2.6PureStretch

Combining Transformations

Year 2 stacks the Year 1 transformations together, often including a modulus. The rules are unchanged — outside the function acts on y in the natural direction, inside acts on x in the opposite/reciprocal way — but the order of operations now matters.

30 min Video by Zeeshan Zamurred Functions and Graphs

Edexcel A level Maths: 2.6 Combination of TransformationsWatch the full walkthrough before the notes below.

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What you'll be able to do

Recall the translation, stretch and reflection rules
Apply two or more transformations in order
Transform specific points
Combine transformations with the modulus

The rules (recap)

Outside the function affects $y$ in the expected direction; inside affects $x$ oppositely/reciprocally.

f (x) + a : up a f (x + a) : left a

Translations.

a f (x) : vertical \times a f (a x) : horizontal \times \frac{1}{a}

Stretches.

- f (x) : x -axis f (- x) : y -axis

Reflections.

Order matters

When combining, apply transformations in a consistent order. A safe approach: deal with the changes $inside$ the bracket as one group and the changes $outside$ as another, sketching each step.

1Inside:

x - 1

⟶ translate right 1.

2Outside:

\times 2

⟶ vertical stretch, scale factor 2.

Answerright 1, then stretch vertically ×2

Tip — Inside changes (with x) often need to be applied carefully relative to stretches — sketch intermediate graphs to stay safe.

Transforming points

Track a known point through each transformation: vertical changes alter the $y$ -coordinate, horizontal changes alter the $x$ -coordinate (opposite/reciprocal direction).

1Right 1:

x : 3 \to 4

2Vertical ×2:

y : 5 \to 10

Answer

(4, 10)

Formula recap

f (x) + a, f (x + a)

Translations (vertical / horizontal).

a f (x), f (a x)

Stretches (×a / ×1/a).

- f (x), f (- x)

Reflections (x-axis / y-axis).

Common mistakes to avoid

Applying combined transformations in a careless order.

Handle inside-bracket and outside changes systematically; sketch each step.

Applying a horizontal change to the y-coordinate of a point.

Inside (horizontal) changes affect x; outside (vertical) changes affect y.

Key takeaways

Same rules as Year 1: outside acts on y, inside acts on x (opposite/reciprocal).
For combinations, work systematically and sketch intermediate steps.
Transform points by applying each change to the right coordinate.

Test yourself

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