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

Answer

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

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

### Step 2: Determine the orientation of the parabola
Since the vertex and focus have the same y-coordinate (3), the parabola opens horizontally. Specifically, because the focus is to the right of the vertex, the parabola opens to the right.

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

### Step 4: Identify \(h\), \(k\), and \(a\)
From the vertex \((-2, 3)\):
- \(h = -2\)
- \(k = 3\)

Next, we calculate \(a\), the distance from the vertex to the focus. The focus is at \((1, 3)\), and the distance \(a\) can be calculated as:
\[ a = |1 - (-2)| = |1 + 2| = 3 \]

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

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

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