Question

use the graph to determine the equation of the circle in center-radius form.
options:

• $(x - 3)^2 + (y + 2)^2 = 16$
• $(x + 3)^2 + (y - 2)^2 = 4$
• $(x + 3)^2 + (y - 2)^2 = 16$
• $(x - 3)^2 + (y + 2)^2 = 4$

Explanation:

Step1: Recall the circle equation

The center - radius form of a circle's equation is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center of the circle and \(r\) is the radius.

Step2: Determine the center of the circle

From the graph, we can observe that the center of the circle is at the point \((3,- 2)\). In the equation \((x - h)^{2}+(y - k)^{2}=r^{2}\), \(h = 3\) and \(k=-2\), so the equation of the circle should be of the form \((x - 3)^{2}+(y+ 2)^{2}=r^{2}\) (since \(y-(-2)=y + 2\)).

Step3: Determine the radius of the circle

We can find the radius by looking at the distance from the center to a point on the circle. Let's take two points on the circle, for example, the top - most point and the center or the right - most point and the center. The distance from the center \((3,-2)\) to a point on the circle (e.g., the right - most point \((7,-2)\)) is \(r=\vert7 - 3\vert=4\). Then \(r^{2}=16\).

So the equation of the circle is \((x - 3)^{2}+(y + 2)^{2}=16\).

Answer:

A. \((x - 3)^{2}+(y + 2)^{2}=16\)