Claire V,
Something tells me you might be unplugging the wrong box. I might be wrong. But I have never seen a modem being unplugged cause a TV to turn off. If your tech plugged the TV power cord into the switched outlet behind the TV descrambler, pulling the cord to the descrambler would cause the TV to turn off. The descrambler is for TV signal, not internet.
I did a Google search and find no information about a combined box that does TV and PC internet both. Asked a couple cable techs I used to work with and a buddy that is a current satellite contractor. They've never heard of it either.
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