In this paper, we propose a weight function to construct a fourth order family of iterative schemes for solving nonlinear equations. This class is parameter-dependent and its numerical performance depends on the value of this free parameter. We analyze the rational function resulting from the fixed point operator applied to a nonlinear polynomial. The dynamics of this rational function allows us to better understand the performance of the iterative methods of the class. In addition, we calculate the critical points and present the parameter spaces dynamical planes, in order to determine the regions with stable and unstable behavior. Finally, parameter values within and outside the stability region are chosen and, with them, numerical tests that confirm the scheme's theoretical convergence and stability are performed, as well as comparisons with other existing methods of the same order of convergence.