To determine which graph corresponds to the equation
[tex]\[
(x-1)^2 + (y-2)^2 = 4
\][/tex]
let's break down the components of the equation and understand what they represent:
1. Standard Form of a Circle's Equation: The given equation is in the standard form of a circle's equation, which 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. Identifying the Center and Radius:
- The terms [tex]\((x - 1)\)[/tex] and [tex]\((y - 2)\)[/tex] indicate that the circle's center is at [tex]\((1, 2)\)[/tex].
- The right side of the equation, [tex]\(4\)[/tex], represents [tex]\(r^2\)[/tex], so we solve for [tex]\(r\)[/tex] by taking the square root of 4. Thus, [tex]\(r = \sqrt{4} = 2\)[/tex].
Hence, the circle has:
- Center: [tex]\((1, 2)\)[/tex]
- Radius: [tex]\(2\)[/tex]
3. Choosing the Correct Graph:
- Look at the provided graphs and identify the one with a circle centered at [tex]\((1, 2)\)[/tex] and a radius of [tex]\(2\)[/tex].
- The correct graph will show a circle that perfectly aligns at the center point [tex]\((1, 2)\)[/tex] with a radius extending [tex]\(2\)[/tex] units in all directions (left, right, up, down).
Check each graph to see which one matches these criteria:
- Graph A: Verify if the center is [tex]\((1, 2)\)[/tex] and the radius is [tex]\(2\)[/tex].
- Graph B: Verify if the center is [tex]\((1, 2)\)[/tex] and the radius is [tex]\(2\)[/tex].
- Graph C: Verify if the center is [tex]\((1, 2)\)[/tex] and the radius is [tex]\(2\)[/tex].
The graph that fits these conditions correctly is the one we are looking for. Carefully analyze each to determine the one centered at [tex]\((1, 2)\)[/tex] with the right radius.
In conclusion, you should identify the graph that aligns with a circle of radius [tex]\(2\)[/tex] centered at [tex]\((1, 2)\)[/tex]. Select that as the correct graph representing
[tex]\[
(x-1)^2 + (y-2)^2 = 4
\][/tex]
