The partial zeta function is a generalization of the zeta function, which is a generating function for the number of solutions to a system of polynomials over a finite field. While zeta functions are well understood, partial zeta functions are not. In this talk, we compare simple examples of zeta functions with their more complicated partial zeta function counter parts. We then find the partial zeta function for \(y - x^n\) in the case when n is an integer not divisible by the characteristic of the ground field.