Every Pre-Calculus I have examined starts with functions in general, then polynomial and rational functions, followed by exponential and logarithmic functions and Trigonometry, and ending with sequences, summations, probability, and limits. Some have vectors and binomial theorem thrown in towards the end. I would rather begin with number theory, sequences, summations and binomial theorem rather than with polynomials. Is there a pedagogical or theoretical reason why this order is so common?
Great question. I think that precalculus is usually a wasteland for some random classical techniques that used to be required to study calculus.
I think that you can take a completely different approach if you're interested and incorporate data and matrices in place of a bunch of algebraic manipulation.
I just taught a precalculus class and rather than the standard topics, we focus on using functions to model data and some introductory linear algebra. I believe this is much more relevant in the computer age.
If you're interested I'm happy to share more(here's a link to a talk I just gave on the class), but we used the MAA textbook by Sheldon on "Functions, Data, and Models" as a foundational text to solve problems with Python in Jupyter notebooks. The material was broken into a few sections:
Introductory work with functions
- Key Features of linear, quadratic, and exponential functions as recursive sequences through population models
- Relate recursive forms to closed forms
- Intro to harmonic motion and key features
- Use trig to model some basic periodic phenomena
Intro with data
- measures of center
- linear and polynomial regression
- non-linear regression and logarithms
- machine learning and regression
- images as matrices through dimension and definition of matrix
- add, subtract, scalar multiply matrices and application to image manipulation
- systems of equations and matrices
- Transpose and regression again
- Trig and rotation by matrix multiplication
Either way, I think that using functions(linear, quadratic, higher polynomials, exponentials, trigonometric) to model real world situations, a little on data analysis, and some introductory linear algebra and applications is an appropriate precalculus course. Take it further depending on how advanced your students are.
Precalculus is not just a course that can be taken before calculus. Almost half a century ago, Precalculus, an American invention, started to replace College Algebra as a preparation for Calculus and in fact, if College Algebra seems to have made a comeback, Precalculus remains almost invariably a requirement for calculus.
In that case then, given that, for instance, "Functions of various kinds" are presented by Wikipedia as ""the central objects of investigation" in most fields of modern mathematics", given that the calculus is the calculus of functions, and given that the calculus is the first tool in the "hard" sciences, it seems to me that the best way to prepare students for the calculus is indeed with a study of power functions followed by "polynomial and rational functions, followed by exponential and logarithmic functions" and followed by trigonometric functions.
The problem I have with Precalculus is that it is a bag of tricks. However, this need not be the case if one is willing to use Laurent Polynomial Approximations (i.e. asymptotic expansions using power functions as gauges) to investigate said functions.
I think the main pedagogical reason is this: Calculus students need to be familiar with the elementary functions, and with the idea of functions and their graphs in general. Not every precal class gets through the desired curriculum, due to various reasons. Thus, if something is going to get cut off at the end, because of snow days or time lost for whatever reason, most teachers would rather miss out on sequences or probability than on trigonometry. Students can succeed in Cal I just fine without sequences, but if they miss out on trig, they're up a creek.