Chapter 3 · Algebraic foundations
Axes, Quadrants, and Ordered Pairs
Learn how two number lines form the coordinate plane, how to name its four quadrants, and how to plot and read points from ordered pairs.
On a number line, one number is enough to locate a point. On a flat surface you can move two ways, left or right and up or down, so you need two numbers.
The coordinate plane is a flat surface with two number lines drawn on it. By definition they cross at right angles, and they cross at zero on both lines.
- The -axis is the horizontal number line. Positive numbers are to the right of zero and negative numbers are to the left.
- The -axis is the vertical number line. Positive numbers are above zero and negative numbers are below.
- The origin is the point where the two axes cross.
In this lesson the axes do not stand for a real-world quantity. They measure position, and both use the same unit: one grid step is unit. Every plane you see runs from to on both axes.
To name a point you give two numbers in a fixed order. This is an ordered pair, written . In this lesson and can be any numbers from to , not only integers.
- The -coordinate comes first. It tells you how far the point is to the right or left of the -axis.
- The -coordinate comes second. It tells you how far the point is above or below the -axis.
A nonzero coordinate carries two facts. Its sign gives the direction. Its absolute value, which is its size without the sign, gives the distance, and a distance is never negative. So means units to the left of the -axis.
In the figure below, Line A is the -axis and Line B is the -axis, with unit per grid step on each. They cross at the origin, . Point C is at : it is units right of the -axis and units above the -axis. The ordered pair is exact, so trust it over how a dot looks.
The axes split the rest of the plane into four regions called quadrants. They are numbered with Roman numerals, starting at the upper right and going counterclockwise. The figure above shows one point in each.
- Quadrant I is right of the -axis and above the -axis, like Point C at .
- Quadrant II is left and above, like Point D at .
- Quadrant III is left and below, like Point E at .
- Quadrant IV is right and below, like Point F at .
The axes are the borders between the quadrants, so a point on an axis is in no quadrant. Every point on the -axis (Line A) has , and every point on the -axis (Line B) has . The origin is on both. So a point is in a quadrant only when both of its coordinates are nonzero.
Decide from the exact coordinates, not from how close a dot looks to a line. The table below turns each location into a sign pattern.
| Location | x-coordinate | y-coordinate | Example |
|---|---|---|---|
Quadrant I | positive | positive | |
Quadrant II | negative | positive | |
Quadrant III | negative | negative | |
Quadrant IV | positive | negative | |
On the -axis, not at the origin | nonzero | ||
On the -axis, not at the origin | nonzero | ||
The origin, on both axes |
Key concept
An ordered pair is an exact address
On this lesson's plane, every ordered pair with and from to names exactly one point, and every point has exactly one ordered pair.
- comes first. It is the horizontal position: positive is right of the -axis and negative is left.
- comes second. It is the vertical position: positive is above the -axis and negative is below.
- The point is units from the -axis and units from the -axis. These distances are never negative, and both axes use the same unit.
Two nonzero coordinates put the point in a quadrant, and their signs tell you which one. A zero puts it on an axis. Go by the exact coordinates, not by how the picture looks.
Worked example
Name the quadrant or the axis
Each grid step is unit on both axes. Using only the exact coordinates, say where each point lies: , and .
Look for a zero first, because a point is in a quadrant only when both of its coordinates are nonzero. Only the third point has a zero, and it is the x-coordinate.
For the point (5, -2), the x-coordinate is positive, so the point is 5 units right of the y-axis. The y-coordinate is negative, so the point is 2 units below the x-axis. Right and below is Quadrant IV.
For the point (-1.5, -4), the x-coordinate is negative, so the point is 1.5 units left of the y-axis. The y-coordinate is negative, so the point is 4 units below the x-axis. Left and below is Quadrant III.
For the point (0, 3), the x-coordinate is 0, so the point is 0 units left or right of the y-axis. That puts it on the y-axis, 3 units above the origin. A point on an axis is in no quadrant.
Answer
The point is in Quadrant IV, the point is in Quadrant III, and the point is on the -axis, so it is in no quadrant.
Worked example
Plot a point from its ordered pair
To plot a point means to mark its exact position. Plot on a plane that runs from to on both axes, with unit per grid step.
Start at the origin, because both coordinates are measured from the point where the axes cross.
Use the first number, the x-coordinate, for the horizontal move. It is negative, so move left. The distance is its absolute value, 3 units, so go 3 grid steps left along the x-axis and stop at -3.
Use the second number, the y-coordinate, for the vertical move. It is positive, so move up 2.5 units: 2 full grid steps and then exactly half of the next one. Moving straight up keeps you directly above -3, so the x-coordinate does not change.
Mark the point where you stop, then check the signs. A negative x-coordinate and a positive y-coordinate mean left of the y-axis and above the x-axis, which is Quadrant II. That matches the moves you made. The ordered pair, not the look of the dot, makes the height exactly 2.5.
Answer
The point is units left of the -axis and units above the -axis, in Quadrant II. It is Point G in the figure below.
Worked example
Read the ordered pair of a marked point
In the figure above, the plane runs from to on both axes and each grid step is unit. Point A is exactly grid steps left of the -axis and grid steps below the -axis. Point B is on the -axis, exactly grid steps right of the origin. Write the ordered pair for each point and say where it lies.
Find the x-coordinate of Point A from its left or right position. Each grid step is 1 unit, so 4 steps is a distance of 4 units from the y-axis. Left means negative, so the x-coordinate is -4. In the figure, the dot labelled -4 on the x-axis is directly above Point A.
Find the y-coordinate from its up or down position. Point A is 3 steps, so 3 units, from the x-axis. Below means negative, so the y-coordinate is -3. The dot labelled -3 on the y-axis is directly to the right of Point A.
Write the pair with x first. Both coordinates are nonzero and negative, so Point A is in Quadrant III.
For Point B, right means positive, so 5 units right gives an x-coordinate of 5. Point B is on the x-axis, so it is 0 units above or below it and the y-coordinate is 0.
A y-coordinate of 0 means Point B is on the x-axis. A point on an axis is in no quadrant, even though this one is on the border between Quadrant I and Quadrant IV.
Answer
Point A is at in Quadrant III. Point B is at on the -axis, so it is in no quadrant.
Takeaway. The coordinate plane turns a location into two numbers. In an ordered pair , comes first and gives the position left or right of the -axis, and comes second and gives the position above or below the -axis. Two nonzero coordinates place the point in a quadrant, and their signs tell you which one. A zero places it on an axis. Work from the exact coordinates, with the same unit on both axes, not from how a dot looks.
Next, you will use ordered pairs to graph whole figures on the coordinate plane.