Exploring the Equation: (x+1)^2 + (y3)^2 = 16
This equation represents a circle in the Cartesian coordinate system. Let's break down its components and understand how to graph it.
Understanding the Equation
The equation is in standard form for a circle, which 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.
In our case:
 (h, k) = (1, 3) This means the center of the circle is at the point (1, 3).
 r^2 = 16 Therefore, the radius of the circle is r = 4 (the square root of 16).
Graphing the Circle

Locate the Center: Plot the point (1, 3) on the coordinate plane.

Draw the Radius: From the center, draw a line segment of length 4 units in all directions (up, down, left, right). This will give you four points on the circle.

Connect the Points: Draw a smooth curve connecting these points to form the circle.
Key Properties of the Circle
 Center: (1, 3)
 Radius: 4
 Diameter: 8
 Circumference: 8π
 Area: 16π
Applications of Circle Equations
Circle equations are used extensively in various fields, including:
 Geometry: Understanding circles and their properties.
 Physics: Describing the motion of objects in circular paths.
 Engineering: Designing circular structures and objects.
 Computer Graphics: Creating circles and other shapes in digital environments.
Understanding the standard form of a circle equation and its components allows us to analyze and visualize circles effectively, paving the way for further exploration and application of this fundamental geometric concept.