Find the equation of the parabola whose vertex is (3,-2) and focus is (3,1).

Answer

To find the equation of the parabola with a vertex at (3, -2) and a focus at (3, 1), we can follow these steps:

### Step 1: Identify the vertex and focus
The vertex of the parabola is given as \( (h, k) = (3, -2) \) and the focus is given as \( (3, 1) \).

### Step 2: Determine the orientation of the parabola
Since the x-coordinates of the vertex and focus are the same (both are 3), the parabola opens vertically. The focus is above the vertex, indicating that the parabola opens upwards.

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

### Step 4: Calculate the value of \( p \)
The distance \( p \) can be calculated as the difference in the y-coordinates of the focus and the vertex:
\[ p = \text{Focus y-coordinate} - \text{Vertex y-coordinate} = 1 - (-2) = 1 + 2 = 3 \]

### Step 5: Substitute values into the equation
Now, substituting \( h = 3 \), \( k = -2 \), and \( p = 3 \) into the standard form:
\[ (x - 3)^2 = 4 \cdot 3 \cdot (y + 2) \]
This simplifies to:
\[ (x - 3)^2 = 12(y + 2) \]

### Final Equation
Thus, the equation of the parabola is:
\[ (x - 3)^2 = 12(y + 2) \]

---