Math worksheet with input/output tables for practicing function rules.

Let's solve each input/output table step by step to find the rule (a mathematical relationship between input $ x $ and output $ y $) and then use it to fill in the missing values.

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🔹 Table 1

| Input (x) | Output (y) |
|----------|------------|
| 3 | 6 |
| 5 | 8 |
| 7 | 10 |
| 9 | ? |

Step 1: Look for a pattern

- $ 3 \to 6 $ → $ 3 + 3 = 6 $
- $ 5 \to 8 $ → $ 5 + 3 = 8 $
- $ 7 \to 10 $ → $ 7 + 3 = 10 $

✔ So, Rule: $ y = x + 3 $

For x = 9:
$ y = 9 + 3 = 12 $

✔️ Answer: Rule = $ y = x + 3 $, Missing output = 12

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🔹 Table 2

| Input (x) | Output (y) |
|----------|------------|
| 0 | 11 |
| 1 | 12 |
| 2 | 13 |
| 3 | ? |

- $ 0 \to 11 $
- $ 1 \to 12 $
- $ 2 \to 13 $

Each time $ x $ increases by 1, $ y $ increases by 1.

So: $ y = x + 11 $

Check:
- $ 0 + 11 = 11 $ ✔
- $ 1 + 11 = 12 $ ✔
- $ 2 + 11 = 13 $ ✔

For $ x = 3 $: $ y = 3 + 11 = 14 $

✔️ Answer: Rule = $ y = x + 11 $, Missing output = 14

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🔹 Table 3

| Input (x) | Output (y) |
|----------|------------|
| 6 | 18 |
| 8 | 20 |
| 10 | 22 |
| 12 | ? |

- $ 6 \to 18 $ → $ 6 \times 3 = 18 $
- $ 8 \to 20 $ → $ 8 \times 2.5 = 20 $? Not consistent.
Try: $ 18 - 6 = 12 $, $ 20 - 8 = 12 $, $ 22 - 10 = 12 $

Ah! $ y = x + 12 $

Check:
- $ 6 + 12 = 18 $ ✔
- $ 8 + 12 = 20 $ ✔
- $ 10 + 12 = 22 $ ✔

So, Rule: $ y = x + 12 $

For $ x = 12 $: $ y = 12 + 12 = 24 $

✔️ Answer: Rule = $ y = x + 12 $, Missing output = 24

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🔹 Table 4

| Input (x) | Output (y) |
|----------|------------|
| 10 | 19 |
| 20 | 29 |
| 30 | 39 |
| 40 | ? |

Look at differences:
- $ 10 \to 19 $: difference = 9
- $ 20 \to 29 $: difference = 9
- $ 30 \to 39 $: difference = 9

So, $ y = x + 9 $

For $ x = 40 $: $ y = 40 + 9 = 49 $

✔️ Answer: Rule = $ y = x + 9 $, Missing output = 49

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🔹 Table 5

| Input (x) | Output (y) |
|----------|------------|
| 10 | 16 |
| 11 | 17 |
| 12 | 18 |
| 13 | ? |

Clearly: $ y = x + 6 $

Check:
- $ 10 + 6 = 16 $ ✔
- $ 11 + 6 = 17 $ ✔
- $ 12 + 6 = 18 $ ✔

So, $ x = 13 $: $ y = 13 + 6 = 19 $

✔️ Answer: Rule = $ y = x + 6 $, Missing output = 19

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🔹 Table 6

| Input (x) | Output (y) |
|----------|------------|
| 2 | 4 |
| 4 | 6 |
| 6 | 8 |
| 8 | ? |

Pattern:
- $ 2 \to 4 $ → $ 2 \times 2 = 4 $
- $ 4 \to 6 $ → $ 4 \times 1.5 = 6 $? No.
But: $ 4 - 2 = 2 $, $ 6 - 4 = 2 $, $ 8 - 6 = 2 $

So $ y = x + 2 $

Check:
- $ 2 + 2 = 4 $ ✔
- $ 4 + 2 = 6 $ ✔
- $ 6 + 2 = 8 $ ✔

So $ x = 8 $: $ y = 8 + 2 = 10 $

✔️ Answer: Rule = $ y = x + 2 $, Missing output = 10

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🔹 Table 7

| Input (x) | Output (y) |
|----------|------------|
| 4 | 9 |
| 8 | 13 |
| 12 | 17 |
| 16 | ? |

Differences:
- $ 4 \to 9 $: $ 9 - 4 = 5 $
- $ 8 \to 13 $: $ 13 - 8 = 5 $
- $ 12 \to 17 $: $ 17 - 12 = 5 $

So, $ y = x + 5 $

For $ x = 16 $: $ y = 16 + 5 = 21 $

✔️ Answer: Rule = $ y = x + 5 $, Missing output = 21

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🔹 Table 8

| Input (x) | Output (y) |
|----------|------------|
| 100 | 101 |
| 150 | 151 |
| 200 | 201 |
| 250 | ? |

Pattern:
- $ 100 \to 101 $: $ +1 $
- $ 150 \to 151 $: $ +1 $
- $ 200 \to 201 $: $ +1 $

So, $ y = x + 1 $

For $ x = 250 $: $ y = 250 + 1 = 251 $

✔️ Answer: Rule = $ y = x + 1 $, Missing output = 251

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✔ Final Answers:

| Table | Rule | Missing Output |
|------|------|----------------|
| 1 | $ y = x + 3 $ | 12 |
| 2 | $ y = x + 11 $ | 14 |
| 3 | $ y = x + 12 $ | 24 |
| 4 | $ y = x + 9 $ | 49 |
| 5 | $ y = x + 6 $ | 19 |
| 6 | $ y = x + 2 $ | 10 |
| 7 | $ y = x + 5 $ | 21 |
| 8 | $ y = x + 1 $ | 251 |

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📝 Summary of Rules:

All these tables follow linear rules of the form $ y = x + c $, where $ c $ is a constant.

You can find the rule by:
1. Checking if the difference between output and input is constant.
2. If so, $ y = x + d $, where $ d $ is that constant difference.

Let me know if you'd like this formatted as a completed worksheet!