Lecture on Parabolas

Introduction

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Definition of a Parabola

• Parabola: The locus of all points equidistant from a fixed point (focus) and a fixed line (directrix)
• Focus: Fixed point
• Directrix: Fixed line not passing through the focus

Key Concepts

• Parabola can open upward, downward, left, or right
• Axis of Symmetry: Line passing through the focus and perpendicular to the directrix
• Lattice Rectum: Line passing through the focus and perpendicular to the axis of symmetry

Understanding Parabola on Cartesian Plane

• Identify focus, directrix, axis of symmetry, and lattice rectum on the graph
• Example: A parabola opening upward
  • Focus: 0, 3
  • Directrix: y = -3
  • Axis of Symmetry: x = 0
  • Lattice Rectum endpoints: (6, 3), (-6, 3)

Finding Components of a Parabola

• Standard Equation: x² = 4cy
• Open Upward: c > 0
• Open Downward: c < 0
• Open Right: y² = 4cx, c > 0
• Open Left: y² = -4cx, c < 0
• Formulas: c is the distance from vertex to focus and directrix, 2c is the distance from focus to endpoints of the lattice rectum

Example Problem

• Given: x² = 12y
  • Opening: Upward
  • Vertex: (0, 0)
  • Focus: (0, 3)
  • Directrix: y = -3
  • Axis of Symmetry: x = 0
  • Lattice Rectum Length: 4c = 12 units

Assignment Example

• Given: y² = -8x
  • Open Left: c < 0
• For a vertex at (h, k): Adjust equations to find components

Transforming Equations

• From Standard to General Form: Expand and rearrange terms
  • Example: y - 3² = -2(x + 1)
    • Expand: y² - 6y + 9 = -2x - 2
    • Rearrange: y² - 6y + 9 + 2x + 2 = 0
• From General to Standard Form: Complete the square and adjust
  • Example: x² + 10x - 2y + 23 = 0
    • Completing the square: (x + 5)² = 2(y - 1)

Additional Practice Problems

• Transform given equations between standard and general forms
  • y - 8² = -3(x + 1)
  • x² = 3(y + 4)³
  • x + 4² = 3(y - 1)

Conclusion

• Next lesson: Ellipse
• Ensure mastery of circles and parabolas before proceeding
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