Here's why:
* It describes a noun: You can say "an extravagant party," "extravagant spending," or "an extravagant lifestyle." It modifies these nouns to tell us more about them.
* It doesn't act as a verb, adverb, or pronoun. It's not an action, doesn't modify a verb or adjective, and doesn't stand in for a noun.
Let me know if you have any other words you'd like to identify the part of speech for!