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

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.