Precalculus Syllabus and Study Guide

Course Overview

This course provides an integrated treatment of the concepts necessary for calculus, including a review of algebraic and transcendental functions, trigonometric functions, and analytic geometry. The course prepares students for further study in calculus and related fields.

• Course Title: Precalculus

• Course Number: 2412

• Term: Fall 2024

• Class Format: Online

• Prerequisites: MATH 1314 and MATH 1316 (or placement by testing)

Required Materials

• Textbook: Precalculus, 8th edition, Michael Sullivan. Includes MyLab Math access.

• Calculator: Graphing calculators are required for some assignments/assessments. Only the following models are allowed:

  • TI-83 or TI-84 series (plus, silver, CE, etc.)

  • TI-30XS scientific calculator (for some assignments/assessments)

• Other: Dry erase markers and whiteboard for scratch work during online assessments.

Student Learning Outcomes

• Demonstrate and apply knowledge of properties of functions.

• Recognize and analyze polynomial, rational, radical, absolute value, and piecewise-defined equations.

• Represent and solve equations and inequalities graphically and algebraically.

• Apply the concept of a function and use function notation.

• Analyze the values of trigonometric functions for key angles in all quadrants of the unit circle measured in both degrees and radians.

• Prove trigonometric identities.

• Apply the binomial theorem.

• Graph conic sections and identify their properties.

• Determine equations of conic sections and graph conics, including translation and identification of vertices, foci, and asymptotes.

• Use properties of arithmetic and geometric sequences and series to identify terms, find sums, and solve application problems.

Main Topics and Subtopics

1. Functions and Their Properties

Functions are fundamental objects in mathematics that assign each input exactly one output. Understanding their properties is essential for analyzing mathematical relationships.

• Definition: A function is a relation in which each element of the domain is paired with exactly one element of the range.

• Types of Functions: Polynomial, rational, radical, absolute value, piecewise-defined, exponential, and logarithmic functions.

• Function Notation: denotes the value of the function at input .

• Domain and Range: The set of all possible inputs (domain) and outputs (range) for a function.

• Example: is a polynomial function with domain and range .

2. Equations and Inequalities

Solving equations and inequalities is a core skill in precalculus, involving algebraic manipulation and graphical interpretation.

• Polynomial Equations: Equations involving terms with variables raised to whole number powers.

• Rational Equations: Equations involving ratios of polynomials.

• Radical Equations: Equations involving roots of variables.

• Absolute Value Equations: Equations involving the absolute value function .

• Example: Solve ; .

3. Trigonometric Functions

Trigonometric functions describe relationships between angles and sides in right triangles and are extended to the unit circle.

• Key Functions: Sine (), Cosine (), Tangent (), and their reciprocals.

• Unit Circle: A circle of radius 1 centered at the origin, used to define trigonometric functions for all real numbers.

• Degrees and Radians: Two units for measuring angles. radians.

• Example: ; .

4. Analytic Geometry and Conic Sections

Analytic geometry involves the study of geometric figures using algebraic equations. Conic sections are curves obtained by intersecting a plane with a cone.

• Conic Sections: Parabolas, ellipses, circles, and hyperbolas.

• Standard Equations:

  • Circle:

  • Parabola:

  • Ellipse:

  • Hyperbola:

• Example: The equation represents a circle centered at the origin with radius 3.

5. Sequences and Series

Sequences are ordered lists of numbers, and series are sums of sequences. Arithmetic and geometric sequences are commonly studied in precalculus.

• Arithmetic Sequence: Each term is obtained by adding a constant difference to the previous term.

  • General term:

  • Sum of first terms:

• Geometric Sequence: Each term is obtained by multiplying the previous term by a constant ratio.

  • General term:

  • Sum of first terms: (for )

• Example: For the arithmetic sequence 2, 5, 8, 11, ..., .

6. Binomial Theorem

The binomial theorem provides a formula for expanding powers of binomials.

• Formula:

• Example:

Calculator Policy Table

The following table summarizes the calculator policy for the course:

Course Policies and Procedures

• Attendance: Regular participation is strongly recommended. Students are responsible for keeping up with assignments and deadlines.

• Homework: Assigned and submitted via MyLab Math. Used for practice and mastery of skills.

• Assessments: All assignments, including homework, quizzes, and tests, are online and submitted through MyLab Math. Tests require a Lockdown Browser and video recording enabled.

• Orientation Quiz: Completion of the Orientation Quiz in MyLab Math by the due date is required to be counted as attending the course at least once.

• Penalty for Not Enrolling in MyLab Math: Students not enrolled in MyLab Math by the specified deadline will be dropped from the class.

Core Curriculum Statement

This course is part of the Texas Higher Education Coordinating Board's Core Curriculum and addresses the following objectives:

• Critical Thinking Skills (CT): Creative thinking, innovation, inquiry, and analysis.

• Communication Skills (COM): Effective development, interpretation, and expression of ideas.

• Empirical & Quantitative Skills (EQS): Manipulation and analysis of numerical data or observable facts.

Additional info: These notes are based on the course syllabus and provide a structured overview of the main topics and policies for a college-level Precalculus course.