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.
$(x + 5)^{2}+(y - 6)^{2}=4$
the center is $(-5,6)$
(type an ordered pair. simplify your answer.)
the radius is 2
(type an integer or a simplified fraction.)
graph the circle.
express the domain of the relation in interval notation.

Explanation:

Step1: Recall the standard form of a circle's 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.
For the given equation \((x + 5)^2+(y - 6)^2=4\), we can rewrite it as \((x-(- 5))^2+(y - 6)^2=2^2\).

Step2: Find the domain

The \(x\) - values of the circle range from \(h - r\) to \(h + r\). Here, \(h=-5\) and \(r = 2\).
So, \(x\) ranges from \(-5-2=-7\) to \(-5 + 2=-3\).
In interval notation, the domain is \([-7,-3]\).

Step3: Find the range

The \(y\) - values of the circle range from \(k - r\) to \(k + r\). Here, \(k = 6\) and \(r=2\).
So, \(y\) ranges from \(6-2 = 4\) to \(6+2=8\).
In interval notation, the range is \([4,8]\).

Answer:

Domain: \([-7,-3]\)
Range: \([4,8]\)