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

Verified step by step guidance

1. Identify the given information: the center of the circle is at the point (0, 4) and the radius is 4.
2. Recall the center-radius form of a circle's equation: \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.
3. Substitute the center coordinates \((h, k) = (0, 4)\) and the radius \(r = 4\) into the formula: \( (x - 0)^2 + (y - 4)^2 = 4^2 \).
4. Simplify the equation: \( x^2 + (y - 4)^2 = 16 \).
5. To graph the circle, plot the center at (0, 4) on the coordinate plane, then draw a circle with radius 4 units extending in all directions from the center.

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. The center is the fixed point from which all points on the circle are equidistant, and the radius is that constant distance.

Graphing a Circle

Graphing a circle involves plotting its center and using the radius to mark points in all directions. Drawing a smooth curve through these points forms the circle, helping visualize its position and size on the coordinate plane.