Question

determine the equation of the circle graphed below.

Explanation:

Step1: Find the center of the circle

The center of the circle \((h,k)\) can be determined from the graph. Looking at the circle, the center is at \((1, -4)\) (since it's 1 unit to the right on the x - axis and 4 units down on the y - axis).

Step2: Find the radius of the circle

The radius \(r\) is the distance from the center to a point on the circle. We can use the distance formula or count the grid units. From the center \((1, -4)\) to the topmost point \((1, -1)\) (or bottommost point \((1, -7)\)), the distance is \(|-1-(-4)| = 3\) (or \(|-7 - (-4)|=3\)). So the radius \(r = 3\).

Step3: Use the standard equation of a circle

The standard equation of a circle is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius. Substituting \(h = 1\), \(k=-4\), and \(r = 3\) into the equation, we get \((x - 1)^2+(y + 4)^2=3^2=9\).

Answer:

\((x - 1)^2+(y + 4)^2 = 9\)