The equation of the parabola is:

$(x + 3)^2 = -6(y - 1)$

This is a parabola that opens downwards because the coefficient of $(y-1)$ is negative.

Step 1: Standard Form

The general form of a parabola that opens up or down is:

$(x - h)^2 = 4p(y - k)$

Here:

• $(h, k)$ is the vertex.
• $p$ is the distance from the vertex to the focus or to the directrix.

For our equation: $(x + 3)^2 = -6(y - 1)$ We can rewrite it as: $(x - (-3))^2 = 4(-\frac{3}{2})(y - 1)$

Thus:

• $h = -3$, $k = 1$
• $4p = -6 \Rightarrow p = -\frac{3}{2}$

Step 2: Vertex

The vertex is: $\boxed{(-3, 1)}$

Step 3: Focus

The focus is $p$ units away from the vertex along the axis of symmetry (the vertical line $x = -3$). Since $p = -\frac{3}{2}$, the focus is: $\text{Focus: } (-3, 1 + p) = (-3, 1 - \frac{3}{2}) = (-3, -\frac{1}{2})$

Step 4: Directrix

The directrix is a horizontal line $p$ units away from the vertex in the opposite direction. The directrix is: $\text{Directrix: } y = 1 - p = 1 + \frac{3}{2} = \frac{5}{2}$

Final Answer:

1. Vertex: $(-3, 1)$
2. Focus: $(-3, -\frac{1}{2})$
3. Directrix: $y = \frac{5}{2}$

Would you like a graph or further details?

Expanding Understanding:

1. How can we verify the parabola’s direction from its equation?
2. What changes if the parabola’s opening is horizontal?
3. How do $p$, focus, and directrix relate geometrically?
4. Can you derive the equation if given the focus and directrix?
5. What happens to the vertex if the equation shifts?

Tip:

The sign of $4p$ determines the direction of the parabola: positive for upwards/right and negative for downwards/left.