Lesson 13

The goal of this warm-up is for students to recall the effects of vertical translations and stretches in the familiar setting of quadratic functions before considering these transformations' effects on trigonometric functions.

Invite students to explain their matches for each of the equations. If needed, encourage them to refine their descriptions of the transformations of \(y=x^2\) using more precise language and mathematical terms (MP6), and connect this matching activity to the work done in a previous unit on transformations of functions.

Ask students, “What other transformations can you think of using the values 1 and 3, starting with the function \(f(x) = x^2\)?” (\(g(x) = 3x^2 + 1\), whose graph is stretched vertically by a factor of 3 and then translated up 1 unit. Or, \(h(x) = (x+3)^2\), whose graph is translated to the left by 3 units.)

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 \(y = 3x^2\) changes the shape of its graph compared to the graph of \(y = x^2\); it stretches the graph vertically so the output increases more quickly, from 0 to 3 instead of from 0 to 1 when \(x\) goes from 0 to 1.

Display for all to see the picture of the windmill and tell students they will be modeling the height of the point \(W\) as the windmill spins. Ask students what kind of function they think would be appropriate and why.

Provide access to Desmos or other graphing technology.

If students are not sure how to start writing a function, suggest that they complete a table of values with columns for \(\theta\) and \(h\) and plot the points. If needed, recommend the table use multiples of \(\frac{\pi}{4}\) from 0 to \(\frac{7\pi}{4}\) for \(\theta\).

Display the graphs of all 3 functions, \(h = 0.5\sin(\theta)\), \(h = \sin(\theta)\), and \(h = 3\sin(\theta)\) on the same coordinate plane for all to see and ask students to identify which is which and 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 \(k\) in the equations \(h = k\cos(\theta)\) and \(h =k\sin(\theta)\) changes the “height” of each graph by a factor of \(k\) and 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. A vertical translation can also be seen in the quadratic equations from the warm-up; the graph of the equation \(y = x^2-1\) looks the same as the graph of \(y = x^2\) but it has been translated downward by one unit.

Provide access to Desmos or other graphing technology.

As in the previous activity, if students struggle getting started writing a function that describes the height of the point on the windmill consider recommending using an abbreviated table of values (just the angles \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\) and \(2\pi\)) or a graph as a scaffold.

Students graphing using technology may not realize that the graph of \(h = \sin(\theta) + 8\) is only a vertical translation of \(h= \sin(\theta)\) if they use different sized graphing windows on the graphs. Changing the visible window of the graphing technology changes the apparent shape of the graph. If the two functions were plotted on the same coordinate plane or graphing window, it would be more obvious that one is a vertical translation of the other. Consider suggesting this for students who struggle making the comparison.

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 \(h = \sin(\theta)+8\) compares to the graph of \(h = \sin(\theta)\). 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 \(h = \sin(\theta)\) and \(h = 3\sin(\theta)\): 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 \(h = \sin(\theta) + 8\) for all to see. 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 \(h=8\) is the "visual center" of the graph and is called the midline. For practice, ask students to consider the equation \(h = \sin(\theta) - 3\). Its graph has a midline of \(h=\text-3\) because its \(h\)-values are centered around -3. A negative midline means that the graph is translated downward rather than upward.