Identifying Points on a Rectangular Coordinate System

Learning Outcomes

Identify points on a graph

In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the x-coordinate of a point on a graph, read the number on the x-axis directly above or below the point. To identify the y-coordinate of a point, read the number on the y-axis directly to the left or right of the point. Remember, to write the ordered pair using the correct order $\left(x,y\right)$ .

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 $(0, 0)$ and move right along the x-axis until you are under the point. Look at the label on the x-axis. The $4$ indicates that, from the origin, you have traveled four units to the right along the x-axis. This is the x-coordinate, the first number in the ordered pair.

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

Let’s look at another example.

Example

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.

Show Solution

Begin at the origin and move along the x-axis. This is the x-coordinate and is written first in the ordered pair.

[latex](5,y)[/latex]

Move from 5 up to the ordered pair and read the number on the y-axis. This is the y-coordinate and is written second in the ordered pair.

$(5,2)$

Answer

The point shown as an ordered pair is $(5,2)$ .

Describe the point shown as an ordered pair

example

Name the ordered pair of each point shown:

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 2, 4” is labeled C. The point “ordered pair -3, 3” is labeled A. The point “ordered pair -1, -3” is labeled B. The point “ordered pair 4, -4” is labeled D.

Solution
Point A is above $-3$ on the $x\text{-axis}$ , so the $x\text{-coordinate}$ of the point is $-3$ . The point is to the left of $3$ on the $y\text{-axis}$ , so the $y\text{-coordinate}$ of the point is $3$ . The coordinates of the point are $\left(-3,3\right)$ .

Point B is below $-1$ on the $x\text{-axis}$ , so the $x\text{-coordinate}$ of the point is $-1$ . The point is to the left of $-3$ on the $y\text{-axis}$ , so the $y\text{-coordinate}$ of the point is $-3$ . The coordinates of the point are $\left(-1,-3\right)$ .

Point C is above $2$ on the $x\text{-axis}$ , so the $x\text{-coordinate}$ of the point is $2$ . The point is to the right of $4$ on the $y\text{-axis}$ , so the $y\text{-coordinate}$ of the point is $4$ . The coordinates of the point are $\left(2,4\right)$ .

Point D is below $4$ on the $x-\text{axis}$ , so the $x\text{-coordinate}$ of the point is $4$ . The point is to the right of $-4$ on the $y\text{-axis}$ , so the $y\text{-coordinate}$ of the point is $-4$ . The coordinates of the point are $\left(4,-4\right)$ .

try it

example

Name the ordered pair of each point shown:

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The point “ordered pair 3, 0” is labeled C. The point “ordered pair 0, 1” is labeled D. The point “ordered pair -4, 0” is labeled A. The point “ordered pair 0, -2” is labeled B.

Show Solution

Solution

Point A is on the x-axis at $x=-4$	The coordinates of point A are $\left(-4,0\right)$
Point B is on the y-axis at $y=-2$	The coordinates of point B are $\left(0,-2\right)$
Point C is on the x-axis at $x=3$	The coordinates of point C are $\left(3,0\right)$
Point D is on the y-axis at $y=1$	The coordinates of point D are $\left(0,1\right)$

try it

Watch the following video for another example of how to determine the ordered pair for points on the coordinate plane.

Module 15: Graphs