Lesson 3

Discover

 

A sine function can be written in the standard form of a sinusoidal function, y = a sin b(x − c) + d, where a, b, c, and d represent real numbers. The values a, b, c, and d are called parameters of this function while x and y are variables. A parameter can be defined as a value that is already built into a function.

 

Parameters can be changed so that the function can be used to model different applications.

 

In Lesson 2 you graphed the function y = sin x. This function can be interpreted as having the parameters a = 1, b = 1, c = 0, and d = 0. The function y = sin x will often be used as a starting point for interpreting functions of the form y = a sin b(x − c) + d in this lesson.

 

This is a graph of the function y = sin x. Some key
points have been highlighted on the graph.

 

In Try This 1, you will explore how a function of the form y = a sin b(x − c) + d changes as the four parameters change.

 

Try This 1

 

Open the piece titled Sine a, b, c, d Explorer.

 

 

Part A

 

How does changing the parameter a affect the graph of the sinusoidal function? For this part of Try This 1, adjust the sliders so that b = 1, c = 0, and d = 0.

1. Complete the following table by changing a on the Sine a, b, c, d Explorer.
   
   

   Action	Value of a	Changes to Graph	Sketch or Screenshot	Midline	Amplitude	Period
   y = sin x	1
   increase a
   decrease a

2.  
   1. Compare the three graphs in the table from question 1. What characteristic of the graph is affected by changing a?
   2. How is the value of a related to the amplitude?

Part B

 

How does changing the parameter b affect the graph of the sinusoidal function? Adjust the sliders so that a = 1, c = 0, and d = 0.

3. Complete the following table by changing b on the Sine a, b, c, d Explorer.
   
   

   Action	Value of b	Changes to Graph	Sketch or Screenshot	Midline	Amplitude	Period
   y = sin x	1
   increase b
   decrease b

4.  
   1. Compare the three graphs in the table from question 3. What characteristic of the graph is affected by changing b?
   2. If b is doubled, how does the period change?

Part C

 

How does changing the parameter c affect the graph of the sinusoidal function? Adjust the sliders so that a = 1, b = 1, and d = 0.

5. Complete the following table by changing c on the Sine a, b, c, d Explorer.
   
   

   Action	Value of c	Changes to Graph	Sketch or Screenshot	Midline	Amplitude	Period
   y = sin x	0
   increase c
   decrease c

6.  
   1. Compare the three graphs in the table from question 5. What characteristic of the graph is affected by changing c?
   2. How is the sign of c related to the direction of the change it causes?

Part D

 

How does changing the parameter d affect the graph of the sinusoidal function? Adjust the sliders so that a = 1, b = 1, and c = 0.

7. Complete the following table by changing d on Sine a, b, c, d Explorer.
   
   

   Action	Value of d	Changes to Graph	Sketch or Screenshot	Midline	Amplitude	Period
   y = sin x	0
   increase d
   decrease d

8.  
   1. Compare the three graphs in the table from question 7. What characteristic of the graph is affected by changing d?
   2. How is the value of d related to the position of the midline?

Save your responses in your course folder.

 

Share 1

 

With a partner or in a group, discuss the following questions based on your responses to Try This 1.

1. Using results gained so far, predict how the graph of  would differ from the graph of y = sin x. Be specific.
2. Use Sine a, b, c, d Explorer to check your prediction.
3. Which parameters will directly affect the range of the sinusoidal function? Show an example to explain your reasoning.

If required, save a record of your discussion in your course folder.