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).

If students assume that the coefficient of is the period in the second problem, consider asking:

• “What do you know about the period of a function?”

• “What value does need to be in order for to be 0? To be ? What does that tell you about the period of ?”

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response by correcting errors, clarifying meaning, and adding details. 

• Display this first draft: “The period of  is 10 because you look at the coefficient to find it.”
  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 

Focus student attention on the two ways of understanding the period of a trigonometric function that appear in this activity: interpreting a graph and interpreting an expression. For the graphical interpretation, ask students questions like:

• "How can you find the period of a trigonometric function from its graph?" (The period is how long it takes the function to complete one full "wave," or cycle of output values.)
• "What is the period of the trigonometric function from the last problem? Explain how you know." (1, because the graph completes one cycle when the input increases by 1.)

For interpreting expressions, ask students questions like:

• “When , what value does take? What about ?” (0 and 1.)
• “When , what value does take? What about ?” ( and 1.)
• “How long does it take the graph of to complete one full period? Explain how you know.” (1 because whenever increases by 1, goes through all values between 0 and .)

Highlight that the function defines the graph in the last question because the period is 1. The function defines the third graph of the first problem, as its period is .

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.

If students are unsure how to interpret the expression inside the sine function, consider saying:

• “Tell me more about the function. How has it been transformed from ?”

• “What value for would you use if you want the input expression to be 0, , , , or ?

Select previously identified students to share their reasoning about the first three questions. Highlight reasoning such as:

• 260 feet is the amplitude. The value of will go between -260 and +260 for different inputs. This means that the diameter of the wheel is 520 feet.
• The expression gives the period. When goes from 0 to 30, this expression goes from 0 to so the Ferris wheel makes one full revolution in 30 minutes, meaning its period is 30 minutes.
• The function has its lowest value when  . This is when , which happens when the input to the sine function is . We can calculate the value of by solving , which is when .
• Since the period of the wheel is 30 minutes, then 22.5 minutes after 8 a.m., and every 30 minutes after that, the passenger seat is at its lowest point.

As students share, highlight how each of these features can be seen on a displayed graph of .