Mixed modes observed in red giants allow for investigation of the stellar interior structures. One important feature in these structures is the buoyancy spike caused by the discontinuity of the chemical gradient left behind during the first dredge-up. The buoyancy spike emerges at the base of the convective zone in low-luminosity red giants and later becomes a glitch when the g-mode cavity expands to encompass the spike. Here, we study the impact of the buoyancy spike on the dipolar mixed modes using stellar models with different properties. We find that the applicability of the asymptotic formalisms for the coupling factor, q, varies depending on the location of the evanescent zone, relative to the position of the spike. Significant deviations between the value of q inferred from fitting the oscillation frequencies and either of the formalisms proposed in the literature are found in models with a large frequency separation in the interval 5-15 mu Hz, with evanescent zones located in a transition region that may be thin or thick. However, it is still possible to reconcile q with the predictions from the asymptotic formalisms, by choosing which formalism to use according to the value of q. For stars approaching the luminosity bump, the buoyancy spike becomes a glitch and strongly affects the mode frequencies. Fitting the frequencies without accounting for the glitch leads to unphysical variations in the inferred q, but we show that this is corrected when properly accounting for the glitch in the fitting.