Question

write the equation of this circle in standard form.

Explanation:

Step1: Recall the standard form of a circle's 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: Identify the center of the circle

From the graph, the center of the circle is at \((0, -6)\), so \(h = 0\) and \(k = -6\).

Step3: Determine the radius of the circle

By counting the grid units from the center to the edge of the circle, we can see the radius \(r = 3\) (since from \(y = -6\) to \(y = -3\) or \(y = -9\) is 3 units, and horizontally it also spans 3 units from \(x = 0\) to \(x = 3\) or \(x = -3\)).

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

Substituting \(h = 0\), \(k = -6\), and \(r = 3\) into \((x - h)^2 + (y - k)^2 = r^2\), we get \((x - 0)^2 + (y - (-6))^2 = 3^2\), which simplifies to \(x^2 + (y + 6)^2 = 9\).

Answer:

\(x^2 + (y + 6)^2 = 9\)