Comprehensive Study Guide: College Algebra Topics and Assignments Overview

Course Overview: College Algebra

This study guide provides a structured overview of the main topics, subtopics, and key concepts covered in a typical College Algebra course, as outlined in the provided assignment list. Each section includes definitions, properties, and examples to support student understanding and exam preparation.

1. Real Numbers and Algebraic Expressions

1.1 Real Numbers

• Definition: Real numbers include all rational and irrational numbers.

• Properties: Closure, commutativity, associativity, distributivity, identity, and inverse properties.

• Example: $\sqrt{2}$ (irrational), $-5$ (integer), $3/4$ (rational).

1.2 Intervals and Set Notation

• Interval Notation: Used to describe subsets of real numbers, e.g., $(a, b)$, $[a, b]$.

• Set Notation: Uses braces and inequalities, e.g., $\{x | a < x < b\}$.

• Example: $[2, 5)$ represents all real numbers $x$ such that $2 \leq x < 5$.

1.3 Algebraic Expressions

• Definition: Combinations of numbers, variables, and operations.

• Simplification: Apply order of operations (PEMDAS).

• Example: Simplify $2(x + 3) - 4x = 2x + 6 - 4x = -2x + 6$.

2. Graphs, Functions, and Models

2.1 Introduction to Graphing, Distance, and Midpoint

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

• Midpoint Formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

• Application: Used to find the distance and midpoint between two points in the coordinate plane.

2.2 Functions and Their Graphs

• Definition: A function is a relation where each input has exactly one output.

• Notation: $f(x)$ denotes the value of function $f$ at $x$.

• Example: $f(x) = x^2$ is a function; $f(2) = 4$.

2.3 Transforming the Graphs of Functions

• Vertical and Horizontal Shifts: $f(x) + k$ (up/down), $f(x - h)$ (left/right).

• Reflections: $-f(x)$ (over x-axis), $f(-x)$ (over y-axis).

• Example: $f(x) = (x - 2)^2 + 3$ is $f(x) = x^2$ shifted right 2 units and up 3 units.

3. Linear and Quadratic Functions

3.1 Linear Equations and Graphs

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Example: Find the equation of a line through $(1,2)$ with slope $3$:$y - 2 = 3(x - 1)$.

3.2 Quadratic Functions and Their Properties

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex: $x = -\frac{b}{2a}$

• Axis of Symmetry: $x = -\frac{b}{2a}$

• Example: For $f(x) = 2x^2 - 4x + 1$, vertex at $x = 1$.

3.3 Solving Quadratic Equations

• Factoring: Set equation to zero and factor.

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

• Completing the Square: Rewrite in the form $(x + p)^2 = q$.

• Example: Solve $x^2 - 5x + 6 = 0$ by factoring: $(x-2)(x-3)=0$ so $x=2,3$.

4. Polynomial and Rational Functions

4.1 Polynomial Functions

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

• Degree: Highest power of $x$.

• End Behavior: Determined by leading term.

• Example: $f(x) = x^3 - 2x^2 + x - 5$ is a cubic polynomial.

4.2 Rational Functions

• Definition: Ratio of two polynomials, $f(x) = \frac{p(x)}{q(x)}$.

• Asymptotes: Vertical (set denominator to zero), horizontal (compare degrees).

• Example: $f(x) = \frac{1}{x-2}$ has a vertical asymptote at $x=2$.

5. Exponential and Logarithmic Functions

5.1 Exponential Functions

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

• Growth and Decay: $b > 1$ (growth), $0 < b < 1$ (decay).

• Example: $f(x) = 2^x$ is exponential growth.

5.2 Logarithmic Functions

• Definition: $y = \log_b x$ is the inverse of $b^y = x$.

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

• Example: $\log_2 8 = 3$ because $2^3 = 8$.

6. Systems of Equations and Matrices

6.1 Solving Systems of Linear Equations

• Methods: Substitution, elimination, and matrix methods (Gaussian elimination).

• Example: Solve $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$ by addition: $2x = 6 \Rightarrow x = 3$, $y = 2$.

6.2 Matrices

• Definition: Rectangular array of numbers.

• Operations: Addition, multiplication, finding determinants and inverses.

• Example: $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, $A + B = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$.

7. Conic Sections

7.1 Circles, Parabolas, Ellipses, and Hyperbolas

• Circle: $(x - h)^2 + (y - k)^2 = r^2$

• Parabola: $y = a(x - h)^2 + k$

• Ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

• Hyperbola: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$

• Example: The equation $x^2 + y^2 = 9$ represents a circle centered at the origin with radius 3.

8. Sequences, Series, and Combinatorics

8.1 Sequences and Series

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

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

• Sum of Arithmetic Series: $S_n = \frac{n}{2}(a_1 + a_n)$

• Sum of Geometric Series: $S_n = a_1 \frac{1 - r^n}{1 - r}$, $r \neq 1$

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

8.2 Combinatorics

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

• Combinations: $nCr = \frac{n!}{r!(n-r)!}$

• Example: Number of ways to choose 3 out of 5: $5C3 = 10$.

Appendix: Sample Assignment Table

The following table summarizes the weekly topics and associated textbook sections as outlined in the assignment list. This can be used to track progress and review key areas for exams.

Week	Section	Topic
1	1.1, 1.2, 1.3	Real Numbers, Intervals, Algebraic Expressions
2	1.4, 1.5, 1.6	Graphing, Distance, Midpoint, Functions
3	2.1, 2.2, 2.3	Linear Equations, Graphs, Applications
4	2.4, 2.5, 2.6	Quadratic Functions, Inequalities
5	3.1, 3.2, 3.3	Polynomial Functions, Rational Functions
6	4.1, 4.2, 4.3	Exponential, Logarithmic Functions
7	5.1, 5.2, 5.3	Systems of Equations, Matrices
8	6.1, 6.2	Conic Sections
9	7.1, 7.2	Sequences, Series, Combinatorics

Additional info: The assignment list covers all major College Algebra topics, including functions, equations, inequalities, polynomials, rational, exponential, and logarithmic functions, systems of equations, matrices, conic sections, and sequences/series.