Final answer:
The equation of the parabola with vertex (10,-14) and focus (10,12) is y = 1/26(x - 10)² - 14, which reveals that the parabola opens upward.
Step-by-step explanation:
The student is tasked with finding the equation of a parabola with a given vertex and focus. Since the vertex is at (10,-14) and the focus is at (10,12), we can determine that this parabola opens upward, as the focus is above the vertex. The distance from the vertex to the focus is 26 units, which is also the value of 4p in the standard form of a parabolic equation (x - h)² = 4p(y - k), where (h,k) is the vertex.
In this equation, h=10, k=-14, and 4p=26, so p=6.5. The equation of the parabola in standard form with the vertex at (10,-14) thus becomes (x - 10)² = 26(y + 14). To express this in standard form yx² + ax + b = 0, we rearrange to get: y = 1/26(x - 10)² - 14.