Transformations

Single-variable calculus from first principles

Once you know the shape of one function, you don't have to re-plot anything to understand a whole family of relatives. Four simple operations move, stretch, and flip a graph in completely predictable ways. Learn to see them and graphing becomes recognition instead of arithmetic.

This is exactly what a photo editor does. You never redraw the picture pixel by pixel; you nudge it sideways, stretch it taller, or flip it horizontally, and the same shape lands somewhere new. Transforming a function is the same handful of one-tap edits applied to a graph instead of a photo.

Starting from a base shape f(x): multiplying the output by a stretches it vertically; multiplying the input by b stretches it horizontally; subtracting c inside shifts it right; adding d outside lifts it up. Put together:

Where this lives in MLThis is not an analogy — batch normalization is literally this transform. A batch-norm layer takes a normalized activation x̂ and outputs γ·x̂ + β, where γ is a learned scale (the a above) and β is a learned shift (the d). The network learns where to place and how to stretch each activation. The shape of an activation function is a transform too: a "steeper" tanh is just b > 1, and temperature in…

▶ Transformations

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