To determine whether the function represented by the table is linear or non-linear, we need to examine the rate of change in the values of \(y\) as \(x\) changes.
Let's calculate the changes in \(y\) for each increase in \(x\):
-
From \(x = 6\) to \(x = 7\):
\(y\) changes from \(4\) to \(2\)
Change in \(y\) = \(2 - 4 = -2\)
Change in \(x\) = \(7 - 6 = 1\)
Rate of change = \(-2/1 = -2\) -
From \(x = 7\) to \(x = 8\):
\(y\) changes from \(2\) to \(0\)
Change in \(y\) = \(0 - 2 = -2\)
Change in \(x\) = \(8 - 7 = 1\)
Rate of change = \(-2/1 = -2\) -
From \(x = 8\) to \(x = 9\):
\(y\) changes from \(0\) to \(-2\)
Change in \(y\) = \(-2 - 0 = -2\)
Change in \(x\) = \(9 - 8 = 1\)
Rate of change = \(-2/1 = -2\)
Now, let's assess the overall changes between each pair of \(x\) and \(y\):
- From \(x=6\) to \(x=9\), the \(y\) values change from \(4\) to \(-2\).
Overall change in \(y\) = \(-2 - 4 = -6\)
Overall change in \(x\) = \(9 - 6 = 3\)
Rate of change = \(-6/3 = -2\)
Since the rate of change is constant (\(-2\)) between every consecutive pair of points, the function does have a constant rate of change and is, therefore, linear.
Thus, the correct answer is: No, because it has a constant rate of change.
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