To determine which set of ordered pairs represents a linear function, we need to check whether the relationship between the $x$-values and $y$-values is consistent with a straight line. In other words, the change in $y$ with respect to $x$ should be constant across all pairs in the set.

Let's calculate the slope (change in $y$ over change in $x$) for each set:

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

For the points $(-6, 0)$ and $(-3, 3)$: $\text{slope} = \frac{3 - 0}{-3 - (-6)} = \frac{3}{3} = 1$

For the points $(-3, 3)$ and $(0, 6)$: $\text{slope} = \frac{6 - 3}{0 - (-3)} = \frac{3}{3} = 1$

For the points $(0, 6)$ and $(3, 9)$: $\text{slope} = \frac{9 - 6}{3 - 0} = \frac{3}{3} = 1$

The slope is consistent (1) for all pairs in set A. Hence, A represents a linear function.

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

For the points $(-6, 5)$ and $(-3, 2)$: $\text{slope} = \frac{2 - 5}{-3 - (-6)} = \frac{-3}{3} = -1$

For the points $(-3, 2)$ and $(0, -2)$: $\text{slope} = \frac{-2 - 2}{0 - (-3)} = \frac{-4}{3} \neq -1$

Since the slope is not constant, B does not represent a linear function.

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

For the points $(2, 8)$ and $(4, 4)$: $\text{slope} = \frac{4 - 8}{4 - 2} = \frac{-4}{2} = -2$

For the points $(4, 4)$ and $(5, 0)$: $\text{slope} = \frac{0 - 4}{5 - 4} = \frac{-4}{1} = -4$

Since the slope is not constant, C does not represent a linear function.

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

For the points $(-4, 5)$ and $(-1, 4)$: $\text{slope} = \frac{4 - 5}{-1 - (-4)} = \frac{-1}{3} = -\frac{1}{3}$

For the points $(-1, 4)$ and $(2, 3)$: $\text{slope} = \frac{3 - 4}{2 - (-1)} = \frac{-1}{3} = -\frac{1}{3}$

For the points $(2, 3)$ and $(6, 2)$: $\text{slope} = \frac{2 - 3}{6 - 2} = \frac{-1}{4} \neq -\frac{1}{3}$

Since the slope is not consistent, D does not represent a linear function.

Conclusion:

Set A represents a linear function.

Would you like further details on how to calculate slopes or have any other questions? Here are some related topics:

1. What is the formula for calculating the slope of a line?
2. How can we recognize if a set of points forms a function?
3. What is the difference between a linear function and a nonlinear function?
4. How do we graph a linear function from a set of ordered pairs?
5. How can the slope-intercept form of a linear equation be derived from a set of points?

Tip: Always check if the slope between different points is consistent to verify whether the relationship is linear!