Identify the predicate of the sentence.

After Mary graduated, she was offered a promotion at her job

In a sentence or clause, the predicate refers to the part which expresses what is said of the subject. It usually consists of a verb with or without objects, complements, or adverbial modifiers.

There are the following types of the predicate: (1) The Simple Predicate, consisted in a verb in some tense, voice, person, number and mood, (2) The Compound Verbal Modal Predicate, which consists of a modal verb plus the Infinitive, (3) The Compound Verbal Aspect Predicate, which consists of a verb denoting the beginning, the continuation or the end of the action plus the Infinitive or the Gerund and (4) The Compound Nominal Predicate, that consists of a link-verb and a predicative (the nominal part) which can be expressed with different parts of speech.

In The particular sentence "After Mary graduated, she was offered a promotion at her job" the predicate would be the underlined: After Mary graduated, she was offered a promotion at her job".

This particular sentence could be considered as an example of the last type of predicate described above, The Compound Verbal Aspect Predicate, since it consists of a link-verb and a nominal part expressed with different parts of speech.