Final answer:
The equation for the parabola with a focus at (2,7) and vertex at (2,3) is (x - 2)^2 = 16(y - 3).
Step-by-step explanation:
To find an equation for a parabola with a given focus and vertex, we can use the general form of a parabola's equation. For a parabola that opens vertically, the equation is of the form (x - h)^2 = 4p(y - k) where (h, k) is the vertex and p is the distance from the vertex to the focus, and the focus is at (h, k+p).
In this question, the vertex is (2,3) and the focus is (2,7). The distance p is the difference in the y-values of the focus and vertex, which is 7 - 3 = 4. Since the parabola opens upwards (because the focus is above the vertex), the equation will be in the form (x - h)^2 = 4p(y - k).
Substituting the known values of h and k, and knowing that p = 4, we get:
(x - 2)^2 = 4*4(y - 3)
(x - 2)^2 = 16(y - 3), which is the equation of the parabola with the given vertex and focus.