Cartesian Coordinates

Definition

Cartesian coordinates are a system used to locate points on a two-dimensional plane. This system uses two perpendicular axes, the x-axis and the y-axis, to define a unique position for any point within the coordinate plane.

5 Must Know Facts For Your Next Test

1. The x-axis represents the horizontal position of a point, while the y-axis represents the vertical position.
2. Positive x-values are to the right of the origin, and negative x-values are to the left of the origin.
3. Positive y-values are above the origin, and negative y-values are below the origin.
4. The origin, where the x-axis and y-axis intersect, is represented by the ordered pair (0, 0).
5. The four quadrants of the coordinate plane are labeled in a counterclockwise direction, starting with the top-right quadrant as Quadrant I.

Review Questions

• Explain how the Cartesian coordinate system is used to represent and locate points on a two-dimensional plane.
  • The Cartesian coordinate system uses two perpendicular axes, the x-axis and the y-axis, to define a unique position for any point within the coordinate plane. Each point is represented by an ordered pair (x, y), where the x-value indicates the horizontal position and the y-value indicates the vertical position. The intersection of the x-axis and y-axis is the origin, represented by the ordered pair (0, 0). The coordinate plane is divided into four quadrants, labeled I, II, III, and IV, which are used to further describe the location of a point based on the signs of its x and y coordinates.
• Describe the relationship between the signs of the x and y coordinates and the location of a point in the four quadrants of the coordinate plane.
  • The signs of the x and y coordinates determine the location of a point in the four quadrants of the coordinate plane. In Quadrant I, both the x and y coordinates are positive. In Quadrant II, the x-coordinate is negative, and the y-coordinate is positive. In Quadrant III, both the x and y coordinates are negative. In Quadrant IV, the x-coordinate is positive, and the y-coordinate is negative. This relationship between the signs of the coordinates and the quadrant location is crucial for understanding and working with Cartesian coordinates, especially when graphing and analyzing the positions of points, lines, and other geometric shapes.
• Analyze how the Cartesian coordinate system is used in the context of polar coordinates and the graphs of polar functions.
  • The Cartesian coordinate system and the polar coordinate system are closely related, as they both provide ways to represent and locate points in a two-dimensional plane. While the Cartesian system uses the x-axis and y-axis to define a point's position, the polar system uses the angle (θ) and the distance from the origin (r) to define a point's location. When working with polar coordinates and graphs of polar functions, it is often necessary to convert between the Cartesian and polar coordinate systems to analyze the properties and behaviors of the functions. Understanding the relationships between these two coordinate systems and how they are used in the context of 8.4 Polar Coordinates: Graphs is crucial for successfully working with and interpreting polar functions.

Related terms

Coordinate Plane:

A two-dimensional plane that is divided into four quadrants by the x-axis and y-axis, used to represent and plot points, lines, and other geometric shapes.

Quadrants:

The four regions created by the intersection of the x-axis and y-axis in a coordinate plane, labeled I, II, III, and IV.

Ordered Pair: A set of two numbers that represents the location of a point on a coordinate plane, written in the form (x, y).