If the focus and vertex of a parabola are the points (0, 2) and (0, 4), respectively, then find the equation

Explanation

To find the equation of the parabola given the focus at (0, 2) and the vertex at (0, 4), we can follow these steps:

### Step 1: Identify the coordinates of the focus and vertex
The focus is at (0, 2) and the vertex is at (0, 4).

### Step 2: Determine the orientation of the parabola
Since the vertex (0, 4) is above the focus (0, 2), the parabola opens downward.

### Step 3: Use the standard form of the equation for a downward-opening parabola
The standard form of the equation for a parabola that opens downward is:
\[ (x - h)^2 = -4a(y - k) \]
where (h, k) is the vertex and "a" is the distance from the vertex to the focus.

### Step 4: Find the values of h, k, and a
From the vertex (0, 4), we have:
- \( h = 0 \)
- \( k = 4 \)

The distance "a" is the distance from the vertex to the focus. The distance between the vertex (0, 4) and the focus (0, 2) is:
\[ a = |4 - 2| = 2 \]

### Step 5: Substitute the values into the equation
Now substituting \( h = 0 \), \( k = 4 \), and \( a = 2 \) into the standard form:
\[ (x - 0)^2 = -4(2)(y - 4) \]
This simplifies to:
\[ x^2 = -8(y - 4) \]

### Step 6: Rearranging the equation
We can rearrange this equation to make it clearer:
\[ x^2 = -8y + 32 \]
or
\[ x^2 + 8y - 32 = 0 \]

### Final Answer
The equation of the parabola is:
\[ x^2 + 8y - 32 = 0 \]