Properties of the Sine Function -- Exploration

Click here to open a new window showing the usual sine function,

Notice that this function goes back-and-forth between -1 and +1 and that it goes through one complete cycle (or period) starting at 0 when t = 0; going up to 1 when t = pi/2; then back down to 0 when t = pi; then down to -1 when t = 3 pi/2; and finally returning to 0 when t = 2 pi. It repeats this same pattern between t = 2 pi and t = 4 pi and between t = -2 pi and t = 0.

As it stands, this function is wonderful for modeling periodic phenomena in which something of interest goes back-and-forth between -1 and +1 and takes 2 pi units of time to go through one complete cycle. Most natural periodic phenomena, however, are not so cooporative -- they might go back-and-forth, for example, between -2 and +2 and they might take 4 units of time to go through one complete cycle.

In order to model phenomena like these we look at a family of functions:

where A and w are two constants called parameters. The values of these parameters change the behavior of the functions in this family. The live graph will enable you to experiment with this family of functions. Notice the two controls at the right side of the live graph. These controls enable you to change the values of these parameters. The first control, labeled A, enables you to vary the value of this parameter in the range -2 <= A <= 2. Click the blue Animate Parameter button under this control to see the effects as the value of this parameter changes. You can also click any place on the scale for this parameter to specify its value. Similarly, the control for the parameter w enables you to experiment with the effects of this parameter.

Questions:

1. Describe the effects of the parameter A. What happens when 0 < A < 1? What happens when 1 < A? What happens when A is negative?
   
   
2. Describe the effects of the parameter w. What happens when 0 < w < 1? What happens when 1 < w?

For modeling, we really need to look at a slightly different and more general family of functions -- the family:

Notice that this family has four parameters:

• the parameter A -- called the amplitude,
  
  
• the parameter p -- called the period,
  
  
• the parameter d called the shift, and
  
  
• the parameter B -- called the center.

Click here to open a new live graph with which you can study this family of functions.

Questions:

3. Describe the effects of the parameter p. Why is the parameter called the period?
   
   
4. Describe the effects of the parameter d. Why is this parameter called the shift?
   
   
5. Describe the effects of the parameter B. Why is this parameter called the center?