College Algebra Course Syllabus and Weekly Study Guide

Course Overview

This study guide summarizes the main topics and weekly structure of a College Algebra course. The course covers fundamental algebraic concepts, functions, equations, inequalities, matrices, and sequences, providing a strong foundation for further mathematical studies.

Weekly Topics and Key Concepts

Week 1: Introduction and Functions

• Functions: Definition, notation, and evaluation of functions.

• Linear and Slope: Understanding linear functions and calculating slope.

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

• Example: Find the slope between points (2,3) and (5,11): $m = \frac{11-3}{5-2} = \frac{8}{3}$

Week 2: More on Functions and Transformations

• Function Transformations: Shifts, stretches, compressions, and reflections of function graphs.

• Combinations of Functions: Addition, subtraction, multiplication, and division of functions.

• Example: If $f(x) = x^2$ and $g(x) = 2x$, then $(f+g)(x) = x^2 + 2x$

Week 3: Combinations, Compositions, and Inverses of Functions

• Compositions: $(f \circ g)(x) = f(g(x))$

• Inverse Functions: A function $f$ has an inverse $f^{-1}$ if $f(f^{-1}(x)) = x$

• Example: If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x-3}{2}$

Week 4: Quadratics and Polynomials

• Quadratic Functions: Standard form $y = ax^2 + bx + c$

• Graphs and Multiplicity: The shape of the graph and the effect of root multiplicity.

• Intermediate Value Theorem (IVT): If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c$ in $[a, b]$ such that $f(c) = N$.

Week 5: Polynomials and Rational Functions

• Dividing Polynomials: Long division and synthetic division.

• Zeros of Functions: Solutions to $f(x) = 0$.

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

• Asymptotes: Vertical and horizontal asymptotes describe end behavior and undefined points.

Week 6: Inequalities and Variation

• Inequalities: Solving linear and polynomial inequalities.

• Variation: Direct, inverse, and joint variation.

• Example: Direct variation: $y = kx$; Inverse variation: $y = \frac{k}{x}$

Week 7: Exponential Functions and Compound Interest

• Exponential Functions: $f(x) = a \cdot b^x$

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

• Example: $P = 1000$, $r = 0.05$, $n = 12$, $t = 3$ years: $A = 1000\left(1 + \frac{0.05}{12}\right)^{36}$

Week 8: Logarithmic Functions

• Logarithms: The inverse of exponentials. $y = \log_b x$ means $b^y = x$

• Properties: $\log_b(xy) = \log_b x + \log_b y$ \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$\log_b(x^r) = r \log_b x$

• Solving Exponential and Logarithmic Equations: Use properties of logarithms and exponentials to isolate variables.

Week 9: Exponential Growth and Decay

• Growth and Decay Models: $y = y_0 e^{kt}$, where $k > 0$ for growth, $k < 0$ for decay.

• Applications: Population growth, radioactive decay.

Weeks 10-11: Systems of Equations and Inequalities

• Solving Systems: Substitution, elimination, and graphical methods.

• Linear Programming: Optimization using systems of inequalities.

• Non-linear Systems: Systems involving quadratic or other non-linear equations.

Weeks 12-13: Matrices and Determinants

• Matrices: Rectangular arrays of numbers used to solve systems of equations.

• Gaussian and Gauss-Jordan Elimination: Methods for solving systems using row operations.

• Matrix Inverses and Determinants: Used to solve systems and analyze properties of matrices.

• Example: For $A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$, $\det(A) = 1 \times 4 - 2 \times 3 = -2$

Weeks 14-15: Sequences and Series

• Sequences: Ordered lists of numbers, such as arithmetic and geometric sequences.

• Summation Notation: $\sum_{i=1}^n a_i$

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Example: Find the 5th term of the sequence $2, 4, 6, 8, ...$ (arithmetic, $d=2$): $a_5 = 2 + (5-1)\times2 = 10$

Final Exam Review

• Comprehensive review of all topics covered.

• Practice problems and applications from each chapter.

Summary Table: Major Topics and Chapters

Week	Main Topic	Key Subtopics
1-2	Functions	Definition, evaluation, transformations, combinations
3	Compositions & Inverses	Function composition, inverse functions
4-5	Polynomials & Rational Functions	Quadratics, zeros, asymptotes, division
6	Inequalities & Variation	Solving inequalities, direct/inverse variation
7-9	Exponential & Logarithmic Functions	Exponents, logs, growth/decay, compound interest
10-11	Systems of Equations	Linear/non-linear systems, linear programming
12-13	Matrices & Determinants	Matrix operations, Gaussian elimination, inverses
14-15	Sequences & Series	Arithmetic, geometric, summation notation