To derive the equation of the parabola, we start by identifying the parameters given:
- The focus of the parabola is at the point \((-2, 1)\).
- The directrix is the line \(y = -1\).
For a parabola, every point on it is equidistant from the focus and the directrix.
Step 1: Identify the vertex
The vertex of a parabola is located midway between the focus and the directrix. Here, the focus is at \((−2, 1)\) and the directrix is the line \(y = −1\).
To find the vertex, we average the y-coordinates of the focus and the directrix:
\[ \text{Vertex y-coordinate} = \frac{1 + (-1)}{2} = \frac{0}{2} = 0. \]
The x-coordinate of the vertex is the same as that of the focus, which is \(-2\). Thus, the vertex \((h, k)\) of the parabola is:
\[ (h, k) = (-2, 0). \]
Step 2: Determine the orientation and distance
Since the focus is above the directrix, the parabola opens upwards. The distance \(p\) from the vertex to the focus (and also from the vertex to the directrix) can be calculated as:
\[ p = \text{Focus y-coordinate} - \text{Vertex y-coordinate} = 1 - 0 = 1. \]
Step 3: Write the standard form of the equation
For a parabola that opens upwards and has its vertex at \((h, k)\), the equation is given by:
\[ (x - h)^2 = 4p(y - k). \]
Substituting \(h = -2\), \(k = 0\), and \(p = 1\):
\[ (x + 2)^2 = 4(1)(y - 0). \]
This simplifies to:
\[ (x + 2)^2 = 4y. \]
Conclusion
Thus, the equation of the parabola that Rosa is drafting is:
\[ \boxed{(x + 2)^2 = 4y}. \]
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