Lesson 3

Lesson 3: Equations of Sinusoidal Functions

 

Focus

 

The distance from Earth to the Sun is not a constant. Earth is closest to the Sun on approximately January 4 and farthest away from the Sun on about June 4. Earth’s distance from the Sun can be approximated by the equation , where D represents the distance in millions of kilometres and n represents the number of days into the year.

 

By just looking at the function, what information can you determine? Can you estimate the maximum distance Earth is from the Sun on about June 4? What about the distance on January 4?

 

In this lesson you will explore how the equation of a sinusoidal function is related to characteristics of that function.

 

Lesson Outcomes

 

At the end of this lesson you will be able to

• describe the characteristics of a sinusoidal function by analyzing its equation
• match a sinusoidal equation to its graph

Lesson Question

 

You will investigate the following question: How are the characteristics of a sinusoidal function related to its equation?

 

Assessment

 

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 3 Assignment (Download the Lesson 3 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms and Formula Sheet

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.

Explore

 

In Discover you explored how a, b, c, and d are related to characteristics of the graph y = a sin b(x − c) + d with the angle measured in radians. You may have found the following:

• The amplitude of the graph is determined by the parameter a of the function.
• The parameter b is related to the period by the equation , where P represents the period and the angle is measured in degrees. Notice that a large b-value results in a shorter period, and a small b-value results in a longer period.

 

The parameters c and d do not change the shape of the graph, but they do move the graph vertically and horizontally.

 

When the graphs are moved up or down, three characteristics are affected: the midline, the maximum, and the minimum.

• The midline will occur at y = d.
• The maximum value = d + a.
• The minimum value = d − a.

In Discover you noticed that changing the parameter c results in the graph being shifted c units horizontally.

 

In Lesson 2 you graphed both y = sin x and y = cos x and saw that they were very similar functions. Just as parameters can be used to interpret a sine function, the parameters a, b, c, and d correspond to characteristics for a function of the form y = a cos b(x − c) + d. Try This 2 explores how these parameters change the graph of the function.

 

Try This 2

1. Predict how the parameters a, b, c, and d will correspond to characteristics of the function y = a cos b(x − c) + d by completing a table similar to the following.
   
   

   	Description of Predicted Change
   Increase a
   Decrease a
   Increase b
   Decrease b
   Increase c
   Decrease c
   Increase d
   Decrease d

2. Following similar instructions as in Discover, open Cosine a, b, c, d Explorer and use this applet to check your predictions from question 1.
   
   
    
   

Save your responses in your course folder.

 

Share 2

 

With a partner or in a group, describe how changing the parameters a, b, c, and d are similar for the function y = a cos b(x − c) + d and the function y = a sin b(x − c) + d.

 

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

In Try This 2, you may have noticed that the parameters a, b, c, and d in y = a cos b(x − c) + d describe the same characteristics as they did for y = a sin b(x − c) + d.

 

So far, you have worked with sine and cosine functions in degrees. It is also possible to use an x-value in radians. All the parameters behave the same, but now c is in radians. The relationship between b and the period also changes slightly: 360° = 2π, so use  instead of when working in radians.

 

The following example shows how an equation in radians can be interpreted.

 

Example

 

Harry examined the function and asked the following questions.

1. What is the equation of the midline?
2. What is the period of the graph?
3. Describe any horizontal translation of the graph from y = cos x.
4. What are the maximum and minimum values?

Solution

1. The midline has the equation y = d. This means the equation of the midline is y = 2.
2. The equation is written in radians because the c-value has no unit.
   
   
    
   360° = 2π rad
   
   So, the equation  can be used to determine the period of the function.
   
   
    
   
3. The c-value determines any horizontal translation from y = cos x. A −4 must have been substituted for c into y = a sin b(x − c) + d. This means the graph moved left 4 units.
4. The amplitude is equal to a and so is . The maximum occurs one amplitude above the midline, so the maximum is . The minimum occurs one amplitude below the midline, so the minimum is .
   
   The graphing calculator can graph  in order to check the solutions.
   
   
   

Self-Check 2

Sometimes it is useful to determine the equation of a function given its graph. Equation from a Graph describes this process.

 

 

From the video, you can summarize the steps for determining the equation of the function from its graph.

 

Step 1: Determine the maximum and minimum values of the graph. Use these values to determine the midline and the amplitude. These are the a- and d-values of the function.

 

 and

 

Step 2: Determine the period of the graph. The parameter b can be calculated by the formula

 

 

Step 3: Determine c by choosing the function y = sin x or y = cos x. If y = sin x, then c is the distance away from the y-axis to the first intersection of the graph with the midline where the graph is increasing. If y = cos x, then c is the distance away from the y-axis to the first maximum point.

 

Step 4: Substitute the parameters into the standard form of the function.

 

Use these steps to help you complete the following Self-Check questions.

 

Self-Check 3

When a sinusoidal function is used to model a situation, the parameters a, b, c, and d can be interpreted to provide information about the model. In Try This 3, you will determine information about a scenario by interpreting the equation of a sinusoidal function.

 

Try This 3

 

The height of a swing over time can be modelled by the function , where h is the height in centimetres above the ground and t is the time in seconds.

1.  
   1. What is the highest point the girl in the photo will reach?
   2. Determine the height of the girl at 3.5 s and at 8.0 s.
2.  
   1. In terms of the movement of the swing, explain what the 65 represents.
   2. Explain what the 15 represents in terms of the movement of the swing.
3. Suppose the girl in the photo is at the highest point. How long will it take her to reach that point again?
   
   
   Ingram Publishing/Thinkstock

Save your responses in your course folder.

 

In Try This 3, you may have noticed that the questions could be answered by interpreting the parameters a, b, c, and d and solving the equation or the questions could be answered by graphing the equation and interpreting the graph.

Connect

 

Lesson 3 Assignment

Lesson 3 Summary

 

A sine function and cosine function can be written in the standard forms y = a sin b(x − c) + d and y = a cos b(x − c) + d, where a, b, c, and d represent real numbers and are called parameters. Parameters can be changed so that the function can be used for many different applications.

 

The characteristics of sinusoidal functions can be summarized as follows.

 

The value of a is the amplitude:

 

 

The value of d determines the midline and helps find the maximum and minimum values.

• equation of the midline is y = d
  
• maximum value = a + d
  
• minimum value = a − d

The value of b is the number of cycles in 360° or 2π. The period can be calculated using this formula:

 

           

 

The value of c designates the horizontal translation.

• A positive c moves the y = sin x or y = cos x function c units to the right.
• A negative c moves the y = sin x or y = cos x function c units to the left.

 

In this diagram, the period is written as , which means this graph is in radians.
If the graph is in degrees, use

 

In Lesson 4 you will investigate how sinusoidal functions can be used to model some real-world scenarios.