AP Precalculus centers on functions modeling dynamic phenomena. This research-based
exploration of functions is designed to better prepare students for college-level calculus
and provide grounding for other mathematics and science courses.
In this course, students study a broad spectrum of function types that are foundational for careers in mathematics,
physics, biology, health science, business, social science, and data science. Furthermore,
as AP Precalculus may be the last mathematics course of a student’s secondary education, the course is structured to provide a coherent capstone experience rather than exclusively
focusing on preparation for future courses.
Throughout this course, students develop and hone symbolic manipulation skills, including
solving equations and manipulating expressions, for the many function types throughout the course. Students also learn that functions and their compositions, inverses, and transformations are understood through graphical, numerical, analytical, and verbal
representations, which reveal different attributes of the functions and are useful for solving problems in mathematical and applied contexts.
AP Precalculus fosters the development of a deep conceptual understanding of functions.
Students learn that a function is a mathematical relation that maps a set of input values—the domain—to a set of output values—the range—such that each input value is uniquely mapped to an output value. Students understand functions and their graphs as embodying dynamic covariation of quantities, a key idea in preparing for calculus. With each function type, students develop and validate function models based on the characteristics of a bivariate data set, characteristics of covarying quantities and their relative rates of change, or a set of characteristics such as zeros, asymptotes, and extrema. These models are used to interpolate, extrapolate, and interpret information with different degrees of accuracy for a given context or data set.
Additionally, students also learn that every model is subject to
assumptions and limitations related to the context. As a result of examining functions from many perspectives, students develop a conceptual understanding not only of specific function types but also of functions in general. This type of understanding helps students to
engage with both familiar and novel contexts.
College Course Equivalent
AP Precalculus is designed to be the equivalent of a first semester college precalculus course. AP Precalculus provides students with an understanding of the concepts of college algebra, trigonometry, and additional topics that prepare students for further collegelevel mathematics courses.
This course explores a variety of function types and their applications—polynomial, rational, exponential, logarithmic, trigonometric, polar, parametric, vector-valued, implicitly defined, and linear transformation functions using matrices.
Prerequisites
Before studying precalculus, all students should develop proficiency in topics typically found in the Algebra1-Geometry-Algebra 2 (AGA) content sequence.
Students should have developed the following:
Proficiency with the skills and concepts related to linear and quadratic functions, including algebraic manipulation, solving equations, and solving inequalities.
Proficiency in manipulating algebraic expressions related to polynomial functions,
including polynomial addition and multiplication, factoring quadratic trinomials, and using the quadratic formula.
Proficiency in solving right triangle problems involving trigonometry.
Proficiency in solving systems of equations in two and three variables.
Familiarity with piecewise-defined functions.
Familiarity with exponential functions and rules for exponents.
Familiarity with radicals (e.g., square roots, cube roots).
Familiarity with complex numbers.
Familiarity with communicating and reasoning among graphical, numerical, analytical, and verbal representations of functions.
AP Calculus AB and AP Calculus BC focus on students’ understanding of calculus concepts and provide experience with methods and applications. Through the use of big ideas of calculus (e.g., modeling change, approximation and limits, and analysis of functions), each course becomes a cohesive whole, rather than a collection of unrelated topics. Both courses require students to use definitions and theorems to build arguments and justify conclusions.
The courses feature a multirepresentational approach to calculus, with concepts, results, and problems expressed graphically, numerically, analytically, and verbally. Exploring connections among these representations builds understanding of how calculus applies limits to develop important ideas, definitions, formulas, and theorems. A sustained emphasis on clear communication of methods, reasoning, justifications, and conclusions is essential.
Teachers and students would regularly use technology to reinforce relationships among functions, to confirm written work, to implement experimentation, and to assist in interpreting results.
College Course Equivalent
AP Calculus AB is designed to be the equivalent of a first semester college calculus course devoted to topics in differential and integral calculus.
AP Calculus BC is designed to be the equivalent to both first and second semester college calculus courses.
AP Calculus BC applies the content and skills learned in AP Calculus AB to parametrically defined curves, polar curves, and vector-valued functions; develops additional integration techniques and applications; and introduces the topics of sequences and series.
Prerequisites
Before studying calculus, all students should complete the equivalent of four years of secondary mathematics designed for college-bound students: courses that should prepare them with a strong foundation in reasoning with algebraic symbols and working with algebraic structures. Prospective calculus students should take courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions. These functions include linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions. In particular, before studying calculus, students must be familiar with the properties of functions, the composition of functions, the algebra of functions, and the graphs of functions.
Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and descriptors such as increasing and decreasing).
Students should also know how the sine and cosine functions are defined from the unit circle and know the values of the trigonometric functions.
Students who take AP Calculus BC should have basic familiarity with sequences and series, as well as some exposure to parametric and polar equations.
AP Calculus AB and AP Calculus BC focus on students’ understanding of calculus concepts and provide experience with methods and applications. Through the use of big ideas of calculus (e.g., modeling change, approximation and limits, and analysis of functions), each course becomes a cohesive whole, rather than a collection of unrelated topics. Both courses require students to use definitions and theorems to build arguments and justify conclusions.
The courses feature a multirepresentational approach to calculus, with concepts, results, and problems expressed graphically, numerically, analytically, and verbally. Exploring connections among these representations builds understanding of how calculus applies limits to develop important ideas, definitions, formulas, and theorems. A sustained emphasis on clear communication of methods, reasoning, justifications, and conclusions is essential.
Teachers and students would regularly use technology to reinforce relationships among functions, to confirm written work, to implement experimentation, and to assist in interpreting results.
College Course Equivalent
AP Calculus AB is designed to be the equivalent of a first semester college calculus course devoted to topics in differential and integral calculus.
AP Calculus BC is designed to be the equivalent to both first and second semester college calculus courses.
AP Calculus BC applies the content and skills learned in AP Calculus AB to parametrically defined curves, polar curves, and vector-valued functions; develops additional integration techniques and applications; and introduces the topics of sequences and series.
Prerequisites
Before studying calculus, all students should complete the equivalent of four years of secondary mathematics designed for college-bound students: courses that should prepare them with a strong foundation in reasoning with algebraic symbols and working with algebraic structures. Prospective calculus students should take courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions. These functions include linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions. In particular, before studying calculus, students must be familiar with the properties of functions, the composition of functions, the algebra of functions, and the graphs of functions.
Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and descriptors such as increasing and decreasing).
Students should also know how the sine and cosine functions are defined from the unit circle and know the values of the trigonometric functions.
Students who take AP Calculus BC should have basic familiarity with sequences and series, as well as some exposure to parametric and polar equations.