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 + 1)² = 36

Key Concepts

Circle Equation

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. In this equation, the left side represents the distance from any point (x, y) on the circle to the center (h, k), squared. Understanding this form is essential for identifying the center and radius from the given equation.

Graphing Circles

Graphing a circle involves plotting its center and using the radius to determine the points that lie on the circle. The radius indicates how far from the center the circle extends in all directions. This visual representation helps in understanding the circle's shape and position in the coordinate plane, which is crucial for identifying the domain and range.

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 its vertical extent. Analyzing the center and radius allows us to calculate these values, providing insight into the limits of the circle's coordinates.