Can I claim a 17 year old who works as a dependent?

Sure. And his income effectively becomes yours, to be taxed at your rate. Under the "kiddie tax", that may actually happen even if you don't claim him and he files on his own...and pays less than you would have). Otherwise, it would be easy for higher earners/tax rate payers, to "pay" their children say 15K a year each and because of their low earnings, effectively pay no tax on it.