Graphene is an ideal candidate for the development of solid state nanopores due to its thickness at the atomic scale and its high chemical and mechanical stabilities. A facile method was adopted to prepare single graphene nanopore supported by PET membrane (G/PET nanopore) within the three steps assisted by the swift heavy ion irradiation and asymmetric etching technology. The inversion of the ion rectification effect was confirmed in G/PET nanopore while comparing with bare PET nanopore in KC1 electrolyte solution. By modifying the wall charge state of PET conical nanopore with hydrochloric acid from negative to positive, the ion rectification effect of G/PET nanopore was found to be greatly enhanced and the large rectification ratio up to 190 was obtained during this work. Moreover, the high ionic flux and high ion separation efficiency was also observed in the G/PET nanopore system. By comparing the "on" and "off' state conductance of G/PET nanopore while immersed in the solution with pH value lower than the isoelectric point of the etched PET (IEP, pH = 3.8), the voltage dependence of the off conductance was established and it was confirmed that the large rectification effect was strongly dependent on the particularly low off conductance at higher applied voltage.