When is a Slope 0 or Undefined?

Date: 03/29/97 at 19:01:33
From: Maya
Subject: When is a Slope 0 or Undefined?

How do I know when the slope of an equation is zero or undefined (no 
slope)? Thank you!

-Maya

Date: 04/03/97 at 20:50:11
From: Doctor Wallace
Subject: Re: When is a Slope 0 or Undefined?

Dear Maya,

Since you didn't say, I'll assume that by "equation" you mean the 
equation of a line, that is, a linear equation.  Talking about slopes 
gets a great deal more complicated when the equation isn't linear.

Do you know about the slope-intercept form of a line?  In general 
terms, we write this as

                 y = mx + b

In this equation, the m stands for the slope of the line, and the b 
for its y-intercept.  So this line has slope m and crosses the y-axis 
at (0, b).

If the slope of the line is 0, then this means that m=0.  But if m is 
zero, then our x term is also 0, since anything times 0 must be 0.  So 
we are left with

                 y = b


This is the only way to have a line with slope 0.  If you graph this 
line, what do we get?  Well, let's take a concrete example:

                 y = 1

This is a perfectly good line.  Notice that x is missing.  But we know 
that y is always equal to 1.  So all the points on this line will have 
1 as their y coordinate.  Since x is missing, it may be any value at 
all:

 (1,1), (2,1), (3,1), (4,1), (1000,1), (-2034, 1)...

All these are on the line.  What does this form if we graph it?  Try 
it and you'll see that it is a horizontal line going through (0,1).  
The x-values can be any number, but y must always be 1.

So what do we conclude?  All horizontal lines have slope 0.  And if 
any line has slope 0, it must be a horizontal line.

This makes sense, given what we know about slope.  A line may have 
positive slope, negative slope, zero slope, or undefined slope.

The way to tell which one it is, is to do this:  Pretend you are 
standing on the line, on its graph, and you are going to walk from 
LEFT to RIGHT along the line.  If you are:

  walking UP hill:      it's a positive slope

  walking DOWN hill:    it's a negative slope

  walking a FLAT line:  it's a zero slope with no hill at all

Easy, huh?

Now, that only leaves the undefined slope.  In the case of a line, 
this means that the line is VERTICAL.  Remember, a horizontal (flat) 
line has a slope of 0.  A Vertical line (one that forms a right angle 
or is perpendicular to a flat line) has an undefined slope.  Think of 
it using the hill analogy again.  If the hill is straight up, we 
couldn't walk up it unless we were Spiderman or something.  If and 
only if the line is perfectly vertical do we say it has no slope, or 
an undefined slope.  If it's almost vertical, but not quite, then it 
will have a very big (steep) slope, but not undefined.

Why is this so?  Why does a vertical line have an undefined slope 
mathematically?  This is all easier to see on a graph, which I can't 
draw for you.  But I'll try to explain.

Slope is defined as the rise over the run, right?  That is, the slope 
or steepness of a line (or hill) is equal to how fast it goes up with 
respect to how far it goes over.  

Think of a very steep hill.  As I move forward, the hill rises very 
fast over a small distance.  So we have a large number over a small 
one, like, say, 100/1.  This is a slope of 100, very steep.

Think of a gentle, small hill.  As I walk along it, it doesn't rise 
very fast over distance.  I can walk a long way and only have gone up 
a few feet.  This is a small number divided by a large one, like 1/25. 
This is a slope of 1/25, very gentle.

When you graph a line, you can use the slope to do it.  Take your 
equation in slope-intercept form, like

                    y = 2x + 1

and graph the y-intercept.  Here I would put a point at (0,1.)
The slope is 2 (the number stuck with the x).  I have to think of the 
slope as a fraction, remember, rise over run.  2 as a fraction is 2/1.  
So the rise is 2, and the run is 1.  So, starting from (0,1) I go up 
2 and over 1 and I arrive at the point (1,3).  I put a point there.  
Now I have 2 points so I can draw my line between them.

Now we can see what an undefined or vertical slope means.  Imagine we 
have drawn a vertical line on our graph paper, through the point 
(1,0).

To get to the next point on the line, which is (1,1), what do I have 
to do? I have to go up 1, and over none.  That is, the rise is 1, and 
the run is 0.

So I have the fraction 1/0.  Aha!  Now we see why a vertical line has 
an undefined slope.  Because we can't divide by zero!

Usually in math, when something is "undefined" it means that 
somewhere, something is being divided by 0.  And that's a no-no.  It 
has no meaning.  So we call it undefined.

I hope this has helped you.  If you have any more questions, don't 
hesitate to write back!

-Doctor Wallace,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/