### Part A: Equation of the Circle (4 points)
To find the equation of a circle, we start with the standard form of the 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.
Given the center [tex]\((3, -6)\)[/tex] and a diameter of 8 units, we can determine the radius as half of the diameter:
[tex]\[
r = \frac{\text{diameter}}{2} = \frac{8}{2} = 4 \text{ units}
\][/tex]
Substituting [tex]\((h, k) = (3, -6)\)[/tex] and [tex]\(r = 4\)[/tex] into the standard form equation, we get:
[tex]\[
(x - 3)^2 + (y - (-6))^2 = 4^2
\][/tex]
Simplifying, we have:
[tex]\[
(x - 3)^2 + (y + 6)^2 = 16
\][/tex]
Therefore, the equation of the circle is:
[tex]\[
(x - 3)^2 + (y + 6)^2 = 16
\][/tex]
### Part B: How to Graph the Circle by Hand (3 points)
To graph the circle by hand on a coordinate plane, follow these steps:
1. Plot the Center: Begin by plotting the center of the circle at the given point [tex]\((3, -6)\)[/tex].
2. Mark the Radius Points: From the center, move [tex]\(4\)[/tex] units (the radius) in each of the four cardinal directions: up, down, left, and right. This gives you four key points on the circle:
- [tex]\((3, -6 + 4) = (3, -2)\)[/tex]
- [tex]\((3, -6 - 4) = (3, -10)\)[/tex]
- [tex]\((3 + 4, -6) = (7, -6)\)[/tex]
- [tex]\((3 - 4, -6) = (-1, -6)\)[/tex]
3. Draw the Circle: Use these four points as guides to draw the circumference of the circle. Ideally, you would use a compass or draw freehand, ensuring that the curve is equidistant from the center at every point.
### Part C: Domain of the Circle (3 points)
The domain of the circle refers to the set of all possible x-values that the circle covers on the coordinate plane.
Given the circle's center at [tex]\((3, -6)\)[/tex] and a radius of [tex]\(4\)[/tex] units:
1. Calculate the Horizontal Extents: Move [tex]\(4\)[/tex] units left and right from the center to find the endpoints of the domain:
- The leftmost point: [tex]\(3 - 4 = -1\)[/tex]
- The rightmost point: [tex]\(3 + 4 = 7\)[/tex]
2. Determine the Domain: The domain is then the interval of x-values from [tex]\(-1\)[/tex] to [tex]\(7\)[/tex]:
[tex]\[
\text{Domain} = [-1, 7]
\][/tex]
To summarize, the domain is found by adding and subtracting the radius to the x-coordinate of the center, covering all x-values from the leftmost to the rightmost point of the circle.
