1. Equation: \((x + 2)^2 + (y - 4)^2 = 9\).
2. Graphing: Plot center at \((-2, 4)\), mark points with radius 3 in all directions, connect for circle.
3. Domain: X-coordinate range from \(-5\) to \(1\), so domain is \([-5, 1]\).
Part A: Equation of the Circle
The equation of a circle with center \((h, k)\) and radius \(r\) is given by \((x - h)^2 + (y - k)^2 = r^2\). In this case, the center is \((-2, 4)\) and the diameter is 6 units, so the radius (\(r\)) is half of the diameter, which is \(r = \frac{6}{2} = 3\).
Substitute the values into the equation:
\[(x + 2)^2 + (y - 4)^2 = 3^2\]
Simplify:
\[(x + 2)^2 + (y - 4)^2 = 9\]
So, the equation of the circle is \((x + 2)^2 + (y - 4)^2 = 9\).
Part B: Graphing the Circle
To graph the circle by hand, follow these steps:
1. Plot the center point \((-2, 4)\).
2. Use the radius of 3 units to mark points in all four directions (up, down, left, and right) from the center.
3. Connect these points smoothly to sketch the circle.
Part C: Domain of the Circle
The domain of a circle is the set of all x-coordinates of its points. In this case, since the center is at \((-2, 4)\), the x-coordinate ranges from \(-2 - 3\) to \(-2 + 3\), i.e., \(-5 \leq x \leq 1\). Therefore, the domain of the circle is \([-5, 1]\).