Question

determine if these are liner relationships or not
input output
x y
-2 4
-1 1
0 0
1 1
2 4
3 9

Answer 1

To determine if the relationship between the input (x) and output (y) values is linear, we can analyze the pairs of (x, y) values given.

A linear relationship can be defined as one that can be described by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In other words, for a linear relationship, the change in y should be consistent for a consistent change in x.

Let's look at the provided input-output pairs:

```
x | y
-------
-2 | 4
-1 | 1
0 | 0
1 | 1
2 | 4
3 | 9
```

We can calculate the differences in y as x changes.

- From x = -2 to -1: \(y\) changes from 4 to 1 (change of -3)
- From x = -1 to 0: \(y\) changes from 1 to 0 (change of -1)
- From x = 0 to 1: \(y\) changes from 0 to 1 (change of +1)
- From x = 1 to 2: \(y\) changes from 1 to 4 (change of +3)
- From x = 2 to 3: \(y\) changes from 4 to 9 (change of +5)

The changes in y are not consistent as x increases, indicating that the relationship is not linear.

Another way to verify this is to check whether the change in y divided by the change in x (the slope) is constant. For non-linear data, the slope will vary between different segments of the x-values.

For example, between:
- x = -2 and x = -1, slope = \((1 - 4) / (-1 + 2) = -3\)
- x = -1 and x = 0, slope = \((0 - 1) / (0 + 1) = -1\)
- x = 0 and x = 1, slope = \((1 - 0) / (1 - 0) = 1\)

The slopes are different, confirming that the relationship is indeed not linear.

In conclusion, the relationship between the given input and output pairs is **not linear**.