AP® Precalculus
Syllabus
Course Essential Questions
- How can we categorize functions by investigating their properties?
- How can we describe functions using multiple representations?
- How can we explore trigonometric functions using right triangles and the unit circle?
Course Learning Objectives
- Compare different types of functions to make general conclusions.
- Solve algebraic and transcendental equations using analytical, numerical, and graphical methods.
- Analyze graphs of elementary functions: linear, quadratic, polynomial, exponential, logarithmic, and trigonometric functions.
- Interpret models of elementary functions: linear, quadratic, polynomial, exponential, logarithmic, and trigonometric functions.
Unit 1: Polynomial and Rational Functions (Weeks 1-11)
In this unit, students will focus on exploring and analyzing polynomial and rational functions, which play a fundamental role in various fields of mathematics and real-life applications. Polynomial functions can have various shapes and characteristics depending on their degree and coefficients. As for rational functions, these are functions that are defined as the quotient of two polynomial functions.
Unit Level Essential Questions
- How can we model real world phenomena using functions?
- How can we use functions to make predictions?
- How can we adjust function models to account for changes in conditions?
Week 1: Introduction to AP Precalculus
Weekly Essential Questions
- How can we model real world phenomena using functions?
Weekly Learning Objectives
- Understand the rules and policies of the course, and expectations of you as a contributing member of our community.
- Begin to build community by sharing important information with your teacher and classmates about yourself and your school.
- Develop your skills for navigating the course environment and using the various course tools.
- Describe how the input and output values of a function vary together by comparing function values.
- Construct a graph representing two quantities that vary with respect to each other in a contextual scenario.
- Compare the rates of change at two points using average rates of change near the points.
- Describe how two quantities vary together at different points and over different intervals of a function.
Week 2: Basic Functions and Building New Functions
Weekly Essential Questions
- How can we model real world phenomena using functions?
- How can we use functions to make predictions?
Weekly Learning Objectives
- Describe how two quantities vary together at different points and over different intervals of a function.
- Construct a graph of an equation involving two variables.
- Determine how the two quantities related in an implicitly defined function vary together.
- Evaluate the composition of two or more functions for given values.
- Construct a representation of the composition of two or more functions.
- Rewrite a given function as a composition of two or more functions.
Week 3: Parametric Functions and Inverse Functions
Weekly Essential Questions
- How can we model real world phenomena using functions?
- How can we use functions to make predictions?
- How can we adjust function models to account for changes in conditions?
Weekly Learning Objectives
- Compare rates of change at two points using average rates of change near the points.
- Describe how two quantities vary together at different points and over different intervals of a function.
- Construct a graph or table of values for a parametric function represented analytically.
- Determine the input-output pairs of the inverse of a function
- Determine the inverse of a function on an invertible domain.
Week 4: Transformations and Modeling with Functions
Weekly Essential Questions
- How can we model real world phenomena using functions?
- How can we use functions to make predictions?
Weekly Learning Objectives
- Describe how the input and output values of a function vary together by comparing function values.
- Construct a graph representing two quantities that vary with respect to each other in a contextual scenario.
- Compare the rates of change at two points using average rates of change near the points.
- Describe how two quantities vary together at different points and over different intervals of a function.
- Construct a function that is an additive and/or multiplicative transformation of another function.
Week 5: Function Families, Linear Functions, and Linear Models
Weekly Essential Questions
- How can we model real world phenomena using functions?
- How can we use functions to make predictions?
Weekly Learning Objectives
- Determine the average rates of change for functions including linear functions.
- Determine the change in the average rates of change for linear functions.
- Analyze linear functions numerically, graphically, and analytically
Week 6: Quadratic Functions and Modeling
Weekly Essential Questions
- How can we model real world phenomena using functions?
- How can we adjust function models to account for changes in conditions?
Weekly Learning Objectives
- Identify key characteristics of power functions related to rates of change.
- Analyze graphs of basic power functions.
Week 7: Polynomial Functions, Modeling, and Real Zeros
Weekly Essential Questions
- How can we model real world phenomena using functions?
- How can we use functions to make predictions?
- How can we adjust function models to account for changes in conditions?
Weekly Learning Objectives
- Identify key characteristics of power functions related to rates of change.
- Identify key characteristics of a polynomial function related to its zeros when suitable factorizations are available or with technology.
- Determine if a polynomial function is even or odd.
- Describe end behaviors of polynomial functions.
- Determine the quotient of two polynomials using long division.
- Rewrite the repeated product of binomials using the binomial theorem.
Week 8: Complex Zeros & the Fundamental Theorem of Algebra
Weekly Essential Questions
- How can we model real world phenomena using functions?
- How can we adjust function models to account for changes in conditions?
Weekly Learning Objectives
- Identify key characteristics of power functions related to rates of change.
- Identify key characteristics of a polynomial function related to its zero when suitable factorizations are available or with technology.
- Rewrite the repeated product of binomials using the binomial theorem.
Week 9: Rational Functions
Weekly Essential Questions
- How can we model real world phenomena using functions?
- How can we adjust function models to account for changes in conditions?
Weekly Learning Objectives
- Describe end behaviors of rational functions.
- Determine the zeros of rational functions.
- Determine vertical asymptotes of graphs of rational functions.
- Determine holes in graphs of rational functions.
Week 10: Rational Equations, Inequalities, and Modeling
Weekly Essential Questions
- How can we model real world phenomena using functions?
- How can we adjust function models to account for changes in conditions?
Weekly Learning Objectives
- Rewrite polynomial and rational expressions in equivalent forms.
- Rewrite the repeated product of binomials using the binomial theorem.
- Solve rational equations.
- Identify extraneous solutions.
Week 11: Unit 1 Summative Review
Weekly Essential Questions
- What knowledge and skills do we need to be successful on the Unit 1 Polynomial and Rational Functions Exam?
Weekly Learning Objectives
- Review your knowledge of basic functions, polynomial functions, and rational functions.
- Demonstrate your knowledge of polynomial and rational functions on the Unit 1 Exam.
Unit 2: Exponential and Logarithmic Functions (Weeks 12-16)
In this unit, students will focus on exploring and analyzing exponential and logarithmic functions, which also play a fundamental role in various fields of mathematics and real-life applications. Exponential functions describe growth and decay processes that occur in nature, economics, science, and in many other areas. Logarithmic functions have a special relationship with exponential functions. That is, logarithmic functions provide a powerful way to understand relationships between quantities that change exponentially.
Unit Level Essential Questions
- How can we model relationships that represent growth and decay?
- How can we model sound and magnitude?
- If different functions can be used to model data, how do we pick which one is best?
Week 12: Arithmetic and Geometric Sequences
Weekly Essential Questions
- How can we model relationships that represent growth and decay?
- How can we model sound and magnitude?
Weekly Learning Objectives
- Express arithmetic sequences found in mathematical and contextual scenarios as functions of the whole numbers.
- Express geometric sequences found in mathematical and contextual scenarios as functions of the whole numbers.
- Construct functions of the real numbers that are comparable to arithmetic and geometric sequences.
- Describe similarities and differences between linear and exponential functions.
- Identify key characteristics of exponential functions.
- Rewrite exponential expressions in equivalent forms.
Week 13: Exponential and Logistic Modeling
Weekly Essential Questions
- How can we model relationships that represent growth and decay?
- How can we model sound and magnitude?
- If different functions can be used to model data, how do we pick which one is best?
Weekly Learning Objectives
- Construct models involving proportional output values over equal-length input-value intervals.
- Apply exponential models to answer questions about a data set or contextual scenario.
- Construct linear, quadratic, and exponential models based on a data set.
- Validate a model constructed from a data set.
Week 14: Logarithmic Functions, Graphs, and Properties
Weekly Essential Questions
- How can we model sound and magnitude?
- If different functions can be used to model data, how do we pick which one is best?
Weekly Learning Objectives
- Determine the input-output pairs of inverse of a function.
- Determine the inverse of a function on an invertible domain.
- Evaluate logarithmic expressions.
- Construct representations of the inverse of an exponential function with an initial value of 1.
- Identify key characteristics of logarithmic functions.
- Rewrite logarithmic expressions in equivalent forms.
Week 15: Semi-Log Plots, Equation Solving, and Modeling
Weekly Essential Questions
- How can we model relationships that represent growth and decay?
- How can we model sound and magnitude?
- If different functions can be used to model data, how do we pick which one is best?
Weekly Learning Objectives
- Solve exponential and logarithmic equations and inequalities.
- Construct the inverse function for exponential and logarithmic functions.
- Construct a logarithmic function model.
Week 16: Unit 2 Summative Review
Weekly Essential Questions
- What knowledge and skills do we need to be successful on the Unit 2 Exponential and Logarithmic Functions Exam?
Weekly Learning Objectives
- Review your knowledge of composite functions, inverse functions, exponential functions, and logarithmic functions..
- Demonstrate your knowledge of exponential and logarithmic functions on the Unit 2 Exam.
Unit 3: Trigonometric and Polar Functions (Weeks 17-26)
This comprehensive unit will explore the fundamental concepts and applications of trigonometric and polar functions. Students will seamlessly integrate classical trigonometry with polar coordinates to explore a diverse range of mathematical tools applicable in various fields.
Unit Level Essential Questions
- How can we use trends in data to predict monthly energy bills?
- How can we model aspects of circular and spinning objects?
- How does right triangle trigonometry from geometry relate to trigonometric functions?
Week 17: Angles and Their Measure
Weekly Essential Questions
- How can we model aspects of circular and spinning objects?
- How does right triangle trigonometry from geometry relate to trigonometric functions?
Weekly Learning Objectives
- Differentiate between degree and radian measure.
- Construct graphs of periodic relationships based on verbal representations.
- Describe key characteristics of a periodic function based on a verbal representation.
Week 18: Trigonometry of Acute Angles and Trigonometric Functions
Weekly Essential Questions
- How can we model aspects of circular and spinning objects?
- How does right triangle trigonometry from geometry relate to trigonometric functions?
Weekly Learning Objectives
- Determine coordinates of points on a circle centered at the origin.
- Construct representations of the tangent function using the unit circle.
- Describe key characteristics of the tangent function.
- Describe additive and multiplicative transformations involving the tangent function.
- Identify key characteristics of functions that involve quotients of the sine and cosine functions.
Week 19: Graphs of Trigonometric Functions
Weekly Essential Questions
- How can we model aspects of circular and spinning objects?
- How does right triangle trigonometry from geometry relate to trigonometric functions?
Weekly Learning Objectives
- Determine the sine, cosine, and tangent of an angle using the unit circle.
- Construct representations of the sine and cosine functions using the unit circle.
- Identify key characteristics of the sine and cosine functions.
- Identify the amplitude, vertical shift, period, and phase shift of a sinusoidal function.
Week 20: Graphs of Composite Trigonometric Functions
Weekly Essential Questions
- How can we model aspects of circular and spinning objects?
- How does right triangle trigonometry from geometry relate to trigonometric functions?
Weekly Learning Objectives
- Combine trigonometric and algebraic functions.
- Analyze the graphs of combinations of sinusoidal functions.
Week 21: Inverse Trigonometric Functions
Weekly Essential Questions
- How can we use trends in data to predict monthly energy bills?
- How can we model aspects of circular and spinning objects?
- How does right triangle trigonometry from geometry relate to trigonometric functions?
Weekly Learning Objectives
- Construct sinusoidal function models of periodic phenomena.
- Construct analytical and graphical representations of the inverse of the sine, cosine, and tangent functions over a restricted domain.
- Solve equations and inequalities involving trigonometric functions.
Week 22: Fundamental Identities and Proving Trigonometric Identities
Weekly Essential Questions
- How does right triangle trigonometry from geometry relate to trigonometric functions?
Weekly Learning Objectives
- Rewrite trigonometric expressions in equivalent forms with the Pythagorean identity.
- Solve equations using equivalent analytic representations of trigonometric functions.
Week 23: Sum and Difference Identities
Weekly Essential Questions
- How does right triangle trigonometry from geometry relate to trigonometric functions?
Weekly Learning Objectives
- Rewrite trigonometric expressions in equivalent forms with sine and cosine sum identities.
- Solve equations using equivalent analytic representations of trigonometric functions.
Week 24: Law of Sines & Law of Cosines
Weekly Essential Questions
- How does right triangle trigonometry from geometry relate to trigonometric functions?
Weekly Learning Objectives
- Distinguish the difference between right triangles and oblique triangles.
- Apply the Law of Sines to calculate the lengths and angles of a triangle.
- Apply the Law of Cosines to calculate the lengths and angles of a triangle.
Week 25: Polar Coordinates and Graphs of Polar Equations
Weekly Essential Questions
- How does right triangle trigonometry from geometry relate to trigonometric functions?
Weekly Learning Objectives
- Determine the location of a point in the plane using both rectangular and polar coordinates.
- Construct graphs of polar functions.
- Describe characteristics of the graph of a polar function.
Week 26: Unit 3 Summative Review
Weekly Essential Questions
- What knowledge and skills do we need to be successful on the Unit 3 Trigonometric and Polar Functions Exam?
Weekly Learning Objectives
- Review your knowledge of trigonometric and polar functions.
- Demonstrate your knowledge of trigonometric and polar functions on the Unit 3 Exam.
Unit 4: Functions Involving Parameters, Vectors, and Matrices (Weeks 27-30)
In this unit, students will explore function types that expand their understanding of the function concept. Parametric functions have multiple dependent variables’ values paired with a single unit variable or parameter. Another major function type in this unit involves matrices mapping a set of input vectors to output vectors. The capacity to map large quantities of vectors instantaneously is the basis for vector-based computer graphics.
Unit Level Essential Questions
- How can we determine when populations will be relatively steady?
- How can we analyze the vertical and horizontal aspects of motion independently?
Week 27: Vectors & Parametric and Implicit Functions Revisited
Weekly Essential Questions
- How can we determine when populations will be relatively steady?
- How can we analyze the vertical and horizontal aspects of motion independently?
Weekly Learning Objectives
- Identify key characteristics of a parametric planar motion function that are related to position.
- Identify key characteristics of a parametric planar motion function that are related to direction and rate of change.
- Express motion around a circle or along a line segment parametrically.
- Identify characteristics of a vector.
- Determine sums and products involving vectors.
- Determine a unit vector for a given vector.
- Determine angle measures between vectors and magnitudes of vectors involved in vector addition.
- Represent planar motion in terms of vector-valued functions.
Week 28: Conic Sections
Weekly Essential Questions
- How can we analyze the vertical and horizontal aspects of motion independently?
Weekly Learning Objectives
- Identify parabolas, circles, ellipses, and hyperbolas.
- Represent conic sections with horizontal or vertical symmetry analytically.
- Represent a curve in the plane parametrically.
- Represent conic sections parametrically.
Week 29: Matrix Algebra and Linear Transformations
Weekly Essential Questions
- How can we determine when populations will be relatively steady?
Weekly Learning Objectives
- Determine the product of two matrices.
- Determine the inverse of a 2x2 matrix.
- Apply the value of the determinant to invertibility and vectors.
- Determine the output vectors of a linear transformation using a 2x2 matrix.
- Determine the association between a linear transformation and a matrix.
- Determine the composition of two linear transformations.
- Determine the inverse of a linear transformation.
- Construct a model of a scenario involving transition between two states using matrices.
- Apply matrix models to predict future and past states for n transition steps.
Week 30: Unit 4 Summative Review
Weekly Essential Questions
- What knowledge and skills do we need to be successful on the Unit 4 Functions Involving Parameters, Vectors, and Matrices Exam?
Weekly Learning Objectives
- Review your knowledge of functions involving parameters, vectors, and matrices.
- Demonstrate your knowledge of functions involving parameters, vectors, and matrices on the Unit 4 Exam.
Unit 5: AP Review, Practice Exam, and Preview of Calculus (Weeks 31-33)
In this final unit of the course, students will focus on AP Review and completing the AP practice exam. In addition, students will also take a look into the start of calculus. This unit is crucial in preparing students for what to expect on the AP exam as well as taking a mock exam.
Unit Level Essential Questions
- How do we apply what we learned to calculus?
Week 31: Practice AP Exam
Weekly Essential Questions
- How can we prepare for the AP Exam?
Weekly Learning Objectives
- Review unit topics.
- Complete practice exam.
Week 32: Practice AP Exam Reflection
Weekly Essential Questions
- How can we prepare for the AP Exam?
Weekly Learning Objectives
- Review graded practice exam.
- Complete reflection activity on practice exam.
Week 33: Final Thoughts and Reflections
Weekly Essential Questions
- How do we apply what we learned to calculus?
Weekly Learning Objectives
- Preview topics in calculus.
- Reflect on course experience.
- Complete end of course activities.