This work investigates the algebraic structure of affine semigroups (additive subsemigroups of Zd 0) using minimal presentations (which encapsulates the minimal trades of a given semigroup). We accomplish this by building on previous research that focused on shifted families of affine semigroups, which are families of affine semigroups parametrized by a shift parameter n. Much is known about the structure of numerical semigroups, that is, affine semigroups that are subsets of Z 0. In particular, prior work has used minimal presentations as a tool to characterize shifted numerical semigroups. The current work generalizes these results to higher dimensional shifted affine semigroups. In particular, we identify a recurrence of minimal presentations of three-generated shifted affine semigroups, as well as give an example of a four-generated shifted affine semigroup where the number of minimal relations is unbounded, which was not the case for families of numerical semigroups.