Calculus×Computing
ES

Transformations of functions

Level FundamentalDifficulty ★★★★★Concept⌖ Open in the map

What is it?

a f(bx+c)+da\,f(bx + c) + d: shifting, stretching and reflecting a graph. A delay in a signal, a zoom in an image or a change of units are all such transformations.

Formulas

g(x)=a f(b(x−h))+kg(x) = a\, f\big(b(x - h)\big) + k

Where it shows up in computing

  • Signal processing★★★★★frequentSignals, media and vision

    Delaying a signal is x(t−t0)x(t - t_0); speeding it up is x(αt)x(\alpha t), which stretches its spectrum.

  • Image processing and computer vision★★★★★frequentSignals, media and vision

    Brightness and contrast adjustments are a I(x,y)+ba\,I(x,y) + b.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

∿ Signals, media and vision

⊕ Cryptography and security

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.

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