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.
- (x-2)2 = -8 (y – 3)
Solution
Here is the graph of the parabola . The key elements shown are:
- Vertex at (in red).
- Focus at (in blue).
- Directrix at (green dashed line).
- Axis of symmetry at (purple dashed line).
- The parabola opens downward.
Let me know if you'd like any more details!
Further exploration questions:
- How does the value of affect the shape of a parabola?
- What happens to the focus and directrix if the parabola opens to the left or right?
- Can you derive the equation of a parabola if only its vertex and focus are given?
- How do the latus rectum and axis of symmetry change for horizontal parabolas?
- 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
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