Stretches of Graphs

How do we stretch graphs of functions?

In relation to y = f(x);

y = f(ax) is a horizontal stretch/ stretch in the x-axis by a scale factor of 1/a
- May also be referred to as a stretch in (or parallel to) the x-direction
- This looks like a “squash” but that is not a technical term – do not use it!
- To perform a stretch in the x-axis given by f(ax);
  1. pick key points on the original graph like end points, turning points and axis intercepts
  2. multiply the x-coordinate of these points by 1/a (or divide by a). Do not change the y-coordinate!
  3. plot the new points and join with a curved or straight line as appropriate

y = af(x) is a vertical stretch/ stretch in the y-axis by a scale factor of a

May also be referred to as a stretch in (or parallel to) the y-direction
To perform a stretch in the y-axis given by af(x);
1. pick key points on the original graph like end points, turning points and axis intercepts
2. multiply the y-coordinate of these points by a. Do not change the x-coordinate
3. plot the new points and join with a curved or straight line as appropriate

edexcel-igcse-3-graphs-transformations-stretches

As with translations and reflections, trig graphs are a common subject for transformation questions

GT Notes fig5, downloadable IGCSE & GCSE Maths revision notes

How are stretches and reflections of graphs related?

We have previously seen that f(-x) and -f(x) are reflections in the x-axis and the y-axis respectively
Both of the above are also special cases of stretches, f(ax) and af(x) where a = -1

How do stretches of graphs work?

There is a logic to these as well but memory and recognition is easier!
With y = af(x) the “a” happens after the function
- So the ‘output’ of the function changes
  ie. y-coordinates change
With y = f(ax) the “a” happens before the function
- So the ‘input’ of the function changes
  ie. x-coordinates change
- This stretch may seem it has the wrong scale factor (1/a)
  - However the ‘input’ is being multiplied by a – so x will need to be divided by a in order for the function to receive the same input

How do I describe stretches of graphs?

Some questions give a transformed function in the form y = f(ax) or y = af(x) and ask you to describe the transformation
To describe a stretch fully, you must include;
- the transformation: "stretch"
- the direction: "in the x-axis" / "in the y-axis"
- the stretch factor: "by 1/a" / "by a"

GT Notes fig8, downloadable IGCSE & GCSE Maths revision notes

Examiner Tip

REMEMBER that;

y = f(ax); "a" next to x, stretches in x-axis by 1/a
y = af(x); "a" not next to x, stretches in y-axis

Worked example

(a)

The graph of

y = f (x)

is shown.

edexcel-igcse-3-graphs-transformations-stretches1

Sketch the graph of

y = 2 f (x)

on the grid.

y = 2f(x) represents a stretch in the y-axis by factor 2

First pick the key points- axis intercepts at (0, 0), a turning point at (1, -1) and integer points at (-1, 3) and (3, 3)
For each point, keep its x-coordinate the same but multiply its y-coordinate by 2
Mark the new points on the grid

edexcel-igcse-3-graphs-transformations-stretches2

Notice that points on the x-axis do not change during a stretch in the y-axis (they are invariant)

Join the new points with a smooth line. You must make sure that your stretched curve passes through the points marked on the answer diagram below

edexcel-igcse-3-graphs-transformations-stretches3

(b)

The diagram shows part of the curve with equation

y = f (x)

.
The coordinates of the maximum point of the curve are

(6, 15)

edexcel-igcse-3-graphs-transformations-stretches4

State the coordinates of the maximum point of the curve with equation

y = f (3 x)

y=f(3x) represents a stretch in the x-axis by a factor of 1/3

The y-coordinate of (6, 15) does not change but the x-coordinate is multiplied by 1/3 (divided by 3)

6 × (1/3) = 2

(2, 15)

Worked example

(a)

Describe the transformation that maps the graph of

y = f (x)

to the graph of

y = f (2 x)

The number is inside the bracket (next to x)

Stretch in x-axis by 1/2

(b)

Describe the transformation that maps the graph of

y = f (x)

to the graph of

y = 3 f (x)

The number is outside the bracket

Stretch in y-axis by 3

Stretches of Graphs (Edexcel IGCSE Maths)

Revision Note