# Periodic Functions

## Graphs of the Sine and Cosine Functions

White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow.
Light waves can be represented graphically by the sine function. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions.

### Graphing Sine and Cosine Functions

Recall that the sine and cosine functions relate real number values to the x- and y-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function. We can create a table of values and use them to sketch a graph.  lists some of the values for the sine function on a unit circle.
Plotting the points from the table and continuing along the x-axis gives the shape of the sine function. See .
Notice how the sine values are positive between 0 and 
  which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between 
  and 
  which correspond to the values of the sine function in quadrants III and IV on the unit circle. See .
Now let’s take a similar look at the cosine function. Again, we can create a table of values and use them to sketch a graph.  lists some of the values for the cosine function on a unit circle.
As with the sine function, we can plots points to create a graph of the cosine function as in .
Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval 
  
In both graphs, the shape of the graph repeats after 
  which means the functions are periodic with a period of 
  A periodic function is a function for which a specific horizontal shift, P, results in a function equal to the original function: 
  for all values of 
  in the domain of 
  When this occurs, we call the smallest such horizontal shift with 
  the period of the function.  shows several periods of the sine and cosine functions.
Looking again at the sine and cosine functions on a domain centered at the y-axis helps reveal symmetries. As we can see in , the sine function is symmetric about the origin. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because 
 
Now we can clearly see this property from the graph.
 shows that the cosine function is symmetric about the y-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that 
 


### Investigating Sinusoidal Functions

As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function is known as a sinusoidal function. The general forms of sinusoidal functions are

### Determining the Period of Sinusoidal Functions

Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.
In the general formula, 
  is related to the period by 
  If 
  then the period is less than 
  and the function undergoes a horizontal compression, whereas if 
  then the period is greater than 
  and the function undergoes a horizontal stretch. For example, 
  
  so the period is 
  which we knew. If 
  then 
  so the period is 
  and the graph is compressed. If 
  then 
  so the period is 
  and the graph is stretched. Notice in  how the period is indirectly related to 
 


### Determining Amplitude

Returning to the general formula for a sinusoidal function, we have analyzed how the variable 
  relates to the period. Now let’s turn to the variable 
  so we can analyze how it is related to the amplitude, or greatest distance from rest. 
  represents the vertical stretch factor, and its absolute value 
  is the amplitude. The local maxima will be a distance 
  above the horizontal midline of the graph, which is the line 
  because 
  in this case, the midline is the x-axis. The local minima will be the same distance below the midline. If 
  the function is stretched. For example, the amplitude of 
  is twice the amplitude of 
  If 
  the function is compressed.  compares several sine functions with different amplitudes.

### Analyzing Graphs of Variations of y = sin x and y = cos x

Now that we understand how 
  and 
  relate to the general form equation for the sine and cosine functions, we will explore the variables 
  and 
  Recall the general form:
The value 
  for a sinusoidal function is called the phase shift, or the horizontal displacement of the basic sine or cosine function. If 
  the graph shifts to the right. If 
  the graph shifts to the left. The greater the value of 
  the more the graph is shifted.  shows that the graph of 
  shifts to the right by 
  units, which is more than we see in the graph of 
  which shifts to the right by 
  units.
While 
  relates to the horizontal shift, 
  indicates the vertical shift from the midline in the general formula for a sinusoidal function. See . The function 
  has its midline at 
  
Any value of 
  other than zero shifts the graph up or down.  compares 
  with 
  which is shifted 2 units up on a graph.

### Graphing Variations of y = sin x and y = cos x

Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations.
Instead of focusing on the general form equations
we will let 
  and 
  and work with a simplified form of the equations in the following examples.

### Using Transformations of Sine and Cosine Functions

We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function.

### Key Equations


### Key Concepts

1. Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of 
  
2. The function 
  is odd, so its graph is symmetric about the origin. The function 
  is even, so its graph is symmetric about the y-axis.
3. The graph of a sinusoidal function has the same general shape as a sine or cosine function.
4. In the general formula for a sinusoidal function, the period is 
  See .
5. In the general formula for a sinusoidal function, 
  represents amplitude. If 
  the function is stretched, whereas if 
  the function is compressed. See .
6. The value 
  in the general formula for a sinusoidal function indicates the phase shift. See .
7. The value 
  in the general formula for a sinusoidal function indicates the vertical shift from the midline. See .
8. Combinations of variations of sinusoidal functions can be detected from an equation. See .
9. The equation for a sinusoidal function can be determined from a graph. See  and .
10. A function can be graphed by identifying its amplitude and period. See  and .
11. A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See .
12. Sinusoidal functions can be used to solve real-world problems. See , , and .

### Section Exercises


### Verbal


### Graphical

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for 
  Round answers to two decimal places if necessary.
For the following exercises, graph one full period of each function, starting at 
  For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for 
  State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

### Algebraic

For the following exercises, let 
 

For the following exercises, let 
 


### Technology


### Real-World Applications
