Tooold: really depends on the private school and the public school. Not all private high schools are traditional prep schools. Some regular old suburban public high schools are very good as well. Some may well offer a better education than some private high schools. Not sure it is accurate to say that private school teachers are more motivated due to no tenure. Teachers in general are more motivated by wanting to be a good teacher, not coasting because they have tenure.
OP: the public high school should be able to answer your question about what is covered but I don’t think common core affects the high school math sequence. Kids still learn algebra, geometry, pre-calc and calculus. I can’t believe California would change that.