The purpose of the verb phrase in the sentence "You are invited to stay for dinner" is the predicate of the sentence.
The predicate refers to the part of a sentence or clause that makes clear what the subject does or is. To put it simply, the predicate is everything that is not the subject. It contains the verb of the sentence, that is, the predicate consists on the action that the person in the subject carries out.
In addition to the verb, a predicate can contain direct objects, indirect objects, and various kinds of phrases.