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)


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

- **Vertex** at \((2, 3)\) (in red).
- **Focus** at \((2, 1)\) (in blue).
- **Directrix** at \(y = 5\) (green dashed line).
- **Axis of symmetry** at \(x = 2\) (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 \(p\) 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.

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)

Solve the parabola (x-2)^2 = -8(y-3) by identifying the vertex, focus, directrix, and length of the latus rectum. This problem covers key concepts such as the focus-directrix property and axis of symmetry. Suitable for high school students (Grades 9-12).

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