Stretches of Graphs (Edexcel IGCSE Maths A (Modular): Higher Unit 2): Revision Note

Stretches of graphs

What are stretches of graphs?

• Stretches of graphs are a type of transformation that pushes points away from, or towards, the $x$-axis or $y$-axis

  • Graphs look like they have been stretched or squashed

    • either horizontally or vertically

How do I stretch graphs?

• Let $y=\mathrm{f}\left(x\right)$ be the equation of the original graph

Vertical stretches: y=af(x)

• $y=a\mathrm{f}\left(x\right)$ is a vertical stretch (in the $y$-direction) of scale factor $a$

  • The $x$-coordinates stay the same but the $y$ coordinates are multiplied by $a$

  • Points appear to move parallel to the $y$-axis

    • either stretching vertically away from the $x$-axis if $a>1$

    • or squashing vertically towards the $x$-axis if $0<a<1$

  • Points on the $x$-axis stay where they are
    

Horizontal stretches: y=f(ax)

• $y=\mathrm{f}\left(ax\right)$ is a horizontal stretch (in the $y$-direction) of scale factor $\frac{1}{a}$ (not $a$)

  • The $y$-coordinates stay the same but the $x$ coordinates are multiplied by $\frac{1}{a}$ (divided by $a$)

  • Points appear to move parallel to the $x$-axis

    • either squashing horizontally towards the $y$-axis if $a>1$

    • or stretching horizontally away from the $y$-axis if $0<a<1$

  • Points on the $y$-axis stay where they are

• This means $y=\mathrm{f}\left(2x\right)$ is more like a horizontal squash of scale factor 2

  • though the correct way to say this is a horizontal stretch of scale factor $\frac{1}{2}$

    • This also means that $\mathrm{f}\left(\frac{x}{2}\right)$ is a horizontal stretch of scale factor 2

What happens to asymptotes when a graph is stretched?

• Any asymptotes of $\mathrm{f}\left(x\right)$ are also stretched

How does a stretch affect the equation of the graph?

• When a graph is stretched, you can change its equation algebraically

  • There is no need to sketch the graph

• Stretching vertically by a scale factor of 3 puts a 3 in front of the whole equation

  • For example, $y=x^2+2x$ becomes $y=3\left(x^2+2x\right)$

    • This simplifies to $y=3x^2+6x$

• Stretching horizontally by a scale factor of $\frac{1}{3}$ ("squashing horizontally" by a scale factor of 3) replaces any $x$ with $\left(3x\right)$ in the equation

  • For example, $y=x^2+2x$ becomes $y={\left(3x\right)}^2+2\left(3x\right)$

    • This simplifies to $y=9x^2+6x$

How do I apply a combined stretch?

• The graph of $y=b\mathrm{f}\left(ax\right)$ is a combined stretch_, both horizontally and vertically

  • It does not matter which order you apply these in

    • For example, a horizontal stretch of scale factor $\frac{1}{a}$ followed by a vertical stretch of scale factor $b$