You can tell if a table is linear by looking at how X and Y change. If X increases by 1, Y increases by a constant rate, then a table is linear. You can find the continuous rate by finding the first difference.
This table is linear. the first difference between Y1 & Y2, Y2 & Y3, and Y3 & Y4 is 2. Therefore, 2 is Y’s constant rate.
This table is NOT linear. The first difference between Y1 & Y2 is 3, Y2 & Y3 is 8, and Y3 & Y4 is 39, which means that Y does not increase at a constant rate as X increases by 1. This proves that this relationship is not linear.