In recent years, modified gravity theories, especially f(R) gravity, have attracted considerable interest. This work derives exact vacuum solutions for Bianchi type-I spacetimes in the framework of metric f(R) gravity. A key assumption is that the expansion scalar \(\theta \) is proportional to the shear scalar \(\sigma ^{2}\) , leading to solutions of the form \(X=Y^{m}\) , where X and Y denote metric coefficients. The Ricci scalar and associated physical quantities are analyzed for these models. It is found that the average Hubble parameter H(T) decreases with cosmic time, approaching equilibrium, consistent with standard cosmology. The expansion scalar \(\theta (T)\) attains very high values near \(T\rightarrow 0\) , corresponding to the early inflationary or radiation-dominated epochs. The shear scalar \(\sigma ^{2}(T)\) diminishes with time, reflecting a trend toward isotropy, while the volume scale factor V(T) grows, signifying continuous expansion. Moreover, the function f(R) behaves regularly at low curvature ( \(R\rightarrow 0\) ) and reduces to \(f(R)\rightarrow R\) in the high-curvature limit. To reinforce the theoretical investigation, we carry out a Markov Chain Monte Carlo (MCMC) analysis employing the latest Baryon Acoustic Oscillations (BAO), Cosmic Chronometer (CC), and Hubble parameter observations to restrict the model parameters. The resulting best-fit estimates demonstrate strong agreement among the different datasets, with the cosmological constant parameter \(\psi \) exhibiting stability across all observational probes.
