  An affine semigroup is a finitely generated subsemigroup of $(\mathbb Z_{\ge
0}^d, +)$, and a numerical semigroup is an affine semigroup with $d = 1$. A
growing body of recent work examines shifted families of numerical semigroups,
that is, families of numerical semigroups of the form $M_n = \langle n + r_1,
\ldots, n + r_k \rangle$ for fixed $r_1, \ldots, r_k$, with one semigroup for
each value of the shift parameter $n$. It has been shown that within any
shifted family of numerical semigroups, the size of any minimal presentation is
bounded (in fact, this size is eventually periodic in $n$). In this paper, we
consider shifted families of affine semigroups, and demonstrate that some, but
not all, shifted families of 4-generated affine semigroups have arbitrarily
large minimal presentations.
