• MathsFirst Home
• Online Maths Help
  • Notation
  • Arithmetic
    • Decimals
    • Fractions
    • HCF and LCM
    • Order of Operations
    • Signed Numbers
  • Algebra
    • Exponents
      • Integer Exponents
      • Fractional Exponents
      • Exponents Worksheet
    • Combining Like Terms
    • Simple Expansions
    • Factorisation
    • Mathematical Formulae
      • Constructing Formulae
      • Rearranging Formulae
        • Order of Operations for Algebraic Expressions
        • Inverse Operations and Functions
        • Rearranging Equations I (Simple Equations)
        • Rearranging Equations II (Quadratic Equations)
        • Rearranging Equations III (Harder Examples)
      • Area, Surface Area and Volume Formulae
    • Coordinate Systems and Graphs
    • Linear Equations & Graphs
    • Simultaneous Linear Equations
    • Polynomials
    • Quadratic Polynomials
    • Rational Functions
    • Logarithms
  • Calculus
    • Differentiation
      • Tangents, Derivatives and Differentiation
      • The Basic Rules
      • Products and Quotients
      • Chain Rule
      • Logarithmic Differentiation
      • Mixed Differentiation Problems
      • Sign of the Derivative
      • Sign of the Second Derivative
    • Integration
      • Basic Integrals
      • Multiple, Sum and Difference Rules
      • Linear Substitution
      • Simpler Integration by Substitution
      • Harder Integration by Substitution
      • Trig Substitution 1
      • Trig Substitution 2
      • Integration by Parts
  • Trigonometry
    • Pythagoras Theorem
    • Trig Functions and Related Topics
• First Year Maths Pathways
• Readiness Quizzes
• Study Maths
  at Massey
  • Undergraduate
  • Postgraduate
• Other Math Links
• About MathsFirst
• Contact Us

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.

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

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

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.

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 1^st, 2^nd, 3^rd, and 4^th quadrant as shown.

• 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.

<< Number Lines | Graphs Index | Cartesian Coordinate System in 3-D >>