Graphing Coordinates

Date: 01/15/97 at 22:07:24
From: Katherine Gallagher
Subject: Complementary and supplementary

When you are graphing numbers, how can you tell if you have a straight 
line just by looking at the numbers that were given to you?

Katherine Gallagher
Fort Collins,CO

Date: 01/16/97 at 09:29:58
From: Doctor Gerald
Subject: Re: Complementary and supplementary

Katherine,

I suppose you are talking about number coordinates like (1,3) etc.  
You must be talking about three or more points that are graphed, 
because just two points always will be a straight line if you connect 
them, right?

Now, if you have three points with coordinates, there are three 
possible straight lines that they might form: 1) horizontal, 
2) vertical, or 3) sloped.  

The first two are pretty easy while the third is a little more 
complicated.  For example: 

1) Here are three points that form a horizontal line: (2,4), (5,4), 
and (10,4).  These three points all have the same y-coordinates (the 
second number in the parentheses), which is 4.  This means that no 
matter what the value of x is, the line will always go through y = 4 
and thus will always be straight.  This means that these points form 
a horizontal line which is parallel to the x-axis and intersects the 
y-axis at y = 4.  The picture below should help:

 y-axis
  /|\
   |
y=4--------------  This is the horizontal line
   |
   |
   |
   |______________\
                  /
       x-axis     

2) Here are three points that form a vertical line: (2,4), (2,5), and 
(2,10).  All three points have the same x-coordinates (the first 
number in the parentheses), which is 2.  This means that no matter 
what the value of y is, the line will always go through x = 2 and will 
always be straight.  This means that these points form a vertical line 
which is parallel to the y-axis and intersects the x-axis as x = 2.  
The picture below should help:

y-axis
  /|\
   |    |   This is the vertical line
   |    |  
   |    | 
   |    |
   |    | 
   |____|__________\
       x=2         /
       x-axis    

3)  Here are three points that form a sloped line: (2,4), (4,7), and 
(8,13).  Look at the changes in the x-coordinate from one point to the 
next.  There's no pattern: it changes by 2, and then by 3.  What about 
the changes in the y-coordinate?  It changes by 3, and then by 6. 

However, the pattern we want to find isn't in the changes themselves,
but in the _ratio_ of the changes:

           change    change     (change in y)
           in x      in y       -------------
                                (change in x)

   (2,4)
             2        3             3/2
   (4,7)
             4        6             6/4 = 3/2
   (8,13)

In fact, for any pair of points on this line, the ratio of the
change in y to the change in x will be the same, 3/2. 

Now, suppose you're given the following set of points:

  A: (1,5)
  B: (2,8)
  C: (5,10)
  D: (7,16)

You might try checking them in pairs:

  A to B:  change in y = 8 - 5 = 3

           change in x = 2 - 1 = 1

           ratio = 3/1 = 3

  C to D:  change in y = 16 - 10 = 6

           change in x = 7 - 5 = 2

           ratio = 6/2 = 3

So far, so good. But let's look at what happens from B to C:

  B to C:  change in y = 10 - 8 = 2

           change in x = 5 - 2 = 3

           ratio = 2/3

Oops!  We get a different ratio.  What happened?  Well, if you 
plot the points, 

   |
   -
   |
   -             D
   |
   -
   |
   -             
   |
   -         C
   |
   -   B     
   |  
   -  
   | A
   -
   |
   -
   |
   +---|---|---|---|---|---|---|

you can see that a line containing A and B is _parallel_ to the line 
containing C and D, but they're not all on the _same_ line.  So in the
worst case, you would want to order all your points from smallest x 
to largest x, and check the ratios for all the adjacent pairs, e.g., 
A to B, B to C, C to D, and so on. 

I say that's the 'worst' case because as soon as you find two pairs 
with different ratios, you can quit!  (In this case, we could quit as
soon as we found that the ratio for B to C was different than the ratio
from A to B.)  Also, there are some easy cases that you might run into.  
For example, if the x coordinates are evenly spaced, e.g., 
 
   (2,__), (5,___), (8,___), (11,___)          x always changes by 3
 
you can check to see if the y coordinates are also evenly spaced.  
If so, e.g., 

   (2,4), (5,6), (8,8), (11,10)          x always changes by 3
                                         y always changes by 2
                                         All on the same line.
     
then you have a line; if not, e.g., 

   (2,4), (5,5), (8,7), (11,8)           x always changes by 3
                                         y changes by 1, then 2, then 1
                                         Not all on the same line. 

you don't.  Do you see why it works out this way? 
    
-Doctors Gerald and Rachel,  The Math Forum
 http://mathforum.org/dr.math/