Question

determine the equation of the circle graphed below.

Explanation:

Step1: Recall the 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: Identify the center \((h, k)\)

From the graph, the center of the circle is at \((-3, 4)\). So \(h = -3\) and \(k = 4\).

Step3: Determine the radius \(r\)

By counting the grid units, the radius (distance from center to a point on the circle) is 2. So \(r = 2\), and \(r^2 = 4\).

Step4: Substitute \(h\), \(k\), and \(r^2\) into the formula

Substitute \(h = -3\), \(k = 4\), and \(r^2 = 4\) into \((x - h)^2 + (y - k)^2 = r^2\):
\[
(x - (-3))^2 + (y - 4)^2 = 4
\]
Simplify \((x - (-3))\) to \((x + 3)\):
\[
(x + 3)^2 + (y - 4)^2 = 4
\]

Answer:

\((x + 3)^2 + (y - 4)^2 = 4\)