Give the center and radius of the circle described by the equation and graph the equation. Use the graph to identify the relation's domain and range. $$ (x-2)^{2}+(y-5)^{2}=16 $$

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To analyze the given equation of the circle and determine its properties, let's follow these steps:

### Given Equation:
$$
(x - 2)^2 + (y - 5)^2 = 16
$$

### 1. **Identify the Center and Radius**

The standard 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:**
- $$h = 2$$
- $$k = 5$$
- $$r^2 = 16 \Rightarrow r = \sqrt{16} = 4$$

**Therefore:**
- **Center:** $$(2, 5)$$
- **Radius:** $$4$$

### 2. **Graphing the Circle**

While I can't provide a visual graph here, I can describe how to plot it:

1. **Plot the Center:** Start by plotting the point $$(2, 5)$$ on the coordinate plane.
2. **Draw the Circle:** Using a compass or a steady hand, draw a circle with a radius of $$4$$ units around the center point.
3. **Key Points on the Circle:**
   - **Rightmost Point:** $$(2 + 4, 5) = (6, 5)$$
   - **Leftmost Point:** $$(2 - 4, 5) = (-2, 5)$$
   - **Topmost Point:** $$(2, 5 + 4) = (2, 9)$$
   - **Bottommost Point:** $$(2, 5 - 4) = (2, 1)$$

### 3. **Determining the Domain and Range**

- **Domain:** Refers to all possible $$x$$-values.
  - Since the center is at $$x = 2$$ and the radius is $$4$$, the circle extends from $$x = 2 - 4 = -2$$ to $$x = 2 + 4 = 6$$.
  - **Domain:** $$[-2, 6]$$

- **Range:** Refers to all possible $$y$$-values.
  - Since the center is at $$y = 5$$ and the radius is $$4$$, the circle extends from $$y = 5 - 4 = 1$$ to $$y = 5 + 4 = 9$$.
  - **Range:** $$[1, 9]$$

### 4. **Summary**

- **Center:** $$(2, 5)$$
- **Radius:** $$4$$
- **Domain:** $$[-2, 6]$$
- **Range:** $$[1, 9]$$

### 5. **Visual Representation (Description)**

Imagine a coordinate plane:

- Plot the center at $$(2, 5)$$.
- From the center, move $$4$$ units right to $$(6, 5)$$, left to $$(-2, 5)$$, up to $$(2, 9)$$, and down to $$(2, 1)$$.
- Connect these points smoothly to form a circle.

Here's a rough ASCII representation:

```
y
|
9|        * (2,9)
8|
7|
6|
5|    *-------*-------* 
4| (-2,5)   (2,5)    (6,5)
3|
2|
1|        * (2,1)
|
+-------------------------- x
```

*Note: The asterisks (*) represent key points on the circle.*

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This analysis provides a comprehensive understanding of the circle described by the equation $$(x - 2)^2 + (y - 5)^2 = 16$$, including its center, radius, graphical representation, and the domain and range derived from the graph.
The circle has a center at (2, 5) and a radius of 4. Its domain is from -2 to 6 on the x-axis and its range is from 1 to 9 on the y-axis.

Give the center and radius of the circle described by the equation and graph the equation. Use the graph to identify the relation's domain and range. \( (x-2)^{2}+(y-5)^{2}=16 \)

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