Respuesta :
Step-by-step explanation:
Part A: To find the equation of a circle with center (-2, 4) and a diameter of 6 units, we can use the standard form of the equation of a circle, which is (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 is (-2, 4), we substitute these values into the equation:
(x - (-2))^2 + (y - 4)^2 = r^2
Simplifying, we have:
(x + 2)^2 + (y - 4)^2 = r^2
Now, we need to find the radius. The diameter is given as 6 units, so the radius is half of the diameter, which is 6/2 = 3 units.
Substituting this value into the equation, we get:
(x + 2)^2 + (y - 4)^2 = 3^2
Simplifying further, we have:
(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 of the circle (-2, 4) on the coordinate plane.
2. Determine the radius, which is 3 units in this case.
3. From the center point, measure 3 units in all directions (up, down, left, and right) and mark those points on the graph. These points will help you draw the circle.
4. Connect the marked points with a smooth curve to form the circle.
Part C: The domain of a circle is the set of all x-coordinates of points on the circle. In other words, it represents the possible values for the x-coordinate of any point on the circle.
To determine the domain of the circle, we consider the x-coordinates of the points on the circle. In this case, the center of the circle is (-2, 4) and the radius is 3 units.
The x-coordinate of the center is -2, and the radius extends 3 units to the left and right of the center. Therefore, the domain of the circle is the range of x-values that can be obtained by subtracting or adding the radius to the x-coordinate of the center.
In this case, the domain would be -2 - 3 to -2 + 3, which simplifies to -5 to 1. Therefore, the domain of the circle is [-5, 1].
Answer:
A) [tex] (x + 2)^2 + (y - 4)^2 = 9 [/tex]
C) Domain: [tex] \boxed{[-5, 1]} [/tex] [tex] \{ x | -5 \leq x \leq 1 \}[/tex]
Step-by-step explanation:
Part A: Equation of the Circle
To find the equation of a circle with center [tex](-2, 4)[/tex] and a diameter of [tex]6[/tex] units, follow these steps:
Identify the Radius:
The diameter of the circle is [tex]6[/tex] units, so the radius [tex]r[/tex] is half of the diameter:
[tex] r = \dfrac{6}{2} = 3 \text{ units} [/tex]
Use the Standard Form of the Circle's Equation:
The standard form of the equation of a circle with center [tex](h, k)[/tex] and radius [tex]r[/tex] is:
[tex] \large\boxed{\boxed{(x - h)^2 + (y - k)^2 = r^2}} [/tex]
Substitute the Given Center and Radius:
Substitute [tex]h = -2[/tex], [tex]k = 4[/tex], and [tex]r = 3[/tex] into the standard form:
[tex] (x + 2)^2 + (y - 4)^2 = 3^2 [/tex]
[tex] (x + 2)^2 + (y - 4)^2 = 9 [/tex]
Therefore, the equation of the circle is:
[tex] \boxed{(x + 2)^2 + (y - 4)^2 = 9} [/tex]
Part B: Graphing the Circle by Hand
To graph the circle on a coordinate plane given its equation [tex](x + 2)^2 + (y - 4)^2 = 9[/tex], follow these steps:
Identify the Center and Radius:
The center of the circle is [tex](-2, 4)[/tex] and the radius is [tex]3[/tex] units.
Plot the Center:
- Plot the point [tex](-2, 4)[/tex] on the coordinate plane. This is the center of the circle.
- Plot the Points on the Circle:
- Use the radius of [tex]3[/tex] units to plot points that are [tex]3[/tex] units away from the center along the x-axis and y-axis. These points will help us to sketch the circle.
Sketch the Circle:
Use the plotted points to draw a smooth curve that connects all the points around the center. This curve represents the circle.
Part C: Domain of the Circle
The domain of a circle is the set of all [tex]x[/tex]-values that lie on the circle.
The domain [tex]D[/tex] of a circle can be determined by considering the [tex]x[/tex]-values where the circle intersects the [tex]x[/tex]-axis.
The general formula for the domain of a circle centered at [tex](h, k)[/tex] with radius [tex]r[/tex] is:
[tex] \large\boxed{\boxed{D = [h - r, h + r]}} [/tex]
In our case:
- Center [tex](h, k) = (-2, 4)[/tex]
- Radius [tex]r = 3[/tex]
Substitute into the Formula
Substitute these values into the formula to find the domain [tex]D[/tex] of the circle:
[tex] D = [-2 - 3, -2 + 3] [/tex]
[tex] D = [-5, 1] [/tex]
Therefore, the domain of the circle defined by [tex](x + 2)^2 + (y - 4)^2 = 9[/tex] is:
[tex] \boxed{[-5, 1]} [/tex]
