Part A: What is the equation of a circle with center (−2, 4) and a diameter of 6 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)
Part A:
The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
In this case, the center is (-2, 4) and the diameter is 6 units. The radius is half of the diameter, so the radius is 6/2 = 3 units.
Plugging in the values into the equation, we get:
(x - (-2))^2 + (y - 4)^2 = 3^2
(x + 2)^2 + (y - 4)^2 = 9
Therefore, the equation of the circle is (x + 2)^2 + (y - 4)^2 = 9.
Part B:
To graph the circle by hand, follow these steps:
1. Plot the center: (-2, 4)
2. Use the radius of 3 units to mark points on the x-axis and y-axis. From the center, move 3 units to the left and right on the x-axis and mark those points. Similarly, move 3 units up and down on the y-axis and mark those points.
3. Connect the points to form a circle.
4. Optionally, label the center and any other relevant points.
Part C:
The domain of the circle is the set of x-coordinates of all points on the circle. In this case, the x-coordinate can vary from -2 - 3 = -5 to -2 + 3 = 1 (considering the center and the radius) because the circle extends 3 units to the left and 3 units to the right of the center.
Therefore, the domain of the circle is -5 ≤ x ≤ 1. To determine the domain, you need to consider the center and the radius of the circle.
Part A: To find the equation of a circle with a center (h, k) and a diameter of d, we can use the formula: (x - h)^2 + (y - k)^2 = (d/2)^2.
Given that the center is (-2, 4) and the diameter is 6 units, we can substitute into the formula:
(x - (-2))^2 + (y - 4)^2 = (6/2)^2
(x + 2)^2 + (y - 4)^2 = 9
Therefore, the equation of the circle is (x + 2)^2 + (y - 4)^2 = 9.
Part B: To graph the circle by hand on the coordinate plane:
1. Plot the center point (-2, 4) on the coordinate plane. This will be the center of the circle.
2. Determine the radius of the circle, which is half of the diameter. In this case, the radius is 6/2 = 3 units.
3. Draw a straight line vertically and horizontally from the center point. The distance along both lines should be equal to the radius (3 units).
4. Plot points along these lines in both directions, up and down, left and right, until you reach the distance equal to the radius.
5. Connect these points smoothly using a curved line. This curve represents the circle.
6. Label the circle with its equation: (x + 2)^2 + (y - 4)^2 = 9.
Part C: The domain of a circle is the set of all x-values that lie on or within the circle. In other words, it is the range of x-values that the circle occupies.
To determine the domain of the circle, we need to look at the x-coordinates of the points on the circle. In this case, the equation is (x + 2)^2 + (y - 4)^2 = 9. By rearranging the equation, we can isolate the x-values:
(x + 2)^2 = 9 - (y - 4)^2
x + 2 = sqrt(9 - (y - 4)^2)
x = -2 + sqrt(9 - (y - 4)^2)
The domain of the circle is the range of x-values that satisfy this equation. Since the square root of a non-negative number is always non-negative, we do not have any restrictions on the x-values. Therefore, the domain of the circle is the set of all real numbers, or (-∞, ∞).
Part A:
To find the equation of a circle with a given center and diameter, we can use the standard form equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center of the circle and r represents the radius.
Given that the center of the circle is (-2, 4) and the diameter is 6 units, we can find the radius by dividing the diameter by 2:
r = 6/2 = 3
Now we can substitute the values into the equation:
(x - (-2))^2 + (y - 4)^2 = 3^2
Simplifying further:
(x + 2)^2 + (y - 4)^2 = 9
Therefore, the equation of the circle is (x + 2)^2 + (y - 4)^2 = 9.
Part B:
To graph the circle by hand on the coordinate plane, follow these steps:
1. Plot the center point: Start by plotting the center of the circle, (-2, 4), on the coordinate plane.
2. Determine the radius: With the given diameter of 6 units, the radius is half of that, which is 3 units. Mark this distance in all four directions from the center point. So, from the center, mark points (-5, 4), (-2, 1), (1, 4), and (-2, 7).
3. Connect the dots: Use a smooth, continuous curve to connect the marked points, creating a circular shape. The curve should pass through all the marked points.
4. Label the circle: You can label the circle with its equation, which is (x + 2)^2 + (y - 4)^2 = 9, as obtained from Part A.
Part C:
The domain of a circle is the set of all possible x-values (horizontal coordinates) that lie on the circle.
To determine the domain of the circle, we need to consider the x-coordinates of the points that lie on the circle.
In this case, since the center of the circle is (-2, 4) and the radius is 3 units, the circle will extend horizontally 3 units to the left and 3 units to the right of the center point.
Therefore, the domain of the circle is the set of x-values that fall within this range, which can be written as:
-2 - 3 ≤ x ≤ -2 + 3
Simplifying,
-5 ≤ x ≤ 1
So, the domain of the circle is -5 to 1, inclusive.