To determine which set of ordered pairs represents a linear function, we need to check whether the change in $y$ values (the vertical change) is consistent with the change in $x$ values (the horizontal change). This means we need to verify if the difference between consecutive $y$-values divided by the difference between consecutive $x$-values remains constant across all pairs. If the ratio of the differences is constant, the function is linear.

Let's check each option:

Option A: $A = \{(0,7),(1,5),(2,2),(3,-1)\}$

• The change in $y$-values: $5-7 = -2$, $2-5 = -3$, $-1-2 = -3$
• The change in $x$-values: $1-0 = 1$, $2-1 = 1$, $3-2 = 1$

Since the changes in $y$ are not consistent (we have -2, -3, -3), Option A is not linear.

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Option B: $B = \{(3,5),(5,2),(7,-1),(8,-4)\}$

• The change in $y$-values: $2-5 = -3$, $-1-2 = -3$, $-4-(-1) = -3$
• The change in $x$-values: $5-3 = 2$, $7-5 = 2$, $8-7 = 1$

Since the change in $x$-values is not consistent (2, 2, 1), Option B is not linear.

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Option C: $C = \{(0,7),(1,3),(3,-1),(5,-5)\}$

• The change in $y$-values: $3-7 = -4$, $-1-3 = -4$, $-5-(-1) = -4$
• The change in $x$-values: $1-0 = 1$, $3-1 = 2$, $5-3 = 2$

Since the change in $x$-values is not consistent (1, 2, 2), Option C is not linear.

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Option D: $D = \{(-3,2),(-1,4),(1,6),(3,8)\}$

• The change in $y$-values: $4-2 = 2$, $6-4 = 2$, $8-6 = 2$
• The change in $x$-values: $-1-(-3) = 2$, $1-(-1) = 2$, $3-1 = 2$

Since the change in both $y$-values and $x$-values is consistent (all differences are 2), Option D is linear.

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Conclusion:

Option D represents a linear function.

Would you like any further explanations on why this is the case, or have any more questions?

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Related questions:

1. How can you graph the linear function represented by option D?
2. What does the slope of a linear function represent geometrically?
3. How do you calculate the equation of a line from a set of points?
4. What are some real-world examples of linear functions?
5. Can a function be linear if the change in the $y$-values is not consistent?

Tip:

When checking for linearity, always calculate the slope between each pair of points. If the slope remains constant, the function is linear!