Create a Function to Visualize Periodicity 

The following program computes the periodicity of a function from the reals to itself. The program then uses that information to create a Manipulate showing the plot and its translation and how a translation by a full period is indistinguishable from no translation.

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X VisualizePeriodicity[f_, x_] := Block[{a}, With[{T = FunctionPeriod[f, x], shiftedF = f /. x -> x - a}, Manipulate[ Plot[{f, shiftedF}, {x, 0, 3 T}, PerformanceGoal -> "Quality", ImageSize -> Medium], {a, 0, T, Appearance -> "Labeled"}]]]

This function has period and results in a plot from to .

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X VisualizePeriodicity[Sin[x] + Cos[3/2 x], x]

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This function has period and generates a plot from to .

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X VisualizePeriodicity[SquareWave[x] + TriangleWave[x/3], x]

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