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.
$(x + 1)^{2}+(y + 2)^{2}=16$
the center is
(type an ordered pair. simplify your answer.)
the radius is
(type an integer or a simplified fraction.)
graph the circle.
click to enlarge graph
express the domain of the relation in interval notation.
express the range 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.

Step2: Rewrite the given equation in the standard form

Given \((x + 1)^2+(y + 2)^2=16\), we can rewrite it as \((x-(-1))^2+(y-(-2))^2 = 4^2\).

Step3: Identify the center

Comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(h=-1\) and \(k = - 2\). So the center is \((-1,-2)\).

Step4: Identify the radius

Since \(r^2=16\), then \(r=\sqrt{16}=4\).

Step5: Find the domain

The domain of a circle \((x - h)^2+(y - k)^2=r^2\) is given by \(h - r\leqslant x\leqslant h + r\). Substituting \(h=-1\) and \(r = 4\), we get \(-1-4\leqslant x\leqslant-1 + 4\), which is \([-5,3]\).

Step6: Find the range

The range of a circle \((x - h)^2+(y - k)^2=r^2\) is given by \(k - r\leqslant y\leqslant k + r\). Substituting \(k=-2\) and \(r = 4\), we get \(-2-4\leqslant y\leqslant-2 + 4\), which is \([-6,2]\).

Answer:

The center is \((-1,-2)\).
The radius is \(4\).
Domain: \([-5,3]\)
Range: \([-6,2]\)