On these questions about slopes on graphs

  a) {C, D}, {B, E}, A . . . . four possible equivalent answers (explanation below)

  b) Line C shows a "directly proportional" relationship

Step-by-step explanation:

Slope is the ratio of "rise" (vertical change) to "run" (horizontal change). Positive changes are "up" and "right".

Line A -- has no vertical change, so its slope is 0.

Line B -- changes 1 unit vertically for each 4 units horizontally, so its slope is 1/4.

Line C -- changes 1 unit vertically for each 2 units horizontally, so its slope is 1/2.

Line D -- changes 1 unit vertically for each 2 units horizontally, so its slope is 1/2. This is the same slope as line C, so neither is greater than the other.

Line E -- changes 1 unit vertically for each 4 units horizontally, so its slope is 1/4. This is the same slope as line B, so neither is greater than the other.

If we use alphabetical order to rank equal slopes, then the slopes (greatest to least) are ...

  C (1/2), D (1/2), B (1/4), E (1/4), A (0)

A "directly proportional" relationship graphs as a straight line through the origin. The only line passing through the point where the x- and y-axes meet is line C.

  Line C represents a proportional function.