Question

determine the equation of the circle graphed below.

Explanation:

Step1: Find the center of the circle

From the graph, the center \((h,k)\) of the circle is at \((-3, -1)\) (by observing the midpoint of the circle's symmetry).

Step2: Find the radius of the circle

The radius \(r\) can be found by calculating the distance from the center \((-3, -1)\) to a point on the circle, say \((2, -1)\) (or other points). Using the distance formula \(r=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), substituting \((x_1,y_1)=(-3,-1)\) and \((x_2,y_2)=(2,-1)\), we get \(r=\sqrt{(2 - (-3))^2+(-1 - (-1))^2}=\sqrt{5^2 + 0^2}=5\).

Step3: Use the standard circle equation

The standard equation of a circle is \((x - h)^2+(y - k)^2=r^2\). Substituting \(h = - 3\), \(k=-1\) and \(r = 5\), we have \((x - (-3))^2+(y - (-1))^2=5^2\), which simplifies to \((x + 3)^2+(y + 1)^2=25\).

Answer:

\((x + 3)^2+(y + 1)^2=25\)