Graph Transformations

This lesson covers graph transformations as required by the AQA A-Level Mathematics specification (7357). You must be able to apply translations, stretches, and reflections to the graphs of functions, and understand the effect of combining transformations. This topic is fundamental and connects to almost every other area of the specification.

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The Standard Transformations

Given a function y = f(x), the following transformations can be applied:

Translations

Transformation	Effect	Description
y = f(x) + a	Vertical translation by a	Shift up by a (or down if a < 0)
y = f(x + a)	Horizontal translation by −a	Shift left by a (or right if a < 0)
y = f(x − a)	Horizontal translation by +a	Shift right by a

Key point: y = f(x + a) shifts left (the opposite direction to what you might expect). Think of it as: the curve reaches each y-value a units sooner.