A leading intellectual challenge in evolutionary genetics is to identify the specific phenotypes that drive adaptation. Enzymes offer a particularly promising opportunity to pursue this question, because many enzymes' contributions to organismal fitness depend on a comparatively small number of experimentally accessible properties. Moreover, on first principles the demands of enzyme thermostability stand in opposition to the demands of catalytic activity. This observation, coupled with the fact that enzymes are only marginally thermostable, motivates the widely held hypothesis that mutations conferring functional improvement require compensatory mutations to restore thermostability. Here, we explicitly test this hypothesis for the first time, using four missense mutations in TEM-1 beta-lactamase that jointly increase cefotaxime Minimum Inhibitory Concentration (MIC) similar to 1500-fold. First, we report enzymatic efficiency (k(cat)/K-M) and thermostability (T-m, and thence Delta G of folding) for all combinations of these mutations. Next, we fit a quantitative model that predicts MIC as a function of k(cat)/K-M and Delta G. While k(cat)/K-M explains similar to 54% of the variance in cefotaxime MIC (similar to 92% after log transformation), Delta G does not improve explanatory power of the model. We also find that cefotaxime MIC rises more slowly in k(cat)/K-M than predicted. Several explanations for these discrepancies are suggested. Finally, we demonstrate substantial sign epistasis in MIC and k(cat)/K-M, and antagonistic pleiotropy between phenotypes, in spite of near numerical additivity in the system. Thus constraints on selectively accessible trajectories, as well as limitations in our ability to explain such constraints in terms of underlying mechanisms are observed in a comparatively "well-behaved" system.