We propose energy-saving algorithms to improve the energy efficiency of hybrid fiber coaxial (HFC) networks that support DOCSIS (Data Over Cable Service Interface Specification) 3.0 standard. The algorithms incorporate a traffic-aware approach and modify the operation statuses of the two primary network elements, the cable modem (CM) and the cable modem termination system (CMTS), dynamically. For the CM-side operation, we first propose a basic algorithm that can optimize the CMs' energy efficiency, regardless of significant increase of the packet delay and the number of operation changes (NoOC). We then propose two modified approaches to reduce the packet delay and NoOC while saving energy. Simulations with these algorithms show 37.5%-42.2% energy saving on the CM side, compared to the traditional case where the CMs' operation statuses are static. Next, we propose a CMTS-side energy-saving algorithm to find the actual connection mapping between the CM channels and CMTS ports. The proposed algorithm tries to support CM connections with minimal numbers of CMTS ports. To further improve the energy efficiency of the CMTS, we design a readjustment approach that can reorganize CM connections on CMTS ports based on their loads. Simulation results show that 31.08%-32.61% energy saving can be achieved on the CMTS in total. Hence, the proposed algorithms achieve effective energy saving on both the CM and CMTS sides.