College Algebra: Core Concepts and Applications Study Guide

Chapter 1: Foundations of Algebra

Types of Numbers

Understanding the classification of numbers is fundamental in algebra. Numbers are grouped based on their properties and forms.

• Rational numbers: Numbers that can be expressed as a fraction of two integers (e.g., $\frac{3}{4}$, $-2$, $0.5$).

• Irrational numbers: Numbers that cannot be written as a simple fraction (e.g., $\sqrt{2}$, $\pi$).

Order of Operations

Calculations must follow a specific sequence to ensure consistent results.

• PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

• Example: $2 + 3 \times (4^2 - 1)$

Evaluating Expressions

Expressions with variables can be evaluated by substituting values.

• Substitution: Replace variables with given numbers to compute the value.

• Example: If $x = 2$, evaluate $3x + 5$: $3(2) + 5 = 11$

One-Variable Data

Analyzing data with a single variable involves measures of central tendency.

• Mean: Average value.

• Median: Middle value when data is ordered.

• Example: Data: 2, 4, 7. Mean: $(2+4+7)/3 = 4.33$; Median: 4.

Two-Variable Data

Data involving two variables can be analyzed for relationships.

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

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

• Example: Points (1,2) and (4,6): Midpoint: $(2.5, 4)$; Distance: $\sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = 5$

Function Notation

Functions describe relationships between variables. Notation helps clarify input and output.

• Function notation: $f(x)$ represents the output when $x$ is the input.

• Example: If $f(x) = 2x + 3$, then $f(4) = 11$

Set-Builder Notation

Set-builder notation is used to describe sets of numbers that satisfy certain conditions.

• Example: $\{x | x > 0\}$ means all $x$ such that $x$ is greater than 0.

Interval Notation

Interval notation provides a concise way to describe ranges of values.

• Example: $(2, 5]$ means all numbers greater than 2 and up to and including 5.

Inequalities

Understanding inequalities is essential for solving and graphing algebraic expressions.

• Signs: $<$, $>$, $\leq$, $\geq$

• Number line representation: Visualizes solution sets.

Types of Functions

Functions can be classified as linear or nonlinear based on their graphical and algebraic properties.

• Linear functions: Graph is a straight line; constant rate of change.

• Nonlinear functions: Graph is not a straight line; rate of change varies.

Rate of Change

Rate of change measures how one variable changes in relation to another.

• Constant rate of change: Slope for linear functions.

• Average rate of change: Used for nonlinear functions; calculated as $\frac{f(b) - f(a)}{b - a}$

Chapter 2: Linear Equations and Modeling

Equations of Lines

Lines can be described using various forms and properties.

• Slope-intercept form: $y = mx + b$

• Point-slope form: $y - y_1 = m(x - x_1)$

• Horizontal line: $y = c$ (slope $m = 0$)

• Vertical line: $x = c$ (undefined slope)

Writing an Equation of a Line

Equations can be written given a point and slope, or two points.

• Parallel lines: Same slope.

• Perpendicular lines: Slopes are negative reciprocals ($m_1 \cdot m_2 = -1$).

• Example: Find the equation of a line passing through (2,3) with slope 4: $y - 3 = 4(x - 2)$

Linear Modeling

Linear models are used to represent real-world relationships with straight lines.

• Application: Predicting costs, population growth, etc.

Absolute Value Equations and Inequalities

Absolute value expressions measure distance from zero and require special techniques to solve.

• Equation: $|x| = a$ has solutions $x = a$ and $x = -a$

• Inequality: $|x| < a$ means $-a < x < a$

Chapter 3: Quadratic Equations and Functions

Equations of Quadratics

Quadratic equations are second-degree polynomials and can be written in several forms.

• Standard form: $y = ax^2 + bx + c$

• Vertex form: $y = a(x - h)^2 + k$

Finding the Vertex

The vertex is the highest or lowest point of a parabola.

• Vertex formula: $x = -\frac{b}{2a}$

• Example: For $y = 2x^2 - 4x + 1$, vertex at $x = 1$

Using Factoring

Factoring is a method to solve quadratic equations when $a = 1$ or other simple cases.

• Example: $x^2 - 5x + 6 = 0$ factors to $(x-2)(x-3) = 0$

Quadratic Formula

The quadratic formula solves any quadratic equation.

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

Discriminant

The discriminant determines the nature of the roots of a quadratic equation.

• Discriminant: $D = b^2 - 4ac$

• Interpretation: $D > 0$ (two real roots), $D = 0$ (one real root), $D < 0$ (no real roots)

Domain and Range of Quadratics

The domain and range describe the set of possible input and output values for quadratic functions.

• Domain: All real numbers ($-\infty, \infty$)

• Range: Depends on the vertex and direction of opening

Quadratic Inequalities

Quadratic inequalities involve finding intervals where the quadratic expression is positive or negative.

• Example: Solve $x^2 - 4 < 0$; solution: $-2 < x < 2$

Transformation of Quadratics

Transformations shift, stretch, or reflect the graph of a quadratic function.

• Vertical shift: $y = x^2 + k$

• Horizontal shift: $y = (x - h)^2$

• Reflection: $y = -x^2$