A NON-TEACHER's ANSWER
(1) I believe that you are correct.
(2) "at the picture" is being used as an adverb.
(3) But books call "at the picture" a prepositional phrase because
it has a preposition ("at") and it has one or more other words ("the
(4) And, as you suggested, "at the picture" is used to modify the
verb "looks." Thus, we can say that the prepositional phrase
"at the picture" is being used as an adverbial modifier of the verb.
(5) If I say, "He looks carefully," then it is correct
to say that "carefully" is a true adverb. I think that your teacher
would be happier if you referred to "at the picture" as a prepositional
phrase. I think that your teacher would also accept "adverbial
phrase." But it might be better NOT to call "at the picture" an
"adverb." Many books suggest that we use the word "adverb" for
single words (such as "carefully"). Oh, by the way, if I say, "He looks
with great care," then we are back to a prepositional/adverbial
phrase, not a true "adverb."
Student or Learner