Bot GPT 4-o mini To determine which set of ordered pairs represents a linear function, we need to check if the y-values change at a constant rate as the x-values change. A linear function is characterized by a constant slope.
Let's analyze each set of ordered pairs:
A: {(3,−4),(5,2),(6,4),(7,6)}
Calculating the slope between each pair:
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Between (3,−4) and (5,2): \[ \text{slope} = \frac{2 - (-4)}{5 - 3} = \frac{2 + 4}{2} = \frac{6}{2} = 3 \]
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Between (5,2) and (6,4): \[ \text{slope} = \frac{4 - 2}{6 - 5} = \frac{2}{1} = 2 \]
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Between (6,4) and (7,6): \[ \text{slope} = \frac{6 - 4}{7 - 6} = \frac{2}{1} = 2 \]
The slopes are not constant; thus, set A does not represent a linear function.
B: {(−2,4),(0,3),(3,2),(5,1)}
Calculating the slope:
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Between (−2,4) and (0,3): \[ \text{slope} = \frac{3 - 4}{0 - (-2)} = \frac{-1}{2} \]
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Between (0,3) and (3,2): \[ \text{slope} = \frac{2 - 3}{3 - 0} = \frac{-1}{3} \]
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Between (3,2) and (5,1): \[ \text{slope} = \frac{1 - 2}{5 - 3} = \frac{-1}{2} \]
The slopes are not constant; thus, set B does not represent a linear function.
C: {(−2,7),(0,5),(1,3),(2,1)}
Calculating the slope:
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Between (−2,7) and (0,5): \[ \text{slope} = \frac{5 - 7}{0 - (-2)} = \frac{-2}{2} = -1 \]
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Between (0,5) and (1,3): \[ \text{slope} = \frac{3 - 5}{1 - 0} = -2 \]
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Between (1,3) and (2,1): \[ \text{slope} = \frac{1 - 3}{2 - 1} = -2 \]
The slopes are not constant; thus, set C does not represent a linear function.
D: {(1,8),(3,0),(4,−4),(5,−8)}
Calculating the slope:
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Between (1,8) and (3,0): \[ \text{slope} = \frac{0 - 8}{3 - 1} = \frac{-8}{2} = -4 \]
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Between (3,0) and (4,−4): \[ \text{slope} = \frac{-4 - 0}{4 - 3} = \frac{-4}{1} = -4 \]
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Between (4,−4) and (5,−8): \[ \text{slope} = \frac{-8 - (-4)}{5 - 4} = \frac{-4}{1} = -4 \]
The slopes are constant (-4); thus, set D represents a linear function.
Therefore, the correct answer is:
D
1 answer