4.6PureStretch

Stretching Graphs

A stretch scales a graph rather than sliding it. As with translations, outside changes behave normally but inside changes act with a reciprocal factor — and a negative multiplier produces a reflection.

25 min Graphs and Transformations

Edexcel AS Level Maths: 4.5–4.7 Transformation of GraphsCurated by Kepler, not produced by us. Watch it before the notes below.

Open on YouTube

What you'll be able to do

Apply vertical stretches a·f(x)
Apply horizontal stretches f(ax)
Understand reflections as negative stretches
Describe the scale factor of a stretch

Vertical stretch: a·f(x)

Multiplying the whole function by $a$ stretches it vertically by scale factor $a$ . Points move away from (or towards) the $x$ -axis; the $x$ -intercepts stay put.

y = a\,f(x) \;\Rightarrow\; \text{vertical stretch, scale factor } a

Outside the function ⟶ vertical stretch by the factor itself.

Horizontal stretch: f(ax)

Multiplying $x$ inside the function by $a$ gives a horizontal stretch of scale factor $\dfrac{1}{a}$ — the $\textbf{reciprocal}$ . So $f(2x)$ squashes the graph to half its width.

y = f(ax) \;\Rightarrow\; \text{horizontal stretch, scale factor } \tfrac{1}{a}

Inside the function ⟶ reciprocal scale factor.

Example 1Describe

y = f(3x)

compared with

y = f(x)

1Inside change, so horizontal.

2Scale factor is the reciprocal:

\tfrac{1}{3}

Answerhorizontal stretch, scale factor

\tfrac{1}{3}

(a squash)

Tip — Inside the bracket = reciprocal factor. f(2x) stretches by ½ (squashes), not by 2.

Reflections

A negative multiplier reflects the graph. $y = -f(x)$ reflects in the $x$ -axis (flips up/down); $y = f(-x)$ reflects in the $y$ -axis (flips left/right).

y = -f(x):\ \text{reflect in } x\text{-axis}; \quad y = f(-x):\ \text{reflect in } y\text{-axis}

A reflection is a stretch with a negative scale factor.

Formula recap

y = a\,f(x)

Vertical stretch, factor a.

y = f(ax)

Horizontal stretch, factor 1/a.

y = -f(x),\; y = f(-x)

Reflections in the x- and y-axes.

Common mistakes to avoid

Saying f(2x) stretches the graph by a factor of 2.

Inside changes use the reciprocal: f(2x) is a stretch of factor ½ (a squash).

Mixing up the two reflections.

−f(x) flips in the x-axis; f(−x) flips in the y-axis.

Key takeaways

a·f(x): vertical stretch by factor a (outside, as expected).
f(ax): horizontal stretch by factor 1/a (inside, reciprocal).
−f(x) reflects in the x-axis; f(−x) reflects in the y-axis.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Stretching Graphs. Use the feedback to decide what needs another look.

1 past-paper question · 1 mark

Practise this topic

Related lessons

All 7 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.