Math Problem Statement

Instruction: Please solve the following Parabola, please find the vertex, Focus, Directrix, length of latus rectum, Axis of Symmetry, identify where the parabola opens, then show your Graph.

  1. (x-2)2 = -8 (y – 3)

Solution

Here is the graph of the parabola (x2)2=8(y3)(x - 2)^2 = -8(y - 3). The key elements shown are:

  • Vertex at (2,3)(2, 3) (in red).
  • Focus at (2,1)(2, 1) (in blue).
  • Directrix at y=5y = 5 (green dashed line).
  • Axis of symmetry at x=2x = 2 (purple dashed line).
  • The parabola opens downward.

Let me know if you'd like any more details!

Further exploration questions:

  1. How does the value of pp affect the shape of a parabola?
  2. What happens to the focus and directrix if the parabola opens to the left or right?
  3. Can you derive the equation of a parabola if only its vertex and focus are given?
  4. How do the latus rectum and axis of symmetry change for horizontal parabolas?
  5. What are the real-world applications of parabolas in physics and engineering?

Tip: The distance between the vertex and focus always equals the distance between the vertex and directrix.

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Math Problem Analysis

Mathematical Concepts

Conic Sections
Parabolas
Coordinate Geometry

Formulas

Standard form of a parabola (x-h)^2 = 4p(y-k)
Vertex: (h, k)
Focus: (h, k + p)
Directrix: y = k - p
Latus rectum length: |4p|

Theorems

Focus-Directrix Property of Parabolas
Symmetry of Parabolas

Suitable Grade Level

Grades 9-12