This paper introduces the approximate solutions of the mathematical model of an elastically restrained tapered beam. At the beginning of the study, the equation of motion is derived in a detailed way. The frequency-amplitude relation is deduced and solved numerically. The nonlinear natural frequencies for the transverse vibrations of an elastically restrained tapered beam are provided using Mathematica software. The max-min approach, the frequency-amplitude method and the parameter-expansion method are applied in order to obtain an approximated solution. The approximate analytical results are further compared with the numerical results for both small and large amplitude oscillations, and very good agreement is observed.