Find the equation of the parabola whose focus is (4,-3) and vertex is (4,-1).

Answer

To find the equation of the parabola with the given focus and vertex, we can follow these steps:

### Step 1: Identify the coordinates of the focus and vertex
The focus is given as \( (4, -3) \) and the vertex is \( (4, -1) \).

### Step 2: Determine the orientation of the parabola
Since the x-coordinates of the focus and vertex are the same (both are 4), the parabola opens either upwards or downwards. Given that the focus is below the vertex, the parabola opens downwards.

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

### Step 4: Identify the vertex coordinates
From the vertex \( (4, -1) \), we have:
- \( h = 4 \)
- \( k = -1 \)

### Step 5: Calculate the distance \( p \)
The distance \( p \) is the distance from the vertex to the focus. The y-coordinates of the vertex and focus are:
- Vertex \( y = -1 \)
- Focus \( y = -3 \)

Thus,
\[ p = -3 - (-1) = -2 \]
Since the parabola opens downwards, \( p = -2 \).

### Step 6: Substitute values into the vertex form
Now substituting \( h \), \( k \), and \( p \) into the vertex form:
\[ (x - 4)^2 = -4(-2)(y + 1) \]
This simplifies to:
\[ (x - 4)^2 = 8(y + 1) \]

### Step 7: Rearranging the equation
To express it in a more standard form, we can expand and rearrange:
\[ (x - 4)^2 = 8y + 8 \]
Thus, we can write:
\[ (x - 4)^2 - 8 = 8y \]
or
\[ 8y = (x - 4)^2 - 8 \]

### Final Equation
The equation of the parabola is:
\[ (x - 4)^2 = 8(y + 1) \]

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