Answer:
(a) linear
(b) non-linear quadratic
(c) non-linear
Explanation:
In order to find first differences, we need to find the differences in the y-values. This only works if the differences in the x-values are constant, so we must first check the x-values. If the differences in the x-values are constant, and the differences in the y-values are constant, the relation is linear. If the differences in the x-values are constant, but the differences in the y-values are NOT constant, then relation is non-linear.
A non-linear relation is quadratic if the second difference of the y-values is constant.
(a) From inspection, we can see that the x-values increase by 10 each time. Therefore, the difference in the x-values is constant.
Comparing the y-values, we can see that they increase by 20 each time, and so their difference is also constant.
Difference in y-values:
41 - 21 = 20
61 - 41 = 20
81 - 61 = 20
Therefore, as the differences in x-values and y-values is constant, we can confirm that this relation is linear.
(b) From inspection, we can see that the x-values increase by 1 each time. Therefore, the difference in the x-values is constant.
Comparing the y-values, we can see that they DO NOT decrease by the same amount each time, and so their difference is NOT constant.
Difference in y-values:
-3 - (-2) = -1
-5 - (-3) = -2
-8 - (-5) = -3
Therefore, as the differences in x-values is constant yet the differences in y-values is not constant, we can confirm that this relation is non-linear.
Second difference in y-values:
-2 - (-1) = -1
-3 - (-2) = -1
Therefore, as the second difference in y-values is constant, the relation is quadratic.
(c) From inspection, we can see that the x-values increase by 1 each time. Therefore, the difference in the x-values is constant.
Comparing the y-values, we can see that they DO NOT increase by the same amount each time, and so their difference is NOT constant.
Difference in y-values:
-1 - (-2) = 1
6 - (-1) = 7
25 - 6 = 19
Therefore, as the differences in x-values is constant yet the differences in y-values is not constant, we can confirm that this relation is non-linear.
Second difference in y-values:
7 - 1 = 6
19 - 7 = 12
Therefore, as the second difference in y-values is NOT constant, the relation is NOT quadratic.