Lecture Notes on Parabolas

Introduction

• Discussion on parabolas following a previous lecture on circles.
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Definition of Parabola

• Parabola: 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 mathematical terms: focus, directrix, axis of symmetry, lattice rectum.

Characteristics of Parabolas

• Parabolas can open in various directions:
  • Upward
  • Downward
  • Left
  • Right
• Axis of Symmetry: Line through focus, perpendicular to the directrix.
• Lattice Rectum: Line through focus, perpendicular to the axis of symmetry.

Graphing a Parabola

• Example parabola graphing:
  • Given parabola: x^2 = 12y
  • Standard Form: x^2 = 4cy
  • Vertex Form: x^2 = 4c(y - k) + h
• Determine the opening:
  • If c > 0, opens upward.
  • If c < 0, opens downward.
  • If in h, k form: determines direction based on sign of c.

Example Analysis: x^2 = 12y

1. Identify Vertex:
   • Vertex is at (0,0).
2. Find Value of c:
   • From equation: 4c = 12 → c = 3.
3. Focus Calculation:
   • Focus located 3 units above the vertex at (0, 3).
4. Directrix Calculation:
   • Directrix is located 3 units below the vertex: y = -3.
5. Axis of Symmetry:
   • Vertical line where x = 0.
6. Endpoints of Lattice Rectum:
   • Calculate length: 2c = 6 (3 units to the left and right of the focus).
   • Endpoints: (-3, 3) and (3, 3).

Length of the Lattice Rectum

• Lattice rectum length = 4c = 12 units.

Assignment

• Students are assigned to analyze the parabola: y^2 = -8.

Conclusion

• Next lesson will cover parabolas in vertex form h, k and how to determine focus, directrix, axis of symmetry, and lattice rectum.