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 + 4)^2+(y + 5)^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 and $r$ is the radius.

Step2: Identify the center

For the equation $(x + 4)^2+(y + 5)^2 = 25$, we have $x-(-4)^2+(y-(-5))^2=5^2$. So the center $(h,k)=(-4,-5)$.

Step3: Identify the radius

Since $r^2 = 25$, then $r = 5$ (we take the positive value as radius is non - negative).

Step4: Find the domain

The left - most $x$ value of the circle is $h - r=-4-5=-9$ and the right - most $x$ value is $h + r=-4 + 5 = 1$. So the domain is $[-9,1]$.

Step5: Find the range

The bottom - most $y$ value of the circle is $k - r=-5-5=-10$ and the top - most $y$ value is $k + r=-5 + 5 = 0$. So the range is $[-10,0]$.

Answer:

Center: $(-4,-5)$; Radius: $5$; Domain: $[-9,1]$; Range: $[-10,0]$