Aug 4, 2024

- Welcome to Senior Pablo TV
- Make sure to subscribe and click the notification bell for updates

- 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

- 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

- 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)

**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

- 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

- Given: y² = -8x
- Open Left: c < 0

- For a vertex at (h, k): Adjust equations to find components

**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

- Example: y - 3² = -2(x + 1)
**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)

- Example: x² + 10x - 2y + 23 = 0

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

- Next lesson: Ellipse
- Ensure mastery of circles and parabolas before proceeding
- Subscribe, click the notification bell, and share the video to help classmates