IGCSEMathsEdexcelMaths AHigherRevision NotesSequences, Functions & GraphsTransformations of GraphsStretches of Graphs

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

Q: How do you know if a horizontal stretch f(ax) stretches or squashes a graph?

For y = f(ax) the scale factor is 1/a, not a — so a > 1 squashes the graph towards the y-axis, and 0 < a < 1 stretches it away from the y-axis. - Every x-coordinate is multiplied by 1/a (divided by a); the y-coordinates stay the same - Points on the y-axis do not move - f(2x): a = 2, so the scale factor is ½ — a squash - f(x/3): a = ⅓, so the scale factor is 3 — a stretch Two different graphs on this page: - On the parabola with minimum (2, −3) and roots at 0 and 4, y = f(2x) sends the minimum to (1, −3) and the root at x = 4 to x = 2, so that curve is half as wide - On the separate cubic with maximum (−2, 6) and minimum (1, −3), y = f(x/3) sends those points to (−6, 6) and (3, −3), so that curve is three times as wide A number inside the bracket does the opposite of what you expect. Even when it looks squashed, the correct wording is still “a horizontal stretch of scale factor ½”.

Q: How does a vertical stretch af(x) affect the y-coordinates of a graph?

A vertical stretch y = af(x) multiplies every y-coordinate by the scale factor a, so a point at height y moves to height a × y. - y = af(x) is a vertical stretch of scale factor a, so (x, y) becomes (x, a × y) - The x-coordinates stay exactly the same - Points on the x-axis stay put, because a × 0 = 0 - A negative y-coordinate gets multiplied too, so it moves further below the x-axis Example using this page's graph: y = f(x) has a minimum at (2, −3) and crosses the x-axis at 0 and 4. For y = 3f(x): - Minimum: (2, 3 × (−3)) = (2, −9), so it drops by 6, not by 3 - The roots stay at (0, 0) and (4, 0) For y = ⅓f(x), the version drawn on this page: - Minimum: (2, ⅓ × (−3)) = (2, −1), squashed towards the x-axis Moving the graph down by 3 would be a different transformation, y = f(x) − 3.

Q: What is the equation of a graph if its width is doubled?

Doubling the width is a horizontal stretch of scale factor 2, so the new graph is y = f(x/2), also written y = f(½x). - For y = f(ax) the scale factor is 1/a - You want a scale factor of 2, so 1/a = 2 and a = ½ - Every x-coordinate is doubled; the y-coordinates stay the same - Do not write f(2x) — that halves the width Example using this page's graph: y = f(x) has a minimum at (2, −3) and roots at x = 0 and x = 4. For y = f(x/2), the minimum is (4, −3) and the roots are x = 0 and x = 8. With an equation: replace every x with (x/2). - y = x² + 2x becomes y = (x/2)² + 2(x/2) = ¼x² + x - Check: the original roots are x = 0 and x = −2; the new roots are x = 0 and x = −4, so the width really has doubled

Exam code: 4MA1

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on 17 May 2026

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Understand this topic with deeper explanations and understanding checks.

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). — **Examples of stretches**

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=f(ax)

$y = f (a x)$ 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 $y = 2 f (\frac{x}{3})$ .

Answer:

$f (\frac{x}{3})$ 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

Graph of the function y = f(x/3) with key points labeled (-6, 6) and (3, -3).

Now apply a vertical stretch of scale factor 2

Graph of the function y=2f(x/3) showing a curve with marked points at (-6,12) and (3,-6) and vertical stretches by a factor of 2 along y-axis.

Show the coordinates of the new points clearly

Graph of the function y = 2f(x/3) with a peak at (-6,12) and a trough at (3,-6), crossing the y-axis at 0. Axes labelled x and y.

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Questions students ask about Stretches of Graphs

Real questions from students revising this topic on Save My Exams

How do you know if a horizontal stretch f(ax) stretches or squashes a graph?

For y = f(ax) the scale factor is 1/a, not a — so a > 1 squashes the graph towards the y-axis, and 0 < a < 1 stretches it away from the y-axis.

Every x-coordinate is multiplied by 1/a (divided by a); the y-coordinates stay the same
Points on the y-axis do not move
f(2x): a = 2, so the scale factor is ½ — a squash
f(x/3): a = ⅓, so the scale factor is 3 — a stretch

Two different graphs on this page:

On the parabola with minimum (2, −3) and roots at 0 and 4, y = f(2x) sends the minimum to (1, −3) and the root at x = 4 to x = 2, so that curve is half as wide
On the separate cubic with maximum (−2, 6) and minimum (1, −3), y = f(x/3) sends those points to (−6, 6) and (3, −3), so that curve is three times as wide

A number inside the bracket does the opposite of what you expect. Even when it looks squashed, the correct wording is still “a horizontal stretch of scale factor ½”.

How does a vertical stretch af(x) affect the y-coordinates of a graph?

A vertical stretch y = af(x) multiplies every y-coordinate by the scale factor a, so a point at height y moves to height a × y.

y = af(x) is a vertical stretch of scale factor a, so (x, y) becomes (x, a × y)
The x-coordinates stay exactly the same
Points on the x-axis stay put, because a × 0 = 0
A negative y-coordinate gets multiplied too, so it moves further below the x-axis

Example using this page's graph: y = f(x) has a minimum at (2, −3) and crosses the x-axis at 0 and 4.

For y = 3f(x):

Minimum: (2, 3 × (−3)) = (2, −9), so it drops by 6, not by 3
The roots stay at (0, 0) and (4, 0)

For y = ⅓f(x), the version drawn on this page:

Minimum: (2, ⅓ × (−3)) = (2, −1), squashed towards the x-axis

Moving the graph down by 3 would be a different transformation, y = f(x) − 3.

What is the equation of a graph if its width is doubled?

Doubling the width is a horizontal stretch of scale factor 2, so the new graph is y = f(x/2), also written y = f(½x).

For y = f(ax) the scale factor is 1/a
You want a scale factor of 2, so 1/a = 2 and a = ½
Every x-coordinate is doubled; the y-coordinates stay the same
Do not write f(2x) — that halves the width

Example using this page's graph: y = f(x) has a minimum at (2, −3) and roots at x = 0 and x = 4. For y = f(x/2), the minimum is (4, −3) and the roots are x = 0 and x = 8.

With an equation: replace every x with (x/2).

y = x² + 2x becomes y = (x/2)² + 2(x/2) = ¼x² + x
Check: the original roots are x = 0 and x = −2; the new roots are x = 0 and x = −4, so the width really has doubled