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which answer identifies the participial phrase and the word it modifies in this sentence?
the dog barking in the yard for the last three hours is giving me a headache.

the participial phrase "for the last three hours" modifies "barking". <
the participial phrase "in the yard" modifies "barking".
the participial phrase "barking in the yard" modifies "dog".
the participial phrase "giving me a headache" modifies "hours".

The 3rd option, The participial phrase "barking in the yard" modifies "dog".

Explanation:

A participal phrase consists in a participle (past participle or present participle <verb+ing>) and a complement to the participle (nouns, pronouns, direct or indirect object, etc.). These phrases fuction as adjectives, it means, are intended to modify a noun or pronoun in the sentence.

In the sentence, "The dog barking in the yard for the last three hours is giving me a headache."

Barking is the participle (verb bark + ing)

in the yard for the last three hours is the complement of the participal phrase

The phrase is explaining why the dog is giving me a headache (because the dog has been barking in the yard for the last three hours), so is directly connected to the noun "The dog"  modifying it.

she is outspoken and direct