Exercises 103–105 will help you prepare for the material covered in the next section. Use a rectangular coordinate system to graph the circle with center (1, -1) and radius 1.

Key Concepts

Rectangular Coordinate System

A rectangular coordinate system, also known as the Cartesian coordinate system, consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point in this system is represented by an ordered pair (x, y), indicating its position relative to these axes. Understanding this system is essential for graphing geometric shapes, such as circles, as it provides a framework for visualizing their locations and dimensions.

Circle Equation

The standard equation of a circle in a rectangular coordinate system is given by (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is its radius. This equation allows us to determine all the points (x, y) that lie on the circle. For the given problem, with a center at (1, -1) and a radius of 1, the equation becomes (x - 1)² + (y + 1)² = 1, which is crucial for accurately plotting the circle.

Graphing Circles

Graphing a circle involves plotting its center and then using the radius to determine the circle's extent in all directions. Starting from the center point, you can mark points at a distance equal to the radius in the four cardinal directions (up, down, left, right) and then sketch a smooth curve connecting these points. This visual representation helps in understanding the properties of the circle and its relationship to the coordinate system.