Final answer:
The equation of a parabola with a vertex at (2,7) and a focus at (2,-1) is represented by (x - 2)^2 = -32(y - 7).
Step-by-step explanation:
The question involves finding the equation of a parabola with a given vertex (2,7) and focus (2,-1). We can use the standard form of a parabola's equation, which is (x - h)^2 = 4p(y - k) for a vertical parabola (opening upwards or downwards), where (h, k) is the vertex and p is the distance between the vertex and the focus.
In this case, the vertex is at (2,7), and the focus at (2,-1) indicates that the parabola opens downwards since the focus is below the vertex. The distance between the vertex and the focus is 7 - (-1) = 8. Since the parabola opens downwards, p should be negative, thus p = -8. Substituting the values into the standard form gives us:
\[(x - h)^2 = 4p(y - k)\]
\[(x - 2)^2 = 4(-8)(y - 7)\]
Therefore, the equation of the parabola is:
\[(x - 2)^2 = -32(y - 7)\]