New graphical methods using the Delano y-ybar diagram are shown to yield simple alternative derivations of the M² beam quality factor for Gaussian-like laser beams. Using M² as a scaler for the Lagrange optical invariant, powerful methods of interpreting and calculating the beam factor become apparent. Elegant methods of determining M² using strategic beam sample locations are also developed, taking advantage of the property that a freely propagating beam trace in the Gaussian beam y-ybar plane is a straight line. Application to non-Gaussian beams is explored.