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 - 3)^2+(y - 1)^2 = 25

Explanation:

Step1: Recall the standard - form of a circle equation

The standard - form of a circle equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius.

Step2: Identify the center

Comparing $(x - 3)^2+(y - 1)^2=25$ with $(x - h)^2+(y - k)^2=r^2$, we have $h = 3$ and $k = 1$. So the center of the circle is $(3,1)$.

Step3: Identify the radius

Since $r^2=25$, then $r = 5$ (we take the positive square root because the radius is a non - negative quantity).

Step4: Find the domain

The left - most $x$ value of the circle is $h - r=3 - 5=-2$ and the right - most $x$ value is $h + r=3 + 5 = 8$. So the domain is $[-2,8]$.

Step5: Find the range

The bottom - most $y$ value of the circle is $k - r=1 - 5=-4$ and the top - most $y$ value is $k + r=1+5 = 6$. So the range is $[-4,6]$.

Answer:

Center: $(3,1)$; Radius: $5$; Domain: $[-2,8]$; Range: $[-4,6]$