AP^® Precalculus

Course Outline

1. Functions
   This unit establishes the foundational language of functions that underpins the entire course. Students explore function notation, domain and range, and rates of change to describe graph behavior. They study composition of functions, transformations including shifts, reflections, and stretches, and absolute value functions. The unit culminates with inverse functions, building students' ability to analyze how functions relate to one another and setting the stage for deeper study of specific function families.
2. Linear Functions
   Building on the foundation of general functions, this unit provides a focused study of linear relationships. Students analyze and graph linear functions, develop models from real-world data, and use linear regression to fit models to data sets. Through this unit, students strengthen their ability to interpret slope as a rate of change and to use linear functions as the simplest yet most fundamental tool for mathematical modeling.
3. Polynomial and Rational Functions
   This unit extends students' understanding to higher-degree polynomials and rational functions. Students begin with complex numbers, then explore quadratic functions, power functions, and polynomial graphs including end behavior and the relationship between zeros and factors. Polynomial division and the Remainder Theorem provide tools for rewriting expressions, while rational functions introduce asymptotic behavior. The unit also covers inverses, radical functions, and modeling using variation.
4. Exponential and Logarithmic Functions
   Students explore exponential functions and their graphs, then are introduced to logarithms as the inverse operation to exponentiation. The unit covers logarithmic properties, solving exponential and logarithmic equations, and building mathematical models for growth and decay. Students learn to fit exponential models to real-world data, developing the ability to recognize and quantify exponential patterns in fields ranging from finance to biology.
5. Trigonometric Functions
   This unit introduces the trigonometric functions by beginning with angles and their measurement. Students study the unit circle and use it to define the sine, cosine, and tangent functions, then extend these definitions through right triangle trigonometry. The unit builds the conceptual foundation for understanding periodicity by connecting circular motion to function values, preparing students for the deeper exploration of trigonometric graphs and identities that follow.
6. Periodic Functions
   Building directly on the trigonometric foundations, this unit focuses on the graphical behavior of periodic functions. Students graph the sine, cosine, and other trigonometric functions, analyzing amplitude, period, phase shift, and vertical translation. The unit also introduces inverse trigonometric functions, which allow students to determine angle measures from known ratios—a key tool applied in later units for solving trigonometric equations.
7. Trigonometric Identities and Equations
   This unit deepens students' mastery of trigonometric relationships through identities and equation-solving techniques. Students work with fundamental identities, sum and difference formulas, double-angle and half-angle formulas, and the optional sum-to-product and product-to-sum formulas. They learn systematic approaches to solving trigonometric equations and apply trigonometric functions to model periodic real-world phenomena such as tides, temperatures, and sound waves.
8. Further Applications of Trigonometry
   This unit extends into advanced topics including the Law of Sines and Law of Cosines for non-right triangles (optional), polar coordinates, polar graphs, and the polar form of complex numbers. Students explore parametric equations and their graphs, gaining a new way to describe curves and motion. The unit also introduces vectors and their operations, connecting algebraic and geometric perspectives and aligning with the AP Precalculus framework's emphasis on parameters, vectors, and non-Cartesian representations.
9. Systems of Equations and Inequalities
   This unit covers methods for solving systems of linear equations in two and three variables, as well as systems of nonlinear equations and inequalities. Students explore partial fractions (optional) and are introduced to matrix operations, including solving systems using matrix inverses and Cramer's Rule. These algebraic tools provide powerful methods for handling multiple-variable problems encountered in science, engineering, and economics.
10. Analytic Geometry
    Students study the conic sections—ellipses, hyperbolas, and parabolas—from an analytical perspective. By deriving and using the standard equations for each curve, students learn to identify key features such as foci, vertices, and asymptotes. This unit connects algebraic representations to geometric shapes, reinforcing the course's emphasis on multiple representations and providing applications in physics, engineering, and astronomy.
11. Sequences and Counting Theory
    The course concludes with an exploration of sequences, series, and introductory combinatorics. Students study arithmetic and geometric sequences, their notations, and associated series formulas. The unit introduces counting principles and the Binomial Theorem, connecting patterns in algebra to foundational ideas in discrete mathematics that are essential for future coursework in calculus and statistics.

The course is 100% online. Students will have access to an online textbook. However, students must have access to a printer at home to print out worksheets and other materials.

Students should have access to a device with a camera to take photos of their work for submission.

In addition to the standard requirements, AP^® Precalculus students will need:

To take any of our courses, students must be familiar with opening a browser, navigating to a website, and joining a Zoom meeting.

Students must have a quiet place to study and participate in the class for the duration of the class. Some students may prefer a headset to isolate any background noise and help them focus in class.

Most course lectures and content may be viewed on mobile devices but programming assignments and certain quizzes require a desktop or laptop computer.

Students are required to have their camera on at all times during the class, unless they have an explicit exception approved by their parent or legal guardian.

Our technology requirements are similar to that of most Online classes.

We encourage (but do not require) students taking AP^® courses to take the AP Exams administered in May by the College Board. Being an Online School, we do not conduct AP^® Exams ourselves yet. See the College Board's website to find a local location near you, if your school doesn't offer these exams.

This course includes several timed tests where you will be asked to complete a given number of questions within a 60-90 minutes limit. These tests are designed to keep you competitively prepared but you can take them as often as you like. We do not proctor these exams, neither do we require that you install special lockdown browser.

In today's environment, when students have access to multiple devices, most attempts to avoid cheating in online exams are symbolic. Our exams are meant to encourage you to learn and push yourself using an honor system.

We do assign a grade at the end of the year based on a number of criteria which includes class participation, completion of assignments, and performance in the tests. We do not reveal the exact formula to minimize students' incentive to optimize for a higher grade.

We believe that your grade in the course should reflect how well you have learnt the skills, and a couple of timed-tests, while traditional, aren't the best way to evaluate your learning.