Analytical and computational solutions are presented for the sliding frictional contact problem between a bidirectional functionally graded material (BFGM) half-plane and an arbitrarily shaped rigid punch. The plane strain elasticity is regarded in the solutions. Independent shear modulus variations are imposed through the lateral and the thickness directions of the elastic medium. Fourier transformation techniques are utilized in the derivation of the field quantities. The quartic characteristic equation is solved using the Ferrari's method. A singular integral equation (SIE) of the second kind is then formulated considering a known displacement gradient on the contact surface. Solution of the SIE is performed for the flat and triangular punch profiles via an expansion-collocation method, quadrature integration techniques and a recursive integration method for the Cauchy integral. Finite element analyses (FEA) the same contact problems are also implemented choosing the augmented Lagrange method as the iterative algorithm. The parametric analyses indicate the reliability and validity of both procedures developed in this study. Extra numerical results are generated to show the influences of the problem parameters on the surface stresses and contact forces. It is observed that the contact responses can be influentially controlled upon imposing a bidirectional gradation in an elastic material.