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Graphs of Step Functions

Date: 11/29/2003 at 18:02:41
From: Soosan
Subject: Steps in graphs  

What does it mean when a graph goes up in steps rather than in a line?
 I don't understand how it's used or what it means.

Date: 11/30/2003 at 14:03:17
From: Doctor Ian
Subject: Re: Steps in graphs  

Hi Soosan,

An example of this would be the amount you pay for postage.  Anything
up to 1 ounce costs 37 cents to mail:

  37 o---*
     +---|---|---|---|---|-- oz

The 'o' means that the segment never gets all the way to zero ounces
(since you can't mail something that weighs nothing), and the '*'
means that the segment goes all the way to 1 ounce, since something
that weighs exactly one ounce still costs 37 cents.  

Anything more than one ounce, but not greater than 2 ounces, costs
more.  I forget how much more, but let's say you pay 20 cents for the
extra.  Then you get

  57 -   o---*
  37 o---*
     +---|---|---|---|---|-- oz
         1   2

Again, the left side of the segment is open, and the right side is
closed.  (Do you see why?)

If the next bit of weight costs another 20 cents, we get 

  77 -       o---*
  57 -   o---*
  37 o---*
     +---|---|---|---|---|-- oz
         1   2   3

And so on.  

So the price steps from one value to another, without ever hitting any
of the intermediate values.  (For example, you never pay 39 cents, or
42 cents, or 68 cents.) 

When would this kind of thing happen?  In general, it happens when a
lot of inputs lead to the same output, often because the outputs can't
be continuous.  In the case of postage, all these inputs (weights)

   0.1 oz
   0.2 oz
   0.3 oz
   1.0 oz

give you the same output: 37 cents, or one stamp.  For something
lighter than an ounce, you can't put on less than a whole stamp!

Similarly, with sales tax, you can't pay in units smaller than a
penny.  When I was growing up in Indiana, we had a sales tax of 4
percent.  What this meant was that anything up to 25 cents had no tax,
anything from 25 to 49 cents had one cent tax, and so on.  The graph
for that looked like

   2 -                                       *-------
   1 -                   *-------------------o
   0 *-------------------o
     +---|---|---|---|---|---|---|---|---|---|---|---|-- price 
         5  10  15  20  25  30  35  40  45  50  55  60  (cents)

So when we wanted to buy, say, 40 cents worth of candy, we'd buy 20
cents worth, and then another 20 cents worth, in two separate
purchases.  That way we'd pay no tax!  :^D

Does this make sense? 

- Doctor Ian, The Math Forum 
Associated Topics:
High School Equations, Graphs, Translations
High School Functions

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