6.1 Plotting Ordered Pairs

Plotting Ordered Pairs

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Plot Points on a Rectangular Coordinate System

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy-plane or the ‘coordinate plane’.

The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The x-axis and the y-axis together form the rectangular coordinate system. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See Figure 6.1.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The top-right portion of the plane is labeled "I", the top-left portion of the plane is labeled "II", the bottom-left portion of the plane is labeled "III" and the bottom-right portion of the plane is labeled "IV". — **Figure 6.1** ‘Quadrant’ has the root ‘quad,’ which means ‘four.’

In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point.

The phrase ‘ordered pair’ means the order is important. What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is $open parenthesis 0, 0 close parenthesis$ . The point $open parenthesis 0, 0 close parenthesis$ has a special name. It is called the origin.

We use the coordinates to locate a point on the xy-plane. Let’s plot the point $open parenthesis 1, 3 close parenthesis$ as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through $x equals 1$ . Then, locate 3 on the y-axis and sketch a horizontal line through $y equals 3$ . Now, find the point where these two lines meet—that is the point with coordinates $open parenthesis 1, 3 close parenthesis$ .

A four-quadrant grid with the point (1, 3) marked in dark blue. A dashed gray vertical line rises from 1 on the x-axis and a dashed gray horizontal line runs across from 3 on the y-axis; the point sits where the two meet. The dashed lines are locating helpers rather than part of the graph. Adjustable parameters: x-coordinate (h) = 1, y-coordinate (k) = 3. Viewing window: x from -9.5 to 9.5, y from -5.87 to 5.88. — The book plots (1, 3): locate 1 on the x-axis and sketch a vertical line through it, locate 3 on the y-axis and sketch a horizontal line, and the point sits where the two helper lines meet. Both helpers are dashed here because they are not part of the graph — only the point is. Drag the x-coordinate to −4 and the point moves into Quadrant II; make both coordinates negative and it lands in Quadrant III; set the y-coordinate to 0 and the point sits exactly on the x-axis.

Rectangular coordinate graph with axes from -6 to 6. The point (1, 3) is marked. Dashed lines from x = 1 and y = 3 meet at the point; there is no arrow from the origin to x = 2.

Notice that the vertical line through $x equals 1$ and the horizontal line through $y equals 3$ are not part of the graph. We just used them to help us locate the point $open parenthesis 1, 3 close parenthesis$ .

Example 1

Plot each point in the rectangular coordinate system and identify the quadrant in which the point is located:

ⓐ $open parenthesis −5, 4 close parenthesis$ ⓑ $open parenthesis −3, −4 close parenthesis$ ⓒ $open parenthesis 2, −3 close parenthesis$ ⓓ $open parenthesis −2, 3 close parenthesis$ ⓔ $(3, \frac{5}{2})$ .

Show solution

The first number of the coordinate pair is the x-coordinate, and the second number is the y-coordinate.

ⓐ Since $x equals −5$ , the point is to the left of the y-axis. Also, since $y equals 4$ , the point is above the x-axis. The point $open parenthesis −5, 4 close parenthesis$ is in Quadrant II.
ⓑ Since $x equals −3$ , the point is to the left of the y-axis. Also, since $y equals −4$ , the point is below the x-axis. The point $open parenthesis −3, −4 close parenthesis$ is in Quadrant III.
ⓒ Since $x equals 2$ , the point is to the right of the y-axis. Since $y equals −3$ , the point is below the x-axis. The point $open parenthesis 2, −3 close parenthesis$ is in Quadrant lV.
ⓓ Since $x equals −2$ , the point is to the left of the y-axis. Since $y equals 3$ , the point is above the x-axis. The point $open parenthesis −2, 3 close parenthesis$ is in Quadrant II.
ⓔ Since $x equals 3$ , the point is to the right of the y-axis. Since $y = \frac{5}{2}$ , the point is above the x-axis. (It may be helpful to write $\frac{5}{2}$ as a mixed number or decimal.) The point $(3, \frac{5}{2})$ is in Quadrant I.

Figure 6.2

How do the signs affect the location of the points? You may have noticed some patterns as you graphed the points in the previous example.

For the point in Figure 6.2 in Quadrant IV, what do you notice about the signs of the coordinates? What about the signs of the coordinates of points in the third quadrant? The second quadrant? The first quadrant?

Can you tell just by looking at the coordinates in which quadrant the point $open parenthesis −2, 5 close parenthesis$ is located? In which quadrant is $open parenthesis 2, −5 close parenthesis$ located?

What if one coordinate is zero as shown in Figure 6.3? Where is the point $open parenthesis 0, 4 close parenthesis$ located? Where is the point $open parenthesis −2, 0 close parenthesis$ located?

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. Points (0, 4) and (negative 2, 0) are plotted and labeled. — **Figure 6.3**

The point $open parenthesis 0, 4 close parenthesis$ is on the y-axis and the point $open parenthesis −2, 0 close parenthesis$ is on the x-axis.

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, when you write the ordered pair use the correct order, $(x, y)$ .

Example 3

Name the ordered pair of each point shown in the rectangular coordinate system.

Show solution

Point A is above $−3$ on the x-axis, so the x-coordinate of the point is $−3$ .

The point is to the left of 3 on the y-axis, so the y-coordinate of the point is 3.
The coordinates of the point are $open parenthesis −3, 3 close parenthesis$ .

Point B is below $−1$ on the x-axis, so the x-coordinate of the point is $−1$ .

The point is to the left of $−3$ on the y-axis, so the y-coordinate of the point is $−3$ .
The coordinates of the point are $open parenthesis −1, −3 close parenthesis$ .

Point C is above 2 on the x-axis, so the x-coordinate of the point is 2.

The point is to the right of 4 on the y-axis, so the y-coordinate of the point is 4.
The coordinates of the point are $open parenthesis 2, 4 close parenthesis$ .

Point D is below 4 on the x-axis, so the x-coordinate of the point is 4.

The point is to the right of $−4$ on the y-axis, so the y-coordinate of the point is $−4.$
The coordinates of the point are $open parenthesis 4, −4 close parenthesis$ .

Point E is on the y-axis at $y equals −2$ . The coordinates of point E are $open parenthesis 0, −2 close parenthesis.$

Point F is on the x-axis at $x equals 3$ . The coordinates of point F are $open parenthesis 3, 0 close parenthesis.$

Practice Makes Perfect

Plot Points in a Rectangular Coordinate System

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.

ⓐ $open parenthesis −4, 2 close parenthesis$
ⓑ $open parenthesis −1, −2 close parenthesis$
ⓒ $open parenthesis 3, −5 close parenthesis$
ⓓ $open parenthesis −3, 5 close parenthesis$
ⓔ $(\frac{5}{3}, 2)$

Show solution

Rectangular coordinate graph with axes from -8 to 8. Points a: (-4, 2), b: (-1, -2), c: (3, -5), d: (-3, 5), and e: (5/3, 2) are marked.

ⓐ $open parenthesis −2, −3 close parenthesis$
ⓑ $open parenthesis 3, −3 close parenthesis$
ⓒ $open parenthesis −4, 1 close parenthesis$
ⓓ $open parenthesis 4, −1 close parenthesis$
ⓔ $(\frac{3}{2}, 1)$

ⓐ $open parenthesis 3, −1 close parenthesis$
ⓑ $open parenthesis −3, 1 close parenthesis$
ⓒ $open parenthesis −2, 2 close parenthesis$
ⓓ $open parenthesis −4, −3 close parenthesis$
ⓔ $(1, \frac{14}{5})$

Show solution

Rectangular coordinate graph with axes from -8 to 8. Points a: (3, -1), b: (-3, 1), c: (-2, 2), d: (-4, -3), and e: (1, 14/5) are marked.

ⓐ $open parenthesis −1, 1 close parenthesis$
ⓑ $open parenthesis −2, −1 close parenthesis$
ⓓ $open parenthesis 1, −4 close parenthesis$
ⓔ $(3, \frac{7}{2})$

In the following exercises, plot each point in a rectangular coordinate system.

ⓐ $open parenthesis −2, 0 close parenthesis$
ⓑ $open parenthesis −3, 0 close parenthesis$
ⓓ $open parenthesis 0, 4 close parenthesis$
ⓔ $open parenthesis 0, 2 close parenthesis$

Show solution

Rectangular coordinate graph with axes from -8 to 8. Points a: (-2, 0), b: (-3, 0), c: (0, 0), d: (0, 4), and e: (0, 2) are marked.

ⓐ $open parenthesis 0, 1 close parenthesis$
ⓑ $open parenthesis 0, −4 close parenthesis$
ⓓ $open parenthesis 0, 0 close parenthesis$
ⓔ $open parenthesis 5, 0 close parenthesis$

ⓐ $open parenthesis 0, 0 close parenthesis$
ⓑ $open parenthesis 0, −3 close parenthesis$
ⓓ $open parenthesis 1, 0 close parenthesis$
ⓔ $open parenthesis 0, −2 close parenthesis$

Show solution

Rectangular coordinate graph with axes from -8 to 8. Points a: (0, 0), b: (0, -3), c: (-4, 0), d: (1, 0), and e: (0, -2) are marked.

ⓐ $open parenthesis −3, 0 close parenthesis$
ⓑ $open parenthesis 0, 5 close parenthesis$
ⓓ $open parenthesis 2, 0 close parenthesis$
ⓔ $open parenthesis 0, 0 close parenthesis$

In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 4, 1) is plotted and labeled “A”. The point (negative 3, negative 4) is plotted and labeled “B”. The point (1, negative 3) is plotted and labeled “C”. The point (4, 3) is plotted and labeled “D”.

Show solution

A: $open parenthesis −4, 1 close parenthesis$ B: $open parenthesis −3, −4 close parenthesis$ C: $open parenthesis 1, −3 close parenthesis$ D: $open parenthesis 4, 3 close parenthesis$

Rectangular coordinate graph with axes from -6 to 6. Points A: (-4, 2), B: (3, 5), C: (-3, -2), and D: (5, -1) are marked.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, negative 2) is plotted and labeled “A”. The point (negative 2, 0) is plotted and labeled “B”. The point (0, 5) is plotted and labeled “C”. The point (5, 0) is plotted and labeled “D”.

Show solution

A: $open parenthesis 0, −2 close parenthesis$ B: $open parenthesis −2, 0 close parenthesis$ C: $open parenthesis 0, 5 close parenthesis$ D: $open parenthesis 5, 0 close parenthesis$

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, negative 1) is plotted and labeled “A”. The point (negative 1, 0) is plotted and labeled “B”. The point (4, 0) is plotted and labeled “C”. The point (0, 4) is plotted and labeled “D”.

Corequisite Skills for Introductory Statistics is a custom course edition compiled for Introductory Statistics corequisite courses from OpenStax Contemporary Mathematics, Elementary Algebra 2e, Intermediate Algebra 2e, Prealgebra 2e. Content by OpenStax is reproduced as published, apart from any corrections recorded for this edition. Sections are renumbered to follow the course outline. The course outline is adapted from Larry Green's Support Course for Elementary Statistics (LibreTexts, https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Support_Course_for_Elementary_Statistics), used under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Lessons 2.4 and 6.7 are written for this edition by XYZ Homework. Every OpenStax lesson is abridged to the subsections and practice exercises this course uses. Lessons are headed with their course titles, not the source section titles. Access the complete originals free at openstax.org. License: CC-BY-NC-SA-4.0.