Once again, there is little I can add to Hanna's high-quality explanation, apart from this:
This example is 100% correct. But to be absolutely clear about it, note that in general, we wouldn't say "Finish off your piece of the roasted pig" or "Finish off your slice of pie".Did you finish off the entire roasted pig? You must be so full!
Let's finish off this pie, have another piece!
However, we might say "Finish up your piece of pie [or pig, or turkey, or cake, etc.], so that we can finish off the whole pie [or pig, turkey, cake, etc.]."
So there is a slight usage difference between "finish up" and "finish off", in the context of eating food. But in both cases, you can omit the preposition "up" or "off" from the phrasal verb, and just say "finish" -- it doesn't really change the meaning.
On the other hand, if the meaning is "to kill", you should only say "finish (someone) off" -- "finish (someone) up" is impossible, and "finish (someone)" is possible as an imperative ("Finish him!" = "Kill him!"), but "Finish him off!" sounds more natural and colloquial.
P.S. One point on which I disagree with Hanna: "Let's finish up this discussion" is totally okay, and it means exactly the same thing as "Let's finish this discussion." The difference here is one of euphony -- i.e., which one "sounds more attractive"? -- and of course, that's very subjective. Some people would say that using "up" improves the rhythm of the sentence, and makes the sentence flow more easily from the lips; others would disagree. But either way, it's a question of aesthetics, not grammar.