Algebra Basics

Algebra

Definition

• Algebra is a branch of mathematics that deals with the study of variables and their relationships, often expressed through the use of symbols, equations, and functions.

Key Concepts

• Variables: Letters or symbols that represent unknown values or quantities.
• Constants: Numbers that do not change value.
• Terms: Parts of an expression separated by addition or subtraction.
• Expressions: Combinations of variables, constants, and mathematical operations.
• Equations: Statements that express the equality of two expressions.
• Inequalities: Statements that express a relationship between two expressions, but not necessarily equal.

Operations

• Addition: Combining two or more expressions by adding their corresponding terms.
• Subtraction: Combining two or more expressions by subtracting their corresponding terms.
• Multiplication: Combining two or more expressions by multiplying their corresponding terms.
• Division: Combining two or more expressions by dividing their corresponding terms.

Equation Solving

• Linear Equations: Equations in which the highest power of the variable is 1.
  • Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
  • Standard Form: ax + by = c, where a, b, and c are constants.
• Quadratic Equations: Equations in which the highest power of the variable is 2.
  • Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a, where a, b, and c are constants.

Functions

• Domain: The set of input values for which a function is defined.
• Range: The set of output values that a function can produce.
• Function Notation: f(x) = ..., where x is the input and f(x) is the output.
• Graphing: Visual representation of a function's relationship between input and output values.

Graphing

• Coordinate Plane: A two-dimensional plane with x and y axes.
• Graphing Linear Equations: Lines with a single slope and y-intercept.
• Graphing Quadratic Equations: Parabolas with a vertex, axis of symmetry, and x-intercepts.

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Algebra

Definition

• Algebra studies variables and their relationships, expressed through symbols, equations, and functions.

Key Concepts

• Variables are letters or symbols representing unknown values or quantities.
• Constants are numbers that do not change value.
• Terms are parts of an expression separated by addition or subtraction.
• Expressions are combinations of variables, constants, and mathematical operations.
• Equations are statements expressing the equality of two expressions.
• Inequalities are statements expressing a relationship between two expressions, but not necessarily equal.

Operations

• Addition combines expressions by adding corresponding terms.
• Subtraction combines expressions by subtracting corresponding terms.
• Multiplication combines expressions by multiplying corresponding terms.
• Division combines expressions by dividing corresponding terms.

Equation Solving

Linear Equations

• Linear equations have a highest power of the variable of 1.
• Slope-Intercept Form is y = mx + b, where m is the slope and b is the y-intercept.
• Standard Form is ax + by = c, where a, b, and c are constants.

Quadratic Equations

• Quadratic equations have a highest power of the variable of 2.
• Quadratic Formula is x = (-b ± √(b² - 4ac)) / 2a, where a, b, and c are constants.

Functions

• Domain is the set of input values for which a function is defined.
• Range is the set of output values that a function can produce.
• Function Notation is f(x) = ..., where x is the input and f(x) is the output.
• Graphing is a visual representation of a function's relationship between input and output values.

Graphing

• Coordinate Plane is a two-dimensional plane with x and y axes.
• Graphing Linear Equations produces lines with a single slope and y-intercept.
• Graphing Quadratic Equations produces parabolas with a vertex, axis of symmetry, and x-intercepts.