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Understanding Linear Functions and Geometry

May 12, 2025

Math Antics: Linear Functions & Basic Geometry Concepts

Introduction to Linear Functions

  • The equation y = mx + b represents any linear function on a 2D coordinate plane.
  • This session focuses on two additional equations:
    • Calculating the slope of a line.
    • Calculating the distance between two points on a line.

Understanding Points and Lines

  • A line can be defined by any two points (Point 1 and Point 2).
  • Connect the dots to create a line segment, extend to form an infinite line.

Right Triangle Formation

  • Use line segments to create right triangles:
    • Draw vertical and horizontal lines from each point to form a right angle.
    • The original line segment becomes the hypotenuse.
  • Define triangle sides:
    • Horizontal side: Change in X or Delta X
    • Vertical side: Change in Y or Delta Y

Calculating Changes in Coordinates

  • Delta X = X2 - X1
  • Delta Y = Y2 - Y1
  • Subscripts distinguish different variables: X1, Y1 for Point 1; X2, Y2 for Point 2.

Slope of a Line

  • Slope formula: Slope = (Delta Y)/(Delta X)
    • Also known as "Rise over Run"
    • Can be expressed as positive or negative depending on line direction.

Distance Between Two Points

  • Distance formula: Distance = sqrt((Delta X)² + (Delta Y)²)
    • Based on the Pythagorean Theorem.
    • Alternative names: Distance Formula or Pythagorean Theorem in Algebra context.

Example Calculations

  1. First Example:

    • Given coordinates: (X1, Y1) = (-2, 0), (X2, Y2) = (4, 3)
    • Calculate deltas:
      • Delta X = 4 - (-2) = 6
      • Delta Y = 3 - 0 = 3
    • Slope: 3/6 = 0.5
    • Distance: sqrt(6² + 3²) = sqrt(45) ≈ 6.708

  2. Second Example:

    • Given coordinates: (X1, Y1) = (-3, 5), (X2, Y2) = (1, -2)
    • Calculate deltas:
      • Delta X = 1 - (-3) = 4
      • Delta Y = -2 - 5 = -7
    • Slope: -7/4 ≈ -1.75
    • Distance: sqrt(4² + (-7)²) = sqrt(65) ≈ 8.062

Conclusion

  • Practice is essential for mastering slope and distance calculations.
  • Utilize the Distance Formula and the concept of Slope in algebraic contexts.
  • Learn and explore more at Math Antics.

Full transcript