The stellar initial mass function (IMF) is commonly assumed to be an invariant probability density distribution function of initial stellar masses. These initial stellar masses are generally represented by the canonical IMF, which is defined as the result of one star formation event in an embedded cluster. As a consequence, the galaxy-wide IMF (gwIMF) should also be invariant and of the same form as the canonical IMF; gwIMF is defined as the sum of the IMFs of all star-forming regions in which embedded clusters form and spawn the galactic field population of the galaxy. Recent observational and theoretical results challenge the hypothesis that the gwIMF is invariant. In order to study the possible reasons for this variation, it is useful to relate the observed IMF to the gwIMF. Starting with the IMF determined in resolved star clusters, we apply the IGIMF-theory to calculate a comprehensive grid of gwIMF models for metallicities, [Fe/H] ∈ (-3, 1), and galaxy-wide star formation rates (SFRs), SFR ∈ (10-5, 105) M⊙ yr-1. For a galaxy with metallicity [Fe/H] < 0 and SFR > 1 M⊙ yr-1, which is a common condition in the early Universe, we find that the gwIMF is both bottom light (relatively fewer low-mass stars) and top heavy (more massive stars), when compared to the canonical IMF. For a SFR < 1 M⊙ yr-1 the gwIMF becomes top light regardless of the metallicity. For metallicities [Fe/H] > 0 the gwIMF can become bottom heavy regardless of the SFR. The IGIMF models predict that massive elliptical galaxies should have formed with a gwIMF that is top heavy within the first few hundred Myr of the life of the galaxy and that it evolves into a bottom heavy gwIMF in the metal-enriched galactic centre. Using the gwIMF grids, we study the SFR-Hα relation and its dependency on metallicity and the SFR. We also study the correction factors to the Kennicutt SFRK - Hα relation and provide new fitting functions. Late-type dwarf galaxies show significantly higher SFRs with respect to Kennicutt SFRs, while star-forming massive galaxies have significantly lower SFRs than hitherto thought. This has implications for gas-consumption timescales and for the main sequence of galaxies. We explicitly discuss Leo P and ultra-faint dwarf galaxies. The IGIMF grid is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/620/A39