One dimensional exponential families on finite sample spaces are studied using the geometry of the simplex Δn°-1  and that of a transformation Vn-1 of its interior. This transformation is the natural parameter space associated with the family of multinomial distributions. The space Vn-1  is partitioned into cones that are used to find one dimensional families with desirable properties for modeling and inference. These properties include the availability of uniformly most powerful tests and estimators that exhibit optimal properties in terms of variability and unbiasedness.