Question

determine the equation of the circle graphed below.

Explanation:

Step1: Recall the circle equation

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: Find the center

From the graph, the center \((h, k)\) is at \((-1, 5)\) (by observing the midpoint of the circle's symmetry).

Step3: Determine the radius

The radius is the distance from the center to a point on the circle. From \((-1, 5)\) to \((0, 0)\) (or other points), we calculate the distance. Using the distance formula or counting grid units, the radius \(r = 5\) (since the vertical distance from \(y = 5\) to \(y = 0\) is 5, and horizontal distance from \(x = -1\) to \(x = 0\) is 1, but actually, by looking at the circle's diameter vertically from \(y = 0\) to \(y = 10\), radius is 5).

Step4: Substitute into the equation

Substitute \(h = -1\), \(k = 5\), and \(r = 5\) into \((x - h)^2 + (y - k)^2 = r^2\). We get \((x - (-1))^2 + (y - 5)^2 = 5^2\), which simplifies to \((x + 1)^2 + (y - 5)^2 = 25\).

Answer:

\((x + 1)^2 + (y - 5)^2 = 25\)