Which table of values represents a linear function?
A
xx	yy
minus, 1−1	77
11	33
33	minus, 1−1
55	minus, 5−5
B
xx	yy
minus, 6−6	minus, 6−6
minus, 2−2	minus, 4−4
22	minus, 2−2
66	11
C
xx	yy
minus, 4−4	44
minus, 1−1	22
33	00
77	minus, 2−2
D
xx	yy
11	33
33	55
55	66
77	88
Answer

To determine which table of values represents a linear function, we need to check if the relationship between \(x\) and \(y\) in the table follows the pattern of a linear equation, which has the form \(y = mx + b\), where \(m\) is the slope.

### Checking each table:

#### Table A:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-1 & 7 \\
1 & 3 \\
3 & -1 \\
5 & -5 \\
\hline
\end{array}
\]
- Differences in \(x\): \(1 - (-1) = 2\), \(3 - 1 = 2\), \(5 - 3 = 2\)
- Differences in \(y\): \(3 - 7 = -4\), \(-1 - 3 = -4\), \(-5 - (-1) = -4\)

Since the differences in \(x\) and \(y\) are consistent, this indicates a constant rate of change, meaning the function is linear.

#### Table B:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-6 & -6 \\
-2 & -4 \\
2 & -2 \\
6 & 1 \\
\hline
\end{array}
\]
- Differences in \(x\): \(-2 - (-6) = 4\), \(2 - (-2) = 4\), \(6 - 2 = 4\)
- Differences in \(y\): \(-4 - (-6) = 2\), \(-2 - (-4) = 2\), \(1 - (-2) = 3\)

The differences in \(y\) are not consistent, so this function is not linear.

#### Table C:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-4 & 4 \\
-1 & 2 \\
3 & 0 \\
7 & -2 \\
\hline
\end{array}
\]
- Differences in \(x\): \(-1 - (-4) = 3\), \(3 - (-1) = 4\), \(7 - 3 = 4\)
- Differences in \(y\): \(2 - 4 = -2\), \(0 - 2 = -2\), \(-2 - 0 = -2\)

The differences in \(x\) are not consistent, so this function is not linear.

#### Table D:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 3 \\
3 & 5 \\
5 & 6 \\
7 & 8 \\
\hline
\end{array}
\]
- Differences in \(x\): \(3 - 1 = 2\), \(5 - 3 = 2\), \(7 - 5 = 2\)
- Differences in \(y\): \(5 - 3 = 2\), \(6 - 5 = 1\), \(8 - 6 = 2\)

The differences in \(y\) are not consistent, so this function is not linear.

### Conclusion:
The table that represents a linear function is **Table A** because both the differences in \(x\) and \(y\) are consistent.

Do you want details or have any questions?

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**Related Questions:**
1. How do you calculate the slope from a table of values?
2. What is the equation of the linear function from Table A?
3. Can a nonlinear function ever have consistent differences in \(x\)?
4. How do you know if a table represents an exponential function?
5. How do you find the y-intercept of a function from a table?

**Tip:** A linear function always has a constant rate of change in both \(x\) and \(y\).

Learn how to identify linear functions from tables of values. Explore different examples and understand the criteria for linear functions based on consistent rate of change in both x and y variables. Suitable for Grades 7-9 students learning about linear equations and functions.

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