Step Functions Are All Around Us!

I've been teaching for many years and I have taught about step functions but it hasn't been until fairly recently that I've been able to find real life situations that illustrate step functions.  To begin with, step functions are defined as a function whose graph looks like a series of steps because the values jump.

If you go by the math textbooks, step functions seem to be completely theoretical because there are never any real examples to show us how step functions operate in real life.  However, with a bit of research, it is easy to find real life examples.

Let's start with postage stamps which are priced for mail weighing up to one ounce.  Back in 1926, it cost 2 cents to mail a letter but the price increased to 3 cents in 1932 where it remained at that price until 1958 when it jumped to a whopping 4 cents for the first ounce.  The next jump came in 1963 when it moved to 5 cents and five years later in 1968, the price changed to 6 cents. The price jumped every few years until 2019 when it cost 55 cents to mail a letter.  The USPS website has a lovely chart with the cost of a stamp from 1885 to 2019.  The jumps are generally from one to three cents except in 2019 when it jumped from 50 cents to 55 cents.

Again, looking at postage stamps, we have a step function for just mailing something first class.  As mentioned earlier, the first ounce costs 55 cents which means that as long as the letter weighs between 0 and .99 of an ounce, it will cost you 55 cents to mail but the minute you go over 1 ounce, the letter will cost an additional 20 cents for the next full ounce and 20 cents each additional ounce.  The cost is not decreased if you are mailing a letter that is 1.5 ounces. It will cost 75 cents if it is 1.1 ounce or 1.99 ounce as the cost if for anything up to the full ounce weight.

Another example of step functions in real life is the minimum wage.  The first minimum wage was set at 25 cents per hour back in 1938 and one year later in October of 1939, the government raised the minimum wage to 30 cents per hour where it remained till 1945 when the minimum wage jumped to 40 cents per hour.  Over the years, the wage was raised periodically until 2009, when it was set to $7.25 per hour. This governmental website has a lovely chart showing the raises over the years.  This lists the federal minimum wage because some states such as Alaska have a higher minimum wage.

Then there is the wonderful Hershey's candy bar.  In 1908, this candy bar cost 2 cents where it stayed till 1930 when they raised the price to 5 cents per bar. The chocolate bar continued to cost 5 cents until 1969 when they raised the price to 10 cents and the price jumped to 15 cents in 1974.  Three years later, in 1977, the price was 20 cents and one year later the cost was 25 cents.  The price continued to rise till today when you can find the standard bar for around $1.00. This site has a nice summary of the price of Hershey's candy bar and the change in weight as the prices changed. 

Here are four examples of step functions because when graphed as a step function, the graphs reflect the data more accurately and give a better representation of the situation.  Let me know what you think, I'd love to hear.  Have a great day.