We study the dynamics of a family of polynomial functions and the relationship to the dynamics of a related entire transcendental family of functions. As the degree of the polynomial approaches infinity, the polynomial functions converge uniformly on compact sets to a function that is a product of a power map and the exponential function. The advantage to the approach in the paper is that we can use the relatively simple, although high degree, polynomial functions to aid our understanding of the dynamics of the related transcendental entire function. We study properties both in the dynamical plane as well as in the parameter plane.

The first author acknowledges the financial support of an Undergraduate Research Assistantship Program (URAP) grant from DePaul University's College of Science and Health. The second author acknowledges the support of the National Science Foundation under Grant No. 1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the "Complex Dynamics: From Special Families to Natural Generalizations in One and Several Variables" semester of 2022.