Algebra Fundamentals

Equations

• An equation is a statement that says two mathematical expressions are equal
• Equations can be:
  • Linear: degree of 1 (e.g., 2x + 3 = 5)
  • Quadratic: degree of 2 (e.g., x^2 + 4x + 4 = 0)
  • Polynomial: degree of 3 or more (e.g., x^3 + 2x^2 - 7x - 12 = 0)
• Types of equations:
  • Simple equations: contain only one variable (e.g., 2x = 6)
  • Simultaneous equations: contain two or more variables (e.g., 2x + 3y = 7, x - 2y = -3)
• Solution to an equation: value(s) that make the equation true
• Methods to solve equations:
  • Addition and subtraction
  • Multiplication and division
  • Substitution
  • Elimination

Functions

• A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range)
• Functions can be represented as:
  • Equations (e.g., f(x) = 2x + 1)
  • Tables
  • Graphs
• Types of functions:
  • Linear functions: straight line (e.g., f(x) = 2x + 1)
  • Quadratic functions: parabola (e.g., f(x) = x^2 + 2x + 1)
  • Exponential functions: rapid growth or decay (e.g., f(x) = 2^x)
• Characteristics of functions:
  • Domain: set of inputs
  • Range: set of possible outputs
  • Increasing or decreasing
  • Maximum or minimum values

Graphing

• Graphing is a visual representation of a function or equation
• Types of graphs:
  • Linear graphs: straight line
  • Quadratic graphs: parabola
  • Exponential graphs: rapid growth or decay
• Graphing methods:
  • Plotting points
  • Using a table of values
  • Using a graphing calculator
• Key features of graphs:
  • Intercepts (x and y)
  • Asymptotes
  • Maxima and minima
  • Inflection points
• Graphing can be used to:
  • Identify patterns and relationships
  • Solve equations and inequalities
  • Model real-world phenomena