Given the equation $(x - 3)^2 = -12(y + 1)$, we can determine the following characteristics of the parabola:

1. Type of Parabola:

The equation is in the form $(x - h)^2 = 4p(y - k)$, which represents a parabola that opens vertically (either up or down). Since the coefficient of $(y + 1)$ is negative, this parabola opens downwards.

2. Opening of Parabola:

The parabola opens downward because of the negative coefficient in front of the $(y + 1)$ term.

3. Vertex:

The vertex of the parabola is given by $(h, k)$. From the equation $(x - 3)^2 = -12(y + 1)$, we have:

• $h = 3$
• $k = -1$

So, the vertex is $(3, -1)$.

4. Focus:

The focus is found using the value of $p$, which is determined from the equation $4p = -12$. Solving for $p$: $p = \frac{-12}{4} = -3$ Since the parabola opens downward, the focus will be $p$ units below the vertex. Therefore, the focus is at $(3, -1 - 3) = (3, -4)$.

5. Directrix:

The directrix is a horizontal line $p$ units above the vertex, so: $y = -1 + 3 = 2$ The directrix is the line $y = 2$.

6. Axis of Symmetry:

The axis of symmetry is the vertical line that passes through the vertex, given by $x = h$. Therefore, the axis of symmetry is: $x = 3$

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Would you like a detailed explanation of any specific part? Here are some related questions for further exploration:

1. How do you find the length of the latus rectum for a parabola?
2. What is the significance of the focus and directrix in a parabola?
3. How would the graph change if the coefficient of $(y + 1)$ were positive?
4. Can you derive the general formula for a vertically opening parabola?
5. How would you transform this equation into its standard form?

Tip: For any conic section, understanding the vertex form of its equation makes identifying the key properties, like focus and directrix, much easier.