Proportional: y ÷ x is the same for every pair, and the graph is a straight line through (0, 0).
1Table test: divide y by x on every row
x
2
5
8
y
5
12.5
20
y ÷ x
2.5
2.5
2.5
Divide on every row. If every quotient is the same, the number is k and the table is proportional. If any quotient differs, it is not.
2Graph test: straight AND through the origin
Both words matter. A curve through the origin fails. A straight line that misses the origin fails.
!Linear but not proportional
proportional ✓
linear, not proportional ✗
Both lines are straight. The $3 starting fee moves the line off the origin, so y ÷ x keeps changing: 5, 3.5, 3 . . . Zero of one must mean zero of the other.
