This activity asks students to interpret cosine and sine equations in the context of a fan. They reason abstractly and quantitatively to connect the properties of the fan with the graphs (MP2). They will need to focus on three different aspects of the equations:

• The amplitude of the sine graph (length of the fan blade)
• The midline of the sine graph (height of the center of the fan blades)
• The direction the fan spins (clockwise or counterclockwise).

A negative amplitude is new for students. If they are unsure how to interpret the final question, consider asking:

• “What do you think it could mean for the amplitude to be negative?”

• “How could it help you to substitute in a few angle values for and plot them?

Display the three equations. Begin the discussion by asking, “What is the midline and amplitude of each function and what do these values mean in this context?” (The midline is the height of the fan blades’ center, and the amplitude is the length of the blades.) Record student responses next to the equations.

Ask students how they interpreted the sign in front of sine on the last equation (it means that the fan is spinning in the clockwise direction as the angle, , increases). To help students visualize the impact of the negative sign in front of the expression, note that while increases from 0 to 1 as increases from 0 to , decreases from 0 to -1 as increases from 0 to . In terms of the fan, this would mean that instead of going toward the highest spot on its circle of rotation, the point begins by going to the lowest spot on its circle of rotation. This means that the fan is rotating in a clockwise direction instead of in a counterclockwise direction.

If students struggle to create a plot or to identify the relationship between the graphs and the equations, consider saying:

• “Tell me more about what the variables and represent in this scenario.”

• “How could a table of values help you? For example, what is the horizontal position when is  How does that compare to at the same angle?”

Begin the discussion by inviting students to share strategies that they used for graphing the functions. Some strategies may include:

• Create a table of values and plot them.
• Use technology to calculate values and then plot them.
• Use transformations on a function to determine the new function and graph it.

Ask students how the new graphs are similar to and different from the graphs of and . Students should understand that while they have the same shape and also the same period () and amplitude (1), they are different because they don’t “start” at the same place. The graph starts at rather than at , and the graph starts at rather than at . Connect this back to the graphs of and from the Warm-up: Both cases represent a horizontal translation to the left (by for the trigonometric functions and by 3 for the parabola).

Display the graphs of and (the equation for the vertical position of the point, , in this activity) on the same axes to help students visualize the horizontal translation.

If equations for the transformations are not brought up by students, invite them to explain how changing the starting position of the blade influences the graph. They should identify that the graph for this problem is a horizontal translation of the graph of and that the translation is to the left by . Tell students that an equation defining the vertical position of the point, , is and that the comes from point  starting at the position.

Time permitting, ask students what the graphs and equations for the vertical position would look like if point started at (They would be translated to the left by , so the equation would be .) What if started at ? (They would be translated to the left by or to the right by , so the equation would be or , respectively.)