A new approach to the investigation of vdW type of equations of state (EOS) is developed by embedding a vapor pressure equation and a saturated liquid volume equation into vdW type EOS, which results in a new function A(S)(T). The A(S)(T) possesses the properties of an attractive parameter A(T), and if an EOS is accurate in the whole PVT space, then its numerical value equals A(T). As a useful tool for investigating EOS, the A(S)(T) has been used to make comparisons among RKS, PRSVII, PT and ALS EOS, and to indicate where the shortcomings of the EOS are coming from. Based on the A(S)(T), a possible way to develop a real predictive equation of state is also suggested.