The evolution of additive manufacturing extends the feasibility of fabricating structures with new complex geometries with desired materials and functionality for different applications. For instance, cellular solids show great potential to be used in many applications. This study extends the classical power-law Gibson-Ashby model for cellular solids for graded lattice structures in two dimensions. Regular hexagonal, square, and triangular unit cells have been considered in which the material varies through the lattice structure based on a defined function. In addition to the material functionally grading, the geometry is also graded by changing the thickness of the unit cell walls making the possibility to improve the weaker material with a stronger geometry. First, the effective mechanical properties of the graded cellular solids are analytically evaluated considering the lattice structure as a beam framework based on the Euler-Bernoulli beam theory. Accordingly, a set of graded functions is fitted to the obtained analytical solution providing approximated continuous mathematical expressions for the variation of the mechanical properties along the graded direction which is useful for the purpose of low-computational-cost optimal designs. Next, by considering the graded lattice structures as continuum homogenized media, the equivalent mechanical properties of the structure are reported. Finally, a correction factor has been introduced to the Gibson-Ashby power law to modify the classical theory for the studied functionally graded lattice structures. While the calibrated Gibson-Ashby power law for the functionally graded lattice structure can predict the relative mechanical properties with good accuracy, the results also show that there is a great agreement between the analytical solution and the numerical analysis performed by the finite element method.