This Warm-up prompts students to carefully analyze and compare four trigonometric functions. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology students know and how they talk about characteristics of the graphs and equations of the functions.

Invite each group to share one reason why a particular set of three go together. Record and display their responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as "scale factor," "horizontal translation," or "period," and to clarify their reasoning as needed. Consider asking:

• “How do you know ?”
• “What do you mean by ?”
• “Can you say that in another way?”

This activity continues to develop the idea of period for a trigonometric function both graphically and from the point of view of expressions and equations. In the previous lesson, students first saw a trigonometric function whose period was scaled. The function has a period of because the expression goes through all values between 0 and when goes from 0 to . For the same reason, has period . From a graphical point of view, decreasing the period means that the wave is compressed horizontally.

The new content in this activity is that students begin to look at functions defined by expressions like . The impact of the coefficient is, again, to change the period, here from to 2, since . Finding trigonometric functions whose period is a whole or rational number is important in the applications that students study for the remainder of this unit. When they model real-world phenomena with trigonometric functions, the period of those phenomena is usually a rational number.

In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).

In this activity, students return to the context of a Ferris wheel with their new understanding of how to transform a periodic function by changing the midline, amplitude, and period. The purpose of this activity is for students to interpret a sine function in context and then make a sketch of the function.

Monitor for students who use clear reasoning about the expression for as they identify the diameter, time per revolution, and times the passenger seat is at its lowest point. For example, students may make and solve equations like  in order to determine the time, , for one full revolution.

A note on the use of as a variable: Until this activity, the input for trigonometric functions has typically been . In a previous course students used to represent angles in different geometric shapes, so its use throughout this unit was meant to remind students of the connection between cosine and sine and the unit circle. Now, however, students are transitioning to working with functions modeling contexts where the input isn't an angle, so it makes sense to choose a variable representative of the input value. Since takes an input of time and gives an output of height, the variables and are reasonable choices. If time allows and students wonder why  is used instead of , invite students to suggest why they think the input variable is , and highlight the reasons listed here for the change.