4.01 The Cartesian plane

A grid reference system is commonly used on maps and makes use of combinations of letters and numbers to locate different positions within an area.

In mathematics, the Cartesian plane (also called the number plane, or coordinate plane) is used to describe the location of points, and provide a means to graphically represent mathematical relationships in a two-dimensional space. It is named after French mathematician Rene Descartes, who invented it in the 17th century.

 

Features of the Cartesian plane

A number plane is made up of a horizontal and a vertical axis.

• The horizontal axis, called the$x$x-axis, is like a number line we have seen previously that runs from left to right
• The vertical axis, called the $y$y-axis, is a number line that runs up and down
• The two lines meet at the origin

 

The $x$x-axis is numbered with negative numbers, zero and positive numbers increasing to the right.

The $y$y-axis is numbered with negative numbers, zero and positive numbers increasing vertically.

 

Points on the Cartesian plane

Points are labelled on the Cartesian plane by an ordered pair of numbers of the form $\left(x,y\right)$(x,y), called coordinates.

The point at which the $x$x-axis and the $y$y-axis intercept is called the origin. The coordinates of the origin are $\left(0,0\right)$(0,0).

All other points on the Cartesian plane are labelled according to where a vertical and horizontal line, drawn from the point, intercept the $x$x-and $y$y axes respectively. For example, consider the point $Q$Q plotted on the number plane.

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.

 

Practice questions

Question 1

Question 2

 

Quadrants of the Cartesian plane

The Cartesian plane has four distinct regions, called quadrants. The applet below shows how these quadrants are defined.

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

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

• 2nd quadrant: $x$x-coordinates are negative, $y$y-coordinates are positive. This is the top-left quadrant.
• 3rd quadrant: both coordinates are negative. This is the bottom-left quadrant.
• 4th quadrant: $x$x-coordinates are positive, $y$y-coordinates are negative. This is the bottom-right quadrant.

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

 

Worked example

Example 1

(a) What quadrant is $Q$Q in?

Think: Earlier in the lesson we found that $Q$Q has a negative $x$x-coordinate and a positive $y$y-coordinate.

Do: This means that it is in the 2nd quadrant, in the top left of the number plane.

(b) What are the coordinates of the point that is $6$6 units to the right and $8$8 units below $Q$Q?

Think: 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).

Do: This gives $\left(-4+6,3-8\right)=\left(2,-5\right)$(−4+6,3−8)=(2,−5).

Practice questions

Question 3

Question 4

QUESTION 5

QUESTION 6

QUESTION 7