Lesson 15

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.

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.

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. 

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. 

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.

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.

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).

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.

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.

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 \(y = \sin(\theta)\).