i was homeschooled , and i went to public school in high school. i promise you its not worth it. i don't know what you believe in , but i would say pray about it. because home schooling is in my opinion alot better.
You could tell them, that while they are doing a good job on your home schooling education, you feel that you also need to be apart of a social learning environment as well. In my opinion, it is very important for teenagers to be in sociable environments as this is where you learn life skills on making friends and social etiquette. Hope this helps.