Questions about Hubble’s Graph

Hi, so recently my teacher showed a graph about redshift vs distance of galaxies. The line was supposed to represent the pattern you would see in a universe where the rate of expansion was constant. This line was vertical and I don't understand why it would be. If we take two points with a distance x between each other, then another two points with distance 2x between them, wouldn't there be more redshift in the second set of points? Because there is more space in-between the points that expands. Am I missing something or is the graph incorrect.

TLDR:

Would 2 pairs of points in the universe, one pair further apart from each other than the other, see differing amounts of redshift if the universe was expanding at a constant rate?