Precalculus Course Overview and Study Guide

Precalculus Course Overview

Course Description

This course prepares students for Calculus by developing foundational skills in algebra, functions, trigonometry, and mathematical reasoning. Students will analyze and graph various types of functions, solve equations and inequalities, and apply mathematical concepts to real-world problems.

• Functions: Study of linear, quadratic, rational, radical, absolute value, and exponential functions.

• Equations and Inequalities: Solving linear, quadratic, rational, and absolute value equations and inequalities.

• Graphing: Techniques for graphing functions and analyzing their properties.

• Trigonometry: Introduction to trigonometric functions, identities, and applications.

• Polynomials: Operations, factoring, and solving polynomial equations.

• Applications: Applying mathematical concepts to solve application problems.

Course Objectives

Upon successful completion of the course, students will be able to:

• Solve linear, quadratic, rational, and absolute value inequalities.

• Graph and analyze functions and relations.

• Graph polynomial, rational, radical, absolute value, and exponential functions.

• Graph and manipulate inverse functions and relations.

• Find domain, range, increasing and decreasing intervals.

• Find rational, irrational roots and solve equations using polynomial division.

• Sketch graphs of trigonometric functions and inverse trigonometric functions.

• Solve trigonometric problems that define functions.

• Develop skills to solve application problems.

• Use software formats in written work.

Key Terms and Definitions

• Function: A relation in which each input has exactly one output.

• Domain: The set of all possible input values for a function.

• Range: The set of all possible output values for a function.

• Polynomial: An expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents.

• Trigonometric Function: Functions related to angles, commonly sine, cosine, and tangent.

Important Formulas

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Slope of a Line: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Standard Form of a Polynomial: $P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$

• Sine and Cosine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Examples and Applications

• Example 1: Solving a quadratic equation Solve $x^2 - 5x + 6 = 0$ using the quadratic formula. $x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2}$ Solutions: $x = 3$ and $x = 2$.

• Example 2: Finding the domain of a function For $f(x) = \frac{1}{x-2}$, the domain is all real numbers except $x = 2$.

• Example 3: Graphing a linear function For $f(x) = 2x + 1$, plot points for $x = -1, 0, 1$ and draw the line through them.

Course Structure and Assessment

• Online Learning: Course is delivered online via CourseCompass/MyMathLab.

• Quizzes and Exams: Regular online quizzes and a cumulative final exam.

• Grading: Quizzes (65%), Final Exam (35%).

• Textbook: MyLab Math with Pearson eText.

Sample Assessment Table

Assignment Name	Available Date	Due Date
Chapter 1 Quiz	9/15	9/15 at 11:59pm
Chapter 2 Quiz	9/15	9/15 at 11:59pm
Chapter 3 Quiz	9/15	9/15 at 11:59pm
Chapter 4 Quiz	9/15	9/15 at 11:59pm
Chapter 5 Quiz	9/15	10/17/15 at 11:59pm

Permitted and Prohibited Uses of AI Tools

Permitted Uses	Prohibited Uses
Supplementary learning (e.g., practice problems, review explanations). Study aid (generating practice problems, step-by-step solutions). Exploring alternative methods to solve a problem.	Completing assignments (using AI to generate or solve graded assignments). Misrepresentation of work (submitting work generated by AI as your own). Unauthorized assistance (using AI during assessments unless explicitly authorized).

Additional Resources

• Math-Science Learning Specialists: Free online support for math and science courses.

• Peer Tutoring: Available for math, science, English, and general study skills.

• Testing Center: For placement exams and other testing needs.

• Library: Access to online and physical resources.

Additional info: These notes summarize the course syllabus and structure, including objectives, assessment, and resources. For detailed mathematical content, refer to the course textbook and online modules.