Combinations of Transformations

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Combinations of Transformations

Edexcel IAL Pure 3

The Big Idea: When multiple transformations are applied to a function, the order is everything. Do horizontal transformations inside the brackets first, then vertical transformations outside—swap the order and you get a completely different graph.

Quick Summary

What you're doing: Applying multiple geometric transformations (stretches, reflections, translations) to a graph, one after another
The golden rule: Horizontal transformations (inside brackets) always come first; vertical transformations (outside brackets) always come second
Within each category: Translations before stretches/reflections for horizontal; stretches/reflections before translations for vertical
Special case: With modulus |f(x)|, transform inside first, apply modulus effect, then transform outside
Why it matters: Doing stretches before translations (instead of vice versa) changes where the graph ends up entirely
In the exam: You'll sketch transformed graphs, track how marked points move, and make sure asymptotes move too

What Are Combinations of Transformations?

A single transformation is one change to a function: y = f(x) + 2 (translate up), or y = 2f(x) (stretch vertically), or y = f(x − 1) (translate right).

A combination of transformations is when you apply multiple changes to the same function. For example: y = 3f(x − 1) + 2 has three transformations baked in: a translation inside the brackets, a vertical stretch outside, and another translation outside.

The key insight is this: you can't do all three at once. You have to apply them in a specific order, one transformation at a time, and track how the graph changes after each step.

Think of it like getting dressed for a formal event: You can't put your jacket on and then try to button your shirt—the order matters. First you button the shirt, then you put the jacket on. The same is true for transformations. Do them in the wrong order and you end up with the wrong result.

In real exam questions, you'll be given an original graph with some marked points (like a maximum at (−2, 6) and a minimum at (1, −3)), and you'll have to sketch what happens when you apply transformations like y = 3f(x) − 2 or y = |f(x + 3)|. The skill is tracking how those marked points move, and where the asymptotes end up.

The Golden Rule: Order of Transformations

This is the most important rule in this topic

Everything else follows from this. If you get the order wrong, your entire answer is wrong.

The Formula Structure

Every combined transformation can be written in the form:

General Form

y = kf(ax + b) + c

Where:

Inside the brackets (the "ax + b" part): horizontal transformations
Outside the brackets (the "k" and "+ c" parts): vertical transformations

Step 1: Do Horizontal Transformations First (Inside the Brackets)

Look at what's happening to the x. In y = f(ax + b), you have two possible transformations:

The b is a translation (usually written as f(x − h), which translates right by h units, or f(x + h), which translates left by h units)
The a is a horizontal stretch (by a scale factor of 1/a)

Exam note

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Also in the full note

Horizontal Transformations (Inside the Brackets)
Vertical Transformations (Outside the Brackets)
Why Order Matters: A Crucial Example
Transformations Involving Modulus
Don't Forget About Asymptotes
Complete Worked Example
Practice Questions
Step 2: Do Vertical Transformations Second (Outside the Brackets)

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Combinations of Transformations