The unique mechanical and electronic properties of graphene, a one-atom-thick sheet of carbon atoms arranged in a honeycomb lattice, make it a promising material for applications in a number of new technologies including ultra capacitors, photvoltaic cells, and sensors. Single layer graphene by itself is a gapless semiconductor, however when two layers of graphene are brought together to form AB-stacked bilayer graphene and an electric field is applied perpendicular to the graphene plane the band structure develops a gap proportional to the magnitude of the electric field. This tunable band gap could have a number of applications in electronics. We have studied the static and dynamic screening of gapped AB-stacked bilayer graphene. Unlike previous works we use the full 4-band model instead of the simplified 2-band model. We find that there are important qualitative differences between the dielectric screening function obtained using the simplified 2-band model and the 4-band model. In particular, within the 4-band model, in the presence of a band-gap, the static screening exhibits Kohn anomalies that are absent within the simplified 2-band model. Moreover, using the 4-band model, we examine the effect of trigonal warping on the screening properties of bilayer graphene. We also find that the plasmon modes have qualitatively different character in the 4-band model compared to the ones obtained using the simplified 2-band model. The work presented here is based on work that has been published in Physical Review B .