We present a determination of the isospin-1/2 elastic πK scattering amplitudes in S and P partial waves using lattice quantum chromodynamics. The amplitudes, constrained for a large number of real-valued energy points, are obtained as a function of light-quark mass, corresponding to four pion masses between 200 and 400 MeV, at a single lattice spacing. Below the first inelastic threshold, the P-wave scattering amplitude is dominated by a single pole singularity that evolves from being a stable bound state at the highest quark mass into a narrow resonance that broadens as the pion and kaon masses are reduced. As in experiment, the S-wave amplitude does not exhibit an obviously resonant behavior, but instead shows a slow rise from threshold, which is not inconsistent with the presence of a κ/K_{0}^{⋆}(700)-like resonance at the considered quark masses. As has been found in analyses of experimental scattering data, simple analytic continuations into the complex energy plane of precisely determined lattice QCD amplitudes on the real energy axis are not sufficient to model-independently determine the existence and properties of this state. The spectra and amplitudes we present will serve as an input for increasingly elaborate amplitude analysis techniques that implement more of the analytic structure expected at complex energies.