Identify Linear and Nonlinear Functions From Tables | Interactive ... - Free Printable

Let's solve each table one by one to determine whether it represents a linear or nonlinear function. A linear function has a constant rate of change (i.e., the difference in $ y $ values is constant for equal differences in $ x $ values). If the rate of change varies, the function is nonlinear.

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Table 1

| x | y |
|---|---|
| 1 | 6 |
| 2 | 10 |
| 3 | 14 |
| 4 | 18 |

- Change in $ x $: +1 each time
- Change in $ y $:
- $ 10 - 6 = 4 $
- $ 14 - 10 = 4 $
- $ 18 - 14 = 4 $

✔ Constant rate of change: +4

➡️ Linear

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Table 2

| x | y |
|---|---|
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |

- $ x $ increases by 1
- $ y $ changes:
- $ 25 - 16 = 9 $
- $ 36 - 25 = 11 $
- $ 49 - 36 = 13 $

✘ Rate of change is increasing: 9, 11, 13 → not constant

➡️ Nonlinear

*(Note: This looks like $ y = x^2 $, which is nonlinear.)*

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Table 3

| x | y |
|---|---|
| 1 | 5 |
| 2 | 4 |
| 3 | 2 |
| 4 | -1 |

- $ x $ increases by 1
- $ y $ changes:
- $ 4 - 5 = -1 $
- $ 2 - 4 = -2 $
- $ -1 - 2 = -3 $

✘ Rate of change: -1, -2, -3 → not constant

➡️ Nonlinear

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Table 4

| x | y |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |

- $ x $ increases by 1
- $ y $ changes:
- $ 2 - 1 = 1 $
- $ 4 - 2 = 2 $
- $ 8 - 4 = 4 $

✘ Rate of change: 1, 2, 4 → doubling → not constant

➡️ Nonlinear

*(This is exponential: $ y = 2^x $)*

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Table 5

| x | y |
|---|---|
| 4 | 9 |
| 5 | 12 |
| 6 | 15 |
| 7 | 18 |

- $ x $ increases by 1
- $ y $ changes:
- $ 12 - 9 = 3 $
- $ 15 - 12 = 3 $
- $ 18 - 15 = 3 $

✔ Constant rate of change: +3

➡️ Linear

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Table 6

| x | y |
|---|---|
| 1 | -6 |
| 2 | -12 |
| 3 | -24 |
| 4 | -33 |

- $ x $ increases by 1
- $ y $ changes:
- $ -12 - (-6) = -6 $
- $ -24 - (-12) = -12 $
- $ -33 - (-24) = -9 $

✘ Rate of change: -6, -12, -9 → not constant

➡️ Nonlinear

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Table 7

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

- $ x $ changes: +1, +1, +3
- $ y $ stays at 4 every time

➡️ Even though $ x $ changes, $ y $ is constant → rate of change is 0 (constant)

✔ Constant rate of change → Linear

*(A constant function is linear!)*

➡️ Linear

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Table 8

| x | y |
|---|---|
| 0 | -1 |
| 1 | 1 |
| 2 | 3 |
| 3 | 5 |

- $ x $ increases by 1
- $ y $ changes:
- $ 1 - (-1) = 2 $
- $ 3 - 1 = 2 $
- $ 5 - 3 = 2 $

✔ Constant rate of change: +2

➡️ Linear

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Table 9

| x | y |
|---|---|
| 21 | -3 |
| 18 | -5 |
| 12 | -6 |
| -12 | -14 |

Let’s check the rate of change. But first, $ x $ is decreasing, so let’s go from top to bottom:

- $ x $: 21 → 18 → 12 → -12
- Changes in $ x $: -3, -6, -24 (not equal intervals!)

We need to check if the rate of change is constant. Since $ x $ changes unevenly, we can compute $ \Delta y / \Delta x $ between consecutive points.

1. From $ x=21 $ to $ x=18 $:
$ \Delta x = -3 $, $ \Delta y = -5 - (-3) = -2 $
$ \text{Rate} = \frac{-2}{-3} = \frac{2}{3} $

2. From $ x=18 $ to $ x=12 $:
$ \Delta x = -6 $, $ \Delta y = -6 - (-5) = -1 $
$ \text{Rate} = \frac{-1}{-6} = \frac{1}{6} $

3. From $ x=12 $ to $ x=-12 $:
$ \Delta x = -24 $, $ \Delta y = -14 - (-6) = -8 $
$ \text{Rate} = \frac{-8}{-24} = \frac{1}{3} $

Rates: $ \frac{2}{3}, \frac{1}{6}, \frac{1}{3} $ → not constant

✘ Not a constant rate of change

➡️ Nonlinear

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✔ Final Answers:

| Table | Answer |
|-------|------------|
| 1 | Linear |
| 2 | Nonlinear |
| 3 | Nonlinear |
| 4 | Nonlinear |
| 5 | Linear |
| 6 | Nonlinear |
| 7 | Linear |
| 8 | Linear |
| 9 | Nonlinear |

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📌 Summary:

- Linear: Tables 1, 5, 7, 8
- Nonlinear: Tables 2, 3, 4, 6, 9

You should circle:
- Linear for tables: 1, 5, 7, 8
- Nonlinear for tables: 2, 3, 4, 6, 9

Let me know if you'd like this formatted as a printable answer sheet!