Final answer:
The equation of the hyperbola is (x - 2)^2 / 3^2 - (y - 1)^2 / 6^2 = 1. The center of the hyperbola is located at (2,1), and the semi-major axis and distance to the foci are 3 and 6 units respectively.
Step-by-step explanation:
A hyperbola is a type of conic section that is defined by the difference of the distances from any point on the curve to two fixed points called foci. In order to find the equation of a hyperbola, we need to determine the center, vertices, and foci.
Given that the vertices are located at (2,-2) and (2,4) and the foci are located at (2,-7) and (2,9), we can determine that the center of the hyperbola is at (2,1) since it is the midpoint of the vertices. The distance between the center and each vertex is called the semi-major axis, which in this case is 3 units. The distance between the center and each focus is called the distance to the foci, which is 6 units.
Therefore, the equation of the hyperbola can be written as (x - 2)^2 / 3^2 - (y - 1)^2 / 6^2 = 1