  In a spatially flat universe and for an interacting cosmology, we have
reconstructed the interaction term, $Q$, between a cold dark matter (DM) fluid
and a dark energy (DE) fluid, as well as a time-varying equation of state (EoS)
parameter $\mathrm{\omega_{DE}}$, and have explored their cosmological impacts
on the amplitudes of the first six cosmographic parameters, which allow us to
extract information about the kinematic state of the universe today. Here, both
$Q$ and $\mathrm{\omega_{DE}}$ have been modeled in terms of the Chebyshev
polynomials. Then, via a Markov-Chain Monte Carlo (MCMC) method, we have
constrained the model parameter space by using a combined analysis of geometric
data. Our results show that the evolution curves of the cosmographic parameters
deviate strongly from those predicted in the standard model when are compared,
namely, they are much more sensitive to $Q$ and $\mathrm{\omega_{DE}}$ during
their cosmic evolution. In this context, we have also found that different DE
scenarios could be compared and distinguished among them, by using the present
values of the highest order cosmographic parameters.
