Question

find the center and the radius of the circle. then graph the circle. ( x^2 = 16 - (y + 3)^2 ) use the graphing tool to graph the circle. click to enlarge graph

Explanation:

Step1: Recall the standard circle equation

The standard 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.

Step2: Rearrange the given equation

Start with \(x^2 = 16 - (y + 3)^2\). Rearrange it to get \(x^2 + (y + 3)^2 = 16\).

Step3: Identify \(h\), \(k\), and \(r\)

Compare \(x^2 + (y + 3)^2 = 16\) with \((x - h)^2 + (y - k)^2 = r^2\). Here, \(h = 0\) (since \(x - 0 = x\)), \(k = - 3\) (since \(y - (-3)=y + 3\)), and \(r^2 = 16\), so \(r=\sqrt{16}=4\).

Answer:

The center of the circle is \((0, - 3)\) and the radius is \(4\).