## Understanding the Graph of (x+3)^2 + (y-2)^2 = 16

The equation (x+3)^2 + (y-2)^2 = 16 represents a circle in the Cartesian coordinate system. Here's how to understand its graph:

### The Standard Form of a Circle Equation

The general equation of a circle with center (h, k) and radius r is:

**(x - h)^2 + (y - k)^2 = r^2**

### Extracting Information from the Equation

Comparing our given equation to the standard form, we can identify the following:

**Center:**(h, k) = (-3, 2)**Radius:**r = √16 = 4

### Graphing the Circle

**Plot the center:**Locate the point (-3, 2) on the coordinate plane.**Mark the radius:**From the center, move 4 units up, down, left, and right. These points will be on the circle's circumference.**Draw the circle:**Connect the points you marked in step 2 to form a smooth circle.

### Key Properties of the Circle

**Center:**The point (-3, 2) is the center of the circle.**Radius:**The circle has a radius of 4 units.**Diameter:**The diameter of the circle is twice the radius, which is 8 units.**Circumference:**The circumference of the circle can be calculated using the formula C = 2πr, where r is the radius. Therefore, the circumference is 8π units.

### Understanding the Equation

The equation itself represents the distance formula applied to every point on the circle. For any point (x, y) on the circle, the distance from that point to the center (-3, 2) will always be 4 units, which is the radius.

### Applications

The equation of a circle finds applications in various fields, including:

**Geometry:**To understand geometric properties like area, circumference, and relationships between points and lines.**Physics:**To model the movement of objects in circular motion.**Engineering:**To design circular structures and systems.**Computer graphics:**To represent and manipulate circular shapes in digital environments.

By understanding the standard form of the circle equation and the key properties of circles, we can effectively graph and interpret equations like (x+3)^2 + (y-2)^2 = 16.