the $x$-axis "compress / squash" downwards

    • below the $x$-axis "compress / squash" upwards

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

• A horizontal stretch of the graph $y=f\left(x\right)$ by a scale factor $a$ parallel to the $x$-axis has the equation

  • $y=f\left(\frac{x}{a}\right)$

    • Any horizontal asymptotes stay the same

    • but vertical asymptotes change

    • e.g. $x=k$ becomes $x=\frac{k}{a}$

• This means the equation $y=f\left(ax\right)$ where $a>1$ represents

  • a horizontal stretch of scale factor $\frac{1}{a}$

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