Question

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which table of values represents a linear function?
a

x	-2	1	4	7
y	8	5	2	-1

b

x	-4	-2	2	3
y	8	4	-2	-4

c

x	-9	-3	3	9
y	-7	-4	-1	3

d

x	-5	-1	3	7
y	8	6	5	4

Explanation:

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

Step 1: Analyze Table A

Points: \((-2, 8)\), \((1, 5)\), \((4, 2)\), \((7, -1)\)

• Slope between \((-2, 8)\) and \((1, 5)\): \( \frac{5 - 8}{1 - (-2)}=\frac{-3}{3}=-1 \)
• Slope between \((1, 5)\) and \((4, 2)\): \( \frac{2 - 5}{4 - 1}=\frac{-3}{3}=-1 \)
• Slope between \((4, 2)\) and \((7, -1)\): \( \frac{-1 - 2}{7 - 4}=\frac{-3}{3}=-1 \)

The slope is constant (\(-1\)) for Table A.

Step 2: Analyze Table B (for verification)

Points: \((-4, 8)\), \((-2, 4)\), \((2, -2)\), \((3, -4)\)

• Slope between \((-4, 8)\) and \((-2, 4)\): \( \frac{4 - 8}{-2 - (-4)}=\frac{-4}{2}=-2 \)
• Slope between \((-2, 4)\) and \((2, -2)\): \( \frac{-2 - 4}{2 - (-2)}=\frac{-6}{4}=-\frac{3}{2} \)

Slope is not constant, so not linear.

Step 3: Analyze Table C (for verification)

Points: \((-9, -7)\), \((-3, -4)\), \((3, -1)\), \((9, 3)\)

• Slope between \((-9, -7)\) and \((-3, -4)\): \( \frac{-4 - (-7)}{-3 - (-9)}=\frac{3}{6}=\frac{1}{2} \)
• Slope between \((-3, -4)\) and \((3, -1)\): \( \frac{-1 - (-4)}{3 - (-3)}=\frac{3}{6}=\frac{1}{2} \)
• Slope between \((3, -1)\) and \((9, 3)\): \( \frac{3 - (-1)}{9 - 3}=\frac{4}{6}=\frac{2}{3} \)

Slope is not constant (last slope differs), so not linear.

Step 4: Analyze Table D (for verification)

Points: \((-5, 8)\), \((-1, 6)\), \((3, 5)\), \((7, 4)\)

• Slope between \((-5, 8)\) and \((-1, 6)\): \( \frac{6 - 8}{-1 - (-5)}=\frac{-2}{4}=-\frac{1}{2} \)
• Slope between \((-1, 6)\) and \((3, 5)\): \( \frac{5 - 6}{3 - (-1)}=\frac{-1}{4} \)

Slope is not constant, so not linear.

Answer:

Table A represents a linear function. (Assuming the question is to identify which table is linear, and from the analysis, Table A has a constant slope of \(-1\) between all consecutive points.)