Graphing Peculiarity

Date: 07/24/2002 at 19:35:02
From: Kartik Trehan
Subject: Graphing Peculiarity

Suppose we have to graph the function 

  y = (x^2 - 4)/(x + 2)

Everything seems fine; there is a hole at -2, etc. But what if we move 
the x+2 to the other side so that 

  (x + 2)y = x^2 - 4. 

Then when we plug in -2 the equation becomes 0=0, and that doesn't 
seem to have a graph. What's happening? I couldn't graph it and no 
online graphing calculators could help me. Either my brain has turned 
against me or there is some simple explanation for this. Thanks a lot.

Date: 07/25/2002 at 13:40:13
From: Doctor Peterson
Subject: Re: Graphing Peculiarity

Hi, Kartik.

The computers have turned against you. This does have a graph, but it 
may take a human brain to recognize it!

The graph of an equation is a graph of THAT PARTICULAR equation, not 
necessarily of any equation you can change it to. In this case, you 
have multiplied the equation by x+2; when x = -2, that means you have 
multiplied by 0 and the new equation is not equivalent to the 
original. The graphs will be the same everywhere except at x = -2.

You found that when you put x = -2 into the new equation, it becomes 
0y = 0; that means that it is true for ALL y! So its graph, rather 
than having a hole at (-2,-4), consists of the vertical line x = -2 
together with the line y = x-2.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/