Geometry worksheet for identifying the center and radius of circles from equations and graphing them.

Problem Overview:

The task involves identifying the center and radius of each circle given its equation and then sketching the graph of the circle. The equations are provided in standard form, which is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Where:
- \((h, k)\) is the center of the circle.
- \(r\) is the radius of the circle.

We will solve each equation step by step and then sketch the corresponding graphs.

---

1. \( x^2 + y^2 = 18 \)

#### Step 1: Identify the center and radius.
The given equation is already in the standard form:
\[
(x - 0)^2 + (y - 0)^2 = 18
\]
Here:
- \( h = 0 \)
- \( k = 0 \)
- \( r^2 = 18 \), so \( r = \sqrt{18} = 3\sqrt{2} \)

#### Center and Radius:
- Center: \((0, 0)\)
- Radius: \(3\sqrt{2}\)

#### Sketch:
- The circle is centered at the origin \((0, 0)\).
- The radius is \(3\sqrt{2} \approx 4.24\).

---

2. \( x^2 + y^2 = 5 \)

#### Step 1: Identify the center and radius.
The given equation is already in the standard form:
\[
(x - 0)^2 + (y - 0)^2 = 5
\]
Here:
- \( h = 0 \)
- \( k = 0 \)
- \( r^2 = 5 \), so \( r = \sqrt{5} \)

#### Center and Radius:
- Center: \((0, 0)\)
- Radius: \(\sqrt{5}\)

#### Sketch:
- The circle is centered at the origin \((0, 0)\).
- The radius is \(\sqrt{5} \approx 2.24\).

---

3. \( (x - 1)^2 + (y - \sqrt{11})^2 = 9 \)

#### Step 1: Identify the center and radius.
The given equation is in the standard form:
\[
(x - 1)^2 + (y - \sqrt{11})^2 = 9
\]
Here:
- \( h = 1 \)
- \( k = \sqrt{11} \)
- \( r^2 = 9 \), so \( r = 3 \)

#### Center and Radius:
- Center: \((1, \sqrt{11})\)
- Radius: \(3\)

#### Sketch:
- The circle is centered at \((1, \sqrt{11})\).
- The radius is \(3\).

---

4. \( (x - 3)^2 + (y + 4)^2 = 2 \)

#### Step 1: Identify the center and radius.
The given equation is in the standard form:
\[
(x - 3)^2 + (y + 4)^2 = 2
\]
Here:
- \( h = 3 \)
- \( k = -4 \)
- \( r^2 = 2 \), so \( r = \sqrt{2} \)

#### Center and Radius:
- Center: \((3, -4)\)
- Radius: \(\sqrt{2}\)

#### Sketch:
- The circle is centered at \((3, -4)\).
- The radius is \(\sqrt{2} \approx 1.41\).

---

5. \( (x + 1)^2 + (y - 2)^2 = 3 \)

#### Step 1: Identify the center and radius.
The given equation is in the standard form:
\[
(x + 1)^2 + (y - 2)^2 = 3
\]
Here:
- \( h = -1 \)
- \( k = 2 \)
- \( r^2 = 3 \), so \( r = \sqrt{3} \)

#### Center and Radius:
- Center: \((-1, 2)\)
- Radius: \(\sqrt{3}\)

#### Sketch:
- The circle is centered at \((-1, 2)\).
- The radius is \(\sqrt{3} \approx 1.73\).

---

6. \( (x + 1)^2 + (y + 1)^2 = 24 \)

#### Step 1: Identify the center and radius.
The given equation is in the standard form:
\[
(x + 1)^2 + (y + 1)^2 = 24
\]
Here:
- \( h = -1 \)
- \( k = -1 \)
- \( r^2 = 24 \), so \( r = \sqrt{24} = 2\sqrt{6} \)

#### Center and Radius:
- Center: \((-1, -1)\)
- Radius: \(2\sqrt{6}\)

#### Sketch:
- The circle is centered at \((-1, -1)\).
- The radius is \(2\sqrt{6} \approx 4.90\).

---

Final Answers:

1. Center: \((0, 0)\), Radius: \(3\sqrt{2}\)
2. Center: \((0, 0)\), Radius: \(\sqrt{5}\)
3. Center: \((1, \sqrt{11})\), Radius: \(3\)
4. Center: \((3, -4)\), Radius: \(\sqrt{2}\)
5. Center: \((-1, 2)\), Radius: \(\sqrt{3}\)
6. Center: \((-1, -1)\), Radius: \(2\sqrt{6}\)

\[
\boxed{
\begin{aligned}
1. & \text{ Center: } (0, 0), \text{ Radius: } 3\sqrt{2} \\
2. & \text{ Center: } (0, 0), \text{ Radius: } \sqrt{5} \\
3. & \text{ Center: } (1, \sqrt{11}), \text{ Radius: } 3 \\
4. & \text{ Center: } (3, -4), \text{ Radius: } \sqrt{2} \\
5. & \text{ Center: } (-1, 2), \text{ Radius: } \sqrt{3} \\
6. & \text{ Center: } (-1, -1), \text{ Radius: } 2\sqrt{6}
\end{aligned}
}
\]