The stability of a graphene field effect transistor (GFET) is important to its performance optimization, and study of hysteresis behavior can propose useful suggestions for GFET fabrication and optimization. In this work, a back-gate GFET has been fabricated and characterized, which is compatible with the CMOS process. The stability of a GFET in air has been studied and it is found that a GFET's electrical performance dramatically changes when exposed to air. The hysteresis characteristic of a GFET depending on time has been observed and analyzed systematically. Hysteresis behavior is reversed at room temperature with the Dirac point positive shifted when the GFET is exposed to air after annealing.