Question

Which table of values represents a linear function? A xx yy minus, 4−4 66 minus, 2−2 22 minus, 1−1 minus, 2−2 11 minus, 6−6 B xx yy minus, 5−5 44 minus, 1−1 11 33 minus, 2−2 77 minus, 5−5 C xx yy minus, 2−2 33 00 22 22 11 44 minus, 1−1 D xx yy 00 22 22 11 44 minus, 1−1 66 minus, 3−3

Answer 1

To determine which set of values represents a linear function, we need to check if the change in \( y \) is constant for a given change in \( x \) (i.e., the slope is constant).

Let's analyze the options one by one:

**A:**
- \( (-4, -4) \)
- \( (6, -2) \)
- \( (2, -1) \)
- \( (1, -2) \)
- \( (6, -6) \)

Calculate the changes between successive points:
1. From \( (-4, -4) \) to \( (6, -2) \):
- Change in \( x = 6 - (-4) = 10 \)
- Change in \( y = -2 - (-4) = 2 \)
- Slope = \( \frac{2}{10} = \frac{1}{5} \)

2. From \( (6, -2) \) to \( (2, -1) \):
- Change in \( x = 2 - 6 = -4 \)
- Change in \( y = -1 - (-2) = 1 \)
- Slope = \( \frac{1}{-4} = -\frac{1}{4} \)

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

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**B:**
- \( (-5, -5) \)
- \( (4, -1) \)
- \( (1, 3) \)
- \( (2, 7) \)
- \( (5, -5) \)

Calculate the changes:
1. From \( (-5, -5) \) to \( (4, -1) \):
- Change in \( x = 9 \)
- Change in \( y = 4 \)
- Slope = \( \frac{4}{9} \)

2. From \( (4, -1) \) to \( (1, 3) \):
- Change in \( x = -3 \)
- Change in \( y = 4 \)
- Slope = \( \frac{4}{-3} = -\frac{4}{3} \)

Since the slopes are not constant, **B is not linear**.

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**C:**
- \( (-2, 3) \)
- \( (0, 2) \)
- \( (2, 2) \)
- \( (1, 4) \)
- \( (-1, -1) \)

Calculate the changes:
1. From \( (-2, 3) \) to \( (0, 2) \):
- Change in \( x = 2 \)
- Change in \( y = -1 \)
- Slope = \( \frac{-1}{2} = -\frac{1}{2} \)

2. From \( (0, 2) \) to \( (2, 2) \):
- Change in \( x = 2 \)
- Change in \( y = 0 \)
- Slope = \( \frac{0}{2} = 0 \)

Since the slopes are not constant, **C is not linear**.

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**D:**
- \( (0, 0) \)
- \( (2, 2) \)
- \( (2, 1) \)
- \( (1, 4) \)
- \( (-1, 6) \)

Calculate the changes:
1. From \( (0, 0) \) to \( (2, 2) \):
- Change in \( x = 2 \)
- Change in \( y = 2 \)
- Slope = \( \frac{2}{2} = 1 \)

2. From \( (2, 2) \) to \( (2, 1) \):
- Change in \( x = 0 \)
- Change in \( y = -1 \)
- Slope = undefined (vertical line)

Since there is a vertical change and not a consistent slope, **D is not linear**.

After checking through all options, none of the tables provided represent a linear function based on the calculations. Please double-check the points or provide corrected values for further evaluation.