Equations of Circles worksheet for identifying center and radius and sketching graphs.

Problem: Identify the center and radius of each circle, then sketch the graph.

The general equation of a circle is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where:
- \((h, k)\) is the center of the circle,
- \(r\) is the radius of the circle.

We will analyze each equation step by step.

---

1. \((x - 1)^2 + (y + 3)^2 = 4\)

#### Step 1: Compare with the standard form
The given equation is:
\[
(x - 1)^2 + (y + 3)^2 = 4
\]
This matches the standard form \((x - h)^2 + (y - k)^2 = r^2\), where:
- \(h = 1\),
- \(k = -3\),
- \(r^2 = 4\).

#### Step 2: Find the radius
Since \(r^2 = 4\), the radius \(r\) is:
\[
r = \sqrt{4} = 2
\]

#### Step 3: Identify the center and radius
- Center: \((h, k) = (1, -3)\)
- Radius: \(r = 2\)

#### Step 4: Sketch the graph
- Plot the center at \((1, -3)\).
- Draw a circle with a radius of 2 units around the center.

---

2. \((x - 2)^2 + (y + 1)^2 = 16\)

#### Step 1: Compare with the standard form
The given equation is:
\[
(x - 2)^2 + (y + 1)^2 = 16
\]
This matches the standard form \((x - h)^2 + (y - k)^2 = r^2\), where:
- \(h = 2\),
- \(k = -1\),
- \(r^2 = 16\).

#### Step 2: Find the radius
Since \(r^2 = 16\), the radius \(r\) is:
\[
r = \sqrt{16} = 4
\]

#### Step 3: Identify the center and radius
- Center: \((h, k) = (2, -1)\)
- Radius: \(r = 4\)

#### Step 4: Sketch the graph
- Plot the center at \((2, -1)\).
- Draw a circle with a radius of 4 units around the center.

---

3. \((x - 1)^2 + (y + 4)^2 = 9\)

#### Step 1: Compare with the standard form
The given equation is:
\[
(x - 1)^2 + (y + 4)^2 = 9
\]
This matches the standard form \((x - h)^2 + (y - k)^2 = r^2\), where:
- \(h = 1\),
- \(k = -4\),
- \(r^2 = 9\).

#### Step 2: Find the radius
Since \(r^2 = 9\), the radius \(r\) is:
\[
r = \sqrt{9} = 3
\]

#### Step 3: Identify the center and radius
- Center: \((h, k) = (1, -4)\)
- Radius: \(r = 3\)

#### Step 4: Sketch the graph
- Plot the center at \((1, -4)\).
- Draw a circle with a radius of 3 units around the center.

---

4. \(x^2 + (y - 3)^2 = 14\)

#### Step 1: Compare with the standard form
The given equation is:
\[
x^2 + (y - 3)^2 = 14
\]
This can be rewritten as:
\[
(x - 0)^2 + (y - 3)^2 = 14
\]
This matches the standard form \((x - h)^2 + (y - k)^2 = r^2\), where:
- \(h = 0\),
- \(k = 3\),
- \(r^2 = 14\).

#### Step 2: Find the radius
Since \(r^2 = 14\), the radius \(r\) is:
\[
r = \sqrt{14}
\]

#### Step 3: Identify the center and radius
- Center: \((h, k) = (0, 3)\)
- Radius: \(r = \sqrt{14}\)

#### Step 4: Sketch the graph
- Plot the center at \((0, 3)\).
- Draw a circle with a radius of \(\sqrt{14}\) units around the center. Note that \(\sqrt{14} \approx 3.74\).

---

Final Answers:

1. Center: \((1, -3)\), Radius: \(2\)
2. Center: \((2, -1)\), Radius: \(4\)
3. Center: \((1, -4)\), Radius: \(3\)
4. Center: \((0, 3)\), Radius: \(\sqrt{14}\)

\[
\boxed{
\begin{aligned}
1. & \text{ Center: } (1, -3), \text{ Radius: } 2 \\
2. & \text{ Center: } (2, -1), \text{ Radius: } 4 \\
3. & \text{ Center: } (1, -4), \text{ Radius: } 3 \\
4. & \text{ Center: } (0, 3), \text{ Radius: } \sqrt{14}
\end{aligned}
}
\]