Precalc Units

This section outlines the eight distinct skills associated with three key mathematical practices that students should build and master throughout the course.

Purpose: To assure skill distribution and repetition throughout the course, the framework supplies skill focus recommendations for each topic.
Instructional Approaches: For detailed teaching practices, refer to the Instructional Approaches section of this publication.

Skill 1.1: Algebraically manipulate functions, equations, and expressions.
Skill 1.2: Translate mathematical information between representations.
Skill 1.3: Communicate with precise language and provide rationales for conclusions.

1.A: Solve equations and inequalities analytically, with and without technology.
1.B: Express functions, equations, or expressions in analytically equivalent forms applicable in specific mathematical or applied contexts.
1.C: Construct new functions using transformations, compositions, inverses, or regressions, beneficial in modeling contexts.

Skill 2.1: Identify information from various representations to construct models.
Skill 2.2: Construct equivalent representations useful in different contexts.

2.A: Identify information from graphical, numerical, analytical, and verbal representations to answer questions.
2.B: Construct equivalent representations beneficial in applied contexts.

Skill 3.1: Describe characteristics of functions with varying levels of precision.
Skill 3.2: Apply numerical results in different situations.
Skill 3.3: Support conclusions and choices with appropriate data or rationale.

This course framework details course requirements essential for student success. It specifies what students are expected to know, understand, and be able to do to prepare for advanced coursework.
The course encourages students to create useful solutions to problems across varying fields engaged in modeling change (such as pure sciences, engineering, or economics).

The AP Precalculus content is organized into four major units:
    1. Unit 1: Polynomial and Rational Functions (30-40% exam weighting)
    2. Unit 2: Exponential and Logarithmic Functions (27-40% exam weighting)
    3. Unit 3: Trigonometric and Polar Functions (30-35% exam weighting)
    4. Unit 4: Functions Involving Parameters, Vectors, and Matrices (not assessed on the AP Exam)
Instructional Pacing: Suggested pacing is provided to help structure teaching over the academic year, accommodating adjustments based on students' needs or local requirements.

Multiple-Choice Sections: Assess knowledge and skills across all units, with specific weightings indicated for each topic:
    - Unit 1: Polynomial and Rational Functions 30-40%
    - Unit 2: Exponential and Logarithmic Functions 27–40%
    - Unit 3: Trigonometric and Polar Functions 30–35%
Free-Response Questions: Allow students to demonstrate understanding through constructed responses in given contexts, typically requiring justification and reasoning for their answers.

Topics Include:
   - 1.1 Change in Tandem
   - 1.2 Rates of Change
   - 1.3 Rates of Change in Linear and Quadratic Functions
   - 1.4 Polynomial Functions and Rates of Change
   - 1.5 Polynomial Functions and Complex Zeros
   - 1.6 Polynomial Functions and End Behavior
   - 1.7 Rational Functions and End Behavior
   - 1.8 Rational Functions and Zeros
   - 1.9 Rational Functions and Vertical Asymptotes
   - 1.10 Rational Functions and Holes
   - 1.11 Equivalent Representations of Polynomial and Rational Expressions
   - 1.12 Transformations of Functions
   - 1.13 Function Model Selection and Assumption Articulation
   - 1.14 Function Model Construction and Application

Topics Include:
   - 2.1 Change in Arithmetic and Geometric Sequences
   - 2.2 Change in Linear and Exponential Functions
   - 2.3 Exponential Functions
   - 2.4 Exponential Function Manipulation
   - 2.5 Exponential Function Context and Data Modeling
   - 2.6 Competing Function Model Validation
   - 2.7 Composition of Functions
   - 2.8 Inverse Functions
   - 2.9 Logarithmic Expressions
   - 2.10 Inverses of Exponential Functions
   - 2.11 Logarithmic Functions
   - 2.12 Logarithmic Function Manipulation
   - 2.13 Exponential and Logarithmic Equations and Inequalities
   - 2.14 Logarithmic Function Context and Data Modeling
   - 2.15 Semi-log Plots

Topics Include:
   - 3.1 Periodic Phenomena
   - 3.2 Sine, Cosine, and Tangent
   - 3.3 Sine and Cosine Function Values
   - 3.4 Sine and Cosine Function Graphs
   - 3.5 Sinusoidal Functions
   - 3.6 Sinusoidal Function Transformations
   - 3.7 Sinusoidal Function Context and Data Modeling
   - 3.8 The Tangent Function
   - 3.9 Inverse Trigonometric Functions
   - 3.10 Trigonometric Equations and Inequalities
   - 3.11 The Secant, Cosecant, and Cotangent Functions
   - 3.12 Equivalent Representations of Trigonometric Functions
   - 3.13 Trigonometry and Polar Coordinates
   - 3.14 Polar Function Graphs
   - 3.15 Rates of Change in Polar Functions

Topics Include:
   - 4.1 Parametric Functions,
   - 4.2 Parametric Functions Modeling Planar Motion,
   - 4.3 Parametric Functions and Rates of Change,
   - 4.4 Parametrically Defined Circles and Lines,
   - 4.5 Implicitly Defined Functions,
   - 4.6 Conic Sections,
   - 4.7 Parametrization of Implicitly Defined Functions,
   - 4.8 Vectors,
   - 4.9 Vector-Valued Functions,
   - 4.10 Matrices,
   - 4.11 The Inverse and Determinant of a Matrix,
   - 4.12 Linear Transformations and Matrices,
   - 4.13 Matrices as Functions,
   - 4.14 Matrices Modeling Contexts

Diverse instructional strategies are recommended to enhance comprehension and engagement in the Precalculus course. Strategies include:
  - Concrete Representations: Use visual aids for theoretical concepts to bridge understanding.
  - Problem-Solving: Implement tasks that break complex problems down into manageable parts.
  - Group Work: Facilitate collaborative learning through peer discussion and debate.
  - Technological Integration: Incorporate tools and software to model functions, enabling practical engagement with algorithms.

Multiple Choice: Contains both calculator and no-calculator sections assessing computational skills and conceptual understanding.
Free-Response: Integrates a calculator-required section allowing for complex function manipulation and modeling.

The AP Precalculus curriculum emphasizes not only mastering mathematical skills but also the ethical implications of data handling, mathematical representation, and communication. These elements are crucial for students as they prepare to engage with advanced coursework and real-world applications.