Problem 9 - Entrance Test

The standard equation of a parabola with vertex (h, k) is y = a(x - h)^2 + k. Given vertex (2, 3), we have y = a(x - 2)^2 + 3. The focus of a parabola in the standard form is (h, k + 1 / 4a). Given focus (2, 5), we can equate the y-coordinates: 5 = 3 + 1 / 4a, 2 = 1 / 4a, 4a = 1 / 2, a = 1 / 8. So the equation of the parabola is y = (1 / 8)(x - 2)^2 + 3, but since this is not an option, we can try another approach using the definition of a parabola as the set of points equidistant from the focus and directrix. The directrix is a horizontal line, so its equation is y = k - 1 / 4a. Since we are given the vertex and focus, we know the parabola opens upward. This information is not sufficient to uniquely determine the equation of the parabola among the given choices without assuming a specific value for 'a', which means we might have misinterpreted the question given the provided answer choices.