will the dvr still record a scheduled show if its powered off?
Depends what you mean by "powered off"
If the DVR is physically unplugged/has no source of power, then no it will not record.
If the DVR is powered, but turned "Off"/aka standby mode, yest it will record, assuming the RG is powered & the broadband signal feeding the RG is good, and that the DVR is still hooked up to the RG