Let's determine whether the function represented by the given table is linear or not.
A function is linear if it has a constant rate of change between all pairs of successive points. The rate of change between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated as [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex].
We have the following points in the table:
1. (10, -6)
2. (11, 1)
3. (12, 6)
4. (13, 12)
Let's calculate the rates of change between successive points:
1. Between (10, -6) and (11, 1):
[tex]\[
\frac{1 - (-6)}{11 - 10} = \frac{1 + 6}{1} = \frac{7}{1} = 7.0
\][/tex]
2. Between (11, 1) and (12, 6):
[tex]\[
\frac{6 - 1}{12 - 11} = \frac{5}{1} = 5.0
\][/tex]
3. Between (12, 6) and (13, 12):
[tex]\[
\frac{12 - 6}{13 - 12} = \frac{6}{1} = 6.0
\][/tex]
Now, compare the rates of change:
- The rate of change between the first two points (10, -6) and (11, 1) is 7.0.
- The rate of change between the next two points (11, 1) and (12, 6) is 5.0.
- The rate of change between the last two points (12, 6) and (13, 12) is 6.0.
Since the rates of change are 7.0, 5.0, and 6.0, they are not consistent. This means the function does not have a constant rate of change.
Therefore, the function is not linear.
The correct answer is:
No, because it does not have a constant rate of change.
