The goal of this task is to introduce the amplitude of a trigonometric function in the context of the position of a point on a spinning windmill. In this case, the amplitude of that position depends on the distance of the point from the center of the windmill. The closer the point is to the center, the smaller the amplitude of the vertical and horizontal positions of the point. The general shape of the graph is the same, rising and falling as the windmill blade makes a complete revolution; what changes is how high and low the graph goes. This is analogous to how the 3 in changes the shape of its graph compared to the graph of ; it stretches the graph vertically so the output increases more quickly, from 0 to 3 instead of from 0 to 1, when goes from 0 to 1.

As students determine what type of model best describes the situation, adjust the trigonometric function to account for the amplitude, and check their predictions using technology, students are engaging in aspects of mathematical modeling (MP4).

If students are not sure how to start writing a function, consider asking:

• “What do you know about the function?”

• “How could a table of values help you write the function? For example, what is the height when is or ?”

Display the graphs of all three functions, , , and on the same coordinate plane, and ask students to identify which is which and to explain how they know.

If students do not use the term "vertical stretch" (or equivalent) to describe the difference between the graphs, do so now, calling back to their work in an earlier unit. Tell them that for these types of functions, the parameter, , in the equations and changes the “height” of each graph by a factor of and that the absolute value is called the amplitude. Highlight where to find the amplitude in the equations for the blades of length 3 meters and 0.5 meters.

The goal of this activity is to introduce the midline of a trigonometric function. Students experiment with changing the vertical position of a trigonometric function, adding a constant within the same context of the height of a windmill. Unlike the scalar multiple in the previous activity, a vertical translation is a rigid transformation and so does not change the shape of the graph, just its location. Students reason quantitatively and abstractly as they consider the effects of changes to the position and shape of the windmill blades on the features of the graph (MP2).

If students are not yet sure how to start writing a function, consider asking:

• “How is this windmill similar to and different from the one in the previous activity?”

• “How could a table of values help you write the function for this new windmill? For example, what is the height when is , , , or ?”

The purpose of this discussion is to introduce students to the term "midline." This is also an opportunity to check and make sure students understand that periodic functions transform just like the other types of functions they have studied previously.

Begin the conversation by asking students to describe how the graph  compares to the graph of . Highlight that the shape is identical but it is translated upward by 8 units. If not mentioned by students, remind them that this is called a vertical translation. Contrast this type of transformation with the difference between the graphs of and : While these have the same general wavelike shape one is not a translation of the other. The coefficient of 3 "stretches” the shape vertically, making the graph steeper as it goes between the larger maximum values and smaller minimum values.

Conclude the discussion by displaying the graph of  . Ask, “What value would you say is the 'middle' value for the outputs of this function?” After a brief quiet think time, invite students to share their thinking. The important takeaway here is that is the "visual center" of the graph and is called the midline. For practice, ask students to consider the equation  . Its graph has a midline of  because its -values are centered around -3. A negative midline means that the graph is translated downward rather than upward.