College Algebra Syllabus and Course Structure: Key Topics and Policies

Course Overview

This course covers fundamental concepts in College Algebra, including polynomial, exponential, logarithmic, and rational functions, with applications in business, economics, and social sciences. Students will also explore systems of linear equations, probability, and financial mathematics.

Course Structure and Main Topics

1. Linear Equations and Functions

Linear equations and functions form the foundation of algebra and are essential for modeling real-world relationships.

• Linear Equations: Equations of the form $ax + b = 0$ where a and b are constants.

• Graphing Linear Equations: The graph of a linear equation is a straight line. The slope-intercept form is $y = mx + c$.

• Function Notation: Expressing functions as $f(x)$ to denote the output for input $x$.

• Piecewise Functions: Functions defined by different expressions over different intervals.

• Applications: Modeling cost, revenue, and other relationships in business and science.

• Example: Solve $2x + 3 = 7$ for $x$.

2. Quadratic Equations and Functions

Quadratic functions are polynomials of degree two and are used to model parabolic relationships.

• Quadratic Equation: $ax^2 + bx + c = 0$

• Factoring: Expressing the quadratic as a product of two binomials.

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

• Graphing: The graph is a parabola, opening upwards if $a > 0$ and downwards if $a < 0$.

• Applications: Projectile motion, area problems.

• Example: Solve $x^2 - 5x + 6 = 0$ by factoring.

3. Polynomial and Rational Functions

Polynomial functions extend linear and quadratic functions to higher degrees, while rational functions involve ratios of polynomials.

• Polynomial Functions: Functions of the form $f(x) = a_nx^n + \ldots + a_1x + a_0$

• Factoring Polynomials: Breaking down polynomials into irreducible factors.

• Rational Functions: Functions of the form $f(x) = \frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$.

• Asymptotes: Lines that the graph approaches but never touches.

• Applications: Rates, proportions, and modeling real-world scenarios.

• Example: Simplify $\frac{x^2 - 9}{x - 3}$.

4. Exponential and Logarithmic Functions

Exponential and logarithmic functions are crucial for modeling growth, decay, and many natural phenomena.

• Exponential Functions: $f(x) = a \cdot b^x$ where $a \neq 0$, $b > 0$, $b \neq 1$.

• Logarithmic Functions: The inverse of exponential functions, $f(x) = \log_b(x)$.

• Properties: Laws of exponents and logarithms, such as $\log_b(xy) = \log_b(x) + \log_b(y)$.

• Converting Forms: $y = b^x \Leftrightarrow x = \log_b(y)$

• Applications: Population growth, radioactive decay, pH in chemistry.

• Example: Solve $2^x = 8$.

5. Financial Mathematics

Financial mathematics applies algebraic concepts to real-world financial problems.

• Simple Interest: $I = Prt$ where $P$ is principal, $r$ is rate, $t$ is time.

• Compound Interest: $A = P\left(1 + \frac{r}{n}\right)^{nt}$

• Annuities: Series of equal payments at regular intervals.

• Loans and Amortization: Calculating payments and interest over time.

• Example: Find the amount after 5 years if $P = 1000$, $r = 5\%$, compounded annually.

6. Systems of Linear Equations and Matrices

Systems of equations and matrices are used to solve multiple equations simultaneously.

• System of Equations: Two or more equations with multiple variables.

• Solving Methods: Substitution, elimination, and matrix methods.

• Matrices: Rectangular arrays of numbers used to represent systems.

• Determinants: Scalar values that can be computed from a square matrix.

• Applications: Business, engineering, and science problems.

• Example: Solve the system: $\begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases}$

7. Inequalities and Linear Programming

Inequalities and linear programming are used for optimization problems.

• Linear Inequalities: Expressions like $ax + by \leq c$.

• Graphing Inequalities: Shading regions on the coordinate plane.

• Linear Programming: Maximizing or minimizing a linear objective function subject to constraints.

• Example: Maximize $P = 3x + 2y$ subject to $x + y \leq 10$, $x \geq 0$, $y \geq 0$.

8. Probability and Counting Principles

Probability and counting principles are essential for analyzing random events and outcomes.

• Sample Spaces: The set of all possible outcomes.

• Counting Principles: Fundamental Counting Principle, permutations, and combinations.

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

• Permutations: Arrangements where order matters: $nPr = \frac{n!}{(n - r)!}$

• Combinations: Selections where order does not matter: $nCr = \frac{n!}{r!(n - r)!}$

• Example: How many ways can 3 students be chosen from a group of 10?

Course Policies and Assessment

Textbook and Materials

• No required textbook. All materials provided online.

• Knewton Alta Access: Required for assignments.

• Scientific Calculator: Recommended models: TI-30, TI-34, TI-83, TI-84, Casio FX-55.

Attendance and Participation

• Attendance recorded every Monday and Thursday.

• Missing assignments or late submissions may result in absences.

Grading Policy

Component	Weight
Homework	20%
Quizzes	10%
Long Tests (Average)	50%
Final Exam	20%

Assessment Details

• Homework: Adaptive, mastery-based, 100% mastery required for perfect score.

• Quizzes: Each major topic includes a short quiz; two attempts allowed.

• Long Tests: Proctored, remote, one attempt per test.

• Final Exam: Comprehensive, mandatory, 20% of final grade.

Academic Integrity

• Cheating results in a zero grade, which cannot be dropped.

Other Policies

• Religious Holidays: Notify by first week for excused absence.

• Incomplete Grade: Apply before final week; must have C or higher; supporting documents required.

• Vacations: Not considered excused absences.

Course Calendar: Major Topics by Week

Week	Main Topics
1-2	Linear Equations, Function Notation, Piecewise Functions
3-4	Factoring Quadratics, Quadratic Equations, Graphs of Quadratic Functions
5-6	Polynomial and Rational Functions, Rational Equations
7-8	Exponential and Logarithmic Functions, Laws of Exponents and Logarithms
9-10	Applications of Exponential and Logarithmic Functions, Financial Mathematics
11-12	Systems of Linear Equations, Matrices, Determinants
13-14	Linear Inequalities, Linear Programming
15-16	Probability, Counting Principles, Permutations, Combinations

Additional info: The above structure is inferred from the syllabus and assignment schedule. Some details, such as specific examples and applications, are added for academic completeness.