Wednesday, September 21, 2016

Step Functions and Graphs

Today in Algebra I, I showed a video on reading graphs.  The guy spoke about step graphs and said this one represented the price per minute and total cost so for the first minute it was 5 cents, then 10 for between 1 and 2 minutes, etc.  Hmmmm, this example made sense but are there other situations which are graphed using step graphs?

When I was in school, we just accepted this was a weird graph you studied but we had no idea what it represented or how it was used.  Today, was the first time I heard any example associated with step graphs.

So the step graph is a visual representation of the step function also known as the Greatest Integer Function which is one step closer to understanding its real world use.  This site has a lovely packet showing some great examples.  For instance, think about the postage on a letter or package.  The amount of postage depends on its weight in terms of whole ounces.  If it weights 5.5 ounces, the clerk will charge you for 6 ounces because that is the next integer.

Another application applies to taxi fees because there is the starting charge and a charge for every 5th of a mile.  So 1.25 is charged as 1.4 miles because its over 1.2 but less than 1.4 miles.  There is also the cell phone where you pay as you go because many of those plans are still set up as so much per minute.

Most tax tables either for sales or income tax are step functions.  If you've ever filled out your own tax tables, you know that at the end you look up on a chart so if you make between $40,000 and $50,000, you pay so much.

If you rent something for so much per hour, half hour, or quarter hour, you are charged for the full amount.  An example might be renting the carpet shampooer from the grocery store.  They charge $3.00 per hour, you use it for 3.5 hours and you pay $12.00 because its over 3 hours so they move up to the next whole hour.

There is bowling shoe rental, laser tax, boating and so many other activities in real life that are step functions. This topic could easily lead into a more specific one on floor and ceiling functions where it goes down instead of up. It opens up so wonderful mathematical discussions.

Let me know what you think.