Question

give the center and radius of the circle described by the equation and graph the equation. use the graph to identify the relations domain and range. (x - 2)^2+(y - 4)^2 = 16. what is the center of the circle? the circle is centered at the point (type an ordered pair.) what is the radius of the circle? the radius of the circle is (type an integer or a fraction.) use the graphing tool to graph the circle. what is the domain of the relation? the domain is -2,6. (type your answer in interval notation.) what is the range of the relation? the range is (type your answer in interval notation.)

Explanation:

Step1: Recall the standard - form of a circle equation

The standard - form of a circle equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center of the circle and \(r\) is the radius. For the equation \((x - 2)^2+(y - 4)^2 = 16\), the center is \((2,4)\) and the radius \(r=\sqrt{16}=4\).

Step2: Find the domain

The \(x\) - values of points on the circle satisfy \((x - 2)^2=16-(y - 4)^2\). Since \((x - 2)^2\geq0\), we know that \(16-(y - 4)^2\geq0\). The left - most and right - most points of the circle occur when \(y = 4\). Then \((x - 2)^2=16\), so \(x-2=\pm4\), and \(x=-2\) or \(x = 6\). The domain is \([-2,6]\).

Step3: Find the range

The \(y\) - values of points on the circle satisfy \((y - 4)^2=16-(x - 2)^2\). Since \((y - 4)^2\geq0\), we know that \(16-(x - 2)^2\geq0\). The bottom - most and top - most points of the circle occur when \(x = 2\). Then \((y - 4)^2=16\), so \(y-4=\pm4\), and \(y = 0\) or \(y = 8\). The range is \([0,8]\).

Answer:

Domain: \([-2,6]\)
Range: \([0,8]\)