Vertex Form To Standard Form Worksheet Answer Key - Fill and Sign ... - Free Printable

You’ve uploaded a worksheet titled “Advanced Math — Worksheet — Vertex Form to Standard Form”, which asks students to convert quadratic equations from vertex form (y = a(x – h)² + k) to standard form (y = ax² + bx + c).

The example provided shows:

> y = -2(x + 5)² - 3
> → Expand: y = -2(x² + 10x + 25) - 3
> → Distribute: y = -2x² - 20x - 50 - 3
> → Combine like terms: y = -2x² - 20x - 53

This is the correct process:
1. Expand the squared binomial using FOIL or the formula (a+b)² = a² + 2ab + b².
2. Distribute the coefficient “a” across all terms inside the parentheses.
3. Combine constant terms.

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Let’s solve all 12 problems step by step.

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Problem 1: y = -6(x + 4)² - 1

Step 1: Expand (x + 4)² = x² + 8x + 16
Step 2: Multiply by -6:
→ -6(x² + 8x + 16) = -6x² - 48x - 96
Step 3: Subtract 1:
→ y = -6x² - 48x - 96 - 1 = -6x² - 48x - 97

✔ Answer: y = -6x² - 48x - 97

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Problem 2: y = ½(x + 4)² + 6

Step 1: (x + 4)² = x² + 8x + 16
Step 2: Multiply by ½:
→ ½(x² + 8x + 16) = ½x² + 4x + 8
Step 3: Add 6:
→ y = ½x² + 4x + 8 + 6 = ½x² + 4x + 14

✔ Answer: y = ½x² + 4x + 14

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Problem 3: y = -3(x - 1)² + 4

Step 1: (x - 1)² = x² - 2x + 1
Step 2: Multiply by -3:
→ -3(x² - 2x + 1) = -3x² + 6x - 3
Step 3: Add 4:
→ y = -3x² + 6x - 3 + 4 = -3x² + 6x + 1

✔ Answer: y = -3x² + 6x + 1

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Problem 4: y = -½(x + 6)² - 1

Step 1: (x + 6)² = x² + 12x + 36
Step 2: Multiply by -½:
→ -½(x² + 12x + 36) = -½x² - 6x - 18
Step 3: Subtract 1:
→ y = -½x² - 6x - 18 - 1 = -½x² - 6x - 19

✔ Answer: y = -½x² - 6x - 19

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Problem 5: y = 4(x + 2)² - 8

Step 1: (x + 2)² = x² + 4x + 4
Step 2: Multiply by 4:
→ 4(x² + 4x + 4) = 4x² + 16x + 16
Step 3: Subtract 8:
→ y = 4x² + 16x + 16 - 8 = 4x² + 16x + 8

✔ Answer: y = 4x² + 16x + 8

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Problem 6: y = ⅔(x - 9)² - 2

Step 1: (x - 9)² = x² - 18x + 81
Step 2: Multiply by ⅔:
→ ⅔(x² - 18x + 81) = ⅔x² - 12x + 54
Step 3: Subtract 2:
→ y = ⅔x² - 12x + 54 - 2 = ⅔x² - 12x + 52

✔ Answer: y = ⅔x² - 12x + 52

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Problem 7: y = -(x - 2)² + 7

Step 1: (x - 2)² = x² - 4x + 4
Step 2: Multiply by -1:
→ -(x² - 4x + 4) = -x² + 4x - 4
Step 3: Add 7:
→ y = -x² + 4x - 4 + 7 = -x² + 4x + 3

✔ Answer: y = -x² + 4x + 3

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Problem 8: y = (x + 5)² - 2

Step 1: (x + 5)² = x² + 10x + 25
Step 2: No coefficient to distribute.
Step 3: Subtract 2:
→ y = x² + 10x + 25 - 2 = x² + 10x + 23

✔ Answer: y = x² + 10x + 23

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Problem 9: y = 18(x - ½)² + 5

Step 1: (x - ½)² = x² - x + ¼
Step 2: Multiply by 18:
→ 18(x² - x + ¼) = 18x² - 18x + 4.5
Step 3: Add 5:
→ y = 18x² - 18x + 4.5 + 5 = 18x² - 18x + 9.5

*(You can write 9.5 as 19/2 if preferred)*
✔ Answer: y = 18x² - 18x + 9.5 or y = 18x² - 18x + 19/2

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Problem 10: y = -2(x + ¼)²

Step 1: (x + ¼)² = x² + ½x + 1/16
Step 2: Multiply by -2:
→ -2(x² + ½x + 1/16) = -2x² - x - 1/8
Step 3: No constant to add/subtract.
→ y = -2x² - x - 1/8

✔ Answer: y = -2x² - x - 1/8

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Problem 11: y = 15(x - 2)² + 16

Step 1: (x - 2)² = x² - 4x + 4
Step 2: Multiply by 15:
→ 15(x² - 4x + 4) = 15x² - 60x + 60
Step 3: Add 16:
→ y = 15x² - 60x + 60 + 16 = 15x² - 60x + 76

✔ Answer: y = 15x² - 60x + 76

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Problem 12: y = 3(x + 8)² + 10

Step 1: (x + 8)² = x² + 16x + 64
Step 2: Multiply by 3:
→ 3(x² + 16x + 64) = 3x² + 48x + 192
Step 3: Add 10:
→ y = 3x² + 48x + 192 + 10 = 3x² + 48x + 202

✔ Answer: y = 3x² + 48x + 202

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## ✔ Final Answers Summary:

1. y = -6x² - 48x - 97
2. y = ½x² + 4x + 14
3. y = -3x² + 6x + 1
4. y = -½x² - 6x - 19
5. y = 4x² + 16x + 8
6. y = ⅔x² - 12x + 52
7. y = -x² + 4x + 3
8. y = x² + 10x + 23
9. y = 18x² - 18x + 9.5 (or 19/2)
10. y = -2x² - x - 1/8
11. y = 15x² - 60x + 76
12. y = 3x² + 48x + 202

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## 📝 Key Tips for Students:
- Always expand the binomial square first: (x ± h)² = x² ± 2hx + h²
- Distribute the leading coefficient carefully — especially with fractions or negatives!
- Combine constants at the end.
- Double-check signs when distributing negative coefficients.

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