To investigate the effects of boundary conditions and geometric properties on dynamic characteristics of axially functionally graded beams, Gauss-Lobatto sampling and Chebyshev polynomials are used to discretize deformation fields of the beams, and the discrete governing equations are obtained by utilizing Chebyshev spectral method and Lagrange's equation. After employing projection matrices, classical as well as elastic boundary conditions are incorporated in the governing equations. The effects of various parameters, such as material gradient index, cross-sectional area, and attached tip mass on the vibration of the beams are analyzed. The results show that these effects differ for different boundary conditions. As the taper ratio increases, the first natural frequency of the cantilever beam increases simultaneously. While for the beams with other boundary conditions, their natural frequencies decrease. With the raising of the material gradient index, the first natural frequency of the cantilever beam increases firstly and then decreases, but other frequencies all increase. But for the fixed-fixed beam, its first two natural frequencies decrease, the third and the fourth increase. For the pinned-pinned beam, all natural frequencies increase with increasing material index. When the elastic support becomes stiffer, all natural frequencies increase with a step. The effect of the rotational spring is more pronounced than the translational spring when the elastic supports are of low stiffness. The attached tip mass makes the natural frequencies smaller and this effect appears more pronounced for higher modes.