Problem 47 - College Algebra

Question

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$(x+2)^{2}+(y+2)^{2}=4$$

Step-by-Step Solution

Answer

The center of the circle is (-2, -2) with a radius of 2. The domain is [-4, 0] and the range is [-4, 0]

Step 1: Finding the center and the radius

The equation \((x+2)^{2}+(y+2)^{2}=4\) is in the form \((x-h)^2 + (y-k)^2 = r^2\). Here, we get \(h = -2\), \(k = -2\) and the radius \(r = \sqrt{4} = 2\). So the circle has center (-2, -2) and radius 2.

Step 2: Determining the domain and range

Once the center (-2, -2) and radius 2 are identified, domain and range can be found. \n\nFor the domain (possible x-values): We can go 2 units (the radius) to the left and right of the x-coordinate of the center. As the x-coordinate of the center is -2, going 2 units to the left gives -2 - 2 = -4 and 2 units to the right gives -2 + 2 = 0. Hence the domain is all x-values between -4 and 0.\n\nSimilarly, the range (possible y-values) are found by going 2 units (the radius) up and down from the y-coordinate of the center. As the y-coordinate of the center is -2, going 2 units down gives -2 - 2 = -4 and 2 units up gives -2 + 2 = 0. Therefore, the range is all y-values between -4 and 0.

Key Concepts

Center and Radius of a Circle

Understanding the center and radius of a circle is fundamental in geometry. For any circle's equation in the format \(x - h)^2 + (y - k)^2 = r^2\), \(h\) and \(k\) represent the x and y coordinates of the center of the circle respectively, and \(r\) is the radius. In the given equation \(x+2)^{2}+(y+2)^{2}=4\), after comparing it with the standard format, it is found that \(h = -2\) and \(k = -2\), which indicates that the circle's center is at the point (-2, -2). The radius \(r\) can be determined by taking the square root of the right-hand side of the equation, yielding \(r = \sqrt{4} = 2\).

Having these calculations in hand give a distinct picture of the circle. The center is a fixed point from which every point on the circle is an equal distance away—the radius. This knowledge helps in understanding not just the size but also the position of the circle on the Cartesian plane.

Domain and Range of a Circle

The domain and range of a circle are references to the set of possible x-values and y-values that the circle covers on the coordinate plane. For the circle described by the equation \(x+2)^{2}+(y+2)^{2}=4\), the center at (-2, -2) serves as the reference point to determine these sets.

The domain is found by moving horizontally from the center's x-coordinate by the distance of the radius; to the left for the smallest x-value, and to the right for the largest. In our case, the domain ranges from -4 to 0. The range is similar but moves vertically from the center's y-coordinate. The y-value also ranges from -4 to 0 in this instance. These values illustrate the horizontal and vertical extents of the circle.

Graphing Circles

To graph circles effectively, one needs the center coordinates and radius as derived from the standard equation of a circle. Using our example \(x+2)^{2}+(y+2)^{2}=4\), with center (-2, -2) and radius 2, begin by plotting the center point on the graph. From this point, mark points that are 2 units in every direction—up, down, left, and right—then continue marking points in a circular path, keeping each equidistant from the center.

Connect these points to form the circle. A proper graph provides a visual representation of the circle's domain and range. The edge of the circle represents the limits of the x and y values, clearly showing which points (x,y) satisfy the circle's equation.

Standard Form of a Circle Equation

The standard form of a circle's equation is vital for identifying key properties like the center and radius. It is expressed as \(x-h)^2 + (y-k)^2 = r^2\), where \( (h, k) \) is the center of the circle and \(r\) is the radius. To convert a given circle equation into this form, one may need to complete the square for both x and y components.

In the exercise, the given equation is already in standard form, which allows immediate identification of the parameters needed to graph the circle and determine its domain and range. Mastery of rewriting the circle equation in standard form is crucial as it simplifies the process of analyzing circles geometrically.