How do I get my family to leave me alone and quit saying I have "mood swings"?

My mom will always say she will do something for me and then change her mind, and my dad will make me think he is going to do something for me and it will never happen. Then they will say I have mood swings and that I'm mean to them just because I get mad at them for things anyone would be mad about. They expect to be able to torture me in any way and me not get mad about it. For example, my mom said she was going to go shopping with me saturday and then changed her mind and said my sister was taking me. My sister wouldn't take me because we are direct opposites who never get along and hate each other and she didn't want me to come. I got mad at my sister and my mom said I was the antagonist because she was only around to see me yelling at my sister. My mom always believes the other person over me, no matter who it is. She never even listens to my side of the story. My mom says she is taking me to the doctor to get medicine for my "mood swings". Same with my brother. He goes "you have mood swings you need medicine for that." I don't have mood swings. Anyone else would be completely insane by now. But someone I manage to keep a little of my composure left. Every day I stay at home with my dad and my brother, because my mom won't let me get a job. At the beginning of the summer she said "I'll take you around town to see if anyone will hire you" when I asked her if I could get a job. But now she says I can't get a job. Every day I have to put up with my brother and my dad while my mom goes to work, and when my mom isn't home my brother and my dad act like total idiots. What am I supposed to do to make my mom see any other 15 year old girl would be crazy by now and that I am peaceful? How do I convince her I don't have mood swings and get her to shut up already?