To determine which of the given graphs represents the equation [tex]\((x-3)^2 + (y+1)^2 = 9\)[/tex], let us analyze the equation in detail.
### Understanding the Equation
1. Standard Form of Circle Equation:
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
- [tex]\(r\)[/tex] is the radius of the circle
2. Identify the Center:
In the given equation [tex]\((x-3)^2 + (y+1)^2 = 9\)[/tex]:
- Comparing it with the general form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we see that:
- [tex]\(h = 3\)[/tex]
- [tex]\(k = -1\)[/tex]
Thus, the center of the circle is [tex]\((3, -1)\)[/tex].
3. Determine the Radius:
- The equation is [tex]\((x-3)^2 + (y+1)^2 = 9\)[/tex].
- Here, [tex]\(r^2 = 9\)[/tex].
- Therefore, [tex]\(r = \sqrt{9} = 3\)[/tex].
### Summary of Circle’s Properties:
- Center: [tex]\((3, -1)\)[/tex]
- Radius: [tex]\(3\)[/tex]
### Analyzing the Graph Options:
To find the correct graph among options A, B, and C, look for a graph that meets the following criteria:
- The circle is centered at [tex]\((3, -1)\)[/tex].
- The circle has a radius of [tex]\(3\)[/tex].
Let's break down what we should see in each graph:
- Center: Look for the point [tex]\((3, -1)\)[/tex].
- Radius: Measure the distance from the center to any point on the circumference of the circle, which should be [tex]\(3\)[/tex] units in all directions (up, down, left, and right).
Therefore, the correct graph will show a circle centered at [tex]\((3, -1)\)[/tex] with a radius of [tex]\(3\)[/tex].
By identifying these characteristics, you can compare them to each of the provided graph options (A, B, C) and select the appropriate matching graph.
