To understand which graph corresponds to the equation [tex]\((x-1)^2 + (y+2)^2 = 4\)[/tex], we need to recognize the form of the equation and identify key characteristics.
### Step-by-Step Analysis
1. Recognize the General Form:
The given equation [tex]\((x-1)^2 + (y+2)^2 = 4\)[/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:
By comparing the given equation with the standard form, we can identify:
- [tex]\(h = 1\)[/tex], which means the x-coordinate of the center is 1.
- [tex]\(k = -2\)[/tex], which means the y-coordinate of the center is -2.
Therefore, the center of the circle is at the point [tex]\((1, -2)\)[/tex].
3. Determine the Radius:
The right side of the equation is 4:
[tex]\[
r^2 = 4
\][/tex]
To find the radius [tex]\(r\)[/tex], we take the square root of 4:
[tex]\[
r = \sqrt{4} = 2
\][/tex]
Hence, the radius of the circle is 2 units.
### Summary
The circle has the following properties:
- Center: [tex]\((1, -2)\)[/tex]
- Radius: [tex]\(2\)[/tex]
### Interpretation for Graph
Given these properties, the graph of the equation [tex]\((x-1)^2 + (y+2)^2 = 4\)[/tex] will be a circle centered at [tex]\((1, -2)\)[/tex] with a radius of 2 units. This circle will include all points that are exactly 2 units away from the center point [tex]\((1, -2)\)[/tex].
When you look at graphs, the one that matches these characteristics—having a center at [tex]\((1, -2)\)[/tex] and a radius of 2—will be the correct graph of the given equation.
