To determine which graph corresponds to the given equation [tex]\((x-3)^2 + (y+1)^2 = 9\)[/tex], let's analyze and understand the equation step-by-step.
1. Recognize the Form of the Equation:
The equation [tex]\((x-3)^2 + (y+1)^2 = 9\)[/tex] is in the standard form of a circle's equation [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. Identify the Center of the Circle:
The center [tex]\((h, k)\)[/tex] of the circle can be identified by comparing [tex]\((x-3)^2 + (y+1)^2\)[/tex] with [tex]\((x-h)^2 + (y-k)^2\)[/tex]. Here, [tex]\(h = 3\)[/tex] and [tex]\(k = -1\)[/tex]. Thus, the center of the circle is at [tex]\((3, -1)\)[/tex].
3. Determine the Radius of the Circle:
The radius [tex]\(r\)[/tex] is found by taking the square root of the constant on the right side of the equation [tex]\( r^2 = 9\)[/tex]. Therefore, [tex]\( r = \sqrt{9} = 3 \)[/tex].
4. Summary of the Circle's Properties:
- Center: [tex]\((3, -1)\)[/tex]
- Radius: [tex]\(3\)[/tex]
To find which graph corresponds to this circle, look for a graph that has:
- A circle centered at the point [tex]\((3, -1)\)[/tex]
- A radius of 3 units
Once you have a graph that matches these properties, you have successfully identified the correct representation of the given equation.
