To determine which table represents a linear function, we need to check for consistent changes between inputs and outputs.
A linear function has a constant rate of change, meaning that the difference in output values divided by the difference in input values remains the same between any two points.
Analysis of Table I:
Input (x) | Output (y)
-----------------------
0 | 1
1 | 0
2 | 1
3 | 4
4 | 9
Calculating the differences in output for each pair of inputs:
- From \(x = 0\) to \(x = 1\): \(y\) goes from \(1\) to \(0\) → change is \(-1\)
- From \(x = 1\) to \(x = 2\): \(y\) goes from \(0\) to \(1\) → change is \(1\)
- From \(x = 2\) to \(x = 3\): \(y\) goes from \(1\) to \(4\) → change is \(3\)
- From \(x = 3\) to \(x = 4\): \(y\) goes from \(4\) to \(9\) → change is \(5\)
The output does not change by a constant amount; therefore, Table I is not a linear function.
Analysis of Table II:
Input (x) | Output (y)
-----------------------
0 | 28
1 | 23
2 | 18
3 | 13
4 | 8
Calculating the differences in output for each pair of inputs:
- From \(x = 0\) to \(x = 1\): \(y\) goes from \(28\) to \(23\) → change is \(-5\)
- From \(x = 1\) to \(x = 2\): \(y\) goes from \(23\) to \(18\) → change is \(-5\)
- From \(x = 2\) to \(x = 3\): \(y\) goes from \(18\) to \(13\) → change is \(-5\)
- From \(x = 3\) to \(x = 4\): \(y\) goes from \(13\) to \(8\) → change is \(-5\)
The outputs change by a constant amount of \(-5\). Therefore, Table II represents a linear function.
Conclusion:
The correct answer is: D. Only Table II