The sentence in which the prepositional phrase functions as an adjective is the first one. "The man with the fancy yellow hat rode an old, beat-up bicycle."
This is the sentence that has a preposition, which is key to notice an prepositional phrase. The preposition is WITH. Prepositional phrases tend to end with a noun, pronoun, gerund, or clause, the "object" of the preposition.In this case it ends with HAT, a noun.
In this case it functions as an adjective as it is describing the subject of the sentence; "the MAN."