3. From the given equation (x-4)² = -6 (y-4)

a. Find the vertex

b.) Find the focus from the vertex

c.) Find the directrix from the vertex

d.) Graph the parabola equation

Here is the graph of the parabola \((x-4)^2 = -6(y-4)\):

- **Vertex** is located at \((4, 4)\) (marked in red).
- **Focus** is at \(\left(4, \frac{5}{2}\right)\) (marked in green).
- **Directrix** is the dashed purple line at \(y = 5.5\).

The graph shows the parabola opening downwards with its vertex, focus, and directrix clearly indicated.

Would you like further details or explanations on any part of this problem?

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Here are five related questions you might find helpful:
1. How does the value of \(p\) affect the shape of a parabola?
2. What happens if the equation of the parabola has a positive coefficient instead of a negative one?
3. How can you derive the equation of a parabola given the focus and directrix?
4. What are the differences between parabolas that open horizontally versus vertically?
5. How does completing the square help in identifying the vertex form of a quadratic equation?

**Tip:** When graphing parabolas, always identify the vertex first; it serves as the central point from which all other characteristics (focus, directrix) are derived.

Explore the graph of the parabola (x-4)^2 = -6(y-4) with detailed solutions for finding the vertex, focus, and directrix. Understand how to plot and analyze parabolas in coordinate geometry.

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