To determine which set of ordered pairs could represent a linear function, we need to check if the slopes between each pair of consecutive points in the set are the same. A consistent slope indicates that the points lie on the same straight line, representing a linear function.

Set A: $\{(-6, 0), (-3, 3), (0, 6), (3, 9)\}$

Calculate the slope between each pair of consecutive points:

1. Between $(-6, 0)$ and $(-3, 3)$: $\text{slope} = \frac{3 - 0}{-3 - (-6)} = \frac{3}{3} = 1$

2. Between $(-3, 3)$ and $(0, 6)$: $\text{slope} = \frac{6 - 3}{0 - (-3)} = \frac{3}{3} = 1$

3. Between $(0, 6)$ and $(3, 9)$: $\text{slope} = \frac{9 - 6}{3 - 0} = \frac{3}{3} = 1$

The slopes are consistent, so Set A does represent a linear function.

Set B: $\{(-6, 5), (-3, 2), (0, -2), (3, -5)\}$

Calculate the slope between each pair of consecutive points:

1. Between $(-6, 5)$ and $(-3, 2)$: $\text{slope} = \frac{2 - 5}{-3 - (-6)} = \frac{-3}{3} = -1$

2. Between $(-3, 2)$ and $(0, -2)$: $\text{slope} = \frac{-2 - 2}{0 - (-3)} = \frac{-4}{3}$

3. Between $(0, -2)$ and $(3, -5)$: $\text{slope} = \frac{-5 - (-2)}{3 - 0} = \frac{-3}{3} = -1$

The slopes are not consistent, so Set B does not represent a linear function.

Set C: $\{(2, 8), (4, 4), (5, 0), (7, -8)\}$

Calculate the slope between each pair of consecutive points:

1. Between $(2, 8)$ and $(4, 4)$: $\text{slope} = \frac{4 - 8}{4 - 2} = \frac{-4}{2} = -2$

2. Between $(4, 4)$ and $(5, 0)$: $\text{slope} = \frac{0 - 4}{5 - 4} = \frac{-4}{1} = -4$

3. Between $(5, 0)$ and $(7, -8)$: $\text{slope} = \frac{-8 - 0}{7 - 5} = \frac{-8}{2} = -4$

The slopes are not consistent, so Set C does not represent a linear function.

Set D: $\{(-4, 5), (-1, 4), (2, 3), (6, 2)\}$

Calculate the slope between each pair of consecutive points:

1. Between $(-4, 5)$ and $(-1, 4)$: $\text{slope} = \frac{4 - 5}{-1 - (-4)} = \frac{-1}{3} = -\frac{1}{3}$

2. Between $(-1, 4)$ and $(2, 3)$: $\text{slope} = \frac{3 - 4}{2 - (-1)} = \frac{-1}{3} = -\frac{1}{3}$

3. Between $(2, 3)$ and $(6, 2)$: $\text{slope} = \frac{2 - 3}{6 - 2} = \frac{-1}{4} = -\frac{1}{4}$

The slopes are not consistent, so Set D does not represent a linear function.

Conclusion:

Set A represents a linear function, as all pairs of consecutive points have the same slope.

Would you like more details on any part of the solution or have any questions?

Here are some related questions for further exploration:

1. What is the equation of the line passing through the points in Set A?
2. How can you determine the slope between two points on a graph?
3. How would you plot the points in Set A to visually confirm linearity?
4. Can a set with different slopes represent a piecewise linear function?
5. How do different slopes between points affect the shape of a function?

Tip: In a linear function, the rate of change (slope) between any two points remains constant, resulting in a straight line graph.