Lesson 2

Lesson 2: Graphs of Sinusoidal Functions

 

Focus

 

iStockphoto/Thinkstock

 

Many natural phenomena are cyclic; they repeat the same pattern over and over. Consider the mean daily temperature in Calgary. You expect warmer temperatures in the summer and then cooler temperatures in the winter, and the cycle repeats. Temperature versus time can be plotted for many years. How would you describe the shape of this extended graph? Are there other phenomena that can be represented by the same shape of graph? Are there similar characteristics that the graphs share?

 

 

In this lesson you will learn the general shape of a sinusoidal function and some of its characteristics. You will also begin to explore models that use sinusoidal functions to represent data.

 

Lesson Outcomes

 

At the end of this lesson you will be able to

• interpret a sinusoidal graph
• draw a sinusoidal graph given characteristics about the function
• compare two different sinusoidal functions

Lesson Questions

 

You will investigate the following questions:

• How can characteristics of a sinusoidal graph be described?
• How can sinusoidal graphs be used to model data?

Assessment

 

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

• completion of the Lesson 2 Assignment (Download the Lesson 2 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

Launch

 

Do you have the background knowledge and skills you need to complete this lesson successfully? Launch will help you find out.

 

Before beginning this lesson, you should be able to identify the equation of a horizontal line.

Are You Ready?

 

Complete these questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1. What is the equation of the following horizontal line?
   
   
   
   Answer
2. Find the equation of the horizontal line that passes through (4, −5). Answer

If you answered the Are You Ready? questions without difficulty, move to Discover.

 

If you found the Are You Ready? questions difficult, complete Refresher.

Refresher

 

The following graph shows how two horizontal lines are graphed. Horizontal lines have equations of the form y = b, where b is a constant. In the graph, all the points on the red line have a y-value of 4 and all the points on the grey line have a y-value of −6.

 

The equations of horizontal lines take the form y = b,
where b represents the y-value the line has at every point.

Go back to the Are You Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

Discover

 

You have practised graphing and finding the equation of horizontal lines. You will now learn how to graph sinusoidal functions.

 

Try This 1

 

When graphed, the function y = sin x displays an interesting pattern. In this activity you will use a table of values to graph this function.

 

1. Use a calculator to complete the following table.
   
   

   x	y = sin x
   0°	0
   30°	0.5
   60°	0.866
   90°	1
   120°
   150°
   180°
   210°	–0.5
   240°
   270°	−1
   300°
   330°
   360°
   390°
   420°
   450°
   480°
   510°	0.5
   540°	0

2. 
   
   1. What patterns do you notice in the table?
   2. Using the patterns, predict the values of sin 570°, sin 600°, sin 630°, sin 660°, sin 690°, and sin 720°.

3. Go to Plotting Template and print a copy. Use the table to graph the function y = sin x to 720° on the template.

4. 1. Describe the shape of the graph.
   2. What is the maximum value of the graph?
   3. What is the minimum value?
   4. Where does the graph start to repeat itself?
   5. How do you expect the graph would look if it also included larger angles?

Save your responses in your course folder.

Explore

 

Throughout previous math courses and in this course, you have learned about different functions: linear, quadratic, polynomial, and rational. In Try This 1 you were introduced to the sine function and discovered that the graph looks like the following.

 

A similar function to the sine function is the cosine function, which is shown with the table of values. Notice that the sine and cosine functions have the same output values, but the inputs or x-values are offset by 90°.

 

 

 

Both the sine and cosine functions are periodic functions. A periodic function is a function that repeats itself at regular intervals.

 

The periodic functions you will study in this course have the following characteristics:

• The maximum and minimum are the highest and lowest points the function reaches.
• The midline is halfway between the maximum and minimum.
• The amplitude is the distance from the midline to a maximum or a minimum.
  
  
  
• Finally, the period is the length of one cycle of the function.
  
  

Look at the y = sin x and the y = cos x graphs shown in the Discover and Explore sections. The following characteristics can be summarized from the graphs.

 

	y = sin x	y = cos x
Amplitude	1	1
Period	360° or 2π	360° or 2π
y-intercept	(0, 0)	(0, 1)
Midline	y = 0	y = 0
Range	−1 ≤ y ≤ 1	−1 ≤ y ≤ 1
Domain	x ∈ R	x ∈ R

Functions that have the same shape as y = sin x or y = cos x are called sinusoidal. From the chart, you learned characteristics of y = sin x and y = cos x. Throughout this lesson and the next lesson, you will investigate how these characteristics change for various sinusoidal functions. In Try This 2 and Try This 3, you will analyze the graphs of different sinusoidal functions.

 

Try This 2

 

 

Use the graphs to answer the following questions.

1. Use Graph 1 and Graph 2 to complete a chart like the following.
   
   

   	Graph 1	Graph 2
   Minimum
   Maximum
   Midline
   Amplitude

2.  
   1. Determine an equation that relates the minimum, the maximum, and the midline. Will your equation work for any sinusoidal graph?
   2. Determine an equation that relates the minimum, the maximum, and the amplitude. Will this equation work for any sinusoidal graph?
3. Determine the range of this graph.

Save your responses in your course folder.

 

Share 1

 

With a partner or group, discuss the following question based on the information from Try This 2.

 

How do the following characteristics relate to the range?

1. the maximum and minimum
2. the amplitude
3. the midline

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

In Try This 2, you may have noticed that the midline and amplitude are related to the maximum and minimum by the following equations. Given the graph of a function, the formulas can be used to determine the amplitude and the midline of the function.

 

          

 

Try This 3

 

 

Use the graphs to answer the following questions.

1. Determine the period of Graph 1.
2. Estimate and then determine the period of Graph 2.
3. Did you use different strategies to determine the period for Graph 1 and for Graph 2? Explain.

Save your responses in your course folder.

 

Share 2

 

With a partner or group, discuss the following question:

 

How do your strategies for determining the period compare?

 

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

In Try This 3, you may have noticed that sometimes it is difficult to determine the period accurately using only one cycle. It is often more accurate to use the length of multiple cycles to determine the length of a single period. For Graph 2 in Try This 3, the graph clearly intersects (0°, 4) and (360°, 4) and takes 5 cycles to move between the two points. This means

 

 

If you tried to determine the period using one cycle, you would have needed to estimate at least one endpoint of the cycle. An estimate of 70° is reasonable, but it is not as accurate as 72°.

 

 

 

So far, you have determined characteristics of a sinusoidal function from a graph. Sometimes, it is also useful to sketch a graph from given information. In the following activity, you will explore strategies to sketch sinusoidal functions.

 

Try This 4

1. Draw at least two cycles of a sinusoidal function with the following characteristics. Click on the hint buttons if you require help.
   
   

   Period	12 rad
   Midline	y = −1
   Maximum	2
   Passes Through the Point	(0, −4)

   
   Graph Size

    

   Midline

    

   Sketching the Graph
   

2. Describe the set of steps you used to sketch the function in question 1. Will these steps work for any sinusoidal function?

Save your responses in your course folder.

 

Share 3

 

With a partner or group, compare your strategies for sketching a sinusoidal function.

 

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

Graphing a sinusoidal function from bits of information, as you did in Try This 4, requires you to think about many things at once. The following example shows how the graph of a function can be drawn given pieces of information. Sketch the graph of a sinusoidal function with the following characteristics:

• The domain is {x | 0° ≤ x ≤ 360°, x ∈ R}.
• The range is {y | −1 ≤ y ≤ 5, y ∈ R}.
• The period is 90°.
• The y-intercept is 2.

Watch Graphing a Sinusoidal Function to see the full solution to the example.

 

 

Self-Check 2

Sinusoidal functions can be used to model many scenarios that involve the regular increase and decrease of a value. In the next activity, you will explore how daily average temperatures can be modelled and compared.

 

Try This 5

 

At the beginning of the lesson, you saw a mean daily temperature graph for Calgary. Use this information to answer the following questions.

 

1.  
   1. Predict two months where the mean daily temperature is 5°C.
   2. What is the mean daily temperature for March?
      
      

2.  
   1. Estimate the maximum and minimum values for the graph. Describe what each of these represents.
   2. Estimate the period of the graph. What does this represent?
   3. Estimate an equation for the midline. What does this represent?
      
      

3. Sketch another cycle of this graph. How is this cycle related to the original?
4. Describe how this graph would differ from one drawn for Whitehorse and one drawn for Miami.

Save your responses in your course folder.

Connect

 

Lesson 2 Assignment

Lesson 2 Summary

 

In this lesson you learned that the sine and cosine functions are considered periodic.

 

You learned the following:

• The maximum and minimum are the highest and lowest points on the graph of the function.
• The midline is halfway between the maximum and minimum.
• The amplitude is the distance from the midline to a maximum or a minimum.
• Finally, the period is the length of one cycle of the function.

 

Before any transformations take place, the characteristics of y = sin x and y = cos x can be summarized as follows:

 

	y = sin x	y = cos x
Amplitude	1	1
Period	360° or 2π	360° or 2π
y-intercept	(0, 0)	(0, 1)
Midline	y = 0	y = 0
Range	−1 ≤ y ≤ 1	−1 ≤ y ≤ 1
Domain	x ∈ R	x ∈ R

In Lesson 3 you will look at how making changes to the characteristics of a graph changes the sinusoidal function.