Question

which table of values represents a linear function?
a

x	y
-5	-5
-2	-4
1	-2
4	0

b

x	y
-2	2
0	1
2	0
4	-1

c

x	y
-2	8
-1	4
0	1
1	-3

d

x	y
-8	9
-3	6
2	3
8	0

Explanation:

To determine which table represents a linear function, we check the rate of change (slope) between consecutive points. For a linear function, the slope should be constant. The slope formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step 1: Analyze Table A

• Between \((-5, -5)\) and \((-2, -4)\): \(m = \frac{-4 - (-5)}{-2 - (-5)} = \frac{1}{3}\)
• Between \((-2, -4)\) and \((1, -2)\): \(m = \frac{-2 - (-4)}{1 - (-2)} = \frac{2}{3}\)
• The slopes are not equal, so A is not linear.

Step 2: Analyze Table B

• Between \((-2, 2)\) and \((0, 1)\): \(m = \frac{1 - 2}{0 - (-2)} = \frac{-1}{2} = -\frac{1}{2}\)
• Between \((0, 1)\) and \((2, 0)\): \(m = \frac{0 - 1}{2 - 0} = \frac{-1}{2} = -\frac{1}{2}\)
• Between \((2, 0)\) and \((4, -1)\): \(m = \frac{-1 - 0}{4 - 2} = \frac{-1}{2} = -\frac{1}{2}\)
• The slope is constant (\(-\frac{1}{2}\)), so B is linear. (We can verify C and D for completeness, but since B has constant slope, we can conclude.)

Step 3: Analyze Table C (for completeness)

• Between \((-2, 8)\) and \((-1, 4)\): \(m = \frac{4 - 8}{-1 - (-2)} = \frac{-4}{1} = -4\)
• Between \((-1, 4)\) and \((0, 1)\): \(m = \frac{1 - 4}{0 - (-1)} = \frac{-3}{1} = -3\)
• Slopes are not equal, so C is not linear.

Step 4: Analyze Table D (for completeness)

• Between \((-8, 9)\) and \((-3, 6)\): \(m = \frac{6 - 9}{-3 - (-8)} = \frac{-3}{5} = -\frac{3}{5}\)
• Between \((-3, 6)\) and \((2, 3)\): \(m = \frac{3 - 6}{2 - (-3)} = \frac{-3}{5} = -\frac{3}{5}\)
• Between \((2, 3)\) and \((8, 0)\): \(m = \frac{0 - 3}{8 - 2} = \frac{-3}{6} = -\frac{1}{2}\)
• The slopes are not equal (last slope differs), so D is not linear.

Answer:

B. The table with \(x\) values \(-2, 0, 2, 4\) and \(y\) values \(2, 1, 0, -1\) (Table B)