Question

determine if these are liner relationships or not
input output
-------
-2 | -2
-1 | 0
0 | 0
1 | 2
2 | 6
3 | 12

Answer 1

To determine if the given pairs represent a linear relationship, we can check if the differences in the output values correspond to a constant difference or if the ratio of the differences for the input values remains constant.

### Given pairs:
- \((-2, -2)\)
- \((-1, 0)\)
- \((0, 0)\)
- \((1, 2)\)
- \((2, 6)\)
- \((3, 12)\)

### Output values:
- For \(x = -2\), \(y = -2\)
- For \(x = -1\), \(y = 0\)
- For \(x = 0\), \(y = 0\)
- For \(x = 1\), \(y = 2\)
- For \(x = 2\), \(y = 6\)
- For \(x = 3\), \(y = 12\)

### Differences:
Let's calculate the differences in the output (y-values) corresponding to the differences in the input (x-values):

1. From \(-2\) to \(-1\): \(y\) changes from \(-2\) to \(0\) → difference of \(+2\) (input change: \(+1\))
2. From \(-1\) to \(0\): \(y\) changes from \(0\) to \(0\) → difference of \(0\) (input change: \(+1\))
3. From \(0\) to \(1\): \(y\) changes from \(0\) to \(2\) → difference of \(+2\) (input change: \(+1\))
4. From \(1\) to \(2\): \(y\) changes from \(2\) to \(6\) → difference of \(+4\) (input change: \(+1\))
5. From \(2\) to \(3\): \(y\) changes from \(6\) to \(12\) → difference of \(+6\) (input change: \(+1\))

### Summary of differences:
- \(2\), \(0\), \(2\), \(4\), \(6\)

### Analysis:
The differences in the output values are not constant. Instead, they are increasing, indicating that the output does not change linearly as the input changes. A linear relationship would show a consistent change in the output for equal changes in the input, which is not the case here.

### Conclusion:
The given pairs do not represent a linear relationship. The relationship appears to be quadratic or nonlinear based on the increasing differences in the output values.