Purpose <p>The present work aims to develop and apply a simple nine-nodded Lagrangian isoparametric element to explore the free vibration characteristics of power-law, exponential, and sigmoid functionally graded (FG) plates under different boundary conditions and material gradient. Additionally, the study aims to comprehend how several parameters, including material distribution, side-to-thickness ratio, aspect ratio, mode shape, and boundary conditions, affect the natural frequency of the P-FGM, E-FGM, and S-FGM plates.</p> Methods <p>To attain this objective, a nine-node quadrilateral isoparametric finite element model with five degrees of freedom is then developed and formulated based on an improved FSDT. The present theory refines the traditional Mindlin–Reissner theory by incorporating a parabolic shear strain distribution, leading to a more accurate depiction of shear strain throughout the structure’s thickness. The material properties of the FG plates are assumed to undergo continuous variation across their thickness, following the power law (P-FGM), exponential (E-FGM), and sigmoid (S-FGM) distributions. The governing equations are obtained through Hamilton’s principle and solved using the finite element method.</p> Results <p>The findings from this study are compared with previously published results in the literature, emphasizing the accuracy, fast rate of convergence, and simplicity of our finite element model. The results demonstrate a significant alignment with solutions derived from other high-order theories, reaffirming the precision of the proposed model. Moreover, a comprehensive parametric study is also conducted to display the impact of the material distribution, side-to-thickness ratio, aspect ratio, mode shape, and boundary conditions on the natural frequency of the P-FGM, E-FGM, and S-FGM plates.</p> Conclusions <p>It is found that the increasing rate of the natural frequency by adding the exponential index (<i>n</i>) belonging to E-FGM is the most noticeable, followed by S-FGM and P-FGM due to the rising <i>n</i> leads to increasing stiffness of E-FGM exponentially, however, the stiffness of P-FGM and S-FGM depend on power law index (<i>k</i>) in the form of power. In addition, the study reveals that, for a square P-FGM (Al/Al₂O₃) plate, the natural frequency decreases with increasing power law index, with a reduction of up to 25% observed as the power law index increased from 0 to 10. The influence of boundary conditions is also notable, as fully clamped plates showed a 50% higher natural frequency than simply supported plates under the same material and geometric conditions. Finally, the results reveal that the material grading function significantly influences the free vibration behavior of functionally graded plates.</p>