In this partner activity, students take turns finding matches of equations, graphs, and descriptions. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and to critique the reasoning of others (MP3). To complete the matching students may:

• Evaluate the functions for strategic inputs such as or 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.

If students are not sure how to begin matching, consider saying:

• “Pick one of the cards. Tell me what you know about the periodic function it represents.”

• “How might substituting in a few values of help you find matches?”

Once all groups have completed the Card Sort, discuss the following: 

• “Which matches were tricky? Explain why.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”

The purpose of this discussion is to make connections between transformations of functions and the period, midline, and amplitude of sine and cosine functions. Select 2–3 groups to share one of their sets of cards and how they matched the graph with an equation and description. Discuss as many different sets of cards as the time allows, making sure that students connect period, amplitude, and midline with transformations of the graph and equation.

Ask students how the graphs of and are related and how this showed up in their matches. If no students suggest that they are horizontal translations of each other, invite them to discuss this idea. For example, shifting the graph of to the right by will make it match up with . Alternatively, shifting the graph of to the right by will make it match up with .)

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Conclude the discussion by inviting 2–3 students to share a norm that they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

• “I noticed our norm ‘’ in action today, and it really helped me/my group because .”

The goal of this activity is to experiment with one final parameter of a trigonometric function, the number, , in . This number determines the period of the trigonometric function. When , this is the normal cosine function whose period is . When the period becomes because as varies from 0 to , the input in the expression takes all values between 0 and . For the same reason, the period of is . The period can also be longer than : For example, the period of is because goes through the values 0 to when goes through the values to .

The work in this activity is an important example of the general idea of horizontal scaling that was studied in a previous unit. Students build on the structure of horizontal scaling in other functions to scale periodic functions (MP7). In general, if is a function, then the graph of the function is a horizontal stretch, by a factor of , of the graph of .

The axes for the graphs are marked in increments of 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).

As students sketch graphs of the functions, it may be necessary to remind them that the graphs of the sine and cosine functions 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 compares to the graph of . Next, ask students what they think the graph of will look like, including midline, amplitude, horizontal translation, and period. After a few students share their predictions, display the graph of , emphasizing the period of , one third the period of .