Lecture Notes on Parabolas

Introduction

• Discussion on identifying parabolas and their equations.
• Focus and directrix as key concepts in parabola equations.

Key Concepts

Definition of Parabola

• Parabolic shapes can open:
  • Upward
  • Downward
  • Leftward
  • Rightward

Identifying Characteristics

1. Focus: Point used to define the parabola.
2. Directrix: Line used to define the parabola, opposite the focus.

Equations of Parabolas

General Forms

1. Opens Upward:
   • Equation: (X - H)² = 4C(Y - K)
2. Opens Downward:
   • Equation: (X - H)² = -4C(Y - K)
3. Opens Right:
   • Equation: (Y - K)² = 4C(X - H)
4. Opens Left:
   • Equation: (Y - K)² = -4C(X - H)

Examples

Example 1

• Vertex: (-3, 4)
• Focus: (0, 4)
• Directrix: Vertical line; opens upward.
• Equation derived from vertex and focus.

Example 2

• Vertex: (-2, -2)
• Directrix: Y = 0
• Opens downward due to the position of the vertex and directrix.

Example 3

• Vertex: (4, 4)
• Focus: (4, 6)
• Opens upward; equation derived from vertex and focus.

Example 4

• Vertex: (0, -8)
• Directrix: Y = -1
• Opens upward; equation derived from vertex and focus.

Example 5

• Focus: (2, 3)
• Directrix: X = 6
• Determines the vertex located between focus and directrix.

Conclusion

• Recap of identifying parabolas, their equations, and key components (focus and directrix).
• Encouragement for students to practice identifying equations.

Call to Action

• Reminder to subscribe and share the lecture notes.