I asked what you though belonged in a precalc course (thank you for your thoughtful responses) and then I explained a bit about our strange course sequencing.
Our Fall term of precalc is for seniors who need to be “calc ready” next year, but will not be taking calc with us. Some may not take calc at all.
Our Spring term is for (1) those same seniors and (2) our advanced juniors who just finished a two-term Algebra II/Trig course in January, and (3) a special group of juniors who were not advanced, but had done very well in all their math courses to this point, and just completed Algebra II, and are concurrently taking trig.
Basic idea: Fall is to repeat, with depth, some of the harder algebra from previous courses. It also hits a few gaps. Spring focuses on topics necessary for calc that are either underemphasized in earlier courses, or necessary but absent.
Both terms weave graphing-algebra connections together freely. Neither term brings in additional topics that do not support calculus. No units belong to probability, statistics, or any area that might be part of a course in “discrete mathematics.”
Topic List Fall: Systems of equations; 3 equations, 3 unknowns; conci sections (not in the algebra II course); polynomials and polynomial functions and their behavior; exponentials; and a chunk of trigonometry.
Topic List Spring: Repeat that chunk of trigonometry, and extend it into a major unit (for the seniors the extra bite let’s them improve. For the advanced juniors they just had some, easier, in the fall in their previous course, arranged for the regents. For the advancing juniors, this is tough. It is their first time, and it comes fast, furious, and at a moderate level of difficulty. Even if they miss some, they will, in their Regents class, get a not-as-hard, not-as-deep, not-as-fast rerun later in the Spring. Second topic will be exponentials and logs, much more detail, with application. Next will be rational functions, emphasizing graphing, asymptotes, endpoint behavior. Final unit will be series, sequences, and limits (about 5-6 weeks).
There is no new idea here. Nothing groundbreaking. The organizational piece of bringing the three groups together is challenging. But other than that, I’d say that this matches 80-90% of what you (readers who commented) thought should be there.
Thoughts? That trig transition needs work. Anything else?