Overview
This lecture discusses the standard equations of parabolas with vertex at (h, k), including cases where they open upward, downward, right, or left, and how to identify their key features.
Parabola Opening Upward
- Standard equation: (x - h)² = 4a(y - k)
- Vertex at (h, k)
- Axis of symmetry: x = h
- Focus: (h, k + a)
- Directrix: y = k - a
- Endpoints of lattice rectum: (h + 2a, k + a) and (h - 2a, k + a)
Example: Upward Parabola
- Given: (x + 5)² = 2(y + 2)
- Opens upward since a > 0
- Vertex: (-5, -2)
- Axis of symmetry: x = -5
- a = 0.5
- Focus: (-5, -1.5)
- Directrix: y = -2.5
- Lattice rectum endpoints: (-4, -1.5) and (-6, -1.5)
Parabola Opening Downward
- Standard equation: (x - h)² = -4a(y - k)
- Vertex at (h, k)
- Axis of symmetry: x = h
- Focus: (h, k + a)
- Directrix: y = k - a
- Endpoints of lattice rectum: (h ± 2|a|, k + a)
Example: Downward Parabola
- Given: (x - 3)² = -8(y + 4)
- Opens downward since a < 0
- Vertex: (3, -4)
- Axis of symmetry: x = 3
- a = -2
- Focus: (3, -6)
- Directrix: y = -2
- Lattice rectum endpoints: (7, -6) and (-1, -6)
Parabola Opening to the Right
- Standard equation: (y - k)² = 4a(x - h)
- Vertex at (h, k)
- Axis of symmetry: y = k
- Focus: (h + a, k)
- Directrix: x = h - a
- Lattice rectum endpoints: (h + a, k ± 2a)
Example: Rightward Parabola
- Given: (y + 6)² = 4(x - 3)
- Opens right since a > 0
- Vertex: (3, -6)
- Axis of symmetry: y = -6
- a = 1
- Focus: (4, -6)
- Directrix: x = 2
- Lattice rectum endpoints: (4, -4) and (4, -8)
Parabola Opening to the Left
- Standard equation: (y - k)² = -4a(x - h)
- Vertex at (h, k)
- Axis of symmetry: y = k
- Focus: (h + a, k)
- Directrix: x = h - a
- Lattice rectum endpoints: (h + a, k ± 2|a|)
Example: Leftward Parabola
- Given: (y - 2)² = -6(x - 1)
- Opens left since a < 0
- Vertex: (1, 2)
- Axis of symmetry: y = 2
- a = -1.5
- Focus: (-0.5, 2)
- Directrix: x = 2.5
- Lattice rectum endpoints: (-0.5, 5) and (-0.5, -1)
Key Terms & Definitions
- Vertex — The turning point (h, k) of the parabola.
- Focus — A fixed point used to define the parabola, at (h, k + a) or (h + a, k).
- Directrix — A line used to define the parabola, y = k - a or x = h - a.
- Axis of Symmetry — The line passing through the vertex dividing the parabola symmetrically.
- Lattice Rectum — The line segment through the focus, perpendicular to the axis of symmetry.
- a — Parameter controlling the "width" and direction of the parabola.
Action Items / Next Steps
- Assignment #4: Sketch the graph and indicate the focus, directrix, vertex, axis of symmetry, and lattice rectum endpoints for 5 given equations in your graphing notebook.