Asking for recommendation letter to a Prof on New Year, deadline in a week I am applying for graduate school. The deadline is in one week. 

I need to ask for one more recommendation letter to a professor who I have not spoken to in a year. 

The problem is that today is December Twenty-Eighth! I thought about asking her over Christmas but I chickened out because I knew for sure that she would be busy. Now we are only three days away from new year and the deadline follows shortly after. For sure, it would be impolite to delay this any further. The thought of having her spending the new years day writing a letter for me really doesn't sit very well!

I am afraid that giving such a sudden short notice over such a busy period would be received very poorly by the professor. 

Should I do it? How do I do it politely? I really hope the prof do not react to this negatively.
Should you ask? Sure -- there's nothing to lose, and it doesn't seem like you have a choice. How to do it politely? Be very apologetic, state that you're aware of how annoying it is to ask for this so close to the deadline (and presumably during holidays), describe any reasons for asking so late, and note that you will understand if your professor is unable to write. 

Personally, I'd refuse such a request unless (i) I knew the student well, (e.g. someone in my lab) or (ii) I had a letter already on-hand that I could easily tweak and submit. ("Easily" doesn't mean it isn't annoying, though -- all the web-based recommendation forms still have a lot of boxes to tick, etc.!)