Does the Graph of the Equation Rise or Fall?

Date: 10/22/2002 at 07:31:39
From: Becky Della Noce
Subject: Linear relations and function (slopes and intercepts)

Hi,  

We are learning about graphs and how to graph the x and y intercepts, 
whether it is rise or fall. The question I don't understand is:

Determine whether the graph of each equation rises to the right or 
falls to the right, is horizontal, or is vertical.

The example question to figure out is 2x + 12 = 0.

Thanks a lot, 
Becca

Date: 10/22/2002 at 11:47:13
From: Doctor Ian
Subject: Re: Linear relations and function (slopes and intercepts)

Hi Becca,

Let's think about the line 

  y = 2x + 12 

Here's one way to think about it. Suppose x is zero. Then 

  y = 2(0) + 12

    = 12

So the line crosses the y-axis at (0,12). We can label that point p on 
the graph below:

              |
              p
              |
              |
              |
     --------- ---------
              |

Now, let's consider what happens if we increase x. The 12 stays the 
same, no matter what value of x we pick. But we're going to add 2x to 
that, and 2x gets bigger as x gets bigger, right?  

So as we move to the right - that is, as we choose bigger values of x 
- the value of y has to increase. So the graph must slope upward as we 
go to the right:

              |          .  more x means more y
              |          .
              p...........
              |
              |
              |
     --------- ---------
              |


Does that make sense?  Now, what if the equation were

  y = -2x + 12

We'd still have our point p in the same place, but now every time we 
make x bigger, we make y _smaller_, because we're subtracting instead 
of adding:

              |          
              |         
              p...........
              |          .
              |          .  more x means less y
              |
     --------- ---------
              |

The only difference between the two equations is the sign of the x 
term. In the case where the sign is positive (more x means more y), 
the line slopes upward. In the case where the sign is negative (more 
x means less y), the line slopes downward.  

What if the slope is zero?  Then we have 

  y = 0x + 12

or just 

  y = 12

which is to say, the value of y is _always_ 12, regardless of what x 
is. This is a horizontal line - a line that has the same value for y 
everywhere. 

And what about a vertical line? That's like a horizontal line, but 
instead of keeping y constant, we keep x constant, e.g., 

  x = 3

Now, the equation you started with can be rearranged to look like 
this: 

   2x + 12 = 0

        2x = -12

         x = -6

So it's the equation of a vertical line. The value of x is -6, no 
matter _what_ value you choose for y. 

Does this help?

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