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Graphs

Ordered Pairs

An ordered pair is simply a pair of elements written in a certain order (order is important). For example, the numbers 5 and 2 can form two ordered pairs:

(5,2) and (2,5).

 

In general, given two numbers x and y, the two ordered pairs (x,y) and (y,x) are different, unless x = y.

The number x is called the first component and y the second component of the ordered pair (x,y).

 

Ordered pairs have many uses. It can be used to represent an object with two distinguishable parts. For example, a complex number x+iy can be represented by the ordered pair (x,y). In an Excel spreadsheet we use a letter and a number to denote a cell on the sheet.

On a street map we use the same method to locate a street.

Cartesian Coordinate System in 2-D

Ordered pairs are useful in representing relationships between two sets of data, in the form of a graph. These can then be represented by points on a grid known as the Cartesian Coordinate System in 2-D.

to see how to draw a grid:

  First we draw two perpendicular number lines.

One horizontal with positive direction to the right, called the x-axis. One vertical with positive direction upward, called the y-axis.

 The intersection of the two axes is called the origin. It represents the point (0,0), usually labelled 0.

 

The following example illustrates how to represent an ordered pair as a point on this grid.

Example 1.

To plot the point represented by (2,4):

First draw a vertical line through x =2.

Next draw a horizontal line through y =4.

The point (2,4) is at the intersection of these two lines, P.

Points are shown on a grid.

More Exercises (opens in another window)

 

If a point P is represented by (a,b) we call a and b the coordinates of P. In 2-D Cartesian Coordinates, the first coordinate in the ordered pair is called the x-coordinate and the second coordinate in the ordered pair is called the y-coordinate.

Exercise 1.

Enter the correct coordinates for the points plotted below, A-L.
Remember to separate your x and y coordinates by a comma.

Points shown visually on a grid.

A: ( )

B: ( )

C: ( )

D: ( )

E: ( )

F: ( )

G: ( )

H: ( )

 I: ( )

J: ( )

K: ( )

L: ( )


 

 

 

 

 

 

 

The two axes partition the plane into 4 regions, called quadrants.
They are referred to as the 1st, 2nd, 3rd, and 4th quadrant as shown.

The first quadrant is located where x and y are both positive. The 2nd quadrant has x negative and y positive, the third has both x and y negative and the fourth has x positive and y negative.

  • If the x and y coordinates are both positive, the point is located in the 1st quadrant.
  • If the x coordinate is negative and the y coordinate is positive, the point is located in the 2nd quadrant.
  • If the x and y coordinates are both negative, the point is located in the 3rd quadrant.
  • If the x coordinate is positive and the y coordinate is negative, the point is located in the 4th quadrant.

Example 2.

Look at the following example to see which quadrants various points are located in. Check the grid below to confirm the solution.



Point
Quadrant
( , )

Grid confirming solution visually.

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