We present a theoretical study of high-order harmonic generation (HHG) driven by; ultrashort optical pulses with different kind of chirps. The goal of the present work is; to perform a detailed study to clarify the relevant parameters in the chirped pulses to; achieve a noticeable cut-off extensions in HHG. These chirped pulses are generated using; both linear and nonlinear dispersive media. The description of the physical mechanisms; origin responsible for this extension is, however, not usually reported with enough detail; in the literature. The study of the behaviour of the harmonic cut-off with this kind of; pulses is carried out in the classical context, by the integration of the Newton-Lorentz; equation complemented with the quantum approach, based on the integration of the time; dependent Schrodinger equation in full dimensions (TDSE-3D).