# Pre-calculus

## GENERAL STUDIES

### Course Description

In this course emphasis is placed on the many trigonometric, geometric and algebraic techniques needed for the preparation of the study of Calculus. The course takes a functional point of view towards topics and is designed to strengthen and enhance conceptual understanding and mathematical reasoning used when solving problems. The course also emphasizes the use of the graphing calculator as a tool to interpret results as well as a method of obtaining an answer. Mastery of algebraic skills is an essential prerequisite. This course surveys a variety of pre-calculus topics including polynomial, rational, exponential, and trigonometric functions over the real and complex numbers. Much emphasis is placed on theory, and breadth and depth of understanding as well as efficacy. A pupil who successfully completes this course is prepared for Calculus AP/AB.

### Course Objectives

• Students will understand the structure of the systems of real and complex numbers, and the concept of functions and their unifying role in mathematics.
• Students will utilize algebraic and trigonometric concepts and skills, will be able to analyze and graph a variety of functions, and will acquire the necessary skills used in calculus such as evaluating limits and the concept of continuity.
• Students are expected to apply the following common skills that are relevant across all curriculum areas and career pathways: students will use technology, apply problem solving and critical thinking skills, and adapt to varied roles and responsibilities.
• Students communicate precisely about quantities, logical relationships, and unknown values through the use of signs, symbols, models, graphs, and mathematical vocabulary. Regular opportunities are provided for students to communicate through oral and written explanations of math concepts.
• Students learn to apply mathematics to everyday life.

### Assessments:

Attendance and participation: 15%

Assignments and homework: 25%

• Nightly homework will be given as a review of what was covered in class

Tests and quizzes: 30%

• Quizzes will be given half way through each unit.
• Tests will be given at the end of each unit.

Midterm or final: 30%

### Scope and Sequence

Linear Relations and Functions

• Relations and Functions
• Compositions of Functions
• Graphing Linear Equations
• Writing Equations
• Equations of Parallel and Perpendicular Lines
• Modeling Real World Data with Linear Functions
• Absolute Value Functions
• Graphing Linear Inequalities

Systems of Linear Equations and Inequalities

• Solving Systems of Equations in Two and Three Variables
• Using Matrices to Model Motion and Real World Data
• Determinants and Multiplicative Inverses of Matrices
• Solving Systems of Inequalities
• Linear Programming

The Nature of Graphs

• Symmetry and Coordinate Graphs
• Families of Graphs
• Graphs of Nonlinear Inequalities
• Inverse Functions and Relations
• Continuity and End Behavior
• Critical Points and Extrema
• Graphs of Rational Functions
• Direct, Indirect, and Joint Variation

Polynomial and Rational Functions

• Polynomial Functions
• The Remainder and Factor Theorems and the Rational Root Theorem
• Locating Zeros of a Polynomial Function
• Rational Equations and Partial Fractions
• Modeling Real World Data with Polynomial Functions

The Trigonometric Functions

• Angles and Degree Measure
• Trigonometric Ratios in Right Triangles
• Trigonometric Functions on the Unit Circle
• Applying Trigonometric Functions
• Solving Right Triangles
• The Law of Sines and The Law of Cosines

Graphs of Trigonometric Functions

• Linear and Angular Velocity
• Graphing Sine and Cosine Functions
• Amplitude and Period of Sine and Cosine Functions
• Horizontal and Vertical Shifts of Sine and Cosine Functions
• Modeling Real World Data with Sinusoidal Functions
• Graphing Other Trigonometric Functions and their Inverses

Trigonometric Identities and Equations

• Basic Trigonometric Identities
• Verifying Trigonometric Identities
• Sum and Difference Identities and Double Angle and Half Angle Identities
• Solving Trigonometric Equations
• Normal Form of a Linear Equation
• Distance From a Point to a Line

Vectors and Parametric Equations

• Geometric and Algebraic Vectors
• Vectors in Three Dimensional Space
• Perpendicular Vectors
• Applications with Vectors
• Vectors and Parametric Equations
• Modeling Motion Using Parametric Equations
• Transformational Matrices in Three Dimensional Space

Polar Coordinates and Complex Numbers

• Polar Coordinates
• Graphs of Polar Equations
• Polar and Rectangular Coordinates
• Polar Form of a Linear Equation
• Simplifying Complex Numbers
• The Complex Plan and Polar Form of Complex Numbers
• Products and Quotients of Complex Numbers in Polar Form
• Powers and Roots of Complex Numbers

Conics

• Introduction to Analytic Geometry
• Circles, Ellipses, Hyperbolas, and Parabolas
• Rectangular and Parametric Forms of Conic Sections
• Transformation of Conics
• Systems of Second Degree Equations and Inequalities

Exponential and Logarithmic Functions

• Real Exponents
• Exponential Functions
• The Number e
• Logarithmic Functions
• Common Logarithms and Natural Logarithms
• Modeling Real World Data with Exponential and Logarithmic Functions

Sequences and Series

• Arithmetic and Geometric Sequences and Series
• Infinite Sequences and Series
• Convergent and Divergent Series
• Sigma Notation and the nth Term
• The Binomial Theorem
• Special Sequences and Series
• Sequences and Iterations
• Mathematical Induction

Combinatorics and Probability

• Permutations and Combinations
• Permutations with Repetitions and Circular Permutations
• Probability and Odds and Probabilities of Compound Events
• Conditional Probabilities
• The Binomial Theorem and Probabilities

Statistics and Data Analysis

• The Frequency Distribution
• Measures of Central Tendency
• Measures of Variability
• The Normal Distribution
• Sample Sets of Data