Jation in vertex form for the parabola with focus $$ (0,-7) $$ and directrix $$ y=9 $$ fractions.

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To find the equation of the parabola in vertex form given the focus and directrix, follow these steps:

### Given:
- **Focus:** $$(0, -7)$$
- **Directrix:** $$y = 9$$

### Step 1: Find the Vertex
The vertex of a parabola is exactly halfway between the focus and the directrix.

$$
\text{Vertex } y\text{-coordinate} = \frac{-7 + 9}{2} = \frac{2}{2} = 1
$$

Since the focus is at $$(0, -7)$$ and directly below the vertex, the vertex is:

$$
\text{Vertex } = (0, 1)
$$

### Step 2: Determine the Parameter $$p$$
The distance $$p$$ between the vertex and the focus (or directrix) determines the "width" and the direction the parabola opens.

$$
p = \text{Vertex } y\text{-coordinate} - \text{Focus } y\text{-coordinate} = 1 - (-7) = 8
$$

Since the focus is below the vertex, the parabola opens downward, so $$p = -8$$.

### Step 3: Write the Vertex Form Equation
The vertex form of a parabola that opens vertically is:

$$
y = a(x - h)^2 + k
$$

Where:
- $$(h, k)$$ is the vertex.
- $$a = \frac{1}{4p}$$

Substituting the values:
$$
a = \frac{1}{4(-8)} = -\frac{1}{32}
$$
$$
h = 0 \quad \text{and} \quad k = 1
$$

So, the equation becomes:

$$
y = -\frac{1}{32}(x - 0)^2 + 1
$$
$$
\boxed{y = -\,\dfrac{1}{32}\,x^{2} + 1}
$$
The equation of the parabola in vertex form is $$ y = -\frac{1}{32}x^{2} + 1 $$.

Jation in vertex form for the parabola with focus \( (0,-7) \) and directrix \( y=9 \) fractions.

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