We theoretically explore consequences of warming for predator-prey dynamics, broadening previous approaches in three ways: we include beyond-optimal temperatures, predators may have a type III functional response, and prey carrying capacity depends on explicitly modeled resources. Several robust patterns arise. The relationship between prey carrying capacity and temperature can range from near-independence to monotonically declining/increasing to hump-shaped. Predators persist in a U-shaped region in resource supply (=enrichment)-temperature space. Type II responses yield stable persistence in a U-shaped band inside this region, giving way to limit cycles with enrichment at all temperatures. In contrast, type III responses convey stability at intermediate temperatures and confine cycles to low and high temperatures. Warming-induced state shifts can be predicted from system trajectories crossing stability and persistence boundaries in enrichment-temperature space. Results of earlier studies with more restricted assumptions map onto this graph as special cases. Our approach thus provides a unifying framework for understanding warming effects on trophic dynamics.