Worksheet for identifying linear and nonlinear functions from tables of values.

To determine whether each table shows a linear or nonlinear function, we need to check if the relationship between the input \( x \) and output \( y \) has a constant rate of change. Here's how to approach this:

Steps to Solve:

1. Calculate the differences in \( x \) and \( y \):
- For each pair of consecutive rows, compute the change in \( x \) (\( \Delta x \)) and the change in \( y \) (\( \Delta y \)).
- The rate of change is given by \( \frac{\Delta y}{\Delta x} \).

2. Check if the rate of change is constant:
- If \( \frac{\Delta y}{\Delta x} \) is the same for all pairs of consecutive rows, the function is linear.
- If \( \frac{\Delta y}{\Delta x} \) varies, the function is nonlinear.

Let's analyze each table:

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#### Table 1:
| \( x \) | \( y \) |
|---------|---------|
| 1 | 6 |
| 2 | 11 |
| 3 | 16 |
| 4 | 21 |

- Calculate \( \Delta x \) and \( \Delta y \):
- From \( (1, 6) \) to \( (2, 11) \): \( \Delta x = 2 - 1 = 1 \), \( \Delta y = 11 - 6 = 5 \), \( \frac{\Delta y}{\Delta x} = 5 \).
- From \( (2, 11) \) to \( (3, 16) \): \( \Delta x = 3 - 2 = 1 \), \( \Delta y = 16 - 11 = 5 \), \( \frac{\Delta y}{\Delta x} = 5 \).
- From \( (3, 16) \) to \( (4, 21) \): \( \Delta x = 4 - 3 = 1 \), \( \Delta y = 21 - 16 = 5 \), \( \frac{\Delta y}{\Delta x} = 5 \).

- The rate of change is constant (\( 5 \)).
- Conclusion: Linear.

---

#### Table 2:
| \( x \) | \( y \) |
|---------|---------|
| 1 | 16 |
| 2 | 8 |
| 3 | 4 |
| 4 | 2 |

- Calculate \( \Delta x \) and \( \Delta y \):
- From \( (1, 16) \) to \( (2, 8) \): \( \Delta x = 2 - 1 = 1 \), \( \Delta y = 8 - 16 = -8 \), \( \frac{\Delta y}{\Delta x} = -8 \).
- From \( (2, 8) \) to \( (3, 4) \): \( \Delta x = 3 - 2 = 1 \), \( \Delta y = 4 - 8 = -4 \), \( \frac{\Delta y}{\Delta x} = -4 \).
- From \( (3, 4) \) to \( (4, 2) \): \( \Delta x = 4 - 3 = 1 \), \( \Delta y = 2 - 4 = -2 \), \( \frac{\Delta y}{\Delta x} = -2 \).

- The rate of change is not constant (\( -8, -4, -2 \)).
- Conclusion: Nonlinear.

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#### Table 3:
| \( x \) | \( y \) |
|---------|---------|
| 1 | 5 |
| 2 | 4 |
| 3 | 2 |
| 4 | -1 |

- Calculate \( \Delta x \) and \( \Delta y \):
- From \( (1, 5) \) to \( (2, 4) \): \( \Delta x = 2 - 1 = 1 \), \( \Delta y = 4 - 5 = -1 \), \( \frac{\Delta y}{\Delta x} = -1 \).
- From \( (2, 4) \) to \( (3, 2) \): \( \Delta x = 3 - 2 = 1 \), \( \Delta y = 2 - 4 = -2 \), \( \frac{\Delta y}{\Delta x} = -2 \).
- From \( (3, 2) \) to \( (4, -1) \): \( \Delta x = 4 - 3 = 1 \), \( \Delta y = -1 - 2 = -3 \), \( \frac{\Delta y}{\Delta x} = -3 \).

- The rate of change is not constant (\( -1, -2, -3 \)).
- Conclusion: Nonlinear.

---

#### Table 4:
| \( x \) | \( y \) |
|---------|---------|
| -1 | 9 |
| 0 | 5 |
| 1 | 11 |
| 2 | 13 |

- Calculate \( \Delta x \) and \( \Delta y \):
- From \( (-1, 9) \) to \( (0, 5) \): \( \Delta x = 0 - (-1) = 1 \), \( \Delta y = 5 - 9 = -4 \), \( \frac{\Delta y}{\Delta x} = -4 \).
- From \( (0, 5) \) to \( (1, 11) \): \( \Delta x = 1 - 0 = 1 \), \( \Delta y = 11 - 5 = 6 \), \( \frac{\Delta y}{\Delta x} = 6 \).
- From \( (1, 11) \) to \( (2, 13) \): \( \Delta x = 2 - 1 = 1 \), \( \Delta y = 13 - 11 = 2 \), \( \frac{\Delta y}{\Delta x} = 2 \).

- The rate of change is not constant (\( -4, 6, 2 \)).
- Conclusion: Nonlinear.

---

#### Table 5:
| \( x \) | \( y \) |
|---------|---------|
| 2 | 1 |
| 4 | 3 |
| 6 | 7 |
| 8 | 13 |

- Calculate \( \Delta x \) and \( \Delta y \):
- From \( (2, 1) \) to \( (4, 3) \): \( \Delta x = 4 - 2 = 2 \), \( \Delta y = 3 - 1 = 2 \), \( \frac{\Delta y}{\Delta x} = 1 \).
- From \( (4, 3) \) to \( (6, 7) \): \( \Delta x = 6 - 4 = 2 \), \( \Delta y = 7 - 3 = 4 \), \( \frac{\Delta y}{\Delta x} = 2 \).
- From \( (6, 7) \) to \( (8, 13) \): \( \Delta x = 8 - 6 = 2 \), \( \Delta y = 13 - 7 = 6 \), \( \frac{\Delta y}{\Delta x} = 3 \).

- The rate of change is not constant (\( 1, 2, 3 \)).
- Conclusion: Nonlinear.

---

#### Table 6:
| \( x \) | \( y \) |
|---------|---------|
| 9 | -3 |
| 11 | -27 |
| 15 | -24 |
| 18 | -33 |

- Calculate \( \Delta x \) and \( \Delta y \):
- From \( (9, -3) \) to \( (11, -27) \): \( \Delta x = 11 - 9 = 2 \), \( \Delta y = -27 - (-3) = -24 \), \( \frac{\Delta y}{\Delta x} = -12 \).
- From \( (11, -27) \) to \( (15, -24) \): \( \Delta x = 15 - 11 = 4 \), \( \Delta y = -24 - (-27) = 3 \), \( \frac{\Delta y}{\Delta x} = 0.75 \).
- From \( (15, -24) \) to \( (18, -33) \): \( \Delta x = 18 - 15 = 3 \), \( \Delta y = -33 - (-24) = -9 \), \( \frac{\Delta y}{\Delta x} = -3 \).

- The rate of change is not constant (\( -12, 0.75, -3 \)).
- Conclusion: Nonlinear.

---

#### Table 7:
| \( x \) | \( y \) |
|---------|---------|
| -2 | 4 |
| -1 | 1 |
| 2 | 5 |
| 3 | 9 |

- Calculate \( \Delta x \) and \( \Delta y \):
- From \( (-2, 4) \) to \( (-1, 1) \): \( \Delta x = -1 - (-2) = 1 \), \( \Delta y = 1 - 4 = -3 \), \( \frac{\Delta y}{\Delta x} = -3 \).
- From \( (-1, 1) \) to \( (2, 5) \): \( \Delta x = 2 - (-1) = 3 \), \( \Delta y = 5 - 1 = 4 \), \( \frac{\Delta y}{\Delta x} = \frac{4}{3} \).
- From \( (2, 5) \) to \( (3, 9) \): \( \Delta x = 3 - 2 = 1 \), \( \Delta y = 9 - 5 = 4 \), \( \frac{\Delta y}{\Delta x} = 4 \).

- The rate of change is not constant (\( -3, \frac{4}{3}, 4 \)).
- Conclusion: Nonlinear.

---

#### Table 8:
| \( x \) | \( y \) |
|---------|---------|
| 9 | -1 |
| 7 | 2 |
| 5 | 5 |
| 3 | 8 |

- Calculate \( \Delta x \) and \( \Delta y \):
- From \( (9, -1) \) to \( (7, 2) \): \( \Delta x = 7 - 9 = -2 \), \( \Delta y = 2 - (-1) = 3 \), \( \frac{\Delta y}{\Delta x} = -\frac{3}{2} \).
- From \( (7, 2) \) to \( (5, 5) \): \( \Delta x = 5 - 7 = -2 \), \( \Delta y = 5 - 2 = 3 \), \( \frac{\Delta y}{\Delta x} = -\frac{3}{2} \).
- From \( (5, 5) \) to \( (3, 8) \): \( \Delta x = 3 - 5 = -2 \), \( \Delta y = 8 - 5 = 3 \), \( \frac{\Delta y}{\Delta x} = -\frac{3}{2} \).

- The rate of change is constant (\( -\frac{3}{2} \)).
- Conclusion: Linear.

---

#### Table 9:
| \( x \) | \( y \) |
|---------|---------|
| 21 | -3 |
| 15 | 5 |
| 12 | -6 |
| 9 | -14 |

- Calculate \( \Delta x \) and \( \Delta y \):
- From \( (21, -3) \) to \( (15, 5) \): \( \Delta x = 15 - 21 = -6 \), \( \Delta y = 5 - (-3) = 8 \), \( \frac{\Delta y}{\Delta x} = -\frac{4}{3} \).
- From \( (15, 5) \) to \( (12, -6) \): \( \Delta x = 12 - 15 = -3 \), \( \Delta y = -6 - 5 = -11 \), \( \frac{\Delta y}{\Delta x} = \frac{11}{3} \).
- From \( (12, -6) \) to \( (9, -14) \): \( \Delta x = 9 - 12 = -3 \), \( \Delta y = -14 - (-6) = -8 \), \( \frac{\Delta y}{\Delta x} = \frac{8}{3} \).

- The rate of change is not constant (\( -\frac{4}{3}, \frac{11}{3}, \frac{8}{3} \)).
- Conclusion: Nonlinear.

---

Final Answers:

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

\[
\boxed{
\text{Linear, Nonlinear, Nonlinear, Nonlinear, Nonlinear, Nonlinear, Nonlinear, Linear, Nonlinear}
}
\]