Periodic functions are special functions that repeat themselves every units. The smallest positive value of that has this property is called the period of the function.
Let's take the following periodic function as an example:
This function repeats itself with a period of To illustrate, first focus on the part of the graph shown in blue below.
If we shift this part units to the right, the graph remains unchanged!
The same would happen if we shifted the part units to the left.
By knowing the period of a periodic function, we can use a value of the function to infer many other values of the function. For example, in the function above, note that Since the period is if we add (or subtract) any multiple of from the input, the output stays the same. We can write this as
where is an integer.
For example, setting gives the following:
The same is true for negative values of Setting gives the following:
What is the period of the function shown below?
Note that if we shift the graph units left or right, the graph remains unchanged.
On the other hand, if we shift the graph horizontally by any distance other than (or a multiple of ), we get a different graph.
Therefore, the period of the function is
The graph of a periodic function $y = f(x)$ is shown above. What is the period of this function?
|
a
|
$\dfrac{1}{4}$ |
|
b
|
$1$ |
|
c
|
$2$ |
|
d
|
$3$ |
|
e
|
$\dfrac{1}{2}$ |
The graph of a periodic function $y = f(x)$ is shown above. What is the period of this function?
|
a
|
$5$ |
|
b
|
$2$ |
|
c
|
$4$ |
|
d
|
$3$ |
|
e
|
$1$ |
Part of the graph of a function with period is shown below. What is the graph of on the interval
The graph above shows the function in the interval
Since this interval has length and the period of the function is also this interval represents a single period of the function.
Therefore, the function will consist of many copies of the shape shown above, shifted horizontally by multiples of
Part of the graph of a function $y=f(x)$ with period $2$ is shown above. What is the graph of $y=f(x)$ on the interval $-4\leq x \leq 4?$
|
a
|
|
|
b
|
 |
|
c
|
|
|
d
|
|
|
e
|
|
Part of the graph of a function $y=f(x)$ with period $2$ is shown above. What is the graph of $y=f(x)$ on the interval $-4\leq x \leq 4?$
|
a
|
|
|
b
|
|
|
c
|
|
|
d
|
|
|
e
|
 |
The graph of is shown below. Given that is periodic with period , what is the value of
Since the period is if we add (or subtract) any multiple of from the input, the output stays the same.
So, we can keep subtracting from the input until we get to an input that we see in the graph:
The graph of $y=f(x)$ is shown above. Given that $f$ is periodic with period $8$, what is the value of $f(111)?$
|
a
|
$-1$ |
|
b
|
$-2$ |
|
c
|
$0$ |
|
d
|
$1$ |
|
e
|
$2$ |
The graph of $y=f(x)$ is shown above. Given that $f$ is periodic with period $4$, what is the value of $f(42)?$
|
a
|
$2$ |
|
b
|
$0$ |
|
c
|
$1$ |
|
d
|
$-2$ |
|
e
|
$-1$ |