Final answer:
The equation for the parabola is y = (-1/8)(x - 1)^2 + 2.
Step-by-step explanation:
To find the equation of a parabola with a known focus and vertex, we can use the standard form equation of a parabola. The general equation of a parabola is given by y = a(x - h)^2 + k, where (h, k) represents the vertex. In this case, the vertex is (1, 2), so our equation becomes y = a(x - 1)^2 + 2. We also know that the focus of the parabola is (1, 4). The distance from the vertex to the focus is given by the equation a = 1 / (4p), where p is the distance from the vertex to the focus. Plugging in the values, we get a = 1 / (4(2 - 4)), which simplifies to a = -1/8. Substituting this value for a, our final equation for the parabola is y = (-1/8)(x - 1)^2 + 2.