A novel design of a dynamic electromagnetically induced transparency (EIT) metamaterial is proposed. The metamaterial consists of two kinds of cut-wire metal resonators with a graphene strip placed between them. The destructive interference between the two resonators gives rise to a transparency window. By varying the Fermi energy of the graphene through external gating, the EIT effect can be manipulated dynamically and the maximum modulation depth can reach up to 81%. The efficient modulation is controlled only in the EIT window with slight changes in transmission dips, which may avert the additional noises at adjacent frequencies in the modulation process. Moreover, the actively controlled slow-light effect and sensing performances can also be realized as the corresponding EIT window is modulated. This work provides a strategy to achieve a tunable EIT effect in a metal-graphene hybrid structure and exhibits potential applications in designing terahertz modulators, environmental sensors and slow-light devices.