What is the point farthest up down to the right or to the left on a parabola on a half of a hyperbola?

The point farthest up, down, to the right, or to the left on a parabola that is part of a hyperbola depends on the specific orientation and equation of the hyperbola. For example, in the case of a hyperbola oriented horizontally, the branches extend infinitely to the left and right, while the vertex of the associated parabola will determine the maximum or minimum point vertically. Therefore, the exact coordinates would require knowing the specific equations involved. In general, the parabola's vertex will provide the extreme vertical points, while the asymptotes of the hyperbola will guide the horizontal extremes.