Periodic functions were discussed in the previous lesson.
Be sure to read that section first!
Consider the periodic function $\,f\,$ shown above.
Assume that enough of the graph is shown to clearly illustrate the repeating part.
(Nothing funny happens outside our viewing window!)
That is, the graph continues in this same way forever and ever, both to the right and to the left.
Here are two different questions (and their answers):
In English, the articles ‘a’ and ’the’ are used in different situations.
The article ‘the’ is used to refer to a unique object.
When you say ‘the cat’, it is clear that you're referring to one particular cat.
The article ‘a’ allows for more than one object.
When you say ‘a cat’, it allows for any cat, not a specific one.
So, you've got to be a bit careful with the language.
If someone stops you on the street,
flashes the graph of a periodic function, and asks
‘Hey! What's the period of this function?’
they want the least positive period—the smallest one.
They don't want you to list (say) all the multiples of a number.
They just want the smallest one that works.
Let's repeat the definition of periodic function from the prior section,
this time including the special
meaning of the word ‘period’ when it is preceded by the article ‘the’:
DEFINITION
periodic function; period
A function $\,f\,$ is periodic
if and only if
there exists a positive number $\,p\,$ with the following properties:
When a function has a least positive period, then it is given a special name: it is called ‘the period’ of the function. 

You might wonder why the definition above didn't simply say:
Consider the constant function, $\,f(x) = 7\,$.
No matter what the input is, the output is $\,7\,$.
(Any constant function would work for this example, but $\,7\,$ is the author's favorite number.)
Is $\,f\,$ periodic?
The answer is a resounding YES.
Indeed, every positive number $\,p\,$ is a period!
Why?
For every real number $\,x\,$, and for every $\,p > 0\,$, we have:
$$\cssId{s54}{7 = f(x) = f(x + p) = f(x  p)}$$
Every number in the interval $\,(0,\infty)\,$ is a period of $\,f\,$.
Is there a least positive period?
In other words, is there a smallest number in the interval $\,(0,\infty)\,$?
Nope.
The number zero is ‘trying to be’ the least positive period, but unfortunately $\,0\,$ isn't a positive number!
So, if you're asked ‘What is the period of $\,f(x) = 7\,$?’ you'll have to give this answer:
at all the integers, as shown at right:  
halfway between all those points:  
halfway between all those points: 
To continue the potential confusion surrounding the word ‘period’, here's yet another use for the word!
Whenever you have a function $\,f\,$ with a least positive period, $\,p\,$,
then the graph of $\,f\,$ on any interval of length $\,p\,$ is often called
a period or a cycle of $\,f\,$.
That is, the word ‘period’ (or ‘cycle’)
is often used to refer to a smallest possible portion of the graph
that could be used as a ‘template’ to produce the entire graph.
For an example, let's return to the graph at the top of the page. 

So, the word ‘period’ can refer to a number or to a piece of a graph.
How's a person to know which is wanted?
If your teacher asks ‘What is the period of $\,f\,$?’ then they want a (unique) number.
If your teacher says ‘Please sketch a period of $\,f\,$’ then they want a piece of the graph.
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
