Lesson 13

Amplitude and Midline

13.1: Comparing Parabolas (5 minutes)

Warm-up

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.

Student Facing

Match each equation to its graph.

  1. \(y = x^2\)
  2. \(y = 3x^2\)
  3. \(y = 3(x-1)^2\)
  4. \(y = 3x^2-1\)
  5. \(y = x^2-1\)

A

parabola opening up with vertex at orgin. narrower than the graph of y = x squared.

B

parabola opening up with vertex at origin.

C

parabola, opening up with x intercepts of -1 and 1 and vertex at -1 comma 0. 

D

parabola opening up with vertex at 0 comma -1, x intercepts between -1 and 0 and between 0 and 1.

E

parabola opening up, with vertex at 1 comma 0.

Be prepared to explain how you know which graph belongs with each equation.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.)

13.2: Blowing in the Wind (15 minutes)

Activity

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.

Launch

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.

Student Facing

A windmill with 5 blades shaped like a tall trapezoid. On the outside end of one of the blade, in the center, a point labeled W.

Suppose a windmill has a radius of 1 meter and the center of the windmill is \((0,0)\) on a coordinate grid.

  1. Write a function describing the relationship between the height \(h\) of \(W\) and the angle of rotation \(\theta\). Explain your reasoning.
  2. Describe how your function and its graph would change if:
    1. the windmill blade has length 3 meters.
    2. The windmill blade has length 0.5 meter.
  3. Test your predictions using graphing technology.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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 \(\theta\).

Activity Synthesis

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.

Three sine curves. 

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.

Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking. As students share their strategies for identifying the graphs of the three functions, scribe their thinking on a visual display. Color code connections students make between the three functions and the graphs of the functions.
Supports accessibility for: Visual-spatial processing; Conceptual processing

13.3: Up, Up, and Away (15 minutes)

Activity

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.

Launch

Provide access to Desmos or other graphing technology.

Representation: Access for Perception. Read the information about the windmill aloud. Support students to visualize or draw the image with labels. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language; Conceptual processing

Student Facing

  1. A windmill has radius 1 meter and its center is 8 meters off the ground. The point \(W\) starts at the tip of a blade in the position farthest to the right and rotates counterclockwise. Write a function describing the relationship between the height \(h\) of \(W\), in meters, and the angle \(\theta\) of rotation.
  2. Graph your function using technology. How does it compare to the graph where the center of windmill is at \((0,0)\)?
  3. What would the graph look like if the center of the windmill were 11 meters off the ground? Explain how you know.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Here is the graph of a different function describing the relationship between the height \(y\), in feet, of the tip of a blade and the angle of rotation \(\theta\) made by the blade. Describe the windmill.

sine function starting at 0 comma 30. maximum at the fraction pi over 2 comma 40. minimum at the fraction 3 pi over 2 comma 20.

 

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.

Conversing, Representing: MLR8 Discussion Supports. Use this routine to amplify mathematical uses of language to communicate about translations and the midline of trigonometric functions. After students share a response, revoice student ideas to demonstrate mathematical language use by applying appropriate language. In this discussion, connect to the concepts multi-modally by using gestures and talking about the context of the windmill when applicable. Some students may benefit from practicing the word “midline” through choral response.  
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

The purpose of this discussion is for students to consider a periodic function where both the amplitude and the midline are not 1. Display for all to see a graph of \(f(x) = 2\cos(\theta)-3\):

A graph of 2 cosine theta minus 3. 

Here are some questions for discussion:

  • “What is the midline of \(f\)? Explain how you know.” (\(y=\text-3\). It is halfway between the highest value of -1 and the lowest value of -5.)
  • “What is the amplitude of \(f\)? Explain how you know.” (2. The coefficient of \(\cos(x)\) is 2, so the graph has a vertical stretch of 2, which is the amplitude. Or, the maximum and minimum values of the graph are 2 away from the midline value.)

Draw the midline onto the graph and indicate where to see the amplitude on the graph, indicated by the length of the dashed vertical line shown here:

A graph of 2 cosine theta minus three with midline labeled. 

13.4: Cool-down - Transforming a Sine Graph (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Suppose a bike wheel has radius 1 foot and we want to determine the height of a point \(P\) on the wheel as it spins in a counterclockwise direction. The height \(h\) in feet of the point \(P\) can be modeled by the equation \(h = \sin(\theta) + 1\) where \(\theta\) is the angle of rotation of the wheel. As the wheel spins in a counterclockwise direction, the point first reaches a maximum height of 2 feet when it is at the top of the wheel, and then a minimum height of 0 feet when it is at the bottom.

A circle, subdivided by 12 congruent central angles. Point P is on the circle on the  right side of the horizontal diagonal. Radius is 1 foot.

The graph of the height of \(P\) looks just like the graph of the sine function but it has been raised by 1 unit:

graph of sine function with midline at the horizontal line h = 1. 

The horizontal line \(h=1\), shown here as a dashed line, is called the midline of the graph.

What if the wheel had a radius of 11 inches instead? How would that affect the height \(h\), in inches, of point \(P\) over time? This wheel can also be modeled by a sine function, \(h = 11\sin(\theta)+11\), where \(\theta\) is the angle of rotation of the wheel. The graph of this function has the same wavelike shape as the sine function but its midline is at \(h=11\) and its amplitude is different:

graph of sine function with midline at the horizontal line h = 11 and amplitude = 11. 

The amplitude of the function is the length from the midline to the maximum value, shown here with a dashed line, or, since they are the same, the length from the minimum value to the midline. For the graph of , the midline value is 11 and the maximum is 22. This means the amplitude is 11 since \(22-11=11\).