To determine the effect of shifting a circle right by 3 units, let's first identify the center of the given circle and understand how the shift impacts its coordinates.
The given equation of the circle is:
[tex]\[
(x+3)^2 + (y-2)^2 = 36
\][/tex]
This is in the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] represents the center of the circle and [tex]\(r\)[/tex] is the radius. By comparing, we can identify the center of the given circle as:
[tex]\[
(h, k) = (-3, 2)
\][/tex]
Now, let's understand what happens when we shift the circle right by 3 units. Shifting a circle to the right impacts the [tex]\(x\)[/tex]-coordinate of the center, but does not change the [tex]\(y\)[/tex]-coordinate. Specifically, the [tex]\(x\)[/tex]-coordinate will increase by the amount of the shift, which is 3 units in this case.
Calculating the new center after shifting:
[tex]\[
(-3 + 3, 2) = (0, 2)
\][/tex]
This means that after shifting the circle to the right by 3 units, the [tex]\(x\)[/tex]-coordinate of the center has increased from [tex]\(-3\)[/tex] to [tex]\(0\)[/tex], while the [tex]\(y\)[/tex]-coordinate remains unchanged at 2.
So, the correct answer is:
[tex]\[
\boxed{\text{C. The } x\text{-coordinate of the center of the circle increases by 3.}}
\][/tex]
