In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x+3)² + (y + 2)² = 4

Verified step by step guidance

1. Identify the standard form of a circle's equation, which is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
2. Compare the given equation \((x+3)^2 + (y+2)^2 = 4\) with the standard form to identify \(h\), \(k\), and \(r^2\).
3. Recognize that \(h = -3\) and \(k = -2\) by comparing \((x+3)\) and \((y+2)\) to \((x-h)\) and \((y-k)\).
4. Determine the radius \(r\) by taking the square root of \(r^2 = 4\), which gives \(r = 2\).
5. The center of the circle is \((-3, -2)\) and the radius is \(2\).

Key Concepts

Equation of a Circle

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. In the given equation, (x + 3)² + (y + 2)² = 4, we can identify the center as (-3, -2) and the radius as the square root of 4, which is 2. Understanding this form is crucial for identifying the circle's properties.

Graphing Circles

To graph a circle, plot the center point and then use the radius to mark points in all directions (up, down, left, right) from the center. Connect these points in a smooth, round shape to represent the circle. This visual representation helps in understanding the spatial relationship of the circle to the coordinate plane.

Domain and Range

The domain of a relation refers to all possible x-values, while the range refers to all possible y-values. For a circle, the domain is determined by the horizontal extent of the circle, and the range is determined by the vertical extent. In this case, the domain is [-5, -1] and the range is [-4, 0], which can be derived from the center and radius.