Stretches of Graphs (Edexcel IGCSE Maths A (Modular))

Revision Note

Flashcards

Written by: Mark Curtis

Reviewed by: Dan Finlay

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

A graph being stretched vertically (on the left) or horizontally (on the right). — **A graph being stretched vertically (left) or horizontally (right)**

How do I stretch graphs?

Let $y = f (x)$ be the equation of the original graph

Vertical stretches: y=af(x)

$y = a f (x)$ 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

Graph showing the function y = f(x) and its transformation y = (1/3)f(x), indicating vertical stretch by a factor of 1/3. Points (2, -3) and (2, -1) shown.

Horizontal stretches: y=fa(x)

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

Stretching a graph horizontally by scale factor one half. y-coordinates stay in same place, x-coordinates change

What happens to asymptotes when a graph is stretched?

Any asymptotes of $f (x)$ are also stretched

A diagram shows transformations of the function y = f(x). It illustrates vertical and horizontal stretches, their effects on asymptotes, and coordinate changes.

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, $y = x^{2} + 2 x$ becomes $y = 3 (x^{2} + 2 x)$
  - This simplifies to $y = 3 x^{2} + 6 x$
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, $y = x^{2} + 2 x$ becomes $y = {(3 x)}^{2} + 2 (3 x)$
  - This simplifies to $y = 9 x^{2} + 6 x$

How do I apply a combined stretch?

The graph of $y = b f (a x)$ 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 $y = f (x)$ .

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

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

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