spot said:
Thank you so much. It makes sense. Is there a test to see if a word is an adverb or not? In another word, if somebody insists that "after" itself in this context is an adverb and not a preposition, what can you say? Can you say that adverb stands alone?
Well, easily enough, an adverb, as its name states, adds to the meaning of a verb. That is, adverbs modify verbs, so for our example, if 'After' is modifying a verb, then it's an adverb, and if it isn't modifying a verb, then it's not an adverb:
After
killing the goose
Question: Is 'killing' a verb?
Answer: No. It's a gerund, a verbal noun.
Question: How do we know it's a gerund, because, you know, it does look like a verb?
Answer: words that end in -ing are not verbs. They are either present participles or gerunds, and only gerunds take objects:
killing (noun, gerund) + the goose (object of 'killing')
Moreover, gerunds, being nouns, act like nouns. They function as subjects and as objects.
Subject:
Killing the goose was tedious work.
Object of the verb: Did you
read Killing the goose?
Nouns also function as objects of prepositions:
Object of a preposition: Tell me
about killing the goose.
The same holds true for our example:
Object of a preposition:
After killing the goose
The thing about prepositional phrases is that they always express Where and When, which is a tell-tale sign that the phrase is functioning as an adverb.
Adverb Test
Q:
When did the farmer gasp? (
Adverb)
A:
After killing the goose. (
Adverb phrase)
In short, every word and phrase has two levels of representation: (1) its form (What it looks like) and (2) its function (What it does in the sentence).
The phrase 'After killing the goose' has the following representation:
Form: prepositional phrase
The phrase gets its name from the word that heads it. 'After' heads the phrase. We know 'After' is not an adverb: (a) it takes an object, and (b) that object is a noun ('killing').
Function: adverbial phrase
It answers the question, When?
Thank you. :-D I shall edit it. 
