Parabolas

Introduction to Parabolas

• Common type of curves in mathematics.
• The term "parabola" has Greek origins, linked to the word for throwing (ballistics).
• Parabolas approximate the trajectory of objects thrown in physics.

Characteristics of Parabolas

• Types of Parabolas:
  • Open upwards (U-shaped).
  • Open downwards (inverted U-shaped).
• Terminology:
  • Vertex: Maximum or minimum point on a parabola.
    • Upward-opening parabola: Vertex is the minimum point.
    • Downward-opening parabola: Vertex is the maximum point.

Vertex Examples

• Yellow Parabola:
  • Vertex coordinates: (3, -3.5).
• Pink Parabola:
  • Vertex is the low point (minimum).

Axis of Symmetry

• Definition: Line that divides the parabola into two mirror-image halves.
• Characteristics:
  • Passes through the vertex.
  • Example Axis of Symmetry:
    • Yellow Parabola: x = 3.5.
    • Pink Parabola: x = -1.
    • Green Parabola: x = -6.

Intercepts

• Y-intercept: Where the parabola intersects the Y-axis.
  • Yellow Parabola Y-intercept: (0, 3).
  • Pink Parabola Y-intercept: Not visible on the graph.
• X-intercept: Where the parabola intersects the X-axis.
  • Yellow Parabola X-intercepts: (1, 0) and (6, 0).
  • Symmetric property of intercepts around the axis of symmetry.
    • Midpoint of X-intercepts gives the axis of symmetry.
• X-intercepts Example:
  • Pink Parabola does not intersect X-axis (no X-intercepts).

Summary of Core Concepts

• Not all parabolas intersect the X-axis (e.g., pink upward-opening parabola).
• Future discussions will involve equations of parabolas, specifically second-degree equations (quadratics).
  • Example of a simple parabola: Y = x^2.
  • More complex forms: Y = 2x^2 - 5x + 7.
• Quadratics are a general representation of parabolas.