Graphing Linear Equations

Date: 07/14/98 at 11:20:20
From: Meagan
Subject: Graphing equations

I am totally lost when it comes to graphing equations. Please tell me 
some ways I can figure it out easier.

Date: 08/01/98 at 14:15:16
From: Doctor Margaret
Subject: Re: Graphing equations

Hi Meagan,

Thanks a lot for writing to us. Although your question was not very 
specific, I will try to give you some answers about graphing in 
general. 

This was actually the area of mathematics that got me really 
interested. A graph of an equation is a picture of it. In the case of 
linear equations, we can tell how fast the equation is increasing or 
decreasing just by looking at the picture of the line it makes in the 
xy plane, that is, the area pictured by the x-axis (horizontal line)  
and the y-axis (vertical line), which intersect each other at zero.  

The equations that we graph in this case will have two variables, an x 
and a y. These variables occur in what we call an "ordered pair,"  that 
is, a set of parentheses like this: (x, y).

Because you sound as if you are just starting out with graphing, let's 
see the easiest equation to graph, which is a straight line that can 
tilt up or down. The equation looks like this:

   y = mx + b   

in writing. 

To graph a linear equation, you have to find the ordered pair solutions 
of the equation. Do this by choosing any value of x and finding the 
corresponding value of y. Repeat this procedure, choosing different 
values for x, until you have found the number of solutions desired. 
Since the graph of a linear equation in two variables is a straight 
line, and a straight line is derermined by two points, it is necessary 
to find at least two solutions. I like to find three and if they all 
line up, then I know I'm right. For example:

   Graph:  y = 2x + 1.

One of the best choices you can make for x is to make it equal to zero. 
This give you the place on the y axis where the line intersects it. 
So:

   y = 2(0) + 1 = 1  

The first ordered pair is (0,1).

Doing this two more times for x = 1 and x = -2 we  have:

   y = 2(1) + 1 =  3, giving us (1,3)

   y = 2(-2) + 1 = -3, giving us (-2,-3)

Now we can graph the line in the xy plane. We have three ordered pairs, 
(0,1), (1,3) and (-2,-3).

For (0,1) we count to zero on the x axis and up one for y. Make a dot. 
(I'll do this with a *.) For (1,3) we count one to the right on the x 
axis and up three for y.  Another dot.  

I'll leave the third pair for you. But you will be counting to the left 
and down because the numbers are negative. Here is the picture for the 
first two:

                         |3   *  (1,3)
                         |    |
                         |2   |
                         |    |
                         *1---  (0,1)
                         |
    _____________________|__________________
     -3    -2    -1     0|    1     2     3
                         |-1
                         | 
                         |-2
                         |
                         |-3

Now draw a line through the three dots you graphed, and you have your 
graph of a straight line.

The easiest way is to practice until you get used to it. Please write 
back if you need more help.
                   
- Doctor Margaret, The Math Forum
Check out our web site! http://mathforum.org/dr.math/