Question

determine the equation of the circle graphed below.

Explanation:

Step1: Recall the standard 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: Identify the center of the circle

From the graph, the center of the circle is at \((-6, -4)\) (since it's 6 units left on the x - axis and 4 units down on the y - axis). So \(h=-6\) and \(k = - 4\).

Step3: Determine the radius of the circle

By looking at the grid, the radius (distance from center to any point on the circle) is 2 units (since from \(x=-6\) to \(x = - 4\) or from \(y=-4\) to \(y=-2\) is 2 units). So \(r = 2\), and \(r^{2}=4\).

Step4: Substitute \(h\), \(k\), and \(r^{2}\) into the equation

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

Answer:

\((x + 6)^{2}+(y + 4)^{2}=4\)