We introduce a simple phenomenological form for the self-energy which allows us to extract important information from angle-resolved photoemission data on the high-T-c superconductor Bi2Sr2CaCu2O8 for binding energies of order the spectral gap. First, we find a rapid suppression of the single particle scattering rate below T-c for all doping levels. Second, we find that in the overdoped materials the gap Delta at all k points on the Fermi surface has significant temperature dependence and vanishes near T-c. In contrast, in the underdoped samples such behavior is found only at k points close to the diagonal. Near (pi,0), Delta is essentially T independent in the underdoped samples. The filling-in of the pseudogap with increasing T is described by a broadening proportional to T-T-c, which is naturally explained by pairing correlations above T-c.