Systematic differences exist between values of the proton's electric form factors in the low-Q2 region extracted by different experimental and theoretical groups, though they are all making use of basically the same electron-proton scattering data. To try to understand the source of these differences, we make use of the analytically well-behaved rational (N=1, M=1) function, a predictive function that can be reasonably used for extrapolations at Q2→0. First, we test how well this deceptively simple two-parameter function describes the extremely complex and state-of-the-art dispersively improved chiral effective field theory calculations. Second, we carry out a complete re-analysis of the 34 sets of electron-proton elastic scattering cross-section data of the Mainz A1 Collaboration with its unconstrained 31 normalization parameters up to Q2=0.5(GeV/c)2. We find that subtle shifts in the normalization parameters can result in relatively large changes in the extracted physical qualities. In conclusion, we show that by simply using a well-behaved analytic function, the apparent discrepancy between recent form-factor extractions can be resolved.