Question

give the center and radius of the circle described by the equation and graph the equation. use the graph to identify the domain and range.
$(x + 2)^2+(y + 6)^2 = 25$
the center is $(-2,-6)$.
(type an ordered pair. simplify your answer)
the radius is 5.
(type an integer or a simplified fraction.)
graph the circle.
click to enlarge graph

Explanation:

Step1: Analyze the standard form of the circle equation

The standard form of a circle equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius.
For the given equation \((x + 2)^2+(y + 6)^2=25\), we can rewrite it as \((x-(- 2))^2+(y-(-6))^2 = 5^2\).

Step2: Determine the domain

The domain of a circle \((x - h)^2+(y - k)^2=r^2\) is given by \(h - r\leqslant x\leqslant h + r\).
Here \(h=-2\) and \(r = 5\), so \(-2-5\leqslant x\leqslant-2 + 5\), which simplifies to \(-7\leqslant x\leqslant3\).

Step3: Determine the range

The range of a circle \((x - h)^2+(y - k)^2=r^2\) is given by \(k - r\leqslant y\leqslant k + r\).
Here \(k=-6\) and \(r = 5\), so \(-6-5\leqslant y\leqslant-6 + 5\), which simplifies to \(-11\leqslant y\leqslant-1\).

Answer:

The domain is \([-7,3]\) and the range is \([-11,-1]\)