Graphing and Properties of Quadratics worksheet with equations and coordinate grids.

Let’s solve each problem step by step. We’re identifying the center and radius of circles from their equations, and then graphing two of them.

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Part 1: Identify center and radius for each equation

The standard form of a circle’s equation is:

> (x – h)² + (y – k)² = r²

Where:
- (h, k) is the center
- r is the radius

We’ll rewrite each given equation into this form if needed.

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#### Problem 1:
x² + y² = 49

This is already in standard form:
(x – 0)² + (y – 0)² = 7²
→ Center: (0, 0)
→ Radius: 7

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#### Problem 2:
x² + y² = 500

Same as above:
(x – 0)² + (y – 0)² = (√500)²
But we simplify √500:
√500 = √(100 × 5) = 10√5
→ Center: (0, 0)
→ Radius: 10√5 (or approx 22.36, but exact form is better)

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#### Problem 3:
(x + 8)² + (y – 2)² = 64

Rewrite to match standard form:
(x – (-8))² + (y – 2)² = 8²
→ Center: (-8, 2)
→ Radius: 8

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#### Problem 4:
(x + 7)² + (y + 3)² = 49

Rewrite:
(x – (-7))² + (y – (-3))² = 7²
→ Center: (-7, -3)
→ Radius: 7

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#### Problem 5:
(x + 1)² + (y + 9)² = 121

Rewrite:
(x – (-1))² + (y – (-9))² = 11²
→ Center: (-1, -9)
→ Radius: 11

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#### Problem 6:
(x + 4)² + (y – 8)² = 36

Rewrite:
(x – (-4))² + (y – 8)² = 6²
→ Center: (-4, 8)
→ Radius: 6

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#### Problem 7:
x² + (y + 1)² = 144

Rewrite:
(x – 0)² + (y – (-1))² = 12²
→ Center: (0, -1)
→ Radius: 12

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#### Problem 8:
(x – 5)² + y² = 25

Rewrite:
(x – 5)² + (y – 0)² = 5²
→ Center: (5, 0)
→ Radius: 5

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#### Problem 9:
9x² + 9y² + 18x + 36y + 36 = 0

First, divide entire equation by 9 to simplify:

→ x² + y² + 2x + 4y + 4 = 0

Now group x and y terms:

→ (x² + 2x) + (y² + 4y) = -4

Complete the square:

For x: x² + 2x → add (2/2)² = 1
For y: y² + 4y → add (4/2)² = 4

Add 1 + 4 = 5 to both sides:

→ (x² + 2x + 1) + (y² + 4y + 4) = -4 + 5
→ (x + 1)² + (y + 2)² = 1

So:
→ Center: (-1, -2)
→ Radius: 1

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#### Problem 10:
x² + y² + 10x + 16y + 80 = 0

Group terms:

→ (x² + 10x) + (y² + 16y) = -80

Complete the square:

x: (10/2)² = 25
y: (16/2)² = 64
Add 25 + 64 = 89 to both sides:

→ (x² + 10x + 25) + (y² + 16y + 64) = -80 + 89
→ (x + 5)² + (y + 8)² = 9

So:
→ Center: (-5, -8)
→ Radius: 3

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#### Problem 11:
4x² + 4y² + 8x – 24y – 32 = 0

Divide entire equation by 4:

→ x² + y² + 2x – 6y – 8 = 0

Group:

→ (x² + 2x) + (y² – 6y) = 8

Complete the square:

x: (2/2)² = 1
y: (-6/2)² = 9
Add 1 + 9 = 10 to both sides:

→ (x² + 2x + 1) + (y² – 6y + 9) = 8 + 10
→ (x + 1)² + (y – 3)² = 18

Radius = √18 = 3√2

→ Center: (-1, 3)
→ Radius: 3√2

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#### Problem 12:
4x² + 4y² – 16x – 8y – 16 = 0

Divide by 4:

→ x² + y² – 4x – 2y – 4 = 0

Group:

→ (x² – 4x) + (y² – 2y) = 4

Complete the square:

x: (-4/2)² = 4
y: (-2/2)² = 1
Add 4 + 1 = 5 to both sides:

→ (x² – 4x + 4) + (y² – 2y + 1) = 4 + 5
→ (x – 2)² + (y – 1)² = 9

→ Center: (2, 1)
→ Radius: 3

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Part 2: Graph the circles

#### Problem 13:
(x + 2)² + (y – 3)² = 4

Center: (-2, 3)
Radius: 2

To graph:
- Plot point at (-2, 3)
- From there, go 2 units up, down, left, right — those are points on the circle
- Draw smooth circle through those points

#### Problem 14:
x² + (y + 1)² = 4

Center: (0, -1)
Radius: 2

To graph:
- Plot point at (0, -1)
- Go 2 units in all directions: (2,-1), (-2,-1), (0,1), (0,-3)
- Draw circle through those

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Final Answer:

Problem 1: Center (0, 0), Radius 7
Problem 2: Center (0, 0), Radius 10√5
Problem 3: Center (-8, 2), Radius 8
Problem 4: Center (-7, -3), Radius 7
Problem 5: Center (-1, -9), Radius 11
Problem 6: Center (-4, 8), Radius 6
Problem 7: Center (0, -1), Radius 12
Problem 8: Center (5, 0), Radius 5
Problem 9: Center (-1, -2), Radius 1
Problem 10: Center (-5, -8), Radius 3
Problem 11: Center (-1, 3), Radius 3√2
Problem 12: Center (2, 1), Radius 3

Graphs (Problems 13 & 14):
- Problem 13: Circle centered at (-2, 3) with radius 2
- Problem 14: Circle centered at (0, -1) with radius 2