Which table of values represents a linear function?

**A**

x*x*

y*y*

00

minus, 6−6

11

minus, 3−3

22

minus, 1−1

33

11

**B**

x*x*

y*y*

minus, 4−4

11

22

44

44

66

66

77

**C**

x*x*

y*y*

11

77

22

11

33

minus, 4−4

44

minus, 9−9

**D**

x*x*

y*y*

minus, 1−1

77

33

55

55

44

99

22

Answer

To determine which table of values represents a linear function, we need to check the consistency of the slope between each pair of points. For a function to be linear, the difference in \( y \) values divided by the difference in \( x \) values (the slope) should be constant.

Let's analyze each table:

---

**Table A:**

\[
\begin{array}{c|c}
x & y \\
\hline
0 & -6 \\
1 & -3 \\
2 & -1 \\
3 & 1 \\
\end{array}
\]

**Changes:**

\[
\text{slope between } (0, -6) \text{ and } (1, -3): \frac{-3 - (-6)}{1 - 0} = \frac{3}{1} = 3
\]

\[
\text{slope between } (1, -3) \text{ and } (2, -1): \frac{-1 - (-3)}{2 - 1} = \frac{2}{1} = 2
\]

\[
\text{slope between } (2, -1) \text{ and } (3, 1): \frac{1 - (-1)}{3 - 2} = \frac{2}{1} = 2
\]

The slopes are not consistent, so **Table A is not linear**.

---

**Table B:**

\[
\begin{array}{c|c}
x & y \\
\hline
-4 & 1 \\
2 & 4 \\
4 & 6 \\
6 & 7 \\
\end{array}
\]

**Changes:**

\[
\text{slope between } (-4, 1) \text{ and } (2, 4): \frac{4 - 1}{2 - (-4)} = \frac{3}{6} = \frac{1}{2}
\]

\[
\text{slope between } (2, 4) \text{ and } (4, 6): \frac{6 - 4}{4 - 2} = \frac{2}{2} = 1
\]

\[
\text{slope between } (4, 6) \text{ and } (6, 7): \frac{7 - 6}{6 - 4} = \frac{1}{2}
\]

The slopes are not consistent, so **Table B is not linear**.

---

**Table C:**

\[
\begin{array}{c|c}
x & y \\
\hline
1 & 7 \\
2 & 1 \\
3 & -4 \\
4 & -9 \\
\end{array}
\]

**Changes:**

\[
\text{slope between } (1, 7) \text{ and } (2, 1): \frac{1 - 7}{2 - 1} = \frac{-6}{1} = -6
\]

\[
\text{slope between } (2, 1) \text{ and } (3, -4): \frac{-4 - 1}{3 - 2} = \frac{-5}{1} = -5
\]

\[
\text{slope between } (3, -4) \text{ and } (4, -9): \frac{-9 - (-4)}{4 - 3} = \frac{-5}{1} = -5
\]

The slopes are not consistent, so **Table C is not linear**.

---

**Table D:**

\[
\begin{array}{c|c}
x & y \\
\hline
-1 & 7 \\
3 & 5 \\
5 & 4 \\
9 & 2 \\
\end{array}
\]

**Changes:**

\[
\text{slope between } (-1, 7) \text{ and } (3, 5): \frac{5 - 7}{3 - (-1)} = \frac{-2}{4} = -\frac{1}{2}
\]

\[
\text{slope between } (3, 5) \text{ and } (5, 4): \frac{4 - 5}{5 - 3} = \frac{-1}{2} = -\frac{1}{2}
\]

\[
\text{slope between } (5, 4) \text{ and } (9, 2): \frac{2 - 4}{9 - 5} = \frac{-2}{4} = -\frac{1}{2}
\]

The slopes are consistent, so **Table D is linear**.

---

**Conclusion:** **Table D** represents a linear function.

Would you like more details or have any questions?

---

**Related Questions:**

1. How do you graph a linear function from a table of values?
2. What is the significance of the slope in a linear function?
3. How can you tell if a function is linear just by looking at its equation?
4. Can a linear function have a slope of zero?
5. How does the slope relate to the steepness of a line?

**Tip:** The slope of a linear function is the same between any two points on the function's graph.

Explore how to identify linear functions using tables of values. Analyze multiple examples and learn about consistent slopes, a key indicator of linearity. Suitable for Grades 7-9 students.

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