The idea

Transformations of graphs

The four graph transformations and the two reflections — why a change "inside" the function behaves backwards — and how to combine them.

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Pure · Algebra & functions

Transformations of graphs

The four graph transformations and the two reflections — why a change "inside" the function behaves backwards — and how to combine them.

Why it works

Outside acts on y, inside acts on x

The whole topic turns on one idea: a change outside $f$ acts on $y$ and behaves as you'd expect; a change inside the bracket acts on $x$ and behaves backwards.

f(x) + a \quad\text{vs}\quad f(x + a)

The six moves

New equation	Effect	Direction
$y = f(x) + a$	translation, $a$ up	outside — as expected
$y = a\,f(x)$	vertical stretch, scale factor $a$	outside — as expected
$y = f(x + a)$	translation, $a$ to the left	inside — backwards
$y = f(ax)$	horizontal stretch, scale factor $\tfrac1a$	inside — backwards
$y = -f(x)$	reflection in the $x$ -axis	outside
$y = f(-x)$	reflection in the $y$ -axis	inside

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