The magnetic correlations within the cuprates have undergone intense scrutiny as part of efforts to understand high-temperature superconductivity. We explore the evolution of the magnetic correlations along the nodal direction of the Brillouin zone in La2−xSrxCuO4, spanning the doping phase diagram from the antiferromagnetic Mott insulator at x=0 to the metallic phase at x=0.26. Magnetic excitations along this direction are found to be systematically softened and broadened with doping, at a higher rate than the excitations along the antinodal direction. This phenomenology is discussed in terms of the nature of the magnetism in the doped cuprates. Survival of the high-energy magnetic excitations, even in the overdoped regime, indicates that these excitations are marginal to pairing, while the influence of the low-energy excitations remains ambiguous.