AP Precalculus Study Guide

The course works through families of functions in order of increasing strangeness. It begins with polynomial and rational functions, where you study end behavior, zeros, asymptotes, and what makes a rate of change increasing or decreasing. It then turns to exponential and logarithmic functions, the tools for modeling growth and decay and for linearizing data — including the key relationship that ${\log }_b(xy)={\log }_{b}x+{\log }_{b}y$ and that logs and exponentials undo each other.

Next comes trigonometric and polar functions, which introduce periodic behavior: the unit circle, sinusoidal models of the form $f(x)=a\sin (b(x-c))+d$, identities, and graphing in polar coordinates. The course closes with functions involving parameters, vectors, and matrices, a wider view in which curves are traced by parametric equations, motion is described by vectors, and matrices act as transformations. Throughout, the recurring question is the same: given how something behaves, which function models it?

AP Precalculus is organized into four units, and the College Board frames the assessment around skills and modeling rather than a fixed per-unit percentage split, so treat the following as relative emphasis.

• Polynomial and Rational Functions — the foundational unit; its ideas about rates of change and function behavior recur everywhere.
• Exponential and Logarithmic Functions — heavily tested because exponential modeling and log manipulation appear across the exam.
• Trigonometric and Polar Functions — a large, technique-rich unit covering periodic models and identities.
• Functions Involving Parameters, Vectors, and Matrices — the capstone unit; on many exam configurations it is assessed more lightly than the first three, but it ties the others together.

The takeaway: the first three units carry most of the exam and reinforce one another, so build them solidly before spending heavily on the fourth.

Precalculus rewards seeing structure across function types. A routine that works:

1. Learn each family by its behavior, not just its formula. Know what a polynomial, an exponential, and a sinusoid each do — where they grow fastest, what stays constant, how they repeat — so you can recognize them in a word problem.
2. Make transformations second nature. Almost every function on the exam is a shifted, stretched, or reflected parent function; reading $a$, $b$, $c$, and $d$ off an equation should be instant.
3. Practice the modeling cycle out loud. The exam asks you to pick a model, justify it, use it to predict, and interpret the result in context. Talking through why a function fits is the skill being tested.
4. Work in mixed sets and review with full solutions. Reading a worked explanation for a problem you missed — including why each wrong choice was tempting — is worth more than three problems you already get right.

• Confusing the rules of logarithms and exponents — for example, treating $\log (x+y)$ as if it equals $\log x+\log y$.
• Losing track of the domain when simplifying rational functions or composing functions, and missing holes versus vertical asymptotes.
• Working in the wrong angle mode or mis-reading a sinusoid's parameters, especially the period $\frac{2\pi }{b}$ and the horizontal shift.
• Describing behavior vaguely. "It goes up" earns little; "the rate of change is positive and increasing, so the graph is concave up" earns the point.
• Ignoring units and context in modeling questions, where an uninterpreted number rarely earns full credit.

Every unit below has a focused practice set with full written explanations and a rationale for every wrong choice, plus a worked-solutions page you can read straight through. Start with a unit you're shaky on, then take a mixed set across the whole subject to pressure-test your ability to recognize which function a situation calls for. It's free and needs no account.