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, 0), radius 6

Verified step by step guidance

1. Identify the center and radius of the circle from the problem statement. The center is given as (0, 0) and the radius is 6.
2. Recall the standard form of the equation of a circle with center (h, k) and radius r: $(x-h{)}^2+(y-k{)}^2=r^2$.
3. Substitute the center coordinates (h = 0, k = 0) and the radius (r = 6) into the standard form equation.
4. Simplify the equation by squaring the radius and writing out the complete equation.
5. To graph the circle, plot the center at (0, 0) on a coordinate plane, and use the radius to measure 6 units in all directions (up, down, left, right) from the center to sketch 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 the center point on a coordinate plane and then using the radius to mark points that are equidistant from the center in all directions. This creates a circular shape, and understanding how to accurately represent the radius is crucial for correct graphing.

Distance Formula

The distance formula, derived from the Pythagorean theorem, is used to calculate the distance between two points in a plane. It is expressed as d = √((x₂ - x₁)² + (y₂ - y₁)²). This concept is essential for understanding how the radius defines the boundary of the circle from its center.