Question

part a: what is the equation of a circle with center (4, -5) and a diameter of 12 units? show all necessary steps. (4 points)
part b: explain how to graph the circle by hand on the coordinate plane. (3 points)
part c: what is the domain of the circle? explain how to determine the domain. (3 points)

Explanation:

Step1: Find the radius

The radius $r$ is half of the diameter. Given diameter $d = 12$, so $r=\frac{d}{2}=\frac{12}{2}=6$.

Step2: Write the standard - form equation of a circle

The standard - form equation of a circle is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle. Here, $h = 4$, $k=-5$ and $r = 6$. So the equation is $(x - 4)^2+(y + 5)^2=36$.

Step3: Explain graphing the circle

1. Locate the center $(4,-5)$ on the coordinate plane.
2. From the center, move 6 units up, down, left and right to mark four points on the circle.
3. Sketch the circle passing through these four points.

Step4: Determine the domain

The $x$ - coordinate of the center is $h = 4$ and the radius is $r = 6$. The left - most $x$ value is $h - r=4 - 6=-2$ and the right - most $x$ value is $h + r=4+6 = 10$. So the domain is $[-2,10]$.

Answer:

Part A: $(x - 4)^2+(y + 5)^2=36$
Part B: Locate the center $(4,-5)$. Move 6 units in all four directions from the center and sketch the circle through the four points.
Part C: Domain: $[-2,10]$. Found by calculating $h - r$ and $h + r$ where $h = 4$ and $r = 6$.