Question

write an equation for 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 of the circle

From the graph, the center of the circle is at \((-4, -2)\) (by observing the grid, the center is 4 units left on the x - axis and 2 units down on the y - axis).

Step3: Determine the radius of the circle

The radius is the distance from the center to any point on the circle. Looking at the graph, the distance from the center \((-4, -2)\) to a point on the circle (e.g., from \(x=-4\) to \(x = - 1\) or \(x=-7\), but more clearly, from the center to the rightmost point: the center's x - coordinate is - 4, and the rightmost point on the circle has x - coordinate - 1, so the radius \(r=3\) (since \(\vert-1-(-4)\vert = 3\)).

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

Substitute \(h=-4\), \(k = - 2\), and \(r = 3\) into \((x - h)^2+(y - k)^2=r^2\). We get \((x-(-4))^2+(y - (-2))^2=3^2\), which simplifies to \((x + 4)^2+(y + 2)^2=9\).

Answer:

\((x + 4)^2+(y + 2)^2 = 9\)