Lesson 15

Features of Trigonometric Graphs (Part 1)

15.1: Notice and Wonder: Musical Notes (5 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that the period of a trigonometric graph can vary, which will be useful when students examine the graph of the position of a point on a wheel in a later activity. While students may notice and wonder many things about these images, the different amplitudes and periods of the two graphs are the important discussion points.

Launch

Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the graphs for all to see. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Student Facing

Here are pictures of sound waves for two different musical notes:

graph of 2 curves. curve a has greater amplitude, curve a has a shorter period, both curves have midline on he x axis. 

What do you notice? What do you wonder?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. Encourage students to respectfully disagree, ask for clarification or point out contradicting information. If the period and amplitude of the graphs do not come up during the conversation, ask students to discuss these ideas.

Tell students that the amplitude of a sound wave relates to how loud the sound is (larger amplitude means a louder sound). The period of the sound wave is related to its pitch. More highs and lows in the wave, that is a shorter period, means a higher pitch. Later in this lesson they will investigate the period of a trigonometric function in a different context. 

15.2: Equations and Graphs (15 minutes)

Activity

The goal of this task is to match equations and graphs of cosine and sine functions. To complete the matching students may:

  • Evaluate the functions for strategic inputs such as \(\theta = 0\) or \(\theta = \frac{\pi}{2}\) and eliminate possibilities helping to identify the correct graph.
  • Identify maximums and minimums of the graphs and trigonometric expressions.
  • Think about the midline, amplitude, and horizontal translation both graphically and in equation form.

Monitor for students who use these different approaches and invite them to share during the discussion.

Students should use the structure of the equations to identify important features of the corresponding graphs. Technology should only be used at the end, to help students verify that their answers are correct. 

Launch

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 4 of the 6 to complete. Chunking this task into more manageable parts may also support students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

Match each equation with its graph. More than 1 equation can match the same graph.

Equations:

  1. \(y = \text-\cos(\theta)\)
  2. \(y = 2\sin(\theta)-3\)
  3. \(y = \cos\left(\theta + \frac{\pi}{2}\right)\)
  4. \(y = 3\sin(\theta) - 2\)
  5. \(y = \sin(\theta-\frac\pi 2)\)
  6. \(y = \sin(\theta+\pi)\)

A

graph of sine function. midline at y = -3. amplitude = 2.

B

graph of sine function with midline of y = -2, amplitude = 3. 

C

graph of cosine function shifted by the fraction pi over 2 to the left. graph could also be the sine function shifted pi to the left. 

D

graph of sine function shifted by the fraction pi over 2 to the right. graph could also be the cosine function reflected over the horizontal axis.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. Find an equation for this graph using the sine function.
  2. Find another equation for the same graph using a cosine function.
Function on a coordinate grid, origin O.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are not sure how to begin matching, suggest they try a few strategic \(\theta\)-values. For example, ask students to name the input values for cosine that have integer outputs.

Activity Synthesis

Select previously identified students to share in this order.

First, invite those who evaluated the trigonometric functions to help identify the proper graph. For example \(3\sin(\theta) - 2\) takes the value -2 when \(\theta = 0\) and 1 when \(\theta = \frac{\pi}{2}\). This means it matches with graph B (in fact, just one of these values is enough to make this match). Similarly for \(\text-\cos(\theta)\), the value when \(\theta = 0\) is -1 and this is enough to make the match with graph D.

Next, invite students who looked specifically for maximums or minimums to share. For example, \(\cos\left(\theta+\frac{\pi}{2}\right)\) takes a maximum value of 1 when \(\theta = \frac{3\pi}{2}\). This means it must match graph C. The expression \(3\sin(\theta)-2\) takes a maximum value of 1 when \(\theta = \frac{\pi}{2}\) so this matches graph B.

Finally, invite students who used the structure of the equations to deduce important properties of the graphs. For example, the graph of \(y = \sin\left(\theta-\frac{\pi}{2}\right)\) is the same as the graph of \(y = \sin(\theta)\) but it is translated right by \(\frac{\pi}{2}\), matching graph D. The graph of \(y = \sin(\theta+\pi)\) is the same as the graph of \(y = \sin(\theta)\) but it has been translated to the left by \(\pi\), matching graph C.

The reasoning in the first two approaches takes advantage of the given fact that each equation matches one of the graphs. The first approach uses values of \(\theta\) for which \(\sin(\theta)\) and \(\cos(\theta)\) are known. The second approach is similar but thinks strategically about where the maximum and minimum values of each graph will be. The third approach is more abstract, thinking about how the graphs relate, via horizontal and vertical translations and vertical stretch, to the graphs of \(y = \sin(\theta)\) and \(y = \cos(\theta)\). This approach could be used not only to identify each match but also to produce the graphs.

Conversing: MLR8 Discussion Supports. Use this routine to prepare students for the whole-class discussion. Arrange students in groups of 2. Invite students to take turns explaining their thinking. Display the following sentence frames for all to see: “_____ matches _____ because. . .”, “I noticed _____, so I matched. . .”, “The strategy that I used was . . . .” Encourage students to challenge each other when they disagree. While monitoring discussions, emphasize students’ use of different strategies, such as evaluating the functions for specific \(\theta\) values or using maximums and minimums, that all led to making the same successful match. This will help students explain their approach to matching a trigonometric equation and graph.
Design Principle(s): Support sense-making; Maximize meta-awareness

15.3: Double the Sine (15 minutes)

Activity

The goal of this activity is to experiment with one final parameter of a trigonometric function, the number \(k\) in \(y = \cos(k\theta)\). This number \(k\) determines the period of the trigonometric function. When \(k = 1\), this is the normal cosine function whose period is \(2\pi\). When \(k = 2\) the period becomes \(\pi\) because as \(\theta\) varies from 0 to \(\pi\), the input \(2\theta\) in the expression \(\cos(2\theta)\) takes all values between 0 and \(2\pi\). For the same reason, the period of \(y = \cos(4\theta)\) is \(\frac{\pi}{2}\). The period can also be longer than \(2\pi\): for example the period of \(y = \cos\left(\frac{\theta}{3}\right)\) is \(6\pi\) because \(\frac{\theta}{3}\) goes through the values 0 to \(2\pi\) when \(\theta\) goes through the values \(0\) to \(6\pi\).

The work in this activity is an important example of the general idea of horizontal scaling studied in a previous unit. In general, if \(f\) is a function then the graph of the function \(y = f(k\theta)\) is a horizontal stretch, by a factor of \(\frac{1}{k}\), of the graph of \(y = f(\theta)\).

The axes for the graphs are marked in increments of \(\frac{\pi}{12}\) so that students can plot all of the values from the table and additional values if they need more points to feel confident about their graphs.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Ask students to predict how they think the graph of \(y = \sin(2\theta)\) compares to the graph of \(y = \sin(\theta)\). How are they alike? How are they different? Give students quiet work time and then time to share their work with a partner. Select students to share their thinking, recording their predictions for all to see. Tell students that this activity is a chance to see how accurate their predictions are.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their verbal and written responses to the question, “How does the graph of \(y=\sin(2\theta)\) compare to the graph of \(y=\sin(\theta)\)?” Give students time to meet with 2–3 partners to share and get feedback on their responses. The listener should press for detail and precision in language by referring to specific features of the graphs such as the midline, amplitude, horizontal translation, and period. Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students use mathematically correct language when describing the effect of the number \(k\) in the sine function.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . . ?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

  1. Complete the table of values for the expression \(\sin(2\theta)\)
    \(\theta\) 0 \(\frac{\pi}{12}\) \(\frac{\pi}{6}\) \(\frac{\pi}{4}\) \(\frac{\pi}{2}\) \(\frac{3\pi}{4}\) \(\pi\) \(\frac{5\pi}{4}\) \(\frac{3\pi}{2}\) \(\frac{7\pi}{4}\) \(2\pi\)
    \(\sin(2\theta)\)                      
  2. Plot the values and sketch a graph of the equation \(y = \sin(2\theta)\). How does the graph of \(y = \sin(2\theta)\) compare to the graph of \(y = \sin(\theta)\)?

    horizontal axis, theta, scale 0 to 2 pi, by the fraction pi over 4. vertical axis, y, scale -1 to 1, by 1's. 
  3. Predict what the graph of \(y = \cos(4\theta)\) will look like and make a sketch. Explain your reasoning.

    horizontal axis, theta, scale 0 to 2 pi, by the fraction pi over 4. vertical axis, y, scale -1 to 1, by 1's. 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

As students sketch graphs of the functions, it may be necessary to remind them that sine and cosine are both smooth, wave-like curves and not a series of "connect-the-dots" segments.

Activity Synthesis

Begin the discussion by inviting students to compare their initial prediction to their answer for how the graph of \(y=\sin(2\theta)\) compares to the graph of \(y=\sin(\theta)\). Next, ask students what they think the graph of \(y = \sin(3\theta)\) will look like, including midline, amplitude, horizontal translation, and period. After a few students share their predictions, display the graph of \(y=\sin(3\theta)\) for all to see, emphasizing the period of \(\frac{2\pi}{3}\), one third the period of \(y = \sin(\theta)\).

Graph of sine 3 theta. 

Lesson Synthesis

Lesson Synthesis

The purpose of this synthesis is to give students an opportunity to put together the different aspects of transforming periodic functions they have learned so far. Since students will work more with period in the following lesson, the focus here on the horizontal scale factor is purposefully straightforward.

Begin by asking students to identify these features for the function \(f(\theta) = 2\sin\left(\theta-\frac{\pi}{4}\right) - 3\):

  • Midline (\(y=\text-3\))
  • Amplitude (2)
  • Horizontal translation (\(\frac{\pi}{4}\) to the right)

Next, ask them to sketch the graph and then show them a picture to verify their thinking. For additional challenge, invite students to make their sketch without referencing a unit circle or any graphs of sine or cosine.

Graph of y equals 2 sine open parentheses theta minus pi over 4 close parentheses minus 3. 

Lastly, display a graph of \(g(\theta)= \cos(2\theta)\):

Graph of y equals cosine 2 theta. 

Here are some questions for discussion:

  • “How is this graph alike and different from the graph of \(y = \cos(\theta)\)?” (It has the same wavy shape but the graph of \(g\) repeats twice between 0 and \(2\pi\) instead of just once for \(y=\cos(\theta)\).)
  • “How is the graph of \(h(\theta)=\cos(3\theta)\) alike and different from the graph of \(g(\theta) = \cos(2\theta)\)?” (It has the same wavy shape but the graph of \(h\) repeats three times between 0 and \(2\pi\) while the graph of \(g\) repeats only twice.)

15.4: Cool-down - To the left, to the right (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

We can find the amplitude and midline of a trigonometric function using the graph or from an equation. For example, let’s look at the function given by the equation \(y = 3\cos\left(\theta+\frac{\pi}{4}\right) + 2\). We can see that the midline of this function is 2 because of the vertical translation up by 2. This means the horizontal line \(y = 2\) goes through the middle of the graph. The amplitude of the function is 3. This means the maximum value it takes is 5, 3 more than the midline value, and the minimum value it takes is -1, 3 less than the midline value. The horizontal translation is \(\frac{\pi}{4}\) to the left, so instead of having, for example, a minimum at \(\pi\), the minimum is at \(\frac{3\pi}{4}\). Here is what the graph looks like:

graph of cosine function. midline at y = 2. amplitude = 3. horizontal translation of the fraction pi over 4 to the left. 

Another type of transformation is one that affects the period and that is when a horizontal scale factor is used. For example, let's look at the equation \(y = \cos(2\theta)\) where the variable \(\theta\) is multiplied by a number. Here, 2 is the scale factor affecting \(\theta\). When \(\theta = 0\), we have \(2\theta = 0\) so the graph of this cosine equation starts at \((0,1)\), just like the graph of \(y = \cos(\theta)\). When \(x = \pi\), we have \(2\theta = 2\pi\) so the graph of \(y = \cos(2\theta)\) goes through two full periods in the same horizontal span it takes \(y = \cos(\theta)\) to complete one full period, as shown in their graphs.

graph of y = cosine of 2 theta. period= pi, amplitude - 1. graph of y = cosine of theta. period = 2 pi, amplitude = 1. 

Notice that the graph of \(y=\cos(2\theta)\) has the same general shape as the graph of \(y =\cos(\theta)\) (same midline and amplitude) but the waves are compressed together. And what if we wanted to give the graph of cosine a stretched appearance? Then we could use a horizontal scale factor between 0 and 1. For example, the graph of \(y=\cos(\frac{\theta}{6})\) has a period of \(12\pi\).