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 + 2)² + (y + 2)² = 4

Verified step by step guidance

1. Identify the standard form of the circle equation, which is \(\left(x - h\right)^2 + \left(y - k\right)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
2. Compare the given equation \(\left(x + 2\right)^2 + \left(y - 2\right)^2 = 4\) to the standard form. Note that \(x + 2\) can be rewritten as \(x - (-2)\), so the center coordinates are \((-2, 2)\).
3. Determine the radius by taking the square root of the right side of the equation: \(r = \sqrt{4}\).
4. Use the center and radius to sketch the circle on a coordinate plane, plotting the center at \((-2, 2)\) and marking points \(r\) units away in all directions.
5. Identify the domain and range of the circle from the graph or equation: the domain is all \(x\) values within \(r\) units of the center's \(x\)-coordinate, and the range is all \(y\) values within \(r\) units of the center's \(y\)-coordinate.

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. By comparing the given equation to this form, you can identify the circle's center and radius directly.

Graphing Circles

Graphing a circle involves plotting its center and using the radius to mark points in all directions. This visual representation helps in understanding the shape and position of the circle on the coordinate plane.

Domain and Range of a Circle

The domain of a circle is the set of all possible x-values, and the range is the set of all possible y-values covered by the circle. These can be found by considering the center coordinates and radius, determining the intervals for x and y.