Cartesian Coordinates

Definition

Cartesian coordinates are a system used to locate points in a two-dimensional or three-dimensional space. This system uses a pair or trio of numerical coordinates to uniquely identify the position of a point relative to a fixed reference frame.

5 Must Know Facts For Your Next Test

1. Cartesian coordinates are essential for graphing functions, including parametric equations.
2. The $x$-coordinate represents the horizontal position, while the $y$-coordinate represents the vertical position of a point.
3. Positive $x$-values are to the right of the origin, and negative $x$-values are to the left.
4. Positive $y$-values are above the origin, and negative $y$-values are below.
5. Cartesian coordinates provide a standardized way to describe the location of points, which is crucial for various mathematical and scientific applications.

Review Questions

• Explain how Cartesian coordinates are used to graph parametric equations.
  • Parametric equations define the $x$ and $y$ coordinates of a point as functions of a third variable, typically denoted as $t$. To graph a parametric equation, you can use Cartesian coordinates to plot the points corresponding to different values of $t$. The $x$-coordinate is determined by the $x$-component function, while the $y$-coordinate is determined by the $y$-component function. By plotting these points on a coordinate plane, you can visualize the shape and behavior of the parametric curve.
• Describe how the sign of the $x$ and $y$ coordinates in a Cartesian coordinate system affects the location of a point.
  • The sign of the $x$ and $y$ coordinates in a Cartesian coordinate system determines the quadrant in which a point is located. Positive $x$ and positive $y$ values place a point in the first quadrant, positive $x$ and negative $y$ values place a point in the second quadrant, negative $x$ and negative $y$ values place a point in the third quadrant, and negative $x$ and positive $y$ values place a point in the fourth quadrant. Understanding the relationship between the sign of the coordinates and the quadrant location is crucial for interpreting and graphing points in the coordinate plane.
• Analyze how the origin and coordinate axes in a Cartesian coordinate system serve as a reference frame for locating points.
  • The origin, where the $x$ and $y$ axes intersect at the coordinates $(0, 0)$, acts as the central reference point in a Cartesian coordinate system. The coordinate axes, with the positive and negative directions, provide a standardized frame of reference for describing the position of a point relative to this origin. The $x$-axis represents the horizontal position, while the $y$-axis represents the vertical position. By specifying the $x$ and $y$ coordinates of a point, you can uniquely identify its location within the coordinate plane, which is essential for graphing functions, including parametric equations.

Related terms

Coordinate Plane: A two-dimensional plane where the position of a point is described by its $x$ and $y$ coordinates.

Coordinate Axes: The perpendicular reference lines, typically labeled $x$ and $y$, that define the coordinate plane.

Origin: The point where the $x$ and $y$ coordinate axes intersect, with coordinates $(0, 0)$.