scaling property of the Fourier transform
Play a recording at double speed and the pitch jumps up an octave; play it at half speed and the pitch drops. Squeezing a signal in time stretches it in frequency, and stretching it in time squeezes it in frequency. The scaling property is the exact statement of this reciprocal trade between how spread-out a function is and how spread-out its spectrum is.
Quantitatively, if f(x) has transform F(k), then the rescaled function f(a x) has transform (1/|a|) F(k/a). Two things happen at once. The argument k/a means the spectrum is stretched by the factor a (so compressing f by making a large widens F), and the prefactor 1/|a| rescales the height so that total area or energy is bookkept correctly. The absolute value handles a < 0, which also reflects the function. There is no way to make a signal narrow in both time and frequency simultaneously: shrinking one necessarily widens the other, and that is the rule, not a limitation of any particular method.
Scaling is why a short pulse needs a wide bandwidth and a narrow spectral line requires a long observation. It governs the resolution of spectrometers, the bandwidth of communication channels, and the diffraction-limited spot of a lens (a small aperture spreads light over a wide angle). Together with the fact that the Gaussian is its own transform, scaling is one face of the Fourier uncertainty principle: the product of a signal's time-spread and frequency-spread cannot be made arbitrarily small.
If e^{-x^2} transforms to sqrt(pi) e^{-k^2/4}, then the narrower e^{-4 x^2} = e^{-(2x)^2} transforms to (1/2) sqrt(pi) e^{-k^2/16} — half the height and four times wider in k.
Compressing the Gaussian by a factor of 2 in x widens and shortens its spectrum, conserving total area.
The prefactor 1/|a| is not optional bookkeeping — dropping it breaks Parseval's energy balance and gives spectra whose total area no longer matches the signal.