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

Identify the general form of the equation of a circle with center \((h, k)\) and radius \(r\), which is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]

Substitute the given center coordinates \((0, 4)\) into the formula, so \(h = 0\) and \(k = 4\). This gives: \[ (x - 0)^2 + (y - 4)^2 = r^2 \]

Substitute the given radius \(r = 4\) into the equation, so the radius squared is \(4^2 = 16\). The equation becomes: \[ x^2 + (y - 4)^2 = 16 \]

This is the center-radius form of the circle's equation. For graphing, plot the center at \((0, 4)\) on the coordinate plane.

From the center, use the radius length of 4 units to mark points up, down, left, and right (and optionally other directions) to sketch the circle accurately.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Equation of a Circle in Center-Radius Form

The center-radius form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form directly relates the geometric properties of the circle to its algebraic representation, making it easier to identify and graph.

Understanding Coordinates of the Center

The center of the circle is given as a point (h, k) in the coordinate plane. Knowing the center allows you to position the circle correctly when graphing and to substitute these values into the equation to define the circle precisely.

Graphing Circles on the Coordinate Plane

Graphing a circle involves plotting its center and using the radius to mark points at equal distances in all directions. This visual representation helps in understanding the circle's size and location relative to the axes.

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