Shafts loaded by axial compressive constant forces constitute an area of considerable technical importance. The transverse vibration of such shafts is the subject of the current work. Occasionally the shafts are tapered and of interest is the effect of employing functionally graded materials (FGM), with properties varying in the axial direction, on the buckling load and lowest natural frequency. The shaft cross section is circular and two types of taper are treated, namely, linear and sinusoidal. All shafts have the same volume and length and are subjected to a constant axial force below the static buckling load. Euler-Bernoulli theory is used with the axial force handled by a buckling type model. The problems that arise are computationally challenging but an efficient strategy employing MAPLE®’s two-point boundary value solver has been developed. Typical results for a linear tapered pin-pin shaft where one end radius is twice the other, and the FGM model varies in a power law fashion with material properties increasing in the direction of increasing area, include doubling of the buckling load and first bending frequency increase of approximately 43%, when compared to a homogeneous tapered shaft. For the same material and boundary conditions, a sinusoidal shaft, with mid-radius twice the value of the end ones, increases the buckling load by about 118% and the first frequency by 26%, when compared to a homogeneous sinusoidal shaft.