Functions: Background You’ll Need – Page 1

  • Identify key parts of the Cartesian coordinate system

 

The Components of the Coordinate Plane

The Cartesian coordinate system, refined by French mathematician René Descartes in 1637, serves as a framework for visualizing algebraic relationships. The coordinate plane can be used to plot points and graph lines. 

This system is comprised of a coordinate plane, which is essentially a grid formed by a horizontal axis, known as the [latex]x[/latex]-axis, and a vertical axis, referred to as the [latex]y[/latex]-axis. These axes intersect perpendicularly at a point called the origin, where both [latex]x[/latex] and [latex]y[/latex] coordinates are zero.

Cartesian coordinate system

In the Cartesian coordinate system, the horizontal axis in the coordinate plane is called the [latex]x[/latex]-axis. The vertical axis is called the [latex]y[/latex]-axis. The point at which the two axes intersect is called the origin. The origin is at [latex]0[/latex] on the [latex]x[/latex]-axis and [latex]0[/latex] on the [latex]y[/latex]-axis.

A graph with an x-axis running horizontally and a y-axis running vertically. The location where these axes cross is labeled the origin, and is the point zero, zero. The axes also divide the graph into four equal quadrants. The top right area is quadrant one. The top left area is quadrant two. The bottom left area is quadrant three. The bottom right area is quadrant four.

Understanding Quadrants

The coordinate plane is partitioned into four distinct regions by the [latex]x[/latex] and [latex]y[/latex] axes. The quadrants can be seen in the coordinate plane above. Each quadrant has different characteristics for the [latex]x[/latex] and [latex]y[/latex] coordinates.

  • Quadrant [latex]I[/latex]: Both [latex]x[/latex] and [latex]y[/latex] coordinates are positive.
  • Quadrant [latex]II[/latex]: [latex]x[/latex] is negative, while [latex]y[/latex] is positive.
  • Quadrant [latex]III[/latex]: Both [latex]x[/latex] and [latex]y[/latex] coordinates are negative.
  • Quadrant [latex]IV[/latex]: [latex]x[/latex] is positive, but [latex]y[/latex] is negative.

To ascertain the quadrant in which a point resides, one must examine the signs of its [latex]x[/latex] and [latex]y[/latex] coordinates.

Locating Points on the Coordinate Plane

In the Cartesian coordinate system, locations are defined by ordered pairs.  An ordered pair tells you the location of a point by relating the point’s location along the [latex]x[/latex]-axis (the first value of the ordered pair) and along the [latex]y[/latex]-axis (the second value of the ordered pair).

In an ordered pair, such as ([latex]x ,y[/latex]), the first value is the [latex]x[/latex]-coordinate and the second value is the [latex]y[/latex]-coordinate. The [latex]x[/latex]-coordinate specifies the horizontal distance from the origin, while the [latex]y[/latex]-coordinate indicates the vertical distance.

Consider the point below.

Grid with x-axis and y-axis. A blue dotted line extends from the origin, which is the point (0,0) along the horizontal x-axis to 4. A red dotted line goes up vertically from 4 on the x-axis to 3 on the y-axis. That point is labeled (4, 3).

 

To identify the location of this point, start at the origin ([latex]0, 0[/latex]) and move right along the [latex]x[/latex]-axis until you are under the point. Look at the label on the [latex]x[/latex]-axis. The [latex]4[/latex] indicates that, from the origin, you have traveled four units to the right along the [latex]x[/latex]-axis. This is the [latex]x[/latex]-coordinate, the first number in the ordered pair.

From [latex]4[/latex] on the [latex]x[/latex]-axis move up to the point and notice the number with which it aligns on the [latex]y[/latex]-axis. The [latex]3[/latex] indicates that, after leaving the [latex]x[/latex]-axis, you traveled [latex]3[/latex] units up in the vertical direction, the direction of the [latex]y[/latex]-axis. This number is the [latex]y[/latex]-coordinate, the second number in the ordered pair. With an [latex]x[/latex]-coordinate of [latex]4[/latex] and a [latex]y[/latex]-coordinate of [latex]3[/latex], you have the ordered pair ([latex]4, 3[/latex]).

This point lies in Quadrant [latex]I[/latex].

Describe the point shown as an ordered pair.

A point that is 2 spaces above the x-axis and 5 spaces to the right of the y-axis.