To determine which table of values represents a nonlinear function, we can check whether the relationship between \(x\) and \(y\) produces a constant rate of change (which would indicate a linear function) or a varying rate of change (which indicates a nonlinear function).
Let's analyze the given tables one by one:
-
Table 1:
x | y -7 | 0 -4 | 2 2 | 6 5 | 8Calculate the differences:
- From \(-7\) to \(-4\): \(2 - 0 = 2\)
- From \(-4\) to \(2\): \(6 - 2 = 4\)
- From \(2\) to \(5\): \(8 - 6 = 2\)
The differences are not constant, so this is nonlinear.
-
Table 2:
x | y -3 | 8 0 | 5 5 | 0 8 | -3Calculate the differences:
- From \(-3\) to \(0\): \(5 - 8 = -3\)
- From \(0\) to \(5\): \(0 - 5 = -5\)
- From \(5\) to \(8\): \(-3 - 0 = -3\)
The differences are not constant, so this is nonlinear.
-
Table 3:
x | y -4 | 0 -2 | -3 0 | -4 6 | 5Calculate the differences:
- From \(-4\) to \(-2\): \(-3 - 0 = -3\)
- From \(-2\) to \(0\): \(-4 + 3 = -1\)
- From \(0\) to \(6\): \(5 - (-4) = 9\)
The differences are not constant, so this is nonlinear.
-
Table 4:
x | y -11 | 22 -6 | 12 1 | -2 9 | -18Calculate the differences:
- From \(-11\) to \(-6\): \(12 - 22 = -10\)
- From \(-6\) to \(1\): \(-2 - 12 = -14\)
- From \(1\) to \(9\): \(-18 - (-2) = -16\)
The differences are not constant, so this is nonlinear.
Conclusion:
All four tables represent nonlinear functions since the differences in \(y\) are not constant as \(x\) changes. However, each individual analysis of changes indicates that variations exist in each.
For identifying nonlinear functions based on specific properties, choosing one could suffice, as all tested indicate nonlinearity. If forced to choose based on typical representations of function behavior:
All tables represent nonlinear functions.
If the intent is to select one which visually stands out more in behavior:
- First table has observable fluctuation — easy to interpret from a raw view instead of deeper algebraic forms.
If more analysis is required or specific characteristics noted, please advise.