Determine if each table represents a linear or nonlinear function by analyzing the x and y values.

To determine if a table represents a linear or nonlinear function, we need to check the rate of change.

* Linear Function: The rate of change is constant. This means for every step you take in $x$, the $y$ value changes by the same amount.
* Formula: $\frac{\text{Change in } y}{\text{Change in } x}$
* Nonlinear Function: The rate of change is not constant. The amount $y$ changes varies between different pairs of points.

Let's analyze each table one by one.

Table 1 (Top Left)

| x | y |
| :--- | :--- |
| -5 | 6 |
| -4 | 13 |
| -3 | 18 |
| -2 | 24 |

1. From $x = -5$ to $-4$ (change of $+1$), $y$ goes from $6$ to $13$ (change of $+7$). Rate = $7/1 = 7$.
2. From $x = -4$ to $-3$ (change of $+1$), $y$ goes from $13$ to $18$ (change of $+5$). Rate = $5/1 = 5$.
3. Since $7 \neq 5$, the rate is not constant.
Conclusion: Nonlinear

Table 2 (Top Middle)

| x | y |
| :--- | :--- |
| 12 | 4 |
| 14 | 8 |
| 16 | 10 |
| 18 | 14 |

1. From $x = 12$ to $14$ (change of $+2$), $y$ goes from $4$ to $8$ (change of $+4$). Rate = $4/2 = 2$.
2. From $x = 14$ to $16$ (change of $+2$), $y$ goes from $8$ to $10$ (change of $+2$). Rate = $2/2 = 1$.
3. Since $2 \neq 1$, the rate is not constant.
Conclusion: Nonlinear

Table 3 (Top Right)

| x | y |
| :--- | :--- |
| 4 | 9 |
| 8 | 6 |
| 12 | 3 |
| 16 | 0 |

1. From $x = 4$ to $8$ (change of $+4$), $y$ goes from $9$ to $6$ (change of $-3$). Rate = $-3/4 = -0.75$.
2. From $x = 8$ to $12$ (change of $+4$), $y$ goes from $6$ to $3$ (change of $-3$). Rate = $-3/4 = -0.75$.
3. From $x = 12$ to $16$ (change of $+4$), $y$ goes from $3$ to $0$ (change of $-3$). Rate = $-3/4 = -0.75$.
4. The rate is constant.
Conclusion: Linear

Table 4 (Middle Left)

| x | y |
| :--- | :--- |
| -4 | -6 |
| -3 | -3 |
| -2 | 0 |
| -1 | 3 |

1. From $x = -4$ to $-3$ (change of $+1$), $y$ goes from $-6$ to $-3$ (change of $+3$). Rate = $3$.
2. From $x = -3$ to $-2$ (change of $+1$), $y$ goes from $-3$ to $0$ (change of $+3$). Rate = $3$.
3. From $x = -2$ to $-1$ (change of $+1$), $y$ goes from $0$ to $3$ (change of $+3$). Rate = $3$.
4. The rate is constant.
Conclusion: Linear

Table 5 (Middle Middle)

| x | y |
| :--- | :--- |
| 5 | 10 |
| 15 | 11 |
| 20 | 12 |
| 35 | 13 |

1. From $x = 5$ to $15$ (change of $+10$), $y$ goes from $10$ to $11$ (change of $+1$). Rate = $1/10 = 0.1$.
2. From $x = 15$ to $20$ (change of $+5$), $y$ goes from $11$ to $12$ (change of $+1$). Rate = $1/5 = 0.2$.
3. Since $0.1 \neq 0.2$, the rate is not constant.
Conclusion: Nonlinear

Table 6 (Middle Right)

| x | y |
| :--- | :--- |
| 0 | 14 |
| 3 | 17 |
| 5 | 19 |
| 12 | 26 |

1. From $x = 0$ to $3$ (change of $+3$), $y$ goes from $14$ to $17$ (change of $+3$). Rate = $3/3 = 1$.
2. From $x = 3$ to $5$ (change of $+2$), $y$ goes from $17$ to $19$ (change of $+2$). Rate = $2/2 = 1$.
3. From $x = 5$ to $12$ (change of $+7$), $y$ goes from $19$ to $26$ (change of $+7$). Rate = $7/7 = 1$.
4. The rate is constant.
Conclusion: Linear

Table 7 (Bottom Left)

| x | y |
| :--- | :--- |
| -3 | -4 |
| 1 | 8 |
| 6 | 23 |
| 15 | 59 |

1. From $x = -3$ to $1$ (change of $+4$), $y$ goes from $-4$ to $8$ (change of $+12$). Rate = $12/4 = 3$.
2. From $x = 1$ to $6$ (change of $+5$), $y$ goes from $8$ to $23$ (change of $+15$). Rate = $15/5 = 3$.
3. From $x = 6$ to $15$ (change of $+9$), $y$ goes from $23$ to $59$ (change of $+36$). Rate = $36/9 = 4$.
4. Since $3 \neq 4$, the rate is not constant.
Conclusion: Nonlinear

Table 8 (Bottom Middle)

| x | y |
| :--- | :--- |
| 5 | 1 |
| 9 | 7 |
| 15 | 16 |
| 23 | 28 |

1. From $x = 5$ to $9$ (change of $+4$), $y$ goes from $1$ to $7$ (change of $+6$). Rate = $6/4 = 1.5$.
2. From $x = 9$ to $15$ (change of $+6$), $y$ goes from $7$ to $16$ (change of $+9$). Rate = $9/6 = 1.5$.
3. From $x = 15$ to $23$ (change of $+8$), $y$ goes from $16$ to $28$ (change of $+12$). Rate = $12/8 = 1.5$.
4. The rate is constant.
Conclusion: Linear

Table 9 (Bottom Right)

| x | y |
| :--- | :--- |
| -10 | 7 |
| -16 | 9 |
| -28 | 13 |
| -31 | 14 |

1. From $x = -10$ to $-16$ (change of $-6$), $y$ goes from $7$ to $9$ (change of $+2$). Rate = $2/-6 = -1/3$.
2. From $x = -16$ to $-28$ (change of $-12$), $y$ goes from $9$ to $13$ (change of $+4$). Rate = $4/-12 = -1/3$.
3. From $x = -28$ to $-31$ (change of $-3$), $y$ goes from $13$ to $14$ (change of $+1$). Rate = $1/-3 = -1/3$.
4. The rate is constant.
Conclusion: Linear

Final Answer:
1. Nonlinear
2. Nonlinear
3. Linear
4. Linear
5. Nonlinear
6. Linear
7. Nonlinear
8. Linear
9. Linear