Algebra Overview in Ten Minutes

What is Algebra?

• Study of symbols and rules
• Math Symbol: Variable (placeholder)
  • Examples: x, y, z, a, b
• Expressions: Multiple symbols and variables combined

Equations

• Adding an equal sign to expressions
• Both sides must stay balanced
• Goal: Isolate the variable and solve for the value

Order of Operations

• Use PEMDAS/M(DAS) rules
• Balanced Equations: Treat 'x's on both sides equally

Functions and Variables

• Function: Value in, value out
• Independent Variable (x): You control the value
• Dependent Variable (y): Depends on the value of x
• Coordinate Plane:
  • x-axis: Independent variable
  • y-axis: Dependent variable
  • **Intercepts: **
    • x-intercept: Function intersects x-axis
    • y-intercept: Graph intersects y-axis

Vertical Line Test

• Determines if a graph represents a function
• Draw a vertical line
• If it intersects the graph in more than one spot, it’s not a function

Domain and Range

• Continuous Functions: No holes or gaps
• Piecewise Functions: May have holes or gaps
• Plotting helps determine domain and range

Linear Functions

• Linear Function: Creates a straight line
• Variables:
  • Independent (e.g., time)
  • Dependent (e.g., distance)
• Slope: Determines steepness
• **Linear Equations: **
  • Forms:
    • Slope-intercept: y = mx + b
    • Point-slope: y - y1 = m(x - x1)\n - Standard: Ax + By = C (A, B, C are constants)
• Systems of Equations: Intersection of two linear functions
  • Methods to solve:
    • Substitution
    • Elimination

Linear Inequalities

• Inequality Signs: >, <, ≥, ≤
• Graphing:
  • If there isn’t an equal sign, use a dashed line
  • Shade according to the sign

Quadratic Functions

• Basic Form: x² (u-shaped graph)
• Domain: Infinite
• Range: Limited (depends on direction of U)
• Forms of Quadratic Functions:
  • Vertex Form
  • Standard Form
  • Factored Form
• Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a
• Discriminant: Determines number of solutions
  • Formula: b² - 4ac

Exponential Functions

• Basic Form: Base number with x as the exponent
• Domain: All real numbers
• Range: Approaches but usually never reaches a certain value
• Asymptotes:
  • Horizontal
  • Vertical
  • Slant
• Rules of Exponents: Important to remember