Give the center and radius of the circle described by the equation and graph the equation. Use the graph to identify the domain and range. $$ (x+2)^{2}+(y-3)^{2}=16 $$

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Certainly! Let's analyze the given equation of the circle, determine its center and radius, graph it, and identify its domain and range.

### Given Equation:
$$
(x + 2)^2 + (y - 3)^2 = 16
$$

### 1. **Center and Radius**

The general form of a circle's equation is:
$$
(x - h)^2 + (y - k)^2 = r^2
$$
where $$(h, k)$$ is the center and $$r$$ is the radius.

**Comparing with the given equation:**
$$
(x + 2)^2 + (y - 3)^2 = 16
$$
$$
\Rightarrow (x - (-2))^2 + (y - 3)^2 = 4^2
$$

- **Center $$(h, k)$$:** $$(-2, 3)$$
- **Radius $$r$$:** $$4$$

### 2. **Graph of the Circle**

Below is a graphical representation of the circle:

```
Y-axis
|
7 |                     *
6 |                   *   *
5 |                 *       *
4 |               *           *
3 |-----*-----------------------*-----
2 |               *           *
1 |                 *       *
0 |                   *   *
-1|                     *
|
------------------------------------------ X-axis
      -6  -4  -2   0   2   4   6
```

- **Center:** The point $$(-2, 3)$$ is marked at the intersection of the horizontal line $$x = -2$$ and the vertical line $$y = 3$$.
- **Radius:** The circle extends 4 units in all directions from the center.

### 3. **Domain and Range**

- **Domain:** The set of all possible $$x$$-values.
  - Since the radius is 4 and the center is at $$x = -2$$, the circle extends from $$x = -2 - 4 = -6$$ to $$x = -2 + 4 = 2$$.
  - **Domain:** $$-6 \leq x \leq 2$$

- **Range:** The set of all possible $$y$$-values.
  - Since the radius is 4 and the center is at $$y = 3$$, the circle extends from $$y = 3 - 4 = -1$$ to $$y = 3 + 4 = 7$$.
  - **Range:** $$-1 \leq y \leq 7$$

### **Summary**

- **Center:** $$(-2, 3)$$
- **Radius:** $$4$$
- **Domain:** $$-6 \leq x \leq 2$$
- **Range:** $$-1 \leq y \leq 7$$

This circle is centered at $$(-2, 3)$$ with a radius of 4 units. It spans from $$x = -6$$ to $$x = 2$$ and from $$y = -1$$ to $$y = 7$$ on the coordinate plane.
The circle has a center at $$(-2, 3)$$ and a radius of 4. Its domain is $$-6 \leq x \leq 2$$ and its range is $$-1 \leq y \leq 7$$.

Give the center and radius of the circle described by the equation and graph the equation. Use the graph to identify the domain and range. \( (x+2)^{2}+(y-3)^{2}=16 \)

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