The purpose of expository writing is to inform, explain, describe, or define the author's subject to the reader.
You'd use this type of writing when you want to explain a certain term or idea to your readers, or just to give them an opportunity to learn something more about it, and for them to be informed/understand something they didn't prior to that.
I think it's a thesis statement
Most of the time is drawing with no purpose but it can mean drawing beautifully as well.
