Converting Quadratic Equations Worksheet: Standard to Vertex ... - Free Printable

Let’s solve each problem step by step.

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Part 1: Convert from vertex form to standard form

Vertex form is:
> y = a(x – h)² + k

Standard form is:
> y = ax² + bx + c

We expand the squared term and simplify.

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Problem 1: y = -(x – 1)² – 1

Step 1: Expand (x – 1)² → x² – 2x + 1
Step 2: Multiply by -1 → -x² + 2x – 1
Step 3: Subtract 1 → -x² + 2x – 1 – 1 = -x² + 2x – 2

✔ Final Answer for #1: y = -x² + 2x – 2

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Problem 2: y = 2(x – 2)² – 3

Step 1: Expand (x – 2)² → x² – 4x + 4
Step 2: Multiply by 2 → 2x² – 8x + 8
Step 3: Subtract 3 → 2x² – 8x + 8 – 3 = 2x² – 8x + 5

✔ Final Answer for #2: y = 2x² – 8x + 5

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Problem 3: y = (x + 4)² + 4

Note: (x + 4)² = (x – (-4))², so it’s still vertex form.

Step 1: Expand (x + 4)² → x² + 8x + 16
Step 2: Add 4 → x² + 8x + 16 + 4 = x² + 8x + 20

✔ Final Answer for #3: y = x² + 8x + 20

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Part 2: Convert from standard form to vertex form

Standard form: y = ax² + bx + c
Vertex form: y = a(x – h)² + k

We complete the square.

Steps:
1. Group x terms: y = (ax² + bx) + c
2. Factor out ‘a’ if needed (here a=1 in all cases)
3. Take half of b, square it, add and subtract inside
4. Rewrite as perfect square trinomial
5. Simplify constants

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Problem 4: y = x² – 8x + 15

Step 1: Focus on x² – 8x
Half of -8 is -4 → square it: 16
Add and subtract 16:
y = (x² – 8x + 16 – 16) + 15
→ y = (x – 4)² – 16 + 15
→ y = (x – 4)² – 1

✔ Final Answer for #4: y = (x – 4)² – 1

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Problem 5: y = x² – 4x

Step 1: Half of -4 is -2 → square it: 4
Add and subtract 4:
y = (x² – 4x + 4 – 4)
→ y = (x – 2)² – 4

✔ Final Answer for #5: y = (x – 2)² – 4

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Problem 6: y = x² + 8x + 18

Step 1: Half of 8 is 4 → square it: 16
Add and subtract 16:
y = (x² + 8x + 16 – 16) + 18
→ y = (x + 4)² + 2

✔ Final Answer for #6: y = (x + 4)² + 2

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Problem 7: y = x² + 4x + 3

Step 1: Half of 4 is 2 → square it: 4
Add and subtract 4:
y = (x² + 4x + 4 – 4) + 3
→ y = (x + 2)² – 1

✔ Final Answer for #7: y = (x + 2)² – 1

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Problem 8: y = x² – 2x + 5

Step 1: Half of -2 is -1 → square it: 1
Add and subtract 1:
y = (x² – 2x + 1 – 1) + 5
→ y = (x – 1)² + 4

✔ Final Answer for #8: y = (x – 1)² + 4

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Problem 9: y = x² – 8x + 17

Step 1: Half of -8 is -4 → square it: 16
Add and subtract 16:
y = (x² – 8x + 16 – 16) + 17
→ y = (x – 4)² + 1

✔ Final Answer for #9: y = (x – 4)² + 1

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Final Answer:

1) y = -x² + 2x – 2
2) y = 2x² – 8x + 5
3) y = x² + 8x + 20
4) y = (x – 4)² – 1
5) y = (x – 2)² – 4
6) y = (x + 4)² + 2
7) y = (x + 2)² – 1
8) y = (x – 1)² + 4
9) y = (x – 4)² + 1