Lecture on Graphs and Equations

Introduction

• Lecturer: Professor Hartfield
• Topics:
  • Graphs and their relationship with equations
  • Interpreting graphs

Graphing in One and Two Variables

• One Variable:
  • Graph on a number line
• Two Variables:
  • Use a surface or plane
  • Cartesian coordinate system (coordinate plane)
  • Consists of two lines (axes) intersecting at a right angle

Cartesian Coordinate System

• Axes:
  • Horizontal axis: x-axis
  • Vertical axis: y-axis
• Origin:
  • Point of intersection (0,0)
• Directions:
  • Right/up is positive
  • Left/down is negative

Plotting Points

• Ordered Pairs: (x, y)
  • Example: (4, 2) is 4 units right and 2 units up

Quadrants

• Divisions: Four quadrants
  • I: Top right, x > 0, y > 0
  • II, III, IV: Labeled counter-clockwise
  • Points on axes are not in a quadrant

Graphing Equations

• Equations with Two Variables: e.g., y = 2x + 4
  • Graph is a line
  • Solutions are points making equation true

Example Problems

• Testing Solutions:
  • Substitute values into equations
  • Check for true statements
• Finding Intercepts:
  • X-intercepts: Set y = 0, solve for x
  • Y-intercepts: Set x = 0, solve for y

Sketching Graphs

• Using Intercepts and Points
• Example: y = x² + 2x - 8
  • Find intercepts and additional points
  • Plot points and draw curve

Analyzing Graphs

• Example: Beef and Chicken Availability
  • Identify trends and highest/lowest points

Graphing with Technology

• Graphing Calculators and Software
  • Enter equations and use standard windows
  • Adjust display settings for better visualization

Solving Equations Graphically

• Example: x³ - x + 3 = 0
  • Find x-intercepts (zeros) using calculator

Case Study: Marriott International

• Net Income Analysis
  • Use graphing calculator to find maximum income
  • Interpretation of results in context