Contemplanda
Contemplanda course syllabus
AP® Precalculus
A Contemplanda online course.
This course follows the College Board's published AP Precalculus framework.
AP® Precalculus is a course about functions as models of change. The College Board's framework covers polynomial, rational, exponential, logarithmic, trigonometric, and polar functions, and, in Unit 4, parametric, implicitly defined, and vector-valued functions and matrices. Each family of functions is used to model how quantities change together.
The College Board designed AP Precalculus to prepare students for college-level calculus and to give a grounding for other mathematics and science courses. Because it may be a student's last high school mathematics course, the College Board also structured it as a coherent capstone in its own right.
The College Board describes AP Precalculus as the equivalent of a first-semester college precalculus course.
The College Board's AP Precalculus framework has four units. Units 1, 2, and 3 cover what colleges and universities typically expect for college credit or placement, and they are assessed on the AP Exam. Unit 4 covers additional topics that schools may include, and it is not assessed on the AP Exam.
This course teaches all four College Board units, with a lesson for every topic.
Course at a glance
- Course title
- AP Precalculus
- Publisher
- Contemplanda
- Framework
- Follows the College Board's published AP Precalculus framework
- Structure
- 12 units over two semesters: 8 content units (1A, 1B, 2A, and 2B in semester 1; 3A, 3B, 4A, and 4B in semester 2) and 4 semester exams
- Lessons
- 70 lessons: 58 topic lessons, one for each College Board topic, 8 unit tests, and 4 semester exams
- Credit
- Not awarded by Contemplanda
- Enrollment
- Purchasing is not open yet
- Course page
- www.contemplanda.com/courses/ap-precalculus
Credit and enrollment
Contemplanda does not award high school credit. A student who wants credit for the course should ask their own school whether it will accept it.
Under College Board rules, a course may carry the AP designation on a student's transcript only if it is authorized through the College Board's AP Course Audit.
Purchasing is not open yet. Questions about enrolling can go to info@contemplanda.com.
Students who are already enrolled sign in at direct.contemplanda.com.
How the course works
- Each topic lesson combines one to three videos, written explanations, practice, and graded checks, and most lessons include graphs.
- Practice comes one question at a time and continues until the student answers four in a row correctly. After a wrong answer, the practice keeps drawing questions.
- Many questions have hints that open one step at a time. After a wrong answer, the next hint is offered when the question has one. Opening a hint costs nothing and never lowers a mastery level, but the top level, mastered, also requires at least one graded question answered correctly without opening a hint.
- Answers take many forms: algebraic expressions checked for mathematical equivalence and for the form a question asks for; points placed on a graph; graphs sketched or transformed; tables filled in; models fitted to data; along with multiple-choice and numeric answers.
- Most figures in the lessons are live graphs that the student can pan and zoom, and two of those graphs let the student change a parameter with a slider.
- Every topic lesson has a graded check whose questions are drawn for each student from a pool.
- Every figure has a written text description.
Recommended preparation
The College Board recommends that students begin the course with the topics of the Algebra 1, Geometry, and Algebra 2 sequence behind them.
Proficiency in
- Linear and quadratic functions: manipulating them algebraically, and solving the related equations and inequalities
- Adding and multiplying polynomials, factoring quadratic trinomials, and using the quadratic formula
- Solving right-triangle problems with trigonometry
- Solving systems of equations in two and three variables
Familiarity with
- Piecewise-defined functions
- Exponential functions and the rules for exponents
- Radicals, such as square roots and cube roots
- Complex numbers
- Communicating and reasoning with graphical, numerical, analytical, and verbal representations of functions
Units and lessons
Every published lesson, by semester and unit. Units 1A to 4B are this course's own units, and each pair of them falls under one College Board unit.
Topic numbers are the College Board's: topic 2.13 is unit 2, topic 13.
Semester 1
College Board unit 1: Polynomial and Rational Functions
Assessed on the AP Exam
How quantities change together and how rates of change are measured, then polynomial and rational functions: zeros, including complex zeros, end behavior, vertical asymptotes and holes, equivalent forms, and transformations. The unit closes by selecting, building, and applying function models.
Polynomial functions
Topics 1.1 to 1.6- 11.1 Change in tandem
- 21.2 Rates of change
- 31.3 Rates of change in linear and quadratic functions
- 41.4 Polynomial functions and rates of change
- 51.5 Polynomial functions and complex zeros
- 61.6 Polynomial functions and end behavior
- 7Unit 1A testUnit test18 questions
Rational functions
Topics 1.7 to 1.14- 11.7 Rational functions and end behavior
- 21.8 Rational functions and zeros
- 31.9 Rational functions and vertical asymptotes
- 41.10 Rational functions and holes
- 51.11 Equivalent representations of polynomial and rational expressions
- 61.12 Transformations of functions
- 71.13 Function model selection and assumption articulation
- 81.14 Function model construction and application
- 9Unit 1B testUnit test18 questions
Semester 1 midterm
- Semester 1 midterm exam24 questions
Covers units 1A and 1B.
College Board unit 2: Exponential and Logarithmic Functions
Assessed on the AP Exam
Arithmetic and geometric sequences alongside linear and exponential change, then exponential functions, composition and inverse functions, logarithmic expressions and functions, exponential and logarithmic equations and inequalities, modeling data and validating competing models, and semi-log plots.
Exponential functions
Topics 2.1 to 2.8- 12.1 Change in arithmetic and geometric sequences
- 22.2 Change in linear and exponential functions
- 32.3 Exponential functions
- 42.4 Exponential function manipulation
- 52.5 Exponential function context and data modeling
- 62.6 Competing function model validation
- 72.7 Composition of functions
- 82.8 Inverse functions
- 9Unit 2A testUnit test18 questions
Logarithmic functions
Topics 2.9 to 2.15- 12.9 Logarithmic expressions
- 22.10 Inverses of exponential functions
- 32.11 Logarithmic functions
- 42.12 Logarithmic function manipulation
- 52.13 Exponential and logarithmic equations and inequalities
- 62.14 Logarithmic function context and data modeling
- 72.15 Semi-log plots
- 8Unit 2B testUnit test18 questions
Semester 1 final
- Semester 1 final exam32 questions
Covers all of semester 1, units 1A to 2B.
Semester 2
College Board unit 3: Trigonometric and Polar Functions
Assessed on the AP Exam
Periodic phenomena and the sine, cosine, and tangent functions, sinusoidal functions, their transformations, and sinusoidal models of data, inverse trigonometric functions, trigonometric equations and inequalities, the secant, cosecant, and cotangent functions, equivalent trigonometric forms, and polar coordinates, polar graphs, and rates of change in polar functions.
Trigonometric functions
Topics 3.1 to 3.7- 13.1 Periodic phenomena
- 23.2 Sine, cosine, and tangent
- 33.3 Sine and cosine function values
- 43.4 Sine and cosine function graphs
- 53.5 Sinusoidal functions
- 63.6 Sinusoidal function transformations
- 73.7 Sinusoidal function context and data modeling
- 8Unit 3A testUnit test18 questions
Polar functions
Topics 3.8 to 3.15- 13.8 The tangent function
- 23.9 Inverse trigonometric functions
- 33.10 Trigonometric equations and inequalities
- 43.11 The secant, cosecant, and cotangent functions
- 53.12 Equivalent representations of trigonometric functions
- 63.13 Trigonometry and polar coordinates
- 73.14 Polar function graphs
- 83.15 Rates of change in polar functions
- 9Unit 3B testUnit test18 questions
Semester 2 midterm
- Semester 2 midterm exam24 questions
Covers units 3A and 3B.
College Board unit 4: Functions Involving Parameters, Vectors, and Matrices
Not assessed on the AP Exam
Parametric functions and planar motion, implicitly defined functions and conic sections, vectors and vector-valued functions, and matrices: inverses and determinants, linear transformations, and matrices as functions and as models of contexts.
Parametric functions
Topics 4.1 to 4.7- 14.1 Parametric functions
- 24.2 Parametric functions modeling planar motion
- 34.3 Parametric functions and rates of change
- 44.4 Parametrically defined circles and lines
- 54.5 Implicitly defined functions
- 64.6 Conic sections
- 74.7 Parametrization of implicitly defined functions
- 8Unit 4A testUnit test18 questions
Vector-valued functions and matrices
Topics 4.8 to 4.14- 14.8 Vectors
- 24.9 Vector-valued functions
- 34.10 Matrices
- 44.11 The inverse and determinant of a matrix
- 54.12 Linear transformations and matrices
- 64.13 Matrices as functions
- 74.14 Matrices modeling contexts
- 8Unit 4B testUnit test18 questions
Semester 2 final
- Semester 2 final exam32 questions
Covers all of semester 2, units 3A to 4B.
Mathematical practices
The College Board organizes the skills of AP Precalculus into three mathematical practices, which it expects students to build throughout the course.
1Procedural and Symbolic Fluency
Working with functions, equations, and expressions algebraically.
- 1.A
- Solve equations and inequalities given in analytical form, with and without technology.
- 1.B
- Rewrite functions, equations, and expressions in analytically equivalent forms suited to a mathematical or applied context.
- 1.C
- Build new functions through transformations, composition, inverses, or regression to model contexts, criteria, or data, with and without technology.
2Multiple Representations
Translating mathematical information from one representation to another.
- 2.A
- Read information from graphical, numerical, analytical, and verbal representations to answer a question or build a model, with and without technology.
- 2.B
- Produce equivalent graphical, numerical, analytical, and verbal representations of a function suited to a mathematical or applied context, with and without technology.
3Communication and Reasoning
Communicating with precise language and giving reasons for conclusions.
- 3.A
- Describe the characteristics of a function as precisely as its representation and the available tools allow.
- 3.B
- Apply numerical results in a mathematical or applied context.
- 3.C
- Support conclusions or choices with logical reasoning or appropriate data.
Function modeling runs through the framework: students select a model from stated criteria or from data, build it, state the assumptions behind it, and check it against competing models (topics such as 1.13, 1.14, 2.5, 2.6, 2.14, and 3.7).
Assessment
Each topic lesson includes graded checks.
Each of the eight content units closes with a unit test of 18 questions.
Each semester has a midterm exam of 24 questions and a final exam of 32 questions.
Unit tests and semester exams are graded automatically, and all graded work counts toward the student's mastery record, which shows each skill as attempted, familiar, proficient, or mastered.
Contemplanda publishes no grade weighting for this course. A school that decides to grant credit for it also decides how the work counts toward a grade.
Academic integrity
Work in the course must be the student's own.
Unit tests and semester exams are closed book: no notes, no lesson look-ups, and no outside help. Each question is worked on paper and the answer entered in the course.
Answer keys for graded questions are never sent to the student's device.
Graphing calculator
The unit tests and semester exams are written to be worked without a calculator.
The College Board treats a graphing calculator as an integral part of AP Precalculus, and one is required on two parts of the AP Exam. It also expects technology to support, not replace, symbolic work by hand.
During the school year, the College Board allows the web or app version of the Desmos graphing calculator in place of, or alongside, a handheld calculator, and it says the Desmos graphing calculator has the capabilities expected for the AP Precalculus Exam. On the AP Exam, the Desmos calculator must be the one built into the testing app; the web or app version cannot be used.
A student preparing for the AP Exam should also practice with a graphing calculator, such as the Desmos graphing calculator, the kind built into the exam.
The AP Exam
The AP Precalculus Exam assesses Units 1, 2, and 3. It lasts 2 hours and 55 minutes and has two sections.
Section I, Part A: multiple choice
- Questions
- 29
- Time
- 65 minutes
- Share of score
- 43.75%
- Calculator
- Not permitted
Section I, Part B: multiple choice
- Questions
- 13
- Time
- 40 minutes
- Share of score
- 18.75%
- Calculator
- Graphing calculator required
Section II, Part A: free response
- Questions
- 2
- Time
- 35 minutes
- Share of score
- 18.75%
- Calculator
- Graphing calculator required
Section II, Part B: free response
- Questions
- 2
- Time
- 35 minutes
- Share of score
- 18.75%
- Calculator
- Not permitted
On the exam, a Desmos graphing calculator is built into the testing app for the calculator parts, and students may also bring up to two handheld graphing calculators from the College Board's approved list. Calculators should be set to radian mode.
Each free-response question is scored out of 6 points, and two of the four set the mathematics in a real-world modeling context.
Students answer the multiple-choice questions and read the free-response questions in the College Board's digital testing app, and they handwrite their free-response answers in a paper booklet.
Scores are reported on a scale of 1 to 5.
This is the exam format in effect from the May 2027 exam.
The AP Exam is given in schools in May.
A student who takes the AP Exam at a school other than the one where they take the course, or who takes the exam without taking the course, gets a join code from the AP coordinator at the school where they will test and joins an exam-only section there.
Materials and technology
- Where the course runs
- In a web browser, through Contemplanda Direct
- Graphing calculator
- Not needed for the unit tests or semester exams; recommended for AP Exam practice outside this course
- Textbook
- None required. The course's own lessons are the text.
- Course framework
- apcentral.collegeboard.org/courses/ap-precalculus