To determine if the relationship between the input (x) and output (y) values is linear, we can look for a constant rate of change in the values of y as x changes. In a linear relationship, the change in y divided by the change in x (often referred to as the slope) should be constant.
Let's analyze the given data:
| x | y | |----|----| | -2 | -7 | | -1 | -3 | | 0 | 2 | | 1 | 8 | | 2 | 15 | | 3 | 23 |
Now, let's calculate the differences in y for each change in x:
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From x = -2 to x = -1:
- Change in y: \(-3 - (-7) = 4\)
- Change in x: \(-1 - (-2) = 1\)
- Slope: \( \frac{4}{1} = 4\)
-
From x = -1 to x = 0:
- Change in y: \(2 - (-3) = 5\)
- Change in x: \(0 - (-1) = 1\)
- Slope: \( \frac{5}{1} = 5\)
-
From x = 0 to x = 1:
- Change in y: \(8 - 2 = 6\)
- Change in x: \(1 - 0 = 1\)
- Slope: \( \frac{6}{1} = 6\)
-
From x = 1 to x = 2:
- Change in y: \(15 - 8 = 7\)
- Change in x: \(2 - 1 = 1\)
- Slope: \( \frac{7}{1} = 7\)
-
From x = 2 to x = 3:
- Change in y: \(23 - 15 = 8\)
- Change in x: \(3 - 2 = 1\)
- Slope: \( \frac{8}{1} = 8\)
Now, let's summarize the slopes calculated:
- From -2 to -1: slope = 4
- From -1 to 0: slope = 5
- From 0 to 1: slope = 6
- From 1 to 2: slope = 7
- From 2 to 3: slope = 8
Since the slopes are not constant (they are increasing), we can conclude that this is not a linear relationship. Instead, it appears to be a quadratic or non-linear relationship.