How to apologize to my mom?

Ok. So last Saturday my Mom and I got into a fight. A big, awful, horrible fight. Let me tell you a little bit about how the day was going. I wake up and everything is going well. My friend already left who had stayed the night before. I had breakfast and my mom and I where getting along. We where watching HGTV, and online looking for homemade headboard ideas together. I'm was getting ready to redo my room, and we where planning on redoing it for a long time now.Well my younger sister and I argue sometimes, just like other normal sisters do. My sister ( even for her young age) is really tough. It sounds silly, but she pushes me around and attacks me sometimes,(gets on top me and I can't get up and hits me or mostly pinches and scratches me). If I lay a hand on her, even to defend myself, I get in trouble. I can't help the way my little sister acts I'm nice to her, and we do get along at times but we fight almost everyday and it gets old. Anyways, my mom was going to take me last Saturday evening to get the paint. I got already to go. Then she yells for me ( in a you're in trouble voice). I go downstairs and her and my sister are sitting on the couch. I sit down across from them, and my mom starts yelling at us, mostly me. Then she gets in my face and starts yelling loudly, I walk away not wanting to fight and go upstairs. Then I have to use tge restroom, and she keeps yelling for me to come down, I told her that I was in the restroom and she didn't believe me. She comes up stairs, and repeatedly " whips my butt" she goes she goes downstairs and keeps saying very rude stuff about me and calling me names a mother shouldn't call her child. Then somehow the name calling and everything begins. I used the "F" word with her, called her a "B" word a dumbass, and only once a whore. She called me a ***** when we where downstairs, well I said sorry yesterday and she said ok but things are still the same how do I meaningfully apologize to her so she will accept it?