This paper centers on what happens to an exponential smoothing forecasting system when the smoothing parameters are changed. It gives sufficient conditions under which several proposed models give numerically identical results, and notes a surprising symmetry in one basic model between the smoothing constants for the smoothed average and the smoothed trend. The paper defines smoothing-constant values that cause oscillatory behavior, discusses the difficulties caused by oscillation, and finds that commonly used values of the smoothing constants lie in a region of oscillation of varying period. It uses these results in an analytical study of the tradeoff between responsiveness and variance; some dominant choices emerge. Finally, on the basis of these results, the paper examines several sample sets of data from the literature, and discusses implications for the choice of smoothing constants in complex situations.