To determine if the function represented by the table is linear, we need to check if there is a constant rate of change between the values of \(y\) as \(x\) changes.
Let's calculate the differences:
-
From \(x = 10\) to \(x = 11\):
- \(y\) changes from \(-6\) to \(1\): \(1 - (-6) = 7\)
- Change in \(x\): \(11 - 10 = 1\)
- Rate of change: \(\frac{7}{1} = 7\)
-
From \(x = 11\) to \(x = 12\):
- \(y\) changes from \(1\) to \(6\): \(6 - 1 = 5\)
- Change in \(x\): \(12 - 11 = 1\)
- Rate of change: \(\frac{5}{1} = 5\)
-
From \(x = 12\) to \(x = 13\):
- \(y\) changes from \(6\) to \(12\): \(12 - 6 = 6\)
- Change in \(x\): \(13 - 12 = 1\)
- Rate of change: \(\frac{6}{1} = 6\)
Now we can summarize the rates of change:
- From \(x = 10\) to \(x = 11\): \(7\)
- From \(x = 11\) to \(x = 12\): \(5\)
- From \(x = 12\) to \(x = 13\): \(6\)
Since the rates of change are not constant (7, 5, and 6 are different), the function represented by the table does not have a linear relationship.
Therefore, the answer is:
No, because it does not have a constant rate of change.
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