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 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.

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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:\ \text{up } a \qquad f(x + a):\ \text{left } a

Translations.

a f(x):\ \text{vertical} \times a \qquad f(ax):\ \text{horizontal} \times \tfrac{1}{a}

Stretches.

-f(x):\ x\text{-axis} \qquad f(-x):\ y\text{-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).

Example 1The point

(2, 5)

is on

y = f(x)

. Where does it move on

y = f(x) - 4

1Outside change of

-4

shifts down 4, affecting only

y

5 - 4 = 1

Answer

(2, 1)

Example 2The point

(6, 3)

is on

y = f(x)

. Where does it move on

y = f(2x)

1Inside change: horizontal stretch by

\tfrac{1}{2}

, so

x

halves.

6 \times \tfrac{1}{2} = 3

;

y

unchanged.

Answer

(3, 3)

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(ax)

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

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

$(3, 7)$ is on $y = f(x)$ . On $y = f(x) + 2$ it becomes:
- $(3, 9)$
- $(5, 7)$
- $(3, 5)$
- $(1, 7)$
Show answer and explanation
Answer: $(3, 9)$
Outside +2 raises y by 2.
$(4, 1)$ is on $y = f(x)$ . On $y = f(x - 1)$ it becomes:
- $(5, 1)$
- $(3, 1)$
- $(4, 0)$
- $(4, 2)$
Show answer and explanation
Answer: $(5, 1)$
f(x − 1) shifts right 1, so x increases by 1.
$(8, 2)$ is on $y = f(x)$ . On $y = f(2x)$ it becomes:
- $(4, 2)$
- $(16, 2)$
- $(8, 4)$
- $(8, 1)$
Show answer and explanation
Answer: $(4, 2)$
Horizontal stretch factor ½ halves x.
$(2, 5)$ is on $y = f(x)$ . On $y = 3f(x)$ it becomes:
- $(2, 15)$
- $(6, 5)$
- $(2, 8)$
- $(6, 15)$
Show answer and explanation
Answer: $(2, 15)$
Vertical stretch factor 3 multiplies y by 3.
$(3, -4)$ is on $y = f(x)$ . On $y = -f(x)$ it becomes:
- $(3, 4)$
- $(-3, -4)$
- $(-3, 4)$
- $(3, -4)$
Show answer and explanation
Answer: $(3, 4)$
Reflection in the x-axis negates y.
A change inside the function brackets always affects the:
- x-coordinate
- y-coordinate
- gradient only
- area
Show answer and explanation
Answer: x-coordinate
Inside changes act horizontally, on x.

Next step

Check your understanding

Work through questions on Transforming Functions. Use the feedback to decide what needs another look.

12 past-paper questions · 20 marks

Practise this topic

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