Are the underlined words an adjective phrase or an adverb phrase? i wrote an essay about the origins of american folk music.
a. adjective phrase
b. adverb phrase

An adverb phrase describes how something is done: when, where, how (quickly, fast, etc). There is no adverb phrase in the following sentence:

I wrote an essay about the origins of American folk music. 

The correct answer is then: adjective phrase, which is : of American folk music.

or, another adjective phrase:  about the origins of American folk music.