Stretches of Graphs (Edexcel IGCSE Maths)

Spec point — 3.3B

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

Examples of stretches

How do I stretch graphs?

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

Vertical stretches: y=af(x)

• $y = a \mathrm{f} ( x )$ 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)

• $y = \mathrm{f} ( a x )$ 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 $y = \mathrm{f} ( 2 x )$ 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 $\mathrm{f} ( \frac{x}{2} )$ is a horizontal stretch of scale factor 2

What happens to asymptotes when a graph is stretched?

• Any asymptotes of $\mathrm{f} ( x )$ 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 $y = 3 ( x^{2} + 2 x )$

    • 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$( 3 x )$ in the equation

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

    • This simplifies to $y=9x^{2}+6x$

How do I apply a combined stretch?

• The graph of $y = b \mathrm{f} ( a x )$ 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 $b$

Worked Example The diagram below shows the graph of $y = \mathrm{f} ( x )$.

Sketch the graph of $y = 2 \mathrm{f} ( \frac{x}{3} )$.

Answer:

$\mathrm{f} ( \frac{x}{3} )$ represents a horizontal stretch of scale factor $\frac{1}{( 1 \frac{1}{3} )} = 3$ And $2 \mathrm{f} (... )$ represents a vertical stretch of scale factor 2

Apply these in any order, e.g. start with the horizontal stretch, scale factor 3

Now apply a vertical stretch of scale factor 2

Show the coordinates of the new points clearly