Standard to vertex form: Fill out & sign online | DocHub - Free Printable

Let's solve each problem step by step. The goal is to write each quadratic equation in standard form:
$$
y = ax^2 + bx + c
$$
and then identify the values of $ a $, $ b $, and $ c $.

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Sample #1:

Given: $ y = -2x + x^2 - 8 $

Step 1: Rearrange terms in descending powers of $ x $:
$$
y = x^2 - 2x - 8
$$

So:
- $ a = 1 $
- $ b = -2 $
- $ c = -8 $

✔ Correct as shown.

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Sample #2:

Given: $ y = -25 + x^2 $

Rearranged:
$$
y = x^2 - 25
$$

So:
- $ a = 1 $
- $ b = 0 $ (no $ x $-term)
- $ c = -25 $

✔ Correct as shown.

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Now let’s solve the problems 1–10:

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1. $ y = x^2 + 3x + 11 $

Already in standard form.

- $ a = 1 $
- $ b = 3 $
- $ c = 11 $

✔ Answer: $ y = x^2 + 3x + 11 $, $ a=1, b=3, c=11 $

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2. $ y = x^2 - 7x - 11 $

Already in standard form.

- $ a = 1 $
- $ b = -7 $
- $ c = -11 $

✔ Answer: $ y = x^2 - 7x - 11 $, $ a=1, b=-7, c=-11 $

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3. $ y = 4x + x^2 - 9 $

Rearrange terms:
$$
y = x^2 + 4x - 9
$$

- $ a = 1 $
- $ b = 4 $
- $ c = -9 $

✔ Answer: $ y = x^2 + 4x - 9 $, $ a=1, b=4, c=-9 $

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4. $ y = 16 - x + 3x^2 $

Rearrange:
$$
y = 3x^2 - x + 16
$$

- $ a = 3 $
- $ b = -1 $
- $ c = 16 $

✔ Answer: $ y = 3x^2 - x + 16 $, $ a=3, b=-1, c=16 $

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5. $ y = x^2 - 9 $

Already in standard form.

- $ a = 1 $
- $ b = 0 $
- $ c = -9 $

✔ Answer: $ y = x^2 - 9 $, $ a=1, b=0, c=-9 $

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6. $ y = 2x^2 + 5x $

No constant term, so $ c = 0 $

- $ a = 2 $
- $ b = 5 $
- $ c = 0 $

✔ Answer: $ y = 2x^2 + 5x $, $ a=2, b=5, c=0 $

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7. $ y = -3 - 4x^2 $

Rewrite in standard form:
$$
y = -4x^2 - 3
$$

- $ a = -4 $
- $ b = 0 $
- $ c = -3 $

✔ Answer: $ y = -4x^2 - 3 $, $ a=-4, b=0, c=-3 $

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8. $ y = 6x^2 $

Only $ x^2 $ term.

- $ a = 6 $
- $ b = 0 $
- $ c = 0 $

✔ Answer: $ y = 6x^2 $, $ a=6, b=0, c=0 $

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9. $ y = -x^2 - 8 + 6x $

Rearrange:
$$
y = -x^2 + 6x - 8
$$

- $ a = -1 $
- $ b = 6 $
- $ c = -8 $

✔ Answer: $ y = -x^2 + 6x - 8 $, $ a=-1, b=6, c=-8 $

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10. $ y = x - x^2 $

Rearrange:
$$
y = -x^2 + x
$$

- $ a = -1 $
- $ b = 1 $
- $ c = 0 $

✔ Answer: $ y = -x^2 + x $, $ a=-1, b=1, c=0 $

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✔ Final Answers Summary:

| Problem | Standard Form | $ a $ | $ b $ | $ c $ |
|--------|-----------------------------|--------|--------|--------|
| 1 | $ y = x^2 + 3x + 11 $ | 1 | 3 | 11 |
| 2 | $ y = x^2 - 7x - 11 $ | 1 | -7 | -11 |
| 3 | $ y = x^2 + 4x - 9 $ | 1 | 4 | -9 |
| 4 | $ y = 3x^2 - x + 16 $ | 3 | -1 | 16 |
| 5 | $ y = x^2 - 9 $ | 1 | 0 | -9 |
| 6 | $ y = 2x^2 + 5x $ | 2 | 5 | 0 |
| 7 | $ y = -4x^2 - 3 $ | -4 | 0 | -3 |
| 8 | $ y = 6x^2 $ | 6 | 0 | 0 |
| 9 | $ y = -x^2 + 6x - 8 $ | -1 | 6 | -8 |
| 10 | $ y = -x^2 + x $ | -1 | 1 | 0 |

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📌 Notes:

- Always write the quadratic in order: $ x^2 $, then $ x $, then constant.
- If a term is missing, its coefficient is zero.
- The standard form is essential for finding the vertex and axis of symmetry later.

Let me know if you'd like help finding the line of symmetry or vertex next!