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**Mapping Rule**

Say I have the function [itex]y = 2 \sin 3(x - 20)[/itex] and the corresponding mapping notation [itex] (x, y) \rightarrow (\frac{1}{3}x + 20, 2y)[/itex] (which I assume is correct.) How come I take the inverse of the amplitude (2) and horizontal "compression" (3), and how come a negative phase shift moves the wave to the right? What is the true purpose of mapping notation?

edit: I guess this is more properly called "mapping rule," true?

Thanks.

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