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Part A: What is the equation of a circle with center (3, −6) and a diameter of 8 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)

1 answer

Part A: Equation of the Circle

To find the equation of a circle given its center and diameter, we follow these steps:

  1. Identify the Center: The center of the circle is given as (3, -6).
  2. Determine the Radius: The diameter of the circle is 8 units. The radius (r) is half the diameter: \[ r = \frac{8}{2} = 4 \text{ units} \]
  3. Use the Circle Equation: The standard form of the equation of a circle with center (h, k) and radius r is: \[ (x - h)^2 + (y - k)^2 = r^2 \] Substituting the center (h = 3, k = -6) and radius (r = 4): \[ (x - 3)^2 + (y + 6)^2 = 4^2 \] Simplifying this gives: \[ (x - 3)^2 + (y + 6)^2 = 16 \]

Thus, the equation of the circle is: \[ \boxed{(x - 3)^2 + (y + 6)^2 = 16} \]

Part B: Graphing the Circle by Hand

To graph the circle on the coordinate plane:

  1. Plot the Center: Start by plotting the center of the circle at the point (3, -6).
  2. Determine the Radius: From the center, measure out 4 units in all directions (up, down, left, right). This will give the points:
    • 4 units up: (3, -2)
    • 4 units down: (3, -10)
    • 4 units left: (-1, -6)
    • 4 units right: (7, -6)
  3. Draw the Circle: Connect these four points in a smooth, round shape to form the circle. It should be symmetrical around the center (3, -6).

Make sure to label the center and the extremes of the radius for clarity.

Part C: Domain of the Circle

The domain of a circle refers to the set of x-values that the points on the circle can take.

  1. Identify the Range of x-values: Since the center of the circle is at x = 3 and the radius is 4, the circle extends 4 units left and right from the center.
    • The leftmost point occurs at \(3 - 4 = -1\).
    • The rightmost point occurs at \(3 + 4 = 7\).
  2. Write the Domain: Therefore, the domain of the circle in interval notation is: \[ [-1, 7] \]

To determine the domain, you analyze how far the circle extends horizontally from its center point. Throughout this process, ensure your calculations adhere to the properties of the circle’s radius and center positioning.

Thus, the domain of the circle is: \[ \boxed{[-1, 7]} \]