The Fundamental Theorem of Algebra is a useful tool in determining the number of zeros of complex-valued polynomials and rational functions. It does not, however, apply to complex-valued harmonic polynomials and rational functions generally. In this thesis, we determine behaviors of the family of complex-valued harmonic functions $f_{c}(z) = z^{n} + \frac{c}{\overline{z}^{k}} - 1$ that defy intuition for analytic polynomials. We first determine the sum of the orders of zeros by using the harmonic analogue of Rouch\'e's Theorem. We then determine useful geometry of the critical curve and its image in order to count winding numbers by applying the harmonic analogue of the Argument Principle. Combining these results, we fully determine the number of zeros of $f_{c}$ for $c > 0$.

Brilleslyper et al. analyzed a one-parameter family of harmonic trinomials, and Brooks and Lee analyzed a one-parameter family of harmonic functions with poles. Each family was explored to find the relationship between the size of the parameter and the number of zeros of the harmonic function. In this thesis, we examine convex combinations of members of these families. We determine conditions under which the critical curves separating the sense-preserving and sense-reversing regions are circular. We show that the number of zeros of a convex combination can be greater than the maximum number of zeros of either part.