I honestly believe that for most kids (although there are exceptions), home schooling isn't the best idea. Especially if your daughter is shy, high school may help her meet different types of people - it may help her fight her own barrier and break out of her shell. Home school may do the opposite.
She doesn't have to attend a regular public school. I am seventeen, and when I was fifteen I was an exchange student to France. I attended an arts & academics magnet school. And now I'm going to start up an accelerated college program early. There are so many schooling options beyond regular public school and homeschooling. You may also want to check private schools, many offer financial aid.
High school may not be easy, but it will help her adapt to different situations as she grows up. She may have to deal with people she does not like (or maybe not - I think the bad high school situations are grossly exaggerated in the media), but she will also encounter that in her future work. I think it will prepare her for her future not only socially, but academically. Learning how to manage her time properly and employ good work habits.
Chances are, she will have to deal with rudeness and snobbiness at some point in her life, and though I'm sure she will rarely encounter these attitudes, perhaps being in such an environment will help her learn how to manage it and grow stronger.
You may also want to think about college and how they will view the different school settings. If she doesn't believe high school gives her enough of an academic challenge, she can speak with a school counselor and she may be able to foster her talents through school or other outlets.
There are LOTS of options. Good luck!
