shifting property of the Fourier transform
Suppose you record the same sound twice, the second recording delayed by a moment. The two clips contain identical frequencies and identical loudness at each frequency; the only difference is timing. The shifting property captures exactly this: delaying or advancing a signal in time does not change how much of each frequency it holds, it only changes the phase — the alignment — of those frequencies.
Stated as a formula, if f(x) has Fourier transform F(k), then the shifted function f(x minus a) has transform e^{-i k a} F(k). The magnitude is untouched because |e^{-i k a}| = 1, so the amplitude spectrum |F(k)| is identical; what the shift inserts is a frequency-dependent phase factor e^{-i k a} that rotates each frequency by an amount proportional to k. There is a mirror-image rule (the modulation property): multiplying f(x) by e^{i a x} shifts the transform to F(k minus a). In words, a shift in one domain is a phase twist in the other — and which way the twist goes follows the duality between x and k.
This rule is a daily tool. It is why a pure time delay shows up in systems as a linear-with-frequency phase and a flat magnitude, the signature of an ideal delay line. It lets you transform a function defined around any point by transforming a centered version and multiplying by a phase, and it pairs with the convolution theorem to handle filters that delay as well as shape a signal. The thing to remember is the asymmetry of effort: a shift, which can be awkward in the original domain, becomes a clean multiplication by a phase in the transform domain.
A pulse centered at the origin with transform F(k) = 2 sin(k)/k, when slid to be centered at x = 3, transforms to e^{-3 i k} (2 sin(k)/k): same magnitude, an added phase that winds three times faster as k grows.
Sliding a pulse leaves its amplitude spectrum unchanged and writes the displacement entirely into the phase.
Discarding the phase (keeping only |F(k)|) throws away all the shift and timing information, which is why a power spectrum alone cannot tell you where in time an event happened.