A headlamp for a motor vehicle, this headlamp comprising at least one optical module capable of emitting a plurality of light beams, including a low beam and at least one additional light beam. The optical module is equipped with a cover plate which can be moved between at least two positions, this movable cover plate being configured along a shaft mounted so as to rotate in the optical module and driven in rotation at least by a motor. The shaft comprises at least one first cut-off member configured as a double-edged cover plate providing an oblique cut-off for emission of the low beam, and at least one second cut-off member configured as a wall of substantially constant height, angularly offset relative to the first cut-off member and providing at least one substantially vertical cut-off line for emission of a high beam.