Find the equation of each of the following parabolas.
vertex at (0,4), focus at (0,2).

Answer

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

### Step 1: Identify the Vertex and Focus
The vertex of the parabola is given as \( V(0, 4) \) and the focus is given as \( F(0, 2) \).

### Step 2: Determine the Orientation of the Parabola
Since the focus is below the vertex (y-coordinate of focus is less than that of the vertex), the parabola opens downwards.

### Step 3: Calculate the Distance from Vertex to Focus
The distance \( VF \) from the vertex to the focus can be calculated as:
\[ VF = |4 - 2| = 2 \]
This distance \( a \) is the distance from the vertex to the focus, which is \( a = 2 \).

### Step 4: Write the Standard Form of the Parabola
For a parabola that opens downwards with vertex at \( (h, k) \), the standard equation is:
\[ (x - h)^2 = -4a(y - k) \]
Here, \( h = 0 \), \( k = 4 \), and \( a = 2 \).

### Step 5: Substitute the Values into the Equation
Substituting \( h \), \( k \), and \( a \) into the equation gives:
\[ (x - 0)^2 = -4(2)(y - 4) \]
This simplifies to:
\[ x^2 = -8(y - 4) \]

### Step 6: Rearrange the Equation
Rearranging the equation, we get:
\[ x^2 + 8y - 32 = 0 \]

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