4.7PureStretch

Transforming Functions

This lesson combines translations, stretches and reflections, and applies them to unfamiliar functions. The skill is tracking how named points and asymptotes move — without ever needing the equation of $f (x)$ .

30 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

Combine multiple transformations in order
Transform specific points on a graph
Apply transformations to an unknown function f(x)
Track turning points and intercepts through a transformation

A summary of the rules

Everything from the previous two lessons in one place. Outside the function acts on $y$ in the natural direction; inside the function acts on $x$ in the opposite/reciprocal direction.

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.

Transforming a point

You can transform a single known point using the same rules. A vertical transformation changes the $y$ -coordinate; a horizontal one changes the $x$ -coordinate (using the reciprocal/opposite rule).

1Outside change of

- 4

shifts down 4, affecting only

y

5 - 4 = 1

Answer

(2, 1)

Tip — When transforming a point, ask: is the change inside (affects x) or outside (affects y)?

Order matters

When combining transformations, apply them in the right order. Generally deal with the changes inside the bracket together and the changes outside together; a sketch of each intermediate step prevents mistakes.

Formula recap

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

Translations (vertical / horizontal).

a f (x), f (a x)

Stretches (factor a / factor 1/a).

- f (x), f (- x)

Reflections (x-axis / y-axis).

Common mistakes to avoid

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

Horizontal (inside) changes affect x; vertical (outside) changes affect y.

Combining transformations in any order.

Order matters — handle inside-bracket and outside changes carefully, step by step.

Key takeaways

Outside changes act on y (natural direction); inside changes act on x (opposite/reciprocal).
Transform points by applying the rule to the relevant coordinate.
For combined transformations, work step by step and sketch as you go.

Test yourself

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