AP PRECALCULUS

Course Overview

AP Precalculus is focused on the study of functions as models for dynamic phenomena. This course, grounded in research, is designed to better equip students for college-level calculus while also providing a strong foundation for other mathematics and science courses. Students explore a wide range of function types, which are essential for various career paths, including mathematics, physics, biology, health science, business, social science, and data science. Recognizing that AP Precalculus might be the final mathematics course in a student’s high school education, the course is structured to offer a cohesive capstone experience, rather than solely preparing students for future coursework.

Throughout the course, students develop and refine their skills in symbolic manipulation, such as solving equations and working with expressions across the different function types. They learn to understand functions and their compositions, inverses, and transformations through various representations—graphical, numerical, analytical, and verbal. Each representation provides unique insights into the functions and aids in solving both mathematical and applied problems. The skills acquired in this course are broadly applicable to situations requiring quantitative reasoning.

AP Precalculus emphasizes a deep conceptual understanding of functions. Students learn that a function is a mathematical relationship mapping a set of input values (the domain) to a set of output values (the range), where each input is uniquely associated with an output. They come to see functions and their graphs as representations of the dynamic covariation of quantities, a crucial concept for calculus preparation. For each function type, students develop and validate models based on the characteristics of a bivariate data set, the behavior of covarying quantities and their relative rates of change, or specific attributes such as zeros, asymptotes, and extrema. These models are then used for interpolation, extrapolation, and interpretation within various contexts, with an understanding that every model comes with assumptions and limitations. Through this multi-perspective approach, students gain a robust conceptual grasp of functions in general, which enables them to engage with both familiar and novel situations.

College Course Equivalent

AP Precalculus is designed to be equivalent to a first-semester college precalculus course. It provides students with a comprehensive understanding of concepts in college algebra, trigonometry, and other topics that prepare them for advanced college-level mathematics courses. The course covers a variety of function types and their applications, including polynomial, rational, exponential, logarithmic, trigonometric, polar, parametric, vector-valued, implicitly defined, and linear transformation functions using matrices. Throughout the course, students develop mathematical practices such as procedural and symbolic fluency, multiple representations, and effective communication and reasoning. They engage with each function type through the lenses of modeling and covariation, exploring these concepts through graphical, numerical, analytical, and verbal representations.

Prerequisites

Before starting precalculus, students should have a solid foundation in the topics typically covered in the Algebra 1-Geometry-Algebra 2 (AGA) sequence. This includes:

• Proficiency in Linear and Quadratic Functions: Students should be skilled in algebraic manipulation, solving equations, and solving inequalities related to linear and quadratic functions.
• Proficiency in Polynomial Functions: Students should be able to manipulate algebraic expressions, including polynomial addition and multiplication, factoring quadratic trinomials, and using the quadratic formula.
• Proficiency in Trigonometry: Students should be comfortable solving right triangle problems involving trigonometric concepts.
• Proficiency in Systems of Equations: Students should be capable of solving systems of equations in both two and three variables.
• Familiarity with Piecewise-Defined Functions: Students should understand how to work with functions that are defined by different expressions over different intervals.
• Familiarity with Exponential Functions and Exponent Rules: Students should know how to work with exponential functions and apply the rules for exponents.
• Familiarity with Radicals: Students should be comfortable with radicals, such as square roots and cube roots.
• Familiarity with Complex Numbers: Students should have a basic understanding of complex numbers.
• Familiarity with Multiple Representations: Students should be able to communicate and reason using graphical, numerical, analytical, and verbal representations of functions.

Technology Needs

Technology should be integrated throughout the course as a tool for exploring concepts. In AP Precalculus, students should practice using technology to:

• Perform calculations, such as exponents, roots, trigonometric values, and logarithms.
• Graph functions and analyze their graphs.
• Generate tables of values for a function.
• Find real zeros of functions.
• Determine points of intersection between graphs of functions.
• Identify minima and maxima of functions.
• Solve equations in one variable numerically.
• Find regression equations to model data (e.g., linear, quadratic, cubic, quartic, exponential, logarithmic, and sinusoidal) and plot the corresponding residuals.
• Perform matrix operations, such as multiplication and finding inverses.

It’s important to emphasize that technology should not substitute the development of symbolic manipulation skills. When algebraic expressions and equations can be handled with precalculus-level algebraic techniques, students are expected to find zeros, solve equations, and compute values without relying on technology. Most of the AP Exam will be completed without the use of technology, though selected multiple-choice and free-response questions will require a graphing calculator to perform the tasks listed above.

AP Precalculus Mathematical Practices

AP Precalculus Mathematical Practices	Practice 1: Procedural and Symbolic Fluency	Practice 2: Multiple Representations	Practice 3: Communication and Reasoning
Description	Algebraically manipulate functions, equations, and expressions.	Translate mathematical information between representations.	Communicate with precise language, and provide rationales for conclusions.
Skills	1.A Solve equations and inequalities represented analytically, with and without technology.	2.A Identify information from graphical, numerical, analytical, and verbal representations to answer a question or construct a model, with and without technology.	3.A Describe the characteristics of a function with varying levels of precision, depending on the function representation and available mathematical tools.
	1.B Express functions, equations, or expressions in analytically equivalent forms that are useful in a given mathematical or applied context.	2.B Construct equivalent graphical, numerical, analytical, and verbal representations of functions that are useful in a given mathematical or applied context, with and without technology.	3.B Apply numerical results in a given mathematical or applied context.
	1.C Construct new functions, using transformations, compositions, inverses, or regressions, that may be useful in modeling contexts, criteria, or data, with and without technology.		3.C Support conclusions or choices with a logical rationale or appropriate data.

Course Content

The AP Precalculus curriculum is divided into four units, with the content from Units 1, 2, and 3 being included on the AP Exam. The topics in Unit 4 are optional and can be taught at the discretion of the school and teacher, depending on state and local requirements. Pacing recommendations are provided at the unit level and in the Course at a Glance to guide teachers on how to effectively cover the required course content. These recommendations assume a school schedule where the class meets five days a week for 45 minutes each day throughout the full school year.

Within Units 1, 2, and 3, many topics are assigned a range of suggested instructional days. The lower end of the range is recommended for teachers who plan to cover all topics across Units 1, 2, 3, and 4. The higher end of the range is suggested for those focusing only on Units 1, 2, and 3. While these pacing guidelines are designed to help with planning, teachers are encouraged to adjust them based on their students’ needs, alternative schedules (e.g., block scheduling), the school’s academic calendar, or the inclusion of Unit 4 topics.

This below form outlines the key units, topics, and exam relevance for each section of the AP Precalculus course:

Unit	Overview	Topics Covered	On The Exam
Unit 1: Polynomial and Rational Functions	Expand your understanding of polynomial and rational functions through the lenses of modeling and various rates of change.	– Describing how quantities change with respect to each other – Describing end behavior of polynomial and rational functions – Identifying asymptotes of and holes in the graphs of rational functions – Modeling aspects of scenarios using polynomial and rational functions – Identifying assumptions and limitations of function models	30%–40% of multiple-choice section score
Unit 2: Exponential and Logarithmic Functions	Deepen your understanding of inverses by exploring the relationship between exponential and logarithmic functions.	– Relating geometric sequences and exponential functions – Modeling data sets with exponential functions – Composing functions and finding inverses – Modeling scenarios with logarithmic functions – Validating a function model using a residual plot	27%–40% of multiple-choice section score
Unit 3: Trigonometric and Polar Functions	Model and explore periodic phenomena using transformations of trigonometric functions.	– Relating right triangle trigonometry to the sine, cosine, and tangent functions – Modeling data and scenarios with sinusoidal functions – Using inverse trigonometric functions to solve trigonometric equations – Graphing functions using polar coordinates – Describing how angles and radii change with respect to each other in a polar graph	30%–35% of multiple-choice section score
Unit 4: Functions Involving Parameters, Vectors, and Matrices	Expand your understanding of the function concept by exploring a variety of new function types.	– Describing how quantities change with respect to each other in a parametric function – Graphing conic sections using implicitly defined functions and parametric functions – Using vectors to describe motion of an object – Describing the impact of a transformation matrix on a graphical object – Modeling change in a context using matrices	Not Assessed on the AP Exam

the exam weighting and the rationale

This table summarizes the exam weighting and the rationale behind the inclusion or exclusion of each unit’s topics on the AP Precalculus Exam.

Unit	Exam Weighting	Notes
Unit 1: Polynomial and Rational Functions	30%–40%	Topics included on the AP Exam, as they cover content and conceptual understandings expected by colleges for credit/placement.
Unit 2: Exponential and Logarithmic Functions	27%–40%	Topics included on the AP Exam, as they cover content and conceptual understandings expected by colleges for credit/placement.
Unit 3: Trigonometric and Polar Functions	30%–35%	Topics included on the AP Exam, as they cover content and conceptual understandings expected by colleges for credit/placement.
Unit 4: Functions Involving Parameters, Vectors, and Matrices	Not assessed on the AP Exam	Topics may be included by teachers based on state or local requirements but are not part of the AP Exam content.

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