The characteristic of ion-acoustic solitary and shock waves propagating in a non-extensive plasma is analysed using the framework of the modified Korteweg–de Vries–Burgers (MKdVB) equation. Employing reductive perturbation technique (RPT), the MKdVB equation is derived from the basic guiding equations andfurther, the equation is converted to a dynamic system using travelling wave transformation. Varying different plasma parameters, phase portraits for the MKdV system are drawn and using bifurcation theory of planar dynamical system, it is observed that the MKdV system may contain shock, solitary and periodic solutions. However, it is evident from the phase portrait analysis of the MKdVB equation that due to the impact of Burgers term, the system includes only the shock and solitary solutions. Initially, different patterned solutions of the MKdV equation are directly derived from the corresponding Hamiltonian of the system, and employing the weighted residual method(WRM), the approximate analytical solutions of the MKdVB equation are explored using the solution of MKdV equation as initial solution. These solutions come as a desired pattern that was predicted by the phase portraits. Finally, some graphs are depicted from a numerical standpoint by which the effects of physical parameters on wave propagation are well understood.