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.
x² + (y - 3)² = 25
the circle is centered at the point (0,3). (type an ordered pair.)
what is the radius of the circle?
the radius of the circle is 5. (type an integer or a fraction.)
use the graphing tool to graph the circle.

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: Identify the center

For the equation $x^{2}+(y - 3)^{2}=25$, comparing with the standard - form, we have $h = 0$ and $k = 3$. So the center is $(0,3)$.

Step3: Identify the radius

Since $r^{2}=25$, taking the square root of both sides (and considering the non - negative value for the radius), we get $r=\sqrt{25}=5$.

Step4: Find the domain

The left - most and right - most points of the circle are found by considering the center $(0,3)$ and radius $r = 5$. The $x$ values range from $0 - 5=-5$ to $0 + 5 = 5$. So the domain is $[-5,5]$.

Step5: Find the range

The bottom - most and top - most points of the circle are found by considering the center $(0,3)$ and radius $r = 5$. The $y$ values range from $3-5=-2$ to $3 + 5 = 8$. So the range is $[-2,8]$.

Answer:

Center: $(0,3)$; Radius: $5$; Domain: $[-5,5]$; Range: $[-2,8]$