Coordinate Geometry (CIE IGCSE Maths: Core)

What are 2D coordinates?

• A graph in two dimensions is also known as a Cartesian plane
• Normally the two dimensions are labelled x and y, where
  • the horizontal axis is x and the vertical axis is y
• The two axes meet where x and y are both 0
  • This point, (0, 0), is known as the origin
• The coordinate (x, y) tells us that the point is x units to the left/ right of the origin and y units below/ above the origin
  • If the x value is negative then the point is to the left of the origin, and if the x value is positive then the point is to the right of the origin
  • If the y value is negative then the point is below the origin, and if the y value is positive then the point is above the origin
• When we want to plot a point on a graph we need to know where to put it
  
  • For example the coordinate (2, 5) means 2 units horizontally to the right and 5 units up from the origin, the coordinate (-1, -4) means 1 unit to the left and 4 units down

What can we do with coordinates?

• At GCSE core, coordinates are usually used with graphs
• To plot a graph, a set of coordinates can be found, often using a table of values and either substitution or your calculator functions 

What is the gradient of a line?

• The gradient is a measure of how steep a 2D line is
• A large value for the gradient means the line is steeper than for a small value of the gradient
• A gradient of 3 is steeper than a gradient of 2
• A gradient of −5 is steeper than a gradient of −4
• A positive gradient means the line goes upwards from left to right ("uphill")
• A negative gradient means the line goes downwards from left to right ("downhill")
• In the equation for a straight line, , the gradient is represented by 
• The gradient of is −3

How do I find the gradient of a line?

• The easiest method to find the gradient is to count how many units up a line goes for each unit across
  • A line that goes up by 2 units for each unit to the right has a gradient of 2
  • A line that goes down by 3 units for each unit to the right has a gradient of -3
• The gradient can be calculated by drawing a right-angled triangle onto the line from two given coordinates and using

• You may see this written as instead, where the vertical length of the triangle is the rise and the horizontal length is the run

STEP 1
Find any two coordinates from a given diagram of a straight line graph (linear graph)

STEP 2
Draw a right-angled triangle onto the straight line graph
The (positive) difference in the y coordinates is the 'rise'
The (positive) difference in the x coordinates is the 'run'

STEP 3
Work out 
If this is a fraction then you can simplify it (your calculator will automatically do this)

STEP 4
Determine if the line has a positive ('uphill') or negative ('downhill') gradient

STEP 5
Putting the results of STEP 3 and STEP 4 together give you the gradient
e.g.  If  and the line was 'downhill' then the gradient of the line would be -2

How do I draw a line with a given gradient?

• A line with a gradient of 4 could instead be written as . 
  • As  , this would mean for every 1 unit to the right ( direction), the line moves upwards ( direction) by 4 units.
  • Notice that this would be the same as "going backwards"
    • i.e. for every 1 unit to the left, the line moves downwards by 4 units
• If the gradient was −4, then the line would move downwards by 4 units for every 1 unit to the right
  • or, "going backwards" it would move upwards by 4 units for every 1 unit to the left
• If the gradient is a fraction, for example , we can think of this as either
  • For every 1 unit to the right, the line moves upwards by , or
  • For every 3 units to the right, the line moves upwards by 2.
    • or for every 3 units to the left, the line moves downwards by 2
• If the gradient was  this would mean the line would move downwards by 2 units for every 3 units to the right
  • we're sure you can do the "going backwards" version for yourself by now!
• Once you know this, you can select a point (usually given, for example the -intercept) and then count across and upwards or downwards to find another point on the line, and then join them with a straight line