A computational study of two morphing, power-extracting flapping foils (a NACA 0015 at [Formula: see text] and a flat plate at [Formula: see text]) is conducted by considering three strategies: morphing of the leading edge (LE) or trailing edge (TE) alone, and morphing of both LE and TE simultaneously. The morphing variables are the phase shift and frequency of the edge deformation. All cases take into account the power required to morph the foils. The independently morphing LE/TE edge cases show regions of phase angle and frequency that yield improved efficiency over the rigid baseline of up to 16.8% for the NACA 0015 foil and 22.6% in the flat-plate case. The combined LE and TE cases further increase the efficiency, up to 29.7% for the NACA 0015 foil and 36.2% in the flat-plate case. Three physical mechanisms are identified that lead to efficiency increases in the various cases through the interaction between the morphing foil and shed vortex structure. These are the change in projected area during the plunging stroke, the variation of leading-edge vortex shedding timing during the cycle, and the proximity of the shed vortex structure to the foil during the pitching portion of the stroke.