We systematically investigated the tunable dynamic characteristics of a broadband surface plasmon polariton (SPP) wave on a silicon-graded grating structure in the range of 10–40 THz with the aid of single-layer graphene. The theoretical and numerical simulated results demonstrate that the SPPs at different frequencies within a broadband range can be trapped at different positions on the graphene surface, which can be used as a broadband spectrometer and optical switch. Meanwhile, the group velocity of the SPPs can be modulated to be several hundred times smaller than light velocity in vacuum. Based on the theoretical analyses, we have predicted the trapping positions and corresponding group velocities of the SPP waves with different frequencies. By appropriately tuning the gate voltages, the trapped SPP waves can be released to propagate along the surface of graphene or out of the graded grating zone. Thus, we have also investigated the switching characteristics of the slow light system, where the optical switching can be controlled as an “off” or “on” mode by actively adjusting the gate voltage. The slow light system offers advantages, including broadband operation, ultracompact footprint, and tunable ability simultaneously, which holds great promise for applications in optical switches.