- The preposition for and of have a special function in English. It can indicate a subject of the action indicated by a to infinitive in a sentence which starts with expletive it (dummy it, dummy pronoun). For example:
A: It is important to attend the seminar. (To attend the seminar is important.)
B: For me or you?
A: It is important for you to attend the seminar.
A's first sentence is ambiguous and doesn't indicate who should attend the seminar. A's second sentence is very clear and indicates who should.
For you in the above example is a prepositional phrase to indicate a subject of to infinitive.
Note: You should use of when you use adjectives (good, nice, brace, stupid, kind, clever, etc.) that describe human nature, e.g., "It was nice of you to to help me."
- "making cakes" in the example is a dangling participle which is
Participles of verbs are often used to introduce subordinate clauses,
which give extra information about the main part of a sentence (known
as the main clause). It’s important to use participles in subordinate
clauses correctly. The participle should always describe an action
performed by the subject of the main part of the sentence.
For example:
A: I am busy.
B: Doing what?
A: I am busy making cakes.
As you can see, A's first sentence is not clear on why A is busy. But A's second sentence clearly states A is busy because A is making cakes. The dangling participle gives extra information about the subject's action or status.
I would not call it "complement of adjective" as it doesn't complement the adjective busy, but actually it complements the verb and its predicate am busy.
Some grammar books call it "participle clause" and "absolute construction (clause)". Whatever it is called, it is very important to note that the participle (making cakes) doesn't have a normal or usual syntactical relation with other words or sentence elements. It should always be a non-finite clause. You can't use make, to make, made, have made, had made, etc. in place of making.
