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.

$(x - 1)^2 + (y - 3)^2 = 16$

what is the center of the circle?
the circle is centered at the point \\(\square\\).
(type an ordered pair.)
what is the radius of the circle?
the radius of the circle is \\(\square\\).
(type an integer or a fraction.)
use the graphing tool to graph the circle.
\\(\text{click to enlarge graph}\\)
what is the domain of the relation?
the domain is \\(\square\\).
(type your answer in interval notation.)
what is the range of the relation?
the range is \\(\square\\).

Explanation:

Step1: Recall circle equation form

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: Find the center

For the equation \((x - 1)^2 + (y - 3)^2 = 16\), compare with the standard form. Here, \(h = 1\) and \(k = 3\), so the center is \((1, 3)\).

Step3: Find the radius

From the equation, \(r^2 = 16\). Taking the square root (radius is non - negative), we get \(r=\sqrt{16} = 4\).

Step4: Find the domain

The domain of a circle is the set of all \(x\) - values. The center is at \(x = 1\) and the radius is \(4\). So the minimum \(x\) - value is \(1-4=-3\) and the maximum \(x\) - value is \(1 + 4=5\). In interval notation, the domain is \([-3,5]\).

Step5: Find the range

The range of a circle is the set of all \(y\) - values. The center is at \(y = 3\) and the radius is \(4\). So the minimum \(y\) - value is \(3-4=-1\) and the maximum \(y\) - value is \(3 + 4=7\). In interval notation, the range is \([-1,7]\).

Answer:

The center of the circle is \((1, 3)\).
The radius of the circle is \(4\).
The domain is \([-3,5]\).
The range is \([-1,7]\).