The Components of the Coordinate Plane

You may have used a coordinate plane before. In this section we’ll review how to identify the coordinates of points in the coordinate plane, and plot points in the coordinate plane.

The coordinate plane consists of a horizontal axis and a vertical axis, number lines that intersect at right angles. (They are perpendicular to each other.)

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.

Locations on the coordinate plane are described as 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 called the x-coordinate and the second value is the y-coordinate. The origin is the ordered pair [latex](0,0)[/latex]. The x-coordinate of a point describes its horizontal position, and the y-coordinate describes its vertical position.

Note: An ordered pair, such as [latex](3,2)[/latex], represents one single point on the graph. It is different than the ordered pair [latex](2,3)[/latex]. Two ordered pairs represent the same point only when their [latex]x[/latex]-coordinates are the same and their [latex]y[/latex]-coordinates are the same.

Consider the point below.

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.

Now move to the point and notice the number 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].

Let’s look at another example.

Example

Describe the point shown as an ordered pair.

Show Answer

From the origin, we need to move [latex]5[/latex] units to the right along the [latex]x[/latex]-axis to line up with the point.  So the [latex]x[/latex]-coordinate is [latex]5[/latex]. Then we need to move up [latex]2[/latex] units vertically (the direction of the [latex]y[/latex]-axis) to get to the point.  Therefore, the ordered pair [latex](5,2)[/latex] describes the point.

Describe the Point Shown as an Ordered Pair

Plotting Points in the Coordinate Plane

You can plot an ordered pair by starting at the origin. Move horizontally the number of units given by the [latex]x[/latex]-coordinate, and then vertically the number of units given by the [latex]y[/latex]-coordinate. The example that follows shows how to graph the ordered pair [latex](1,3)[/latex].

In the previous example, both the x- and y-coordinates were positive. When one (or both) of the coordinates of an ordered pair is negative, you will need to move in the negative direction along one or both axes. Consider the example below in which both coordinates are negative.

Example

Plot the point [latex](−4,−2)[/latex].

Show Answer

The x-coordinate is [latex]−4[/latex] because it comes first in the ordered pair. Start at the origin and move [latex]4[/latex] units in a negative direction (left) along the x-axis.

The y-coordinate is [latex]−2[/latex] because it comes second in the ordered pair. Now move [latex]2[/latex] units in a negative direction (down). If you look over to the y-axis, you should be lined up with [latex]−2[/latex] on that axis.

Now draw a point at that location and label it.

The steps for plotting a point are summarized below.

Plotting Points on the Coordinate Plane