A high-performance top-gated graphene field-effect transistor (FET) with excellent mechanical flexibility is demonstrated by implementing a surface-energy-engineered copolymer gate dielectric via a solvent-free process called initiated chemical vapor deposition. The ultrathin, flexible copolymer dielectric is synthesized from two monomers composed of 1,3,5-trimethyl-1,3,5-trivinyl cyclotrisiloxane and 1-vinylimidazole (VIDZ). The copolymer dielectric enables the graphene device to exhibit excellent dielectric performance and substantially enhanced mechanical flexibility. The p-doping level of the graphene can be tuned by varying the polar VIDZ fraction in the copolymer dielectric, and the Dirac voltage (V-Dirac) of the graphene FET can thus be systematically controlled. In particular, the V-Dirac approaches neutrality with higher VIDZ concentrations in the copolymer dielectric, which minimizes the carrier scattering and thereby improves the charge transport of the graphene device. As a result, the graphene FET with 20 nm thick copolymer dielectrics exhibits field-effect hole and electron mobility values of over 7200 and 3800 cm(2) V-1 s(-1), respectively, at room temperature. These electrical characteristics remain unchanged even at the 1 mm bending radius, corresponding to a tensile strain of 1.28%. The formed gate stack with the copolymer gate dielectric is further investigated for high-frequency flexible device applications.