In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 1 and 2. center (0, 4), radius 4

Verified step by step guidance

1. Identify the center and radius of the circle from the problem statement. The center is given as (0, 4) and the radius is 4.
2. Recall the standard form of the equation of a circle: \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.
3. Substitute the center coordinates \(h = 0\) and \(k = 4\) into the equation, replacing \(h\) and \(k\) respectively.
4. Substitute the radius \(r = 4\) into the equation and square it to find \(r^2\).
5. Simplify the equation to get the center-radius form of the circle.

Key Concepts

Center-Radius Form of a Circle

The center-radius form of a circle's equation is expressed as (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. This format allows for easy identification of the circle's center and radius, facilitating both graphing and analysis.

Graphing a Circle

Graphing a circle involves plotting its center on a coordinate plane and using the radius to determine the points that lie on the circle. From the center, you can move r units in all directions (up, down, left, right) to find key points, which helps in sketching the circle accurately.

Coordinate Geometry

Coordinate geometry is the study of geometric figures using a coordinate system, typically the Cartesian plane. It provides a framework for representing shapes like circles with equations, allowing for the analysis of their properties and relationships with other geometric figures.