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Point (0, 1) is a single point and it would be in the coordinate (not cordinate) plane. And I have no idea what "you" is doing at the end of the question!
A translation moves every point on a shape in the same distance and in the same direction on the Cartesian plane
Ordered pairs are used to locate points on the coordinate plane.
It is the straight line equation that can be used to locate coordinates of x and y on the Cartesian plane
There are infinitely many possible locations.