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### Part A: What is the equation of a circle with center (-2, 4) and a diameter of 6 units? (4 points)
To derive the equation of the circle, follow these steps:
1. Identify the center and diameter of the circle.
- Center: [tex]\((-2, 4)\)[/tex]
- Diameter: 6 units
2. Calculate the radius of the circle, which is half of the diameter.
[tex]\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{6}{2} = 3 \][/tex]
3. Recall the general equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex]:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
4. Substitute the given center and radius into the general equation.
- [tex]\(h = -2\)[/tex]
- [tex]\(k = 4\)[/tex]
- [tex]\(r = 3\)[/tex]
5. Plug in the values:
[tex]\[ (x - (-2))^2 + (y - 4)^2 = 3^2 \][/tex]
Thus, the equation of the circle is:
[tex]\[ (x + 2)^2 + (y - 4)^2 = 9 \][/tex]
### Part B: Explain how to graph the circle by hand on the coordinate plane. (3 points)
To graph the circle, follow these steps:
1. Plot the Center:
Begin by plotting the center of the circle at the point [tex]\((-2, 4)\)[/tex] on the coordinate plane.
2. Determine the Radius:
The radius of the circle is 3 units. From the center [tex]\((-2, 4)\)[/tex], measure a distance of 3 units in all four directions—up, down, left, and right.
3. Mark Key Points:
- Move 3 units up from [tex]\((-2, 4)\)[/tex] to plot the point [tex]\((-2, 7)\)[/tex].
- Move 3 units down from [tex]\((-2, 4)\)[/tex] to plot the point [tex]\((-2, 1)\)[/tex].
- Move 3 units left from [tex]\((-2, 4)\)[/tex] to plot the point [tex]\((-5, 4)\)[/tex].
- Move 3 units right from [tex]\((-2, 4)\)[/tex] to plot the point [tex]\((1, 4)\)[/tex].
4. Draw the Circle:
Connect these points with a smooth, continuous curve, ensuring it is an equal distance (3 units) from the center in all directions, forming a circle.
### Part C: What is the domain of the circle? Explain how to determine the domain. (3 points)
To find the domain of the circle, which is the set of all x-values that the circle encompasses:
1. Understand the Horizontal Span:
The domain is determined by how far the circle extends horizontally from the center of the circle.
2. Use the Center and Radius:
- The center of the circle is at [tex]\((-2, 4)\)[/tex].
- The radius is 3 units.
3. Calculate the Horizontal Extremes:
- The furthest point to the left is: [tex]\[x = -2 - 3 = -5\][/tex]
- The furthest point to the right is: [tex]\[x = -2 + 3 = 1\][/tex]
Thus, the domain of the circle is:
[tex]\[ (-5, 1) \][/tex]
You can state this as:
"The domain is determined by how far the circle extends horizontally from the center. The radius is 3, so the circle extends from [tex]\(x = -2 - 3\)[/tex] to [tex]\(x = -2 + 3\)[/tex], thus the domain is [tex]\((-5, 1)\)[/tex]."
To derive the equation of the circle, follow these steps:
1. Identify the center and diameter of the circle.
- Center: [tex]\((-2, 4)\)[/tex]
- Diameter: 6 units
2. Calculate the radius of the circle, which is half of the diameter.
[tex]\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{6}{2} = 3 \][/tex]
3. Recall the general equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex]:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
4. Substitute the given center and radius into the general equation.
- [tex]\(h = -2\)[/tex]
- [tex]\(k = 4\)[/tex]
- [tex]\(r = 3\)[/tex]
5. Plug in the values:
[tex]\[ (x - (-2))^2 + (y - 4)^2 = 3^2 \][/tex]
Thus, the equation of the circle is:
[tex]\[ (x + 2)^2 + (y - 4)^2 = 9 \][/tex]
### Part B: Explain how to graph the circle by hand on the coordinate plane. (3 points)
To graph the circle, follow these steps:
1. Plot the Center:
Begin by plotting the center of the circle at the point [tex]\((-2, 4)\)[/tex] on the coordinate plane.
2. Determine the Radius:
The radius of the circle is 3 units. From the center [tex]\((-2, 4)\)[/tex], measure a distance of 3 units in all four directions—up, down, left, and right.
3. Mark Key Points:
- Move 3 units up from [tex]\((-2, 4)\)[/tex] to plot the point [tex]\((-2, 7)\)[/tex].
- Move 3 units down from [tex]\((-2, 4)\)[/tex] to plot the point [tex]\((-2, 1)\)[/tex].
- Move 3 units left from [tex]\((-2, 4)\)[/tex] to plot the point [tex]\((-5, 4)\)[/tex].
- Move 3 units right from [tex]\((-2, 4)\)[/tex] to plot the point [tex]\((1, 4)\)[/tex].
4. Draw the Circle:
Connect these points with a smooth, continuous curve, ensuring it is an equal distance (3 units) from the center in all directions, forming a circle.
### Part C: What is the domain of the circle? Explain how to determine the domain. (3 points)
To find the domain of the circle, which is the set of all x-values that the circle encompasses:
1. Understand the Horizontal Span:
The domain is determined by how far the circle extends horizontally from the center of the circle.
2. Use the Center and Radius:
- The center of the circle is at [tex]\((-2, 4)\)[/tex].
- The radius is 3 units.
3. Calculate the Horizontal Extremes:
- The furthest point to the left is: [tex]\[x = -2 - 3 = -5\][/tex]
- The furthest point to the right is: [tex]\[x = -2 + 3 = 1\][/tex]
Thus, the domain of the circle is:
[tex]\[ (-5, 1) \][/tex]
You can state this as:
"The domain is determined by how far the circle extends horizontally from the center. The radius is 3, so the circle extends from [tex]\(x = -2 - 3\)[/tex] to [tex]\(x = -2 + 3\)[/tex], thus the domain is [tex]\((-5, 1)\)[/tex]."
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