In this paper a Metal-Insulator-Metal (MIM) plasmonic absorber consisting of asymmetric double bars with bent arms located on top of a silica layer coated on a metal film is proposed, and its resonant features are analyzed. The suggested structure supports both Fano and dipole perfect absorption resonances at the Near-Infrared Region (NIR). The asymmetry introduced into the structure can be induced by changing the bending angle or rotation angle of one of the antennas while the other one remains fixed. Simulation results demonstrate that by applying both asymmetry factors to the structure, one can have two individual Fano peaks at the same time. It is shown that the magnitude, central wavelength and line-width of the Fano peaks are adjustable by controlling the geometrical parameters of the structure. It is also indicated that, the quality factor (Q-factor) of the Fano resonance is inversely related to the degree of asymmetry introduced into the structure. According to the simulations, an ultra-narrow resonance peak with a bandwidth of 1.87 nm at the wavelength of 710 nm (corresponding to a Q-factor of 387) can be obtained by controlling the geometrical parameters. It is also discussed that, the absorptivity of Fano and the dipole peaks can be adjusted inversely, by manipulating the grapheme chemical potential. The ratio of the absorptivity to the chemical potential of graphene about-275 %/eV and 226 %/eV is calculated for the Fano peak and dipole peak, respectively. Accordingly, the presented structure is an adjustable NIR absorber with a fully tunable absorption spectrum which can be utilized in various applications from tunable reflectors and photo-detectors to ultra-narrowband and broadband optical modulators.