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 Video by Zeeshan Zamurred Graphs and Transformations

Edexcel AS Level Maths: 4.5–4.7 Transformation of GraphsWatch the full walkthrough before the notes below.

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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 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 $\frac{1}{a}$ — the $reciprocal$ . So $f (2 x)$ squashes the graph to half its width.

y = f (a x) \Rightarrow horizontal stretch, scale factor \frac{1}{a}

Inside the function ⟶ reciprocal scale factor.

1Inside change, so horizontal.

2Scale factor is the reciprocal:

\frac{1}{3}

Answerhorizontal stretch, scale factor

\frac{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) : reflect in x -axis; y = f (- x) : reflect in y -axis

A reflection is a stretch with a negative scale factor.

Formula recap

y = a f (x)

Vertical stretch, factor a.

y = f (a x)

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.

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