What is the vertex form of the equation of a parabola with directrix x = -3 and focus at (1, 2)?

To find the vertex form of the equation of the parabola given its focus and directrix, we need to first determine the vertex's coordinates. The vertex of a parabola lies midway between its focus and directrix.

In this case, the focus is at (1, 2) and the directrix is the line x = -3. The midpoint, which is the vertex, can be calculated by averaging the x-coordinates of the focus and the directrix. The x-coordinate of the vertex is:

[

\text{Vertex x-coordinate} = \frac{1 + (-3)}{2} = \frac{-2}{2} = -1.

]

The y-coordinate of the vertex is the same as the y-coordinate of the focus since both the focus and vertex share the same vertical line (the vertex line). Thus, the vertex is at (-1, 2).

Now, to find the equation of the parabola, we note that it opens horizontally, toward the focus (to the right) since the focus is to the right of the directrix. The general vertex form of a horizontally opening parabola is given by:

[

(y - k)^2 = 4p(x - h),