Question

determine the equation of the circle graphed below.

Explanation:

Step1: Recall circle equation formula

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 \((h, k)\)

From the graph, the center of the circle is at \((6, 1)\) (by observing the grid, the center is at \(x = 6\), \(y = 1\)).

Step3: Determine the radius \(r\)

The circle spans from \(x = 4\) to \(x = 8\) (or \(y = -1\) to \(y = 3\), but horizontal/vertical distance is easier). The radius is half the diameter. The diameter is \(8 - 4 = 4\), so \(r = \frac{4}{2}= 2\).

Step4: Substitute \(h\), \(k\), and \(r\) into the formula

Substitute \(h = 6\), \(k = 1\), and \(r = 2\) into \((x - h)^2 + (y - k)^2 = r^2\). We get \((x - 6)^2 + (y - 1)^2 = 2^2\), which simplifies to \((x - 6)^2 + (y - 1)^2 = 4\).

Answer:

\((x - 6)^2 + (y - 1)^2 = 4\)