The Monte Carlo technique is applied to a study of the phase transitions and the critical behavior of the spin- Ising model on an fcc lattice with mixtures of two- (J2) and four - (J4) spin interactions. In the limit J2=0 the model exhibits a first-order transition. The transition remains of first order for J4J212, but a crossover to continuous transitions is found around J4J214-12 indicating that the model exhibits tricritical behavior. A modified mean-field theory is presented leading to an approximate description of the tricritical behavior in agreement with the Monte Carlo calculations. In the region of continuous transitions. 0<~J4J214, the critical exponent pertaining to the order parameter derived from the Monte Carlo data retains the Ising value, in accordance with the universality hypothesis. Our findings show that the four-spin interactions do not lead to nonuniversal critical behavior, contrary to the conclusions made by Griffiths and Wood from a series analysis.