Pre-calculus weaves together concepts of algebra and geometry into a preparatory course for calculus. The course focuses on the mastery of critical skills and exposure to new skills necessary for success in subsequent math courses. Topics include quadratic, exponential, logarithmic,radical, polynomial, and rational functions; matrices; and conic sections in the first semester. The second semester covers an introduction to infinite series, trigonometric ratios, functions, and equations; inverse trigonometric functions; applications of trigonometry, including vectors; polar equations and polar form of complex numbers; arithmetic of complex numbers; and parametric equations. Connections are made throughout the course to calculus and a variety of other fields related to mathematics. Purposeful concentration is placed on how the concepts covered relate to each other. Demonstrating the connection between the algebra and the geometry of concepts highlights the interwoven nature of the study of mathematics. Prerequisite: Geometry and Algebra II (or equivalents)
REQUIRED BUT NOT PROVIDED-
TI-84 Plus Graphing Calculator
Unit 1: Introduction to Precalculus and Polynomial Functions
Students extend their understanding of quadratic, radical and rational equations to encompass inequalities of the same. Concepts and theorems of polynomial and rational equations and functions are reviewed and extended to include holes, slant asymptotes and limit notation to end behavior.
- Solving quadratic, radical and rational inequalities
- Decomposing fractions
- Applying remainder, factor, rational root, intermediate value, upper and lower bound theorems
- Drawing conclusions from Descartes’ rule of sign
- Graphing polynomial and rational functions
- Using limit notation to describe end behavior of polynomial and rational functions
Unit 2: Matrices
For many students, this may be their first exposure to matrices. Students explore solving a system of equations using matrices through elementary row operations and reduced row echelon form, Cramer’s rule and the inverse of a coefficient matrix. Students explore the use of matrices in business through the completion of a graded assignment.
- Transforming a matrix using elementary row operations
- Solving a system of equations by transforming a matrix into reduced row echelon form
- Arithmetic operations and algebraic properties of matrices
- Determining the determinant of a 2 2 matrix
- Applying Cramer’s rule to solve a system of two linear equations
- Determining the inverse of a 2 2 matrix
- Solving a system of equations using a matrix equation
Unit 3: Conic Sections
Conic sections are real-life phenomena that are found in architecture, space, nature, and more. Students explore how and where they occur as they progress through this unit, learning about four conic sections—circles, ellipses, hyperbolas, and parabolas. Derivations of the standard form of these conic sections are included to help foster student appreciation for the relationship that exists between algebra and geometry.
- Graphing and writing the equation of circles, ellipses, hyperbolas and parabolas
- Solving a systems of conic sections
- Solving real world problems involving conic sections
Unit 4: Exponential and Logarithmic Functions
A thorough understanding of logarithmic expressions and functions is necessary for success in higher mathematics. Students deepen their understanding of the relationship between exponential and logarithmic functions as they explore how the properties of logarithms allow for equivalent forms of logarithmic expressions, and for a variety of logarithmic and exponential equations to be solved. Concentration is on real-world application of exponential equations.
- Graphing and solving exponential equations using the one-to-one property of exponents
- Graphing logarithmic functions as the inverse of a related exponential function
- Exploring applications of the properties of logarithms
- Solving logarithmic equations
- Solving exponential equations using logarithms
- Solving applications of logarithmic and exponential functions found in the real world
Unit 5: Semester Review
Students review what they have learned and take the semester exam.
Unit 1: Discrete Mathematics
Students build on their understanding of finite arithmetic and geometric sequences and series to explore infinite sequences and series. Students learn sigma notation, and properties of limits, as well as the ratio test as a method of determining if an infinite series converges or diverges. Students explore two different, but closely related, methods of expanding binomials.
- Solving problems related to finite arithmetic and geometric sequences and series
- Writing in sigma notation
- Determining the sum of an infinite geometric series
- Applying properties of limits to determine the limit of a term in a sequence or limit of a series
- Applying the ratio test as a method of determining if a series converges or diverges
- Expanding binomial expressions using Pascal’s triangle and by applying the binomial theorem
Unit 2: Trigonometric Ratios
Students begin a three unit exploration into trigonometry. In this unit, students review their understanding of the three basic trigonometric functions learned in geometry; sine, cosine and tangent; and extend this to include their reciprocal functions. Students learn radian measure and are introduced to the unit circle, the foundation of all of trigonometry.
- Applying definitions of trigonometric ratios to mathematical and real world problems
- Converting between degree and radian measure
- Interpreting information provided on the unit circle
- Determining exact values of angle measures using special right triangles
Unit 3: Graph Trigonometric Functions
Students learn the graphs of the parent functions of the six trigonometric functions, as an extension of the unit circle, and explore transformations of these graphs. Students learn about inverse trigonometric functions and restrictions placed on them. Through applications of real-world problems involving trigonometric functions, students form connections between the algebra, the graph, and the description of scenarios that can be modeled with trigonometric functions.
- Graphing, and writing equations of transformations of trigonometric function
- Graphing inverse trigonometric functions and restrictions placed on them
- Evaluating inverse trigonometric expressions
- Solving real-world problems that can be modeled with trigonometric function
Unit 4: Trigonometric Laws and Identities
The study of trigonometry provides an opportunity to investigate the algebra of trigonometry. This extends to the verification of trigonometric identities, applications of sum, difference, double, and half angle formulas, derivations and applications of the laws of sine and cosine, alternate methods of determining the area of a triangle, and an exploration into angular and linear velocities and how they are related to one another.
- Simplifying trigonometric expressions and verifying identities
- Applying sum, difference, double and half angle formulas
- Deriving and applying the Law of sines and the Law of cosines
- Deriving and applying alternate methods of determining the area of a triangle, including Heron’s formula
- Deriving the formulas for linear and angular velocity, and applying these formulas in real-world scenarios
Unit 5: Complex Numbers and Vectors
Students learn to plot points and express coordinates in the polar coordinate system; convert between polar and rectangular coordinates; graph polar equations; and add, subtract, multiply, and divide complex numbers in both polar and rectangular coordinate systems. They learn to calculate powers of complex numbers using De Moivre's theorem, to calculate roots of complex numbers, and to understand roots of unity and their graphical interpretation. Students learn about the different forms used to represent vectors, and the connections between vectors and matrices. Students also explore parametric equations, and learn to convert between parametric and rectangular form of equations, both graphically and algebraically. Real-world applications of parametric equations are highlighted..
- Plotting points and graphing equations in the polar coordinate system
- Converting between polar and rectangular forms
- Performing arithmetic on complex numbers both algebraically and graphically
- Representing vectors in a variety of ways, and transforming vectors with matrix multiplication
- Converting between parametric and rectangular forms of equations
- Graphing parametric equations, and using parametric equations to solve real-world problems
Unit 6: Semester Review
Students review what they have learned and take the semester exam.