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We study the influence of material parameters on elastic band gaps of one-dimensional functionally graded phononic crystals (FGPCs). By using plane-wave expansion, we calculate the first four band structures of FGPCs consisting of functionally graded materials (FGMs). These structures vary exponentially. We systematically study the influence of material parameters for four different FGPC models. Compared with traditional phononic crystals (PCs), the FGPC band gaps are clearly changed by FGMs. We also consider the influence of material composition, material properties and geometrical parameters on band gaps. Results show that different FGM properties can change the band structures remarkably. Our work can facilitate the design of vibration filters and noise insulators and provide more design freedom in engineering.