Precalculus does seem to be a hodge-podge of different ideas, doesn’t it? In talking to students, I’ve explained the reason for this as this is the last chance a math student has of picking up certain skills before moving on fully to calculus. I taught 6th grade math one year, and I noticed that the book (Saxon Math 76) was all over the place in the subject matter. But I realized that the reason for this was that the students would be starting prealgebra next, and this was the last chance to hone the skills they’d learned before moving on. I always pitch the hodge-podge nature of precalculus as a positive to my students: “Hey, if you don’t like what we’re doing, there’s always a chance you’ll like what we do next.” And this has been true of a number of students. A few of them who don’t like the insight required for algebra absolutely love the rigor and plodding nature of matrices and/or statistics.
But there are some major themes running through precalculus which give us a few categories to classify the disparate subjects into. I talk about precalculus as being a little bit Algebra III (into which I group matrices), a little bit Trigonometry (which I also call Geometry’s big brother), a little bit Complex Numbers (into which I group polar graphing and vectors), and a little bit Statistics/Probability. So those are the four major themes I’d identify.