What's your question?
So my mom and I were arguing cause I was in charge of my brother, and I made him do the dishes. My mom was mad because he had already cleaned his room and done it without being asked, but I said, "Do it anyways." We were arguing, and I said some very offensive stuff, and she just yelled, "Go away!" She threatened to ground me from hanging out with friends, and I want to apologize, but I don't know what to say.
First of all, good on you for wanting to apologize first. Apologizing to someone is always hard because you are, in a way, admitting you were wrong. But just getting the words out is all you need to do. Just say you're sorry. It may be awkward at first, but it is important to let her know that you do care. Just say the words.