%5E2%2B(y%2B1)%5E2%3D16.jpeg "(x-4)^2+(y+1)^2=16")
Exploring the Equation: (x-4)^2 + (y+1)^2 = 16
This equation represents a circle in the Cartesian coordinate system. Let's delve into its characteristics and how to interpret it.
Understanding the Standard Form of a Circle Equation
The general equation for a circle is:
(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 Given Equation: (x-4)^2 + (y+1)^2 = 16
Comparing it to the general form, we can identify:
- Center: (h, k) = (4, -1)
- Radius: r = √16 = 4
Therefore, the equation describes a circle centered at (4, -1) with a radius of 4.
Visualizing the Circle
To visualize the circle, follow these steps:
- Locate the center: Plot the point (4, -1) on the coordinate plane.
- Mark the radius: From the center, move 4 units to the right, left, up, and down. These points will be on the circle's circumference.
- Connect the points: Connect the points you marked to form a smooth circle.
Key Takeaways
- The equation (x-4)^2 + (y+1)^2 = 16 defines a circle with a specific center and radius.
- Understanding the standard form of a circle equation allows you to readily extract information about its properties.
- Visualizing the circle helps in understanding its location and size within the coordinate system.
This simple equation encapsulates a wealth of information about a circle, highlighting the power of mathematical equations to describe geometric shapes and their characteristics.