We present new measurements of the inclusive forward-backward tt¯ production asymmetry, AFB, and its dependence on several properties of the tt¯ system. The measurements are performed with the full Tevatron data set recorded with the CDF II detector during pp¯ collisions at s√=1.96 TeV, corresponding to an integrated luminosity of 9.4 fb−1. We measure the asymmetry using the rapidity difference Δy=yt−yt¯. Parton-level results are derived, yielding an inclusive asymmetry of 0.164±0.047(stat+syst). We establish an approximately linear dependence of AFB on the top-quark pair mass Mtt¯ and the rapidity difference |Δy| at detector and parton levels. Assuming the standard model, the probabilities to observe the measured values or larger for the detector-level dependencies are 7.4×10−3 and 2.2×10−3 for Mtt¯ and |Δy| respectively. Lastly, we study the dependence of the asymmetry on the transverse momentum of the tt¯ system at the detector level. These results are consistent with previous lower-precision measurements and provide additional quantification of the functional dependencies of the asymmetry.