To determine which graph represents the equation [tex]\((x - 3)^2 + (y + 1)^2 = 9\)[/tex], let's analyze the characteristics of this equation step by step.
1. Identify the type of curve:
The given equation [tex]\((x - 3)^2 + (y + 1)^2 = 9\)[/tex] is in the standard form of the equation of a circle. The general form of a circle's equation is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
2. Determine the center:
By comparing the given equation [tex]\((x - 3)^2 + (y + 1)^2 = 9\)[/tex] with the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we can identify the center [tex]\((h, k)\)[/tex]:
[tex]\[
h = 3, \quad k = -1
\][/tex]
Therefore, the center of the circle is [tex]\((3, -1)\)[/tex].
3. Determine the radius:
The term on the right side of the equation is [tex]\(9\)[/tex], which is [tex]\(r^2\)[/tex]. Thus, [tex]\(r^2 = 9\)[/tex], and taking the square root of both sides, we get:
[tex]\[
r = \sqrt{9} = 3
\][/tex]
Therefore, the radius of the circle is [tex]\(3\)[/tex].
4. Graph characteristics:
- The center of the circle is at the point [tex]\((3, -1)\)[/tex].
- The radius of the circle is [tex]\(3\)[/tex].
- The circle is centered at [tex]\((3, -1)\)[/tex] and extends [tex]\(3\)[/tex] units in all directions from this center point.
Given these characteristics, you should look for a graph that shows a circle with its center at [tex]\((3, -1)\)[/tex] and a radius of [tex]\(3\)[/tex].
