Exact closed-form solutions of the time-dependent Schrodinger equation are obtained, describing the propagation of wavepackets in the neighbourhood of a potential. Examples given include zero reflection, total reflection and partial reflection of the wavepacket, for the sech[superscript 2]x/a, 1/x[superscript 2] and delta(x) potentials, respectively. In the first two of these cases the results are obtained using the methods of elementary supersymmetric quantum mechanics. This gives an introduction to supersymmetry in its simplest form, suitable for graduate or advanced undergraduate students. Animations of wavepacket propagation are provided. (Contains 4 figures.)