Transmission rate adaptation in wireless devices provides a unique opportunity to trade off data service rate with energy consumption. In this paper, we study optimal rate control to minimize transmission energy expenditure subject to strict deadline or other quality-of-service (QoS) constraints. Specifically, the system consists of a wireless transmitter with controllable transmission rate and with strict QoS constraints on data transmission. The goal is to obtain a rate-control policy that minimizes the total transmission energy expenditure while ensuring that the QoS constraints are met. Using a novel formulation based on cumulative curves methodology, we obtain the optimal transmission policy and show that it has a simple and appealing graphical visualization. Utilizing the optimal ldquoofflinerdquo results, we then develop an online transmission policy for an arbitrary stream of packet arrivals and deadline constraints, and show, via simulations, that it is significantly more energy-efficient than a simple head-of-line drain policy. Finally, we generalize the optimal policy results to the case of time-varying power-rate functions.National Science Foundation (U.S.) (NSF ITR Grant CCR-0325401)United States. Defense Advanced Research Projects Agency (University of Illinois Grant F49620-02-1-0325)United States. Office of Naval Research (ONR Grant N000140610064)United States. Army Research Office (ARO Muri Grant W911NF-08-1-0238)