Let's analyze and graph the given circle equation:
[tex]\[ (x+6)^2 + (y-3)^2 = 9 \][/tex]
### Step-by-Step Solution:
1. Identify the Center of the Circle:
The general form of the equation of a circle is [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
Comparing the given equation [tex]\((x+6)^2 + (y-3)^2 = 9\)[/tex] with the general form, we can identify:
- [tex]\(h = -6\)[/tex]
- [tex]\(k = 3\)[/tex]
So, the center of the circle is:
[tex]\[\text{Center} = (-6, 3)\][/tex]
2. Find the Radius of the Circle:
From the equation [tex]\((x+6)^2 + (y-3)^2 = 9\)[/tex], we can see that the right side of the equation is [tex]\(9\)[/tex].
Comparing this with the general form [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], where [tex]\(r^2 = 9\)[/tex]:
[tex]\[r = \sqrt{9} = 3\][/tex]
So, the radius of the circle is:
[tex]\[\text{Radius} = 3\][/tex]
3. Graph the Equation:
To graph this circle:
- Plot the center at [tex]\((-6, 3)\)[/tex].
- Draw a circle with radius [tex]\(3\)[/tex] around this center point.
4. Determine the Domain:
The domain of a circle is all [tex]\(x\)[/tex]-values that fall within the circle, which ranges from the leftmost point to the rightmost point of the circle.
- The leftmost point (or minimum [tex]\(x\)[/tex]-value): [tex]\(-6 - 3 = -9\)[/tex]
- The rightmost point (or maximum [tex]\(x\)[/tex]-value): [tex]\(-6 + 3 = -3\)[/tex]
Hence, the domain in interval notation is:
[tex]\[\text{Domain} = [-9, -3]\][/tex]
5. Determine the Range:
The range of a circle is all [tex]\(y\)[/tex]-values that fall within the circle, which ranges from the bottommost point to the topmost point of the circle.
- The bottommost point (or minimum [tex]\(y\)[/tex]-value): [tex]\(3 - 3 = 0\)[/tex]
- The topmost point (or maximum [tex]\(y\)[/tex]-value): [tex]\(3 + 3 = 6\)[/tex]
Hence, the range in interval notation is:
[tex]\[\text{Range} = [0, 6]\][/tex]
### Summary:
- Center: [tex]\((-6, 3)\)[/tex]
- Radius: [tex]\(3\)[/tex]
- Domain: [tex]\([-9, -3]\)[/tex]
- Range: [tex]\([0, 6]\)[/tex]
To accurately graph the circle, plot the point [tex]\((-6, 3)\)[/tex] as the center and draw a circle with radius [tex]\(3\)[/tex] units. From this graph, verify the domain and range as above.
