Math worksheet for completing function tables with equations and input/output values.

It looks like you've uploaded a completed function table worksheet with various linear equations and their corresponding input-output tables. The task was to complete the function table for each equation, meaning to calculate the output value (y) for each given input (x) using the provided equation.

Let me go through this step-by-step, verify the completed tables, and explain how each one works.

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🔍 Understanding the Task

For each equation of the form $ y = \text{expression involving } x $, we substitute the given values of $ x $ into the equation to compute the corresponding $ y $-values.

We'll check each table one by one.

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✔ 1. $ y = x - 9 $

| x | y |
|---|---|
| 7 | -2 |
| 3 | -6 |
| -8 | -17 |
| -5 | -14 |
| 5 | -4 |

✔ All correct:
- $ 7 - 9 = -2 $
- $ 3 - 9 = -6 $
- $ -8 - 9 = -17 $
- $ -5 - 9 = -14 $
- $ 5 - 9 = -4 $

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✔ 2. $ y = -9x + 1 $

| x | y |
|---|---|
| -5 | 46 |
| 0 | 1 |
| 5 | -44 |
| -8 | 73 |
| -1 | 10 |

Check:
- $ -9(-5) + 1 = 45 + 1 = 46 $
- $ -9(0) + 1 = 1 $
- $ -9(5) + 1 = -45 + 1 = -44 $
- $ -9(-8) + 1 = 72 + 1 = 73 $
- $ -9(-1) + 1 = 9 + 1 = 10 $

✔ All correct.

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✔ 3. $ y = 4x $

| x | y |
|---|---|
| -8 | -32 |
| -2 | -8 |
| -1 | -4 |
| -4 | -16 |
| -6 | -24 |

Check:
- $ 4(-8) = -32 $
- $ 4(-2) = -8 $
- $ 4(-1) = -4 $
- $ 4(-4) = -16 $
- $ 4(-6) = -24 $

✔ All correct.

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✔ 4. $ y = \frac{1}{4}x + 7 $

| x | y |
|---|---|
| -6 | 7.75 |
| 4 | 6.5 |
| 0 | 7 |
| 7 | 6.13 |
| 8 | 6 |

Wait — let's check:

- $ \frac{1}{4}(-6) + 7 = -1.5 + 7 = 5.5 $ → but table says 7.75 ✘

Uh-oh! Let's double-check.

Actually:
- $ \frac{1}{4} \times (-6) = -1.5 $, so $ y = -1.5 + 7 = 5.5 $, not 7.75 → Error

But wait — maybe it’s $ y = \frac{1}{4}x + 7 $, so let's recompute:

| x | Calculation | y |
|---|-----------|---|
| -6 | $ 0.25 × (-6) + 7 = -1.5 + 7 = 5.5 $ | Should be 5.5, but table says 7.75 ✘
| 4 | $ 0.25×4 + 7 = 1 + 7 = 8 $ → table says 6.5 ✘
| 0 | $ 0 + 7 = 7 $ ✔
| 7 | $ 0.25×7 + 7 = 1.75 + 7 = 8.75 $ → table says 6.13 ✘
| 8 | $ 0.25×8 + 7 = 2 + 7 = 9 $ → table says 6 ✘

So many errors here. This table is incorrect.

Let me recalculate correctly:

| x | y = 0.25x + 7 |
|---|----------------|
| -6 | $ -1.5 + 7 = 5.5 $ |
| 4 | $ 1 + 7 = 8 $ |
| 0 | $ 0 + 7 = 7 $ |
| 7 | $ 1.75 + 7 = 8.75 $ |
| 8 | $ 2 + 7 = 9 $ |

✘ So this table has errors.

But wait — perhaps the equation is different?

Looking at the table:
- At x = 0, y = 7 → matches $ y = \frac{1}{4}x + 7 $
- But others don’t match.

Alternatively, maybe it's $ y = -\frac{1}{4}x + 7 $? Try:

- x = -6: $ -0.25×(-6) + 7 = 1.5 + 7 = 8.5 $ ≠ 7.75
- x = 4: $ -0.25×4 + 7 = -1 + 7 = 6 $ → close to 6.5? No.

Hmm.

Wait — maybe the equation is $ y = \frac{1}{4}x + 7 $, but the table is wrong.

But look at x = -6 → y = 7.75

Try solving: $ 7.75 = \frac{1}{4}(-6) + 7 = -1.5 + 7 = 5.5 $ → no.

Try $ y = -\frac{1}{4}x + 7 $:

- x = -6: $ -0.25×(-6) + 7 = 1.5 + 7 = 8.5 $
- x = 4: $ -1 + 7 = 6 $
- x = 0: 7
- x = 7: $ -1.75 + 7 = 5.25 $
- x = 8: $ -2 + 7 = 5 $

Still doesn't match.

Wait — maybe the equation is $ y = \frac{1}{4}x + 7 $, but the values are misprinted.

But looking at the next table: $ y = \frac{1}{4}x + 7 $ again?

Let’s move on and come back.

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✔ 5. $ y = x - 2 $

| x | y |
|---|---|
| -1 | -3 |
| 2 | 0 |
| -5 | -7 |
| -2 | -4 |
| 0 | -2 |

Check:
- $ -1 - 2 = -3 $ ✔
- $ 2 - 2 = 0 $ ✔
- $ -5 - 2 = -7 $ ✔
- $ -2 - 2 = -4 $ ✔
- $ 0 - 2 = -2 $ ✔

✔ All correct.

---

✔ 6. $ y = 6x $

| x | y |
|---|---|
| 8 | 48 |
| -4 | -24 |
| 0 | 0 |
| 4 | 24 |
| -8 | -48 |

All correct: $ 6×8=48 $, etc.

✔ Correct.

---

✔ 7. $ y = 6x - 5 $

| x | y |
|---|---|
| -1 | -12 |
| -6 | -42 |
| 4 | 19 |
| -8 | -54 |
| 1 | 1 |

Check:
- $ 6(-1) - 5 = -6 - 5 = -11 $ → but table says -12 ✘
- $ 6(-6) - 5 = -36 - 5 = -41 $ → table says -42 ✘
- $ 6(4) - 5 = 24 - 5 = 19 $ ✔
- $ 6(-8) - 5 = -48 - 5 = -53 $ → table says -54 ✘
- $ 6(1) - 5 = 6 - 5 = 1 $ ✔

So only x = 4 and x = 1 are correct. Others are off by 1.

So likely typo or calculation error.

Correct values:
- x = -1 → y = -11
- x = -6 → y = -41
- x = -8 → y = -53

Table shows -12, -42, -54 → all off by 1 → possibly mistake in subtraction.

So this table has errors.

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✔ 8. $ y = \frac{1}{4}x + 7 $

| x | y |
|---|---|
| -3 | 6.4 |
| 0 | 7 |
| 1 | 7.2 |
| 2 | 7.4 |
| 3 | 7.6 |

Now check:
- $ \frac{1}{4}(-3) + 7 = -0.75 + 7 = 6.25 $ → table says 6.4 ✘
- $ 0 + 7 = 7 $ ✔
- $ 0.25 + 7 = 7.25 $ → table says 7.2 ✘
- $ 0.5 + 7 = 7.5 $ → table says 7.4 ✘
- $ 0.75 + 7 = 7.75 $ → table says 7.6 ✘

So again, incorrect.

But if it were $ y = \frac{1}{4}x + 7 $, then:
- x = -3 → 6.25
- x = 1 → 7.25
- x = 2 → 7.5
- x = 3 → 7.75

But table says:
- -3 → 6.4 → maybe it's $ y = \frac{1}{4}x + 7 $ but rounded? No, 6.4 ≠ 6.25

Wait — what if the equation is $ y = \frac{1}{4}x + 7 $, but the table is wrong?

Alternatively, maybe it's $ y = \frac{1}{4}x + 7 $, and values are approximated?

No — 6.4 is not close to 6.25.

Wait — could it be $ y = \frac{1}{4}x + 7 $, but with decimal rounding?

No — differences are significant.

Wait — maybe it's $ y = \frac{1}{4}x + 7 $, but the values are corrected manually?

Alternatively, maybe there's a typo in the equation.

But notice: in both #4 and #8, the same equation appears, but different values.

Let’s check #4 again.

Wait — in #4: $ y = \frac{1}{4}x + 7 $

Given:
- x = -6 → y = 7.75 → $ 0.25×(-6) = -1.5 $, $ -1.5 + 7 = 5.5 $ → not 7.75

But 7.75 = 7 + 0.75 = 7 + 3/4

So maybe $ y = \frac{1}{4}x + 7 $, but x = -6 → should be 5.5

Unless it's $ y = -\frac{1}{4}x + 7 $

Try that:
- x = -6 → $ -0.25×(-6) = 1.5 $, $ 1.5 + 7 = 8.5 $ → not 7.75

Or $ y = \frac{1}{4}x + 7 $, but x = -6 → 5.5

No match.

Wait — maybe the equation is $ y = \frac{1}{4}x + 7 $, and the table is correct, but I'm missing something.

Wait — perhaps the equation is wrong.

But looking at the pattern:

In table #8:
- x = 0 → y = 7 → so constant term is 7
- x = 1 → y = 7.2 → increase of 0.2
- x = 2 → 7.4 → +0.2
- x = 3 → 7.6 → +0.2
- x = -3 → 6.4 → from 7 down by 0.6 → 0.2 per unit

So slope = 0.2 = $ \frac{1}{5} $

So maybe equation is $ y = \frac{1}{5}x + 7 $

Check:
- x = -3 → $ -0.2 + 7 = 6.8 $ → but table says 6.4 ✘
- x = 1 → $ 0.2 + 7 = 7.2 $ ✔
- x = 2 → $ 0.4 + 7 = 7.4 $ ✔
- x = 3 → $ 0.6 + 7 = 7.6 $ ✔
- x = -3 → $ -0.6 + 7 = 6.4 $ ✔

Ah! So the equation should be $ y = \frac{1}{5}x + 7 $, not $ \frac{1}{4}x $

Similarly, in table #4:
- x = -6 → y = 7.75
- x = 4 → y = 6.5
- x = 0 → y = 7
- x = 7 → y = 6.13
- x = 8 → y = 6

Check:
- x = 0 → y = 7 → good
- x = 4 → y = 6.5 → so change: from 7 to 6.5 over +4 → Δy = -0.5 → slope = -0.5 / 4 = -0.125 = -1/8

Try $ y = -\frac{1}{8}x + 7 $

- x = -6 → $ -(-6)/8 + 7 = 0.75 + 7 = 7.75 $ ✔
- x = 4 → $ -0.5 + 7 = 6.5 $ ✔
- x = 0 → 7 ✔
- x = 7 → $ -7/8 + 7 = -0.875 + 7 = 6.125 ≈ 6.13 $ ✔
- x = 8 → $ -1 + 7 = 6 $ ✔

So this table uses $ y = -\frac{1}{8}x + 7 $, not $ \frac{1}{4}x + 7 $

So there are errors in the printed equations!

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🚨 Conclusion: There are inconsistencies between the equations and the tables.

Let me now correctly identify what the actual equations should be based on the tables.

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## ✔ Corrected Analysis

Let’s go through each table and determine the correct equation.

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1. $ y = x - 9 $ → ✔ Correct

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2. $ y = -9x + 1 $ → ✔ Correct

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3. $ y = 4x $ → ✔ Correct

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4. Table says $ y = \frac{1}{4}x + 7 $, but data suggests $ y = -\frac{1}{8}x + 7 $

From earlier:
- x = -6 → y = 7.75 → $ y = -\frac{1}{8}(-6) + 7 = 0.75 + 7 = 7.75 $
- x = 4 → $ -0.5 + 7 = 6.5 $
- x = 0 → 7
- x = 7 → $ -0.875 + 7 = 6.125 $
- x = 8 → $ -1 + 7 = 6 $

So correct equation: $ y = -\frac{1}{8}x + 7 $

But printed as $ y = \frac{1}{4}x + 7 $ → ✘ Error

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5. $ y = x - 2 $ → ✔ Correct

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6. $ y = 6x $ → ✔ Correct

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7. $ y = 6x - 5 $

But:
- x = -1 → y = -11 (not -12)
- x = -6 → y = -41 (not -42)
- x = -8 → y = -53 (not -54)

So correct values:
| x | y |
|---|---|
| -1 | -11 |
| -6 | -41 |
| 4 | 19 |
| -8 | -53 |
| 1 | 1 |

So table has off-by-one errors — likely typo.

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8. $ y = \frac{1}{4}x + 7 $ → but data suggests $ y = \frac{1}{5}x + 7 $

From:
- x = -3 → y = 6.4 → $ 6.4 = 7 + (-0.6) = 7 - 0.6 $ → $ \frac{1}{5}(-3) = -0.6 $
- x = 1 → 7.2 → $ 7 + 0.2 $
- x = 2 → 7.4 → $ 7 + 0.4 $
- x = 3 → 7.6 → $ 7 + 0.6 $

So slope = $ \frac{1}{5} $

Thus, correct equation: $ y = \frac{1}{5}x + 7 $

But printed as $ \frac{1}{4}x + 7 $ → ✘ Error

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9. $ y = 4x $

| x | y |
|---|---|
| -6 | -24 |
| -2 | -8 |
| 0 | 0 |
| -9 | -36 |
| 6 | 24 |

All correct: $ 4×(-6) = -24 $, etc.

✔ Correct

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10. $ y = \frac{1}{4}x + 3 $

| x | y |
|---|---|
| 6 | 4.5 |
| -1 | 2.75 |
| -2 | 2.5 |
| -5 | 1.75 |
| -8 | 1 |

Check:
- $ \frac{1}{4}(6) + 3 = 1.5 + 3 = 4.5 $ ✔
- $ \frac{1}{4}(-1) + 3 = -0.25 + 3 = 2.75 $ ✔
- $ \frac{1}{4}(-2) + 3 = -0.5 + 3 = 2.5 $ ✔
- $ \frac{1}{4}(-5) + 3 = -1.25 + 3 = 1.75 $ ✔
- $ \frac{1}{4}(-8) + 3 = -2 + 3 = 1 $ ✔

✔ Correct

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11. $ y = 2x - 7 $

| x | y |
|---|---|
| -9 | -25 |
| 6 | 5 |
| 8 | 9 |
| -2 | -11 |
| 7 | 7 |

Check:
- $ 2(-9) - 7 = -18 - 7 = -25 $ ✔
- $ 2(6) - 7 = 12 - 7 = 5 $ ✔
- $ 2(8) - 7 = 16 - 7 = 9 $ ✔
- $ 2(-2) - 7 = -4 - 7 = -11 $ ✔
- $ 2(7) - 7 = 14 - 7 = 7 $ ✔

✔ Correct

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12. $ y = x - 5 $

| x | y |
|---|---|
| 2 | -4 |
| 6 | 0 |
| -9 | -15 |
| -1 | -7 |
| 7 | 1 |

Check:
- $ 2 - 5 = -3 $ → but table says -4 ✘
- $ 6 - 5 = 1 $ → table says 0 ✘
- $ -9 - 5 = -14 $ → table says -15 ✘
- $ -1 - 5 = -6 $ → table says -7 ✘
- $ 7 - 5 = 2 $ → table says 1 ✘

All off by 1 → likely typo.

Correct values:
- x = 2 → y = -3
- x = 6 → y = 1
- x = -9 → y = -14
- x = -1 → y = -6
- x = 7 → y = 2

So table has systematic error — maybe it’s $ y = x - 6 $? Try:
- $ 2 - 6 = -4 $ ✔
- $ 6 - 6 = 0 $ ✔
- $ -9 - 6 = -15 $ ✔
- $ -1 - 6 = -7 $ ✔
- $ 7 - 6 = 1 $ ✔

Ah! So the equation should be $ y = x - 6 $, not $ y = x - 5 $

So another error in the printed equation

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## ✔ Final Summary

✔ Correct Tables:

- 1, 2, 3, 5, 6, 9, 10, 11 → Correct
- 4, 7, 8, 12 → Have errors in equations or values

✘ Errors Found:

| Table | Issue |
|------|-------|
| 4 | Equation printed as $ y = \frac{1}{4}x + 7 $, but data fits $ y = -\frac{1}{8}x + 7 $ |
| 7 | Values are off by 1; correct equation $ y = 6x - 5 $, but values should be: -11, -41, 19, -53, 1 |
| 8 | Equation printed as $ y = \frac{1}{4}x + 7 $, but data fits $ y = \frac{1}{5}x + 7 $ |
| 12 | Equation printed as $ y = x - 5 $, but data fits $ y = x - 6 $ |

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## ✔ How to Solve These Problems (General Method)

To complete a function table:
1. Identify the equation: $ y = f(x) $
2. For each x-value, plug into the equation.
3. Compute $ y = f(x) $
4. Fill in the table.

Example: $ y = 2x - 7 $, x = 6
→ $ y = 2(6) - 7 = 12 - 7 = 5 $

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## 📝 Recommendation

This worksheet has several printing or transcription errors. If used for learning, the student should be told to trust the data in the table and recheck the equation, or vice versa.

But since the task is to complete the table, and the tables are already filled, the student likely just needed to verify.

However, in this case, the tables are mostly correct except for the equations being mislabeled.

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✔ Final Answer:

The function tables are mostly completed correctly, but there are errors in the equations listed:

- Table 4: Should be $ y = -\frac{1}{8}x + 7 $
- Table 7: Values are off by 1 (e.g., x = -1 → y = -11, not -12)
- Table 8: Should be $ y = \frac{1}{5}x + 7 $
- Table 12: Should be $ y = x - 6 $

Otherwise, the computations are correct.

> ⚠️ Note: This worksheet may have been copied incorrectly or contains typos. Always verify the equation against the data when possible.

Let me know if you'd like a corrected version of the worksheet!