To understand and graph the equation [tex]\((x-4)^2 + (y-3)^2 = 36\)[/tex], let's break it down step-by-step.
### Step 1: Recognize the Equation
The equation [tex]\((x-4)^2 + (y-3)^2 = 36\)[/tex] represents a circle in the coordinate plane.
### Step 2: Identify the Components of the Circle
This circle equation is in the standard form [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.
- The center of this circle is [tex]\((h, k) = (4, 3)\)[/tex].
- The radius [tex]\(r\)[/tex] is given by the square root of 36, which means [tex]\(r = 6\)[/tex].
### Step 3: Determine the Domain and Range
The domain and range for a circle are determined by the extent of the circle along the [tex]\(x\)[/tex]-axis and [tex]\(y\)[/tex]-axis, respectively.
#### Domain:
- The circle is centered at [tex]\(x = 4\)[/tex] and has a radius of 6.
- Therefore, the [tex]\(x\)[/tex]-values will extend from [tex]\(4 - 6 = -2\)[/tex] to [tex]\(4 + 6 = 10\)[/tex].
Hence, the domain is:
[tex]\[
[-2, 10]
\][/tex]
#### Range:
- The circle is centered at [tex]\(y = 3\)[/tex] and has a radius of 6.
- Therefore, the [tex]\(y\)[/tex]-values will extend from [tex]\(3 - 6 = -3\)[/tex] to [tex]\(3 + 6 = 9\)[/tex].
Hence, the range is:
[tex]\[
[-3, 9]
\][/tex]
### Step 4: Graph the Circle
To graph the circle:
1. Plot the center of the circle at [tex]\((4, 3)\)[/tex].
2. Use the radius [tex]\(6\)[/tex] to mark points that are 6 units away from the center in all directions (up, down, left, right).
3. Draw a smooth circular curve through these points.
### Final Answer
- Equation: [tex]\((x-4)^2 + (y-3)^2 = 36\)[/tex]
- Domain: [tex]\([-2, 10]\)[/tex]
- Range: [tex]\([-3, 9]\)[/tex]
The graph of this equation is a circle centered at [tex]\((4, 3)\)[/tex] with a radius of 6, extending from [tex]\([-2, 10]\)[/tex] along the [tex]\(x\)[/tex]-axis and [tex]\([-3, 9]\)[/tex] along the [tex]\(y\)[/tex]-axis.
