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

Verified step by step guidance

1. Identify the given information: the center of the circle is at the point (0, 0) and the radius is 6.
2. Recall the center-radius form of a circle's equation: \(\left(x - h\right)^2 + \left(y - k\right)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
3. Substitute the center coordinates \((h, k) = (0, 0)\) and the radius \(r = 6\) into the formula to get the equation of the circle.
4. Simplify the equation by squaring the radius: \(r^2 = 6^2\).
5. For graphing, plot the center at the origin \((0, 0)\), then mark points 6 units away in all directions (up, down, left, right) to represent the radius, and sketch the circle through these points.

Key Concepts

Equation of a Circle in Center-Radius Form

The center-radius form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. This form directly shows the circle's location and size, making it easier to graph and analyze.

Identifying the Center and Radius

To write the equation of a circle, you must know its center coordinates (h, k) and radius r. In this problem, the center is (0, 0) and the radius is 6, so these values are substituted into the standard form to create the equation.

Graphing a Circle

Graphing a circle involves plotting its center point and then drawing all points that are a distance r from the center. For a circle centered at the origin with radius 6, plot (0,0) and mark points 6 units away in all directions to sketch the circle.