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### When is a Slope 0 or Undefined?

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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
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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/
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Associated Topics:
High School Coordinate Plane Geometry
High School Equations, Graphs, Translations
High School Linear Equations

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