11.02 Integers on the number plane

We built the Cartesian plane using two copies of the number line and we explored how it can be used to describe the location of shapes and points in a 2D space.

Now we can extend this coordinate system using integers, which will allow us to describe the location of points in any direction from the origin.

The applet below demonstrates the main features of this extended Cartesian plane. Each axis now has positive and negative numbers, and this means we can talk about four distinct regions of the plane, called quadrants.

Notice that the 1st quadrant in the top right is equivalent to the Cartesian plane that we looked at last chapter. The $x$x-coordinate and $y$y-coordinate of a point in the 1st quadrant are both positive.

Moving around counterclockwise we cover the other three quadrants, which have the following features:

• 2nd quadrant: $x$x-coordinates are negative, $y$y-coordinates are positive
• 3rd quadrant: both coordinates are negative
• 4th quadrant: $x$x-coordinates are positive, $y$y-coordinates are negative

Points that lie on an axis, like $\left(-5,0\right)$(−5,0) or $\left(0,4\right)$(0,4), are not in any quadrant.

The advantage of using integers on the Cartesian plane is that we no longer have boundaries for the coordinates. If an object begins at some point on the plane, we can move it any which way we like, as far as we like, and still be able to describe its location with respect to the origin.

Worked example

Consider the point $Q$Q plotted on the Cartesian plane.

 

a. What are the coordinates of $Q$Q?
To find the $x$x-coordinate we can draw a vertical line from $Q$Q and read off the number at the point where this line intercepts the $x$x-axis (the horizontal axis). This gives us the number $-4$−4. Similarly, we can find the $y$y-coordinate by drawing a horizontal line from $Q$Q and reading the number at the point where this line intercepts the $y$y-axis (the vertical axis), which gives the number $3$3. So the coordinates of $Q$Q are $\left(-4,3\right)$(−4,3).

Notice the order of this process. We draw a vertical line toward the horizontal axis to find the $x$x-coordinate, and we draw a horizontal line toward the vertical axis to find the $y$y-coordinate.

 

b. What quadrant is $Q$Q in?
In the previous part we found that $Q$Q has a negative $x$x-coordinate and a positive $y$y-coordinate. This means that it is in the 2nd quadrant, in the top left of the Cartesian plane.

 

c. What are the coordinates of the point that is $6$6 units to the right and $8$8 units below $Q$Q?
Starting at point $Q$Q with coordinates $\left(-4,3\right)$(−4,3), we add $6$6 units to the $x$x-coordinate (because we are moving to the right), and subtract $8$8 units from the $y$y-coordinate (because we are moving downward). This gives $\left(-4+6,3-8\right)=\left(2,-5\right)$(−4+6,3−8)=(2,−5).

 

d. If point $R$R has the coordinates $\left(-4,-7\right)$(−4,−7), what is the distance between $Q$Q and $R$R?
Notice that point $R$R has the same $x$x-coordinate as point $Q$Q. This means that the distance between the two points is given by the difference in the two $y$y-coordinates. So the distance is $3-\left(-7\right)=10$3−(−7)=10 units.

Practice questions

Question 1

Question 2

Question 3