Question

Which of these tables represents a linear function?
A 2-column table with 4 rows. Column 1 is labeled x with entries 3, 4, 5, 6. Column 2 is labeled y with entries 3, 4, 6, 7.
A 2-column table with 4 rows. Column 1 is labeled x with entries 3, 4, 5, 6. Column 2 is labeled y with entries 6, 5, 4, 3.
A 2-column table with 4 rows. Column 1 is labeled x with entries 3, 4, 5, 6. Column 2 is labeled y with entries 7, 6, 5, 3.
A 2-column table with 4 rows. Column 1 is labeled x with entries 3, 4, 5, 6. Column 2 is labeled y with entries 2, 4, 5, 6.

Answer 1

To determine which of these tables represents a linear function, we need to check if the change in \( y \) values is consistent as the \( x \) values change. This means the differences between consecutive \( y \) values should be constant.

Let's analyze each table:

1. **First table**:
- \( x \): 3, 4, 5, 6
- \( y \): 3, 4, 6, 7
- Changes in \( y \):
- \( 4 - 3 = 1 \)
- \( 6 - 4 = 2 \)
- \( 7 - 6 = 1 \)
- This does not represent a linear function since the changes in \( y \) are not consistent.

2. **Second table**:
- \( x \): 3, 4, 5, 6
- \( y \): 6, 5, 4, 3
- Changes in \( y \):
- \( 5 - 6 = -1 \)
- \( 4 - 5 = -1 \)
- \( 3 - 4 = -1 \)
- The changes in \( y \) are consistent (-1), so this represents a linear function.

3. **Third table**:
- \( x \): 3, 4, 5, 6
- \( y \): 7, 6, 5, 3
- Changes in \( y \):
- \( 6 - 7 = -1 \)
- \( 5 - 6 = -1 \)
- \( 3 - 5 = -2 \)
- The changes are not consistent, so this does not represent a linear function.

4. **Fourth table**:
- \( x \): 3, 4, 5, 6
- \( y \): 2, 4, 5, 6
- Changes in \( y \):
- \( 4 - 2 = 2 \)
- \( 5 - 4 = 1 \)
- \( 6 - 5 = 1 \)
- The changes in \( y \) are also not consistent, so this does not represent a linear function.

### Conclusion:
The **second table** represents a linear function.