To determine the graph of the given equation [tex]\((x + 3)^2 + (y - 1)^2 = 9\)[/tex], we need to recognize that this equation represents a circle in standard form. Here's the detailed step-by-step analysis:
### Step 1: Identifying the Center and Radius
The standard form of a circle's equation is given by:
[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.
Comparing [tex]\((x + 3)^2 + (y - 1)^2 = 9\)[/tex] with the standard form, we can identify the following:
- The expression [tex]\((x + 3)\)[/tex] implies [tex]\((x - (-3))\)[/tex]. Therefore, the x-coordinate of the center, [tex]\(h\)[/tex], is [tex]\(-3\)[/tex].
- The expression [tex]\((y - 1)\)[/tex] matches directly with the standard form, so the y-coordinate of the center, [tex]\(k\)[/tex], is [tex]\(1\)[/tex].
- On the right side of the equation, we have [tex]\(9\)[/tex], which represents [tex]\(r^2\)[/tex]. So, [tex]\(r^2 = 9\)[/tex]. Taking the square root of both sides, the radius [tex]\(r\)[/tex] is [tex]\(3\)[/tex].
### Step 2: Assembling the Information
- The center of the circle is [tex]\((-3, 1)\)[/tex].
- The radius of the circle is [tex]\(3\)[/tex].
### Step 3: Graph Description
With these details, we can describe the graph of the circle:
- The circle is centered at the point [tex]\((-3, 1)\)[/tex], meaning it is shifted 3 units to the left and 1 unit up from the origin.
- The radius of the circle is [tex]\(3\)[/tex], so the circle extends 3 units in all directions from the center.
Therefore, the graph of the equation [tex]\((x + 3)^2 + (y - 1)^2 = 9\)[/tex] is a circle centered at [tex]\((-3, 1)\)[/tex] with a radius of [tex]\(3\)[/tex].
This detailed step-by-step explanation confirms that the circle described by the given equation has the specified center and radius.
