To address the problem, we'll break it down into two main parts: graphing the circle and determining its domain and range.
### Graphing the Circle
1. Identify the Components of the Circle's Equation:
The given equation is [tex]\((x - 4)^2 + (y - 3)^2 = 36\)[/tex].
This is the standard form of a circle's equation, written as [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.
2. Determine the Center and Radius:
- The center of the circle [tex]\((h, k)\)[/tex] can be identified from the equation where the terms inside the parentheses are set equal to 0:
- [tex]\(h = 4\)[/tex]
- [tex]\(k = 3\)[/tex]
- The radius [tex]\(r\)[/tex] is the square root of 36:
- [tex]\(r = \sqrt{36} = 6\)[/tex]
3. Sketch the Circle:
- Draw the coordinate axes.
- Plot the center of the circle at the point [tex]\((4, 3)\)[/tex].
- With a radius of 6 units, draw a circle around this center. This circle includes all points that are exactly 6 units away from the center.
### Determine the Domain and Range
1. Domain:
- The circle extends horizontally from the leftmost point to the rightmost point.
- The horizontal span (domain) can be determined by considering the center and the radius:
- The leftmost point is at [tex]\(4 - 6 = -2\)[/tex].
- The rightmost point is at [tex]\(4 + 6 = 10\)[/tex].
- Thus, the domain of the circle is all [tex]\(x\)[/tex]-values between [tex]\(-2\)[/tex] and [tex]\(10\)[/tex]:
[tex]\[
\text{Domain: } -2 \leq x \leq 10
\][/tex]
2. Range:
- The circle extends vertically from the bottommost point to the topmost point.
- The vertical span (range) can be determined similarly:
- The bottommost point is at [tex]\(3 - 6 = -3\)[/tex].
- The topmost point is at [tex]\(3 + 6 = 9\)[/tex].
- Thus, the range of the circle is all [tex]\(y\)[/tex]-values between [tex]\(-3\)[/tex] and [tex]\(9\)[/tex]:
[tex]\[
\text{Range: } -3 \leq y \leq 9
\][/tex]
### Summary
- Center: [tex]\((4, 3)\)[/tex]
- Radius: [tex]\(6\)[/tex]
- Domain: [tex]\([-2, 10]\)[/tex]
- Range: [tex]\([-3, 9]\)[/tex]
By plotting and using the properties of the circle, we determine that:
- The circle centered at [tex]\((4, 3)\)[/tex] with a radius of [tex]\(6\)[/tex] has a domain of [tex]\(-2 \leq x \leq 10\)[/tex] and a range of [tex]\(-3 \leq y \leq 9\)[/tex].
