The present work considers two types of functionally graded material with the through-the-thickness distributions in the core of a sandwich beam. A numerical study of free vibrations of a sandwich beam was carried out using the finite element method (using ANSYS) and corresponding modal parameters (natural frequencies and mode shapes) were determined. Deflection of a simply-supported sandwich beam was investigated additionally. The effects of material parameters were discussed in detail. Two models developed enable to determine a single material parameter using the frequency or displacement responses, if the remaining material data or distribution laws are available. By means of the model I, the inverse problem solution allows to calculate the values of material properties at the bottom surface of the core if the properties are linearly distributed through-the-thickness of the core material and are known at the top surface. Application of the model II allows to determine the law of distribution of material properties through-the-thickness of the core if material properties are known only on the top and bottom surfaces.