A two-parameter semiempirical theory of positron scattering and annihilation is developed and used to investigate the behavior of positrons interacting with the rare gases and metal vapors. The two-parameter theory is able to do a reasonable job of reproducing existing cross section and annihilation data for the rare gases. A model-potential calculation that correctly predicts the behavior of the phase shifts will also predict the energy dependence of Z(eff)(k) even if the magnitude is incorrect. Analysis of the Z(eff) versus temperature data of Kurz [Phys. Rev. Lett. 77, 2929 (1996)] suggests scattering lengths of -5.6+/-1.0a(0), -10.3+/-2.0a(0), and -56+/-15a(0) for Ar, Kr, and Xe, respectively. Existing bound-state calculations can be used to fix the values of the semi-empirical parameters for a number of metal vapors, resulting in predicted Z(eff) of 119, 36, and 94 for Be, Mg, and Cu at threshold. In addition to the calculations, expressions relating the threshold form of Z(eff)(k) to the complex scattering length are presented.