A new mathematical periodic function, called VAP function, is proposed for the design of cam-follower mechanisms. With the VAP function, a given variable increases and decreases almost linearly from its minimum and maximum values, respectively, while the function and its derivate are smooth. As a case study, the design of a disc cam with a translating roller follower is considered, using four displacement laws: one based on a 4-5-6-7 polynomial displacement function, one based on a harmonic function, and two based on the VAP function. A comparative dynamic and kinematic analysis is shown. It is concluded that the cam-follower mechanism designed with the VAP function has smaller input torques, as well as lower pressure angles, than the one designed with harmonic function and that the one with 4-5-6-7 polynomial function