  We present the first three-parameter fit to a 3+1 sterile neutrino model
using 7.634 years of data from the IceCube Neutrino Observatory on
$\nu_\mu+\overline{\nu}_\mu$ charged-current interactions in the energy range
500-9976 GeV. Our analysis is sensitive to the mass-squared splitting between
the heaviest and lightest mass state ($\Delta m_{41}^2$), the mixing matrix
element connecting muon flavor to the fourth mass state ($|U_{\mu4}|^2$), and
the element connecting tau flavor to the fourth mass state ($|U_{\tau4}|^2$).
Predicted propagation effects in matter enhance the signature through a
resonance as atmospheric neutrinos from the Northern Hemisphere traverse the
Earth to the IceCube detector at the South Pole. The result is consistent with
the no-sterile neutrino hypothesis with a probability of 4.3 %. Profiling the
likelihood of each parameter yields the 90 % confidence levels: $
2.4\,\mathrm{eV}^{2} < \Delta m_{41}^2 <9.6\,\mathrm{eV}^{2} $ , $0.0081 <
|U_{\mu4}|^2 < 0.10$ , and $|U_{\tau4}|^2< 0.035$, which narrows the allowed
parameter-space for $|U_{\tau4}|^2$. However, the primary result of this
analysis is the first map of the 3+1 parameter space exploring the
interdependence of $\Delta m_{41}^2$, $|U_{\mu4}|^2$, and $|U_{\tau4}|^2$.
