To find the coordinates of the vertex given the focus and directrix, we can use the formula:
Vertex = (h, k) = (h, (p + k'))
where h is the x-coordinate of the vertex, k is the y-coordinate of the vertex, p is the distance between the focus and vertex, and k' is the y-coordinate of the directrix.
First, we need to find the value of p. The distance between the focus and directrix is given by:
p = |k' - y-coordinate of focus|
In this case, k' = 1 and y-coordinate of focus = 7. Therefore,
p = |1 - 7| = 6
Next, we can use the x-coordinate of the focus to find the x-coordinate of the vertex. Since the parabola is symmetric about its axis (which is parallel to the directrix), we know that the x-coordinate of the vertex is halfway between the x-coordinate of the focus and the directrix. Therefore,
h = (x-coordinate of focus + x-coordinate of directrix) / 2
= (-5 + 0) / 2
= -2.5
Finally, we can substitute h, k', and p into our formula for Vertex to find k:
Vertex = (h, k) = (-2.5, (p + k'))
= (-2.5, (6 + 1))
= (-2.5, 7)
Therefore, the coordinates of the vertex are (-2.5, 7).