The coordinate plane is an organized way to identify locations in two dimensions, using a grid and coordinates on the x-axis and y-axis. Learning how to move around, and how to graph points on the coordinate plane, will help you understand geometry.

Table Of Contents

  1. Coordinate Plane Definition
  2. Coordinate Plane Quadrants
  3. Plotting Points on a Coordinate Plane
  4. Graph Points On A Coordinate Plane Using Equations

Coordinate Plane Definition

The coordinate plane has two axes (plural of axis). These are called the x-axis and y-axis. To ensure understanding, all mathematicians label points on the coordinate plane first with the x-axis value, then the y-axis value.

  • The x-axis goes left and right, horizontally
  • The y-axis goes up and down, vertically

These two axes divide a coordinate grid (an overlay of intersecting perpendicular lines on the coordinate plane) into four regions, called quadrants.

Only one quadrant, Quadrant I, will have positive numbers for both the x-value and the y-value. One quadrant, Quadrant III, will have negative numbers for both x-values and y-values. The other two quadrants, II and IV, will have negative and positive numbers.

Coordinate Plane Quadrants

With the x-axis cutting horizontally across the coordinate plane, and the y-axis cutting vertically, the coordinate plane divides neatly into four quadrants, labeled with Roman numerals in a counterclockwise direction from the top right (I) to bottom right (IV).

  1. Quadrant I, top right, will have two positive numbers, like this: (+, +)
  2. Quadrant II, top left, will have a negative x-value and a positive y-value, like this: (-, +)
  3. Quadrant III, bottom left, will have two negative numbers, like this: (-, -)
  4. Quadrant IV, bottom right, will have a positive x-value and a negative y-value, like this: (+, -)

Here are some self-checks. See if you can tell each ordered pair's quadrant:

  • (-3, -6)
  • (3, -6)
  • (-3, 6)
  • (3, 6)

We hope you said (-3, -6) is in Quadrant III, (3, -6) is in Quadrant IV, (-3, 6) is in Quadrant II, and (3, 6) is in Quadrant I.

The y-axis (the vertical axis) is a divider between left and right. All x-values to the left of the y-axis are always negative; x-values to the right of the y-axis are always positive.

Similarly, the x-axis is a divider between above and below: y-values above the x-axis are always positive, and y-values below the x-axis are always negative.

Plotting Points on a Coordinate Plane

Hold your dominant hand up in front of you, index finger extended and thumb up, like this:

[insert cartoon hand pointing like a gun]

See the long index finger, longer like it is more important? That's the first coordinate, the x-axis value. The short thumb is the y-axis value. So you always put them in that order: (x, y)

To graph a particular point on a coordinate grid, start at the origin, (0, 0). Every grid has an origin, the point where the two axes intersect.

Look at the ordered pair; say it is (3, 6).

  1. Go to your right three units (because 3, the x-value, is positive).
  2. Go up six units (because 6, the y-value, is also positive).
  3. Mark the intersection of the perpendicular lines you see at that spot, 3 positive units right and six positive units up. That is the point on the coordinate grid (3, 6).

Let's try another from our earlier list, (-3, -6):

[insert coordinate grid showing Quadrant III]

  1. Go to your left three units, because -3 is the negative x-value.
  2. Go down six units, because -6 is the negative y-value.
  3. Mark the intersection of the perpendicular lines you see at the spot 3 negative units left and 6 negative units down. That is (-3, -6).

Remember to always mark the intersection of lines, not the empty space in between the lines. Think of the grid as a net; you only "catch" what is on the lines, until you get into fussy coordinate pairs like (-3.5, -6.75), where you may have to estimate those fractions of units inside the boxes.

Most coordinate grids are marked off in whole numbers. They can also be in multiples of 10s, or they can be in decimal values so you could still have perpendicular lines for points like -3.5 or 6.5 and so on.

Graph Points On A Coordinate Plane Using Equations

An equation requiring graphing on a coordinate grid will come in one of two types:

  1. Coordinates appear in the equation, or,
  2. Coordinates do not appear in the equation

If you are given an equation with no coordinates, you have to pick a random value for one of the two numbers, put it into the equation, and see what the other value becomes. Repeat this, putting in, say, a random x-value and seeing what y-value the equation produces. You need enough passes through the equation to graph all the (x, y) pairs and connect them.

Here is a linear equation:

y = x + 3

Pick some value for x, like 1, and out pops the y-value: 4.

Repeat this with another x-value, 2, and another, 3:

y = 2 + 3 = 5

y = 3 + 3 = 6

You now have three ordered pairs:

  • (1, 4)
  • (2, 5)
  • (3, 6)

Plotting those in Quadrant I gives you a straight line.

[insert Quadrant I drawing with this line]

Here is something a bit more complex, a quadratic equation for a parabola:

y = x2 + 3

Keep the same strategy. Start with 0 (that is the easiest first step):

y = 02 + 3 = 3

This gives you (0, 3) for the minimum of the parabola, so you know its direction is up into Quadrants I and II. Plot some other points using small x-values:

y = 12 + 3 = 4

y = 22 + 3 = 7

y = 32 + 3 = 12

Plotting just a few points on the coordinate grid will give you the layout of at least the right side of the parabola. Throw in some negative x-values to ensure you find the points in Quadrant II, too. (They will be mirror negatives of the Quadrant I x-values.)

[insert drawing of this parabola if possible]

Linear equations produce lines; quadratic equations produce curves. If plotting a circle, you will need two points if you are given a center, and three points if you are not given the center. Parabolas also require three points, while hyperbolas require six points (three on the x-axis, three on the y-axis).

Next Lesson:

Geometric Constructions

Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.
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