%5E2%2By%5E2%3D9.jpeg "(x-3)^2+y^2=9")
Exploring the Equation: (x-3)^2 + y^2 = 9
The equation (x-3)^2 + y^2 = 9 represents a circle in the coordinate plane. Let's break down why and understand its key features.
Understanding the Standard Form
The equation is in the standard form of a circle's equation:
(x - h)^2 + (y - k)^2 = r^2
Where:
- (h, k) represents the center of the circle
- r represents the radius of the circle
Analyzing the Equation
In our case:
- (h, k) = (3, 0), indicating the center of the circle is at the point (3, 0).
- r^2 = 9, so r = 3, indicating the radius of the circle is 3 units.
Key Features
- Center: The center of the circle is located at (3, 0).
- Radius: The radius of the circle is 3 units.
- Symmetry: The circle is symmetrical about both the x-axis and the y-axis.
- Domain and Range: The domain of the circle is [0, 6] and the range of the circle is [-3, 3].
Graphing the Circle
To graph the circle, follow these steps:
- Locate the center: Plot the point (3, 0) on the coordinate plane.
- Draw the radius: From the center, draw a line segment of length 3 units in all four directions (up, down, left, and right).
- Connect the points: Connect the endpoints of these radius lines to form a smooth circle.
Conclusion
The equation (x-3)^2 + y^2 = 9 describes a circle with a center at (3, 0) and a radius of 3 units. By understanding the standard form of a circle's equation, we can easily extract key information like the center, radius, and graph the circle accurately.