College Algebra Syllabus & Core Concepts Study Guide

College Algebra: Course Overview

Course Structure and Key Information

This course, MATH 1111: College Algebra, covers fundamental and advanced algebraic concepts essential for college-level mathematics. The course is delivered online, utilizing MyMathLab for assignments and assessments. Students are expected to engage with digital resources, participate in discussions, and use a TI-84+ graphing calculator for coursework.

• Textbook: Algebra and Trigonometry: Graphs and Models, Bittinger, Marvin, 7th Edition

• Software: MyMathLab (ML) is required for all homework and tests

• Calculator: TI-84+ Graphing Calculator (or equivalent)

• Assessment: Tests (40%), Homework (30%), Participation (10%), Final Exam (20%)

Core Competencies and Learning Outcomes

Review of Algebra

This section establishes foundational skills necessary for success in algebra, including set theory, exponents, radicals, polynomials, rational expressions, and complex numbers.

• Sets and Set Notation: Understanding sets, elements, and notation such as $A = \{1, 2, 3\}$.

• Set Operations: Complements, unions ($A \cup B$), and intersections ($A \cap B$).

• Laws of Exponents: Simplifying expressions using rules like $a^m \cdot a^n = a^{m+n}$.

• Radicals: Simplifying and performing arithmetic with roots, e.g., $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.

• Polynomials: Addition, subtraction, multiplication, and factoring.

• Rational Expressions: Operations with fractions involving polynomials.

• Complex Numbers: Definition ($i = \sqrt{-1}$), arithmetic, and properties.

Equations and Inequalities

Solving various types of equations and inequalities is a central skill in algebra, including linear, quadratic, rational, exponential, and logarithmic forms.

• Linear Equations: Solve for $x$ in $ax + b = 0$.

• Quadratic Equations: Use factoring, completing the square, or the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

• Linear Inequalities: Solve and graph solutions on a number line.

• Quadratic and Rational Inequalities: Analyze sign charts and intervals.

• Exponential and Logarithmic Equations: Apply properties such as $\log_b(x) = y \implies b^y = x$.

Functions and Graphs

Functions are mathematical relationships between inputs and outputs. Graphing functions helps visualize their behavior and properties.

• Ordered Pairs: Plotting $(x, y)$ on the Cartesian plane.

• Relations and Functions: A function assigns each input exactly one output.

• Graphing Linear Functions: $y = mx + b$; slope-intercept form.

• Graphing Quadratic Functions: $y = ax^2 + bx + c$; parabolic shape.

• Graphing Exponential Functions: $y = a \cdot b^x$; rapid growth or decay.

• Graphing Logarithmic Functions: $y = \log_b(x)$; inverse of exponential.

Systems of Equations and Matrices

Systems of equations involve solving for multiple variables simultaneously. Matrices provide a structured way to represent and solve these systems.

• Linear Systems: Solve two equations with two unknowns using substitution or elimination.

• Application Problems: Model real-world scenarios with systems.

• Matrices: Optional topic; use matrix methods for solving systems.

Optional Topics

These topics extend algebraic concepts to more advanced applications, including variation, absolute value, conic sections, sequences, induction, binomial theorem, combinatorics, and probability.

• Variation: Direct ($y = kx$), inverse ($y = \frac{k}{x}$), and joint ($y = kxz$) relationships.

• Absolute Value Equations/Inequalities: $|x| = a$ or $|x| < a$.

• Conic Sections: Circles, ellipses, parabolas, hyperbolas.

• Sequences: Arithmetic ($a_n = a_1 + (n-1)d$) and geometric ($a_n = a_1 r^{n-1}$).

• Mathematical Induction: Prove statements for all integers $n$.

• Binomial Theorem: Expand $(a + b)^n$ using $\sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$.

• Permutations and Combinations: $n!$ for permutations, $\binom{n}{r}$ for combinations.

• Probability: $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$.

Course Schedule and Weekly Topics

Weekly Progression

The course is organized by weeks, each focusing on specific sections and competencies. Assignments are completed in MyMathLab, with readiness quizzes and chapter tests marking progress.

• Weeks 1-4: Review of Algebra (Sets, Exponents, Radicals, Polynomials, Rational Expressions, Complex Numbers)

• Weeks 5-8: Equations and Inequalities (Linear, Quadratic, Rational, Exponential, Logarithmic)

• Weeks 9-11: Functions and Graphs (Linear, Quadratic, Exponential, Logarithmic)

• Weeks 12-17: Polynomial and Rational Functions, Systems of Equations, Optional Topics

Grading and Evaluation

Grade Breakdown

Grades are determined by performance in tests, homework, participation, and the final exam. The grading scale is standard, with a minimum grade of "C" required for progression.

Category	Weight
Tests	40%
Homework	30%
Participation	10%
Final Exam	20%

Grading Scale:

Grade	Percentage
A	90-100%
B	80-89%
C	70-79%
D	60-69%
F	59% and below

Academic Integrity and Course Policies

Ethical Standards

Students are expected to adhere to strict academic integrity, including policies on cheating, plagiarism, and ethical use of AI. Misconduct may result in disciplinary action.

• Cheating: Unauthorized use of materials or information.

• Plagiarism: Presenting others' work as your own.

• AI Use: Only permitted when explicitly allowed by the instructor.

Support and Resources

Academic support, tutoring, library services, and accessibility accommodations are available to all students. Students are encouraged to utilize these resources for success.

• Academic Support Centers: Free tutoring for math and other subjects.

• Library Services: Access to resources and research support.

• Accessibility Services: Accommodations for students with disabilities.

Summary Table: College Algebra Core Topics

Main Topic	Subtopics
Review of Algebra	Sets, Exponents, Radicals, Polynomials, Rational Expressions, Complex Numbers
Equations & Inequalities	Linear, Quadratic, Rational, Exponential, Logarithmic Equations & Inequalities
Functions & Graphs	Relations, Linear, Quadratic, Exponential, Logarithmic Functions
Systems of Equations	Linear Systems, Matrices (optional)
Optional Topics	Variation, Absolute Value, Conic Sections, Sequences, Induction, Binomial Theorem, Combinatorics, Probability

Example: Solving a Quadratic Equation

Problem: Solve $x^2 - 5x + 6 = 0$.

• Factoring: $x^2 - 5x + 6 = (x - 2)(x - 3) = 0$

• Solutions: $x = 2$ or $x = 3$

Quadratic Formula:

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

For $a = 1$, $b = -5$, $c = 6$:

$x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2}$

Thus, $x = 3$ or $x = 2$.

Additional info:

• Some context and explanations were expanded for clarity and completeness.

• Tables were recreated to summarize grading and core topics.

• Examples and formulas were provided to illustrate key concepts.