To determine which table represents a linear function, let's analyze the rate of change for each table. A linear function has a constant rate of change between $x$ and $y$ values.

Table A:

• $x$ values: 3, 5, 7, 9
• $y$ values: 9, 5, 1, -4

Changes in $x$: $+2$ each time Changes in $y$: $-4$ each time

Since the changes in $y$ are consistent as $x$ changes, this table has a constant rate of change. Table A represents a linear function.

Table B:

• $x$ values: -2, 0, 4, 6
• $y$ values: -3, -2, 0, 1

The changes in $y$ are not consistent, so Table B does not represent a linear function.

Table C:

• $x$ values: -3, 0, 3, 6
• $y$ values: 5, 3, 0, -3

The changes in $y$ are consistent as $x$ changes, indicating a linear function. Table C also represents a linear function.

Table D:

• $x$ values: -1, 2, 6, 9
• $y$ values: -8, -5, -2, 1

The changes in $y$ are consistent with changes in $x$, meaning this table has a constant rate of change. Table D represents a linear function.

Answer: Tables A, C, and D represent linear functions.

Would you like further details on how to calculate the rate of change?

Related Questions

1. What makes a function linear in terms of rate of change?
2. How do we determine the rate of change from a table of values?
3. Can a nonlinear function ever have a constant rate of change?
4. What are examples of nonlinear functions?
5. How do we represent a linear function graphically?

Tip

When determining if a table represents a linear function, always look for a constant difference in $y$ for equal changes in $x$.