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Mathematical Expression Editor
Motivating Questions
What do we mean when we say a function is periodic?
Periodic functions
Certain naturally occurring phenomena eventually repeat themselves, especially when
the phenomenon is somehow connected to a circle. For example, suppose that you are
taking a ride on a Ferris wheel and we consider your height, , above the ground
and how your height changes in tandem with the distance, , that you have
traveled around the wheel. We can see a full animation of this situation at
http://gvsu.edu/s/0Dt.
Because we have two quantities changing in tandem, it is natural to wonder if it is
possible to represent one as a function of the other.
In the context of the ferris wheel mentioned above, assume that the height, , of the
moving point (the cab in which you are riding), and the distance, , that the point has
traveled around the circumference of the ferris wheel are both measured in
meters.
Further, assume that the circumference of the ferris wheel is 24 meters (it’s a
pretty short ferris wheel). In addition, suppose that after getting in your
cab at the lowest point on the wheel, you traverse the full circle several
times.
(a)
Recall that the circumference, , of a circle is connected to the circle’s
radius, , by the formula . What is the radius of the ferris wheel? How high
is the highest point on the ferris wheel?
(b)
How high is the cab after it has traveled of the circumference of the circle?
(c)
How much distance along the circle has the cab traversed at the moment
it first reaches a height of meters?
(d)
Can be thought of as a function of ? Why or why not?
(e)
Can be thought of as a function of ? Why or why not?
Let’s consider a point traversing a circle of circumference 24 and examine how the
point’s height, , changes as the distance traversed, , changes. Note particularly that
each time the point traverses of the circumference of the circle, it travels a distance
of units, as seen below, where each noted point lies 3 additional units along the circle
beyond the preceding one.
Note that we know the exact heights of certain points. Since the circle has
circumference , we know that and therefore . Hence, the point where (located of
the way along the circle) is at a height of . Doubling this value, the point where has
height . Other heights, such as those that correspond to and (identified on the
figure by the green line segments) are not obvious from the circle’s radius, but can be
estimated from the grid in the figure above as (for ) and (for ). Using all of
these observations along with the symmetry of the circle, we can construct a
table..
Moreover, if we now let the point continue traversing the circle, we observe that the
-values will increase accordingly, but the -values will repeat according to the
already-established pattern, resulting in the data in the table below.
It is apparent that each point on the circle corresponds to one and only one height,
and thus we can view the height of a point as a function of the distance the
point has traversed around the circle, say . Using the data from the two
tables and connecting the points in an intuitive way, we get the graph shown
below
Notice that the graph above resembles the graph of the sine function. As it turns out,
the sine function exhibits some of the same oscillatory behavior as . This shared
property turns out to be very important, especially when looking at functions that
are related to circles.
Let be a function whose domain and codomain are each the set of all real numbers.
We say that is periodic provided that there exists a real number such that for
every possible choice of . The smallest positive value for which for every choice of is
called the period of .
For our ferris wheel example above, the period is the circumference of the circle that
generates the curve. In the graph, we see how the curve has completed one
full cycle of behavior every 24 units, regardless of where we start on the
curve.
Two important periodic functions are the sine function, which you have seen, and the
cosine function, which resembles the sine function. You will study these functions and
learn about their relationship with circles in trigonometry.
As a reminder, here is a graph of the sine function along with a table listing some of
its values.
-
-−1
0 0
π___2 1
π 0
−1
2 π 0
Notice that and as well. In fact, the sine function is periodic with period
.
Now, here is a graph of the cosine function along with a table listing some of its
values.
-
- 0
0 1
π___2 0
π − 1
0
2 π 1
Notice that and as well. In fact, the cosine function is also periodic with period
.
For a function defined on the real numbers, we say is periodic if there exists
some such that
for all possible choices of . The smallest value of for which for all possible
choices of is called the period of .