Question

which set of ordered pairs represents a linear relationship?

first table (top left):

x	y
1	2
2	4
3	8
4	16

second table (top right):

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

third table (bottom left):

x	y
1	1
2	4
3	9
4	16

fourth table (bottom right):

x	y
-1	1
2	2
7	3
10	4

Explanation:

To determine which set of ordered pairs represents a linear relationship, we check if the rate of change (slope) between consecutive points is constant. The slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step 1: Analyze the first table

• Points: \((0,1)\), \((1,2)\), \((2,4)\), \((3,8)\), \((4,16)\)
• Slope between \((0,1)\) and \((1,2)\): \(\frac{2 - 1}{1 - 0}=1\)
• Slope between \((1,2)\) and \((2,4)\): \(\frac{4 - 2}{2 - 1}=2\)
• The slopes are not constant, so not linear.

Step 2: Analyze the second table

• Points: \((-4,6)\), \((-1,4)\), \((2,2)\), \((5,0)\), \((8,-2)\)
• Slope between \((-4,6)\) and \((-1,4)\): \(\frac{4 - 6}{-1 - (-4)}=\frac{-2}{3}=-\frac{2}{3}\)
• Slope between \((-1,4)\) and \((2,2)\): \(\frac{2 - 4}{2 - (-1)}=\frac{-2}{3}\)
• Slope between \((2,2)\) and \((5,0)\): \(\frac{0 - 2}{5 - 2}=\frac{-2}{3}\)
• Slope between \((5,0)\) and \((8,-2)\): \(\frac{-2 - 0}{8 - 5}=\frac{-2}{3}\)
• The slope is constant (\(-\frac{2}{3}\)), so this is linear.

Step 3: Analyze the third table

• Points: \((0,0)\), \((1,1)\), \((2,4)\), \((3,9)\), \((4,16)\)
• Slope between \((0,0)\) and \((1,1)\): \(\frac{1 - 0}{1 - 0}=1\)
• Slope between \((1,1)\) and \((2,4)\): \(\frac{4 - 1}{2 - 1}=3\)
• The slopes are not constant, so not linear (this is a quadratic relationship, \(y = x^2\)).

Step 4: Analyze the fourth table

• Points: \((-2,0)\), \((-1,1)\), \((2,2)\), \((7,3)\), \((10,4)\)
• Slope between \((-2,0)\) and \((-1,1)\): \(\frac{1 - 0}{-1 - (-2)} = 1\)
• Slope between \((-1,1)\) and \((2,2)\): \(\frac{2 - 1}{2 - (-1)}=\frac{1}{3}\)
• The slopes are not constant, so not linear.

Answer:

The set of ordered pairs in the second table (with \(x\) values \(-4, -1, 2, 5, 8\) and \(y\) values \(6, 4, 2, 0, -2\)) represents a linear relationship.