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 Functions and Graphs

Edexcel A level Maths: 2.6 Combination of TransformationsCurated by Kepler, not produced by us. Watch it 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:\ \text{up } a \quad f(x + a):\ \text{left } a

Translations.

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

Stretches.

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

Reflections.

Order matters

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

Example 1Describe

y = 2f(x - 1)

as transformations of

f

1Inside:

x - 1

⟶ translate right 1.

2Outside:

\times 2

⟶ vertical stretch, scale factor 2.

Answer

right 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).

Example 1

(3, 5)

is on

y = f(x)

. Find its image on

y = 2f(x - 1)

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(ax)

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

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.

$y = f(x - 1)$ translates the graph:
- Right 1
- Left 1
- Up 1
- Down 1
Show answer and explanation
Answer: Right 1
Inside −1 ⟶ right.
$y = 2f(x)$ is a:
- Vertical stretch ×2
- Horizontal stretch ×2
- Translation up 2
- Reflection
Show answer and explanation
Answer: Vertical stretch ×2
Outside factor.
For $y = 2f(x - 1)$ , the transformations are:
- Right 1, vertical stretch ×2
- Left 1, stretch ×2
- Right 1, up 2
- Down 2, left 1
Show answer and explanation
Answer: Right 1, vertical stretch ×2
Inside then outside.
$(3, 5)$ on $y = f(x)$ maps under $y = 2f(x-1)$ to:
- $(4, 10)$
- $(2, 10)$
- $(4, 5)$
- $(3, 10)$
Show answer and explanation
Answer: $(4, 10)$
x: 3→4, y: 5→10.
$y = f(2x)$ is a horizontal stretch of factor:
- $\tfrac{1}{2}$
- $2$
- $-2$
- $4$
Show answer and explanation
Answer: $\tfrac{1}{2}$
Inside factor → reciprocal.
Inside-the-bracket changes affect the:
- x-coordinate
- y-coordinate
- gradient
- area
Show answer and explanation
Answer: x-coordinate
Horizontal changes act on x.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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