Which of the following phrases is an example of an adverbial clause?

An adverbial clause is a dependent clause that functions as an adverb in a sentence, providing information about when, where, why, how, or to what extent an action takes place. The phrase "While she works" is an example of an adverbial clause because it begins with the subordinating conjunction "while," indicating the time during which the action of the main clause occurs. This clause cannot stand alone as a complete sentence and relies on additional information to convey a full thought, making it a dependent clause. The presence of "while" indicates that it modifies an action related to the main clause, fulfilling the role of an adverb in expressing the timing of the action.

In contrast, the other phrases either do not contain a subject and a verb in a subordinate structure or do not provide the necessary context to function as an adverbial clause. For instance, "She runs quickly" is a simple sentence with an adverb modifying the verb, "The dog barks" is also a complete sentence, and "To dance beautifully" is a non-finite clause that does not provide the same grammatical function or structure as an adverbial clause.