Math Antics: Slope and Distance on the Coordinate Plane

Introduction

• Equation for Linear Functions: y = mx + b
• New Equations Covered:
  • Slope of a line
  • Distance between two points

Basics of a Line Segment

• Defined by any two points: Point 1 (X1, Y1) and Point 2 (X2, Y2)
• Line segment can form a right triangle:
  • Vertical side: "change in y"
  • Horizontal side: "change in x"

Delta Notation

• Delta (Δ) used for "change in"
  • ΔX: Change in X = X2 - X1
  • ΔY: Change in Y = Y2 - Y1

Calculating Slope

• Slope Formula: Slope = ΔY / ΔX
  • Simplified as "rise over run"
  • Can be positive or negative

Calculating Distance

• Distance Formula: Distance = √[(ΔX)^2 + (ΔY)^2]
  • Derived from the Pythagorean Theorem: a^2 + b^2 = c^2
  • Used to find distance between two points

Examples

1. Example 1:

   • Points: (X1, Y1) = (-2, 0), (X2, Y2) = (4, 3)
   • ΔX = 6, ΔY = 3
   • Slope = 3/6 = 0.5
   • Distance = √(6^2 + 3^2) = √45 ≈ 6.708

2. Example 2:

   • Points: (X1, Y1) = (-3, 5), (X2, Y2) = (1, -2)
   • ΔX = 4, ΔY = -7
   • Slope = -7/4 = -1.75
   • Distance = √(4^2 + (-7)^2) = √65 ≈ 8.062

Important Notes

• Negative Deltas: Indicate direction of change, not length
• Absolute Values: Used for lengths, but signs affect slope

Conclusion

• Practice using these formulas for better understanding
• Remember the historical context and contribution of Pythagoras
• Further resources available at mathantics.com