In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 5 and 6.center (0, 4), radius 4

Verified step by step guidance

1. insert step 1: Start by recalling the standard form of the equation of a circle, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
2. insert step 2: Identify the center (h, k) and the radius r from the problem. Here, the center is (0, 4) and the radius is 4.
3. insert step 3: Substitute the center (0, 4) and the radius 4 into the standard form equation. This gives you (x - 0)^2 + (y - 4)^2 = 4^2.
4. insert step 4: Simplify the equation. Since (x - 0)^2 is just x^2, the equation becomes x^2 + (y - 4)^2 = 16.
5. insert step 5: To graph the circle, plot the center at (0, 4) on the coordinate plane, and draw a circle with a radius of 4 units around this center.

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

To graph a circle, one starts by plotting its center on the coordinate plane. From the center, the radius is used to mark points in all directions (up, down, left, right) to outline the circle. Connecting these points smoothly creates the circular shape, ensuring that all points are equidistant from the center.

Distance Formula

The distance formula, derived from the Pythagorean theorem, calculates the distance between two points in a plane. It is given by d = √((x₂ - x₁)² + (y₂ - y₁)²). This concept is essential for verifying that points on the graph maintain a constant distance (the radius) from the center of the circle.