In the classical growth curve setting, individuals are repeatedly measured over time on an outcome of interest. The objective of statistical modeling is to fit some function of time, generally a polynomial, that describes the outcome's behavior. The polynomial coefficients are assumed drawn from a multivariate normal mixing distribution. At times, it may be known that each individual's polynomial must follow a restricted form. When the polynomial coefficients lie close to the restriction boundary, or the outcome is subject to substantial measurement error, or relatively few observations per individual are recorded, it can be advantageous to incorporate known restrictions. This paper introduces a class of models where the polynomial coefficients are assumed drawn from a restricted multivariate normal whose support is confined to a theoretically permissible region. The model can handle a variety of restrictions on the space of random parameters. The restricted support ensures that each individual's random polynomial is theoretically plausible. Estimation, posterior calculations, and comparisons with the unrestricted approach are provided.