Which table of values represents a linear function?

To determine which table of values represents a linear function, we need to check if the change in y (Δy) over the change in x (Δx) is constant, which is the definition of a linear relationship.

Let's analyze each option:

Option A:

x | y
-------
-2 | 0
 1 | 2
 5 | 4
 8 | 6

Calculating the slopes:

• From (-2, 0) to (1, 2): Δy = 2 - 0 = 2; Δx = 1 - (-2) = 3; slope = 2/3
• From (1, 2) to (5, 4): Δy = 4 - 2 = 2; Δx = 5 - 1 = 4; slope = 2/4 = 1/2
• From (5, 4) to (8, 6): Δy = 6 - 4 = 2; Δx = 8 - 5 = 3; slope = 2/3

Since the slopes are not constant, Option A is not linear.

Option B:

x | y
-------
 1 | 2
 2 | 0
 3 | -2
 5 | -5

Calculating the slopes:

• From (1, 2) to (2, 0): Δy = 0 - 2 = -2; Δx = 2 - 1 = 1; slope = -2/1 = -2
• From (2, 0) to (3, -2): Δy = -2 - 0 = -2; Δx = 3 - 2 = 1; slope = -2/1 = -2
• From (3, -2) to (5, -5): Δy = -5 - (-2) = -3; Δx = 5 - 3 = 2; slope = -3/2

Since the slopes vary, Option B is not linear.

Option C:

x | y
-------
-5 | -6
-1 | 0
 3 | 6
 6 | 9

Calculating the slopes:

• From (-5, -6) to (-1, 0): Δy = 0 - (-6) = 6; Δx = -1 - (-5) = 4; slope = 6/4 = 3/2
• From (-1, 0) to (3, 6): Δy = 6 - 0 = 6; Δx = 3 - (-1) = 4; slope = 6/4 = 3/2
• From (3, 6) to (6, 9): Δy = 9 - 6 = 3; Δx = 6 - 3 = 3; slope = 3/3 = 1

Since the slopes vary, Option C is not linear.

Option D:

x | y
-------
-5 | -6
-3 | -4
 0 | -1
 2 | 1

Calculating the slopes:

• From (-5, -6) to (-3, -4): Δy = -4 - (-6) = 2; Δx = -3 - (-5) = 2; slope = 2/2 = 1
• From (-3, -4) to (0, -1): Δy = -1 - (-4) = 3; Δx = 0 - (-3) = 3; slope = 3/3 = 1
• From (0, -1) to (2, 1): Δy = 1 - (-1) = 2; Δx = 2 - 0 = 2; slope = 2/2 = 1

Here, slope remains consistent at 1. Thus, Option D is a linear function.

Final Answer:

Option D represents a linear function.