Identify the center and radius of each circle, then graph. Also state the domain and range of th

Identify the center and radius of each circle, then graph....

Key Concepts

Graphical Representation of Circles

Graphing a circle involves plotting the center (h, k), measuring out the radius in all directions, and drawing the set of all points that lie at this fixed distance from the center. This visual representation helps in understanding the structure and properties of the circle within the coordinate plane.

Radius of a Circle

The radius is the constant distance from the center to any point on the circle. In the standard form equation, r is found by taking the square root of the constant on the right side, and it determines the size of the circle as well as its dimensions on a graph.

Domain and Range of a Circle

The domain and range of a circle describe the set of possible x-values and y-values for points on the circle. They are determined by the center and radius: the domain extends from h - r to h + r, and the range extends from k - r to k + r, reflecting the horizontal and vertical spans of the circle.

Standard Form of a Circle Equation

This is a fundamental representation of a circle in coordinate geometry, typically written as (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents its radius. This form makes it straightforward to read off these key properties of the circle.

Center of a Circle

The center is the fixed point equidistant from every point on the circle. In the standard equation (x - h)^2 + (y - k)^2 = r^2, the coordinates (h, k) directly provide the position of the center, which is critical for graphing and analysis of the circle.

Transcript

00:01 The general equation of a circle in standard form is x minus h squared plus y minus k squared equals r squared, where the point hk is the center of the circle, and r is the radius.

00:20 So take for example if we have the equation x minus 7 squared plus 5 plus 4 squared equals 20, and if we want to know the center and the radius of the circle.

00:41 So we know the point hk is the center, and we know that h is equal to 7, and k is equal to negative 4.

00:51 And it might be a little confusing because this is y plus 4.

00:55 But notice up here, the equation says y minus k.

00:58 So y minus a negative 4 is y plus 4.

01:04 So we know our center is at the point 7.

01:07 Minus 4.

01:10 Now we also want to know our radius.

01:13 So r squared is equal to 20.

01:17 So r is equal to the square root of 20.

01:21 Now we want to graph this equation.

01:25 So our first step in doing that is to plot our center point.

01:36 So our center point of our circle is right there at 7 -9 -4.

01:40 And then we know the radius is uniform across the entire circle.

01:45 So we can plot four points a distance of the square root of 20 away from our center, and by connecting those four points, we can get a rough sketch of our circle.

01:56 So the square root of 20 is approximately equal to 4 .47...

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