What do i do when my mom is disappointed in me?

i went to the movies today with my friend and since im not allowed to go alone (even though im 14) my mom came with us. we saw a movie and i told my mom i thought she wasnt watching the same moive and that i didnt really want to her to , and she got really mad. i know that was kind of rude, but i am 14 and i don't want to hang out with my mom at the mall. now my mom is disappointed in me and is "hurt" and she wont accept my apologies. on the fourth of july i want to spend the night at my best friends house. ann im supposed to get cake mix and stuff so we can bake..but my mom tends to over react so i don't think she will say yes. how can i get her to forgive me?? i really want to do ths on the fourth of july, and if i ask while she is angry i know she will say no.