The differential scalar polarizability Δα₀(ω) of the Ba⁺ S_(1/2)-to-D_(5/2) clock transition has a zero crossing near 481nm, which is measured to be 623.603 13(17) THz. From this measurement, we infer a ratio of reduced matrix elements ⟨ P_(3/2)|r|S_(1/2)⟩/⟨ P_(1/2)|r|S_(1/2)⟩=1.411 81(13), which provides a stringent test of atomic structure calculations and experimental determination of matrix elements. Additionally, it enables the construction of an accurate approximation to Δα₀(ω), valid for frequencies up to 450 THz, with only one reduced matrix element, ⟨ P_(1/2)|r|S_(1/2)⟩, appearing in the model's parameterization. We discuss the achievable accuracy of the model, the application to the assessment of blackbody radiation (BBR) shifts in ion-based clocks, and the applicability of the approach to other alkaline-earth ions.