Final answer:
The boldfaced prepositional phrase 'within the designated swimming area' in the given sentence is an adjectival phrase because it modifies the noun 'area'.
Step-by-step explanation:
The prepositional phrase in the sentence 'The sign cautioned swimmers to stay within the designated swimming area' functions as an adjectival phrase.
It modifies the noun 'area' by specifying which area the swimmers should stay within. A prepositional phrase acts as an adjectival phrase when it modifies a noun or a pronoun, giving more information about which one, what kind, or which one.
In comparison, an adverbial phrase will modify a verb, an adjective, or another adverb, answering questions like where, when, why, or under what conditions.The prepositional phrase "to stay within the designated swimming area" is an adverbial phrase in the sentence.
It modifies the verb "cautioned" by providing information about where swimmers should stay. Adverbial phrases answer questions like "where?" or "when?" and provide additional details about the verb, adjective, or adverb in the sentence.