Lesson 16

Features of Trigonometric Graphs (Part 2)

16.1: Which One Doesn't Belong: Graph Periods (5 minutes)

Warm-up

This warm-up prompts students to compare four trigonometric functions. It gives students a reason to use language precisely (MP6) and gives the opportunity to hear how they use terminology and talk about characteristics of the items in comparison to one another.

Launch

Arrange students in groups of 2–4. Display the graphs for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

Student Facing

Which one doesn't belong?

A. \(y=\sin(\pi \theta)\)

sine graph with period = 2. horizonta axis, theta, 0 to 2, by the fraction 1 over 4. 

B. \(y=\sin(\theta)\)

graph of y = sine of theta that has not been transformed. 

C. \(y=\sin (\frac 12 \theta)\)

only graph where period is not less than or equal to 2 pi. only equation with a fractional coefficent of theta. 

D. \(y=\sin(\theta+\frac \pi 2)\)

only graph that does not pass through the origin and the only graph with a horizontal translation. 

 

Student Response

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

Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as scale factor, horizontal translation, or period. Also, press students on unsubstantiated claims.

16.2: Any Period (15 minutes)

Activity

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 \(\cos(2\theta)\) has period \(\pi\) because the expression \(2\theta\) goes through all values between 0 and \(2\pi\) when \(\theta\) goes from 0 to \(\pi\). For the same reason, \(\sin(5\theta)\) has period \(\frac{2\pi}{5}\). 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 \(\cos(\pi \theta)\). The impact of the coefficient \(\pi\) is again to change the period, here from \(2\pi\) to 2, since \(\frac{2\pi}{\pi} = 2\). 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.

Student Facing

  1. For each graph of a trigonometric function, identify the period.

    A

    horizontal axis = theta. vertical axis = y. graph of sine function. amplitude = 1. period = 1.

    B

    horizontal axis = theta. vertical axis = y. graph of sine function. amplitude = 1. period = 2.

    C

    horizontal axis = theta. vertical axis = y. graph of cosine function. amplitude = 1. period = the fraction 1 over 2.

    D

    horizontal axis = theta. vertical axis = y. graph of cosine function. amplitude = 1. period = the fraction 2 over 3.
  2. Here are some trigonometric functions. Find the period of each function.
    function period
    \(y=\cos(\theta)\)  
    \(y=\cos(3\theta)\)  
    \(y=\sin(6\theta)\)  
    \(y=\sin(10\theta)\)  
    \(y=\cos\left(\frac{1}{3}\theta\right)\)  
  3. What is the period of the function \(y=\cos(\pi \theta)\)? Explain your reasoning.
  4. Identify a possible equation for a trigonometric function with this graph.

    graph of cosine function with period = 1, amplitude = 1. horizontal axis = theta, vertical axis = y. 

Student Response

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

Anticipated Misconceptions

As students calculate the period for the 5 functions listed, some may take the coefficient of \(\theta\) as the period. For example, they may think that the period of \(y=\cos(10\theta)\) is 10. Ask, “What value does \(\theta \) need to be in order for \(10\theta\) to be 0? To be \(2\pi\)?” (0 and \(\frac{2\pi}{10}\), respectively.) “What does that tell you about the period of \(\cos(10\theta)\)?” (That it repeats every \(\frac{2\pi}{10}\), so that's the period of \(y=\cos(10\theta)\).)

Activity Synthesis

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 \(\theta = 0\), what value does \(2\pi \theta\) take? What about \(\cos(2\pi \theta)\)?” (0 and 1.)
  • “When \(\theta = 1\) what value does \(2\pi \theta\) take? What about \(\cos(2\pi \theta)\)?” (\(2\pi\) and 1.)
  • “How long does it take the graph of \(y=\cos(2\pi \theta)\) to complete one full period? Explain how you know.” (1 because whenever \(\theta\) increases by 1, \(2\pi \theta\) goes through all values between 0 and \(2\pi\).)

Highlight that the function \(y = \cos(2\pi \theta)\) defines the graph in the last question because the period is 1. The function \(y=\cos(4\pi \theta)\) defines the third graph of the first problem as its period is \(\frac{1}{2}\).

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share how they calculated the period for the functions in the table, present an incorrect statement that represents an incomplete understanding of the coefficient of \(\theta\). For example, “The period of \(y=\cos(10 \theta)\) is 10 because you look at the coefficient to find it.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. For example, the author could improve their statement by stating that the period is determined by finding the value that makes \(10 \theta\) equal \(2\pi\). This helps students evaluate, and improve upon, the written mathematical arguments of others, as they clarify how to find the period of a function written as an equation.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking. As students share their strategies for interpreting trigonometric functions presented graphically and as functions, scribe their thinking on a visual display. Highlight connections students make between the graph and the corresponding function.
Supports accessibility for: Visual-spatial processing; Conceptual processing

16.3: Around the World’s Largest Ferris Wheel (15 minutes)

Activity

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 using clear reasoning about the expression for \(f\) 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 \(\frac{2\pi t}{30}=2\pi\) in order to determine the time \(t\) for one full revolution.

A note on the use of \(t\) as a variable: Until this activity, the input for trigonometric functions has typically been \(\theta\). In a previous course students used \(\theta\) 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 \(f\) takes an input of time and gives and output of height, the variables \(t\) and \(h\) are reasonable choices. If time allows and students wonder why \(t\) is used instead of \(\theta\), invite students to suggest why they think the input variable is \(t\) and highlight the reasons listed here for the change.

Launch

Tell students that they will be examining the position of a seat on a very large Ferris wheel. The Ferris wheel moves slowly enough that passengers can get on and off while the Ferris wheel continues its motion.

It time allows, recommend students try making a sketch of \(f\) before they graph the function using technology.

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students interpret a sine function in context of a Ferris wheel. Display the image and Student Task Statement, leaving out the questions. Ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving features of the Ferris wheel, the height of passenger seat \(f(t)\), or the graph of the function \(f\). This will help students create the language of mathematical questions before feeling pressure to produce solutions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “ Why did you . . . ?”, “How do you know . . . ?”, “First, I _____ because. . .”, “Then/next, I . . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

A photograph of a large Ferris wheel

The world’s tallest Ferris wheel is in Las Vegas. The height \(h\) in feet of one of the passenger seats on the Ferris wheel can be modeled by the function \(f(t) = 275+ 260 \sin\left(\frac{2\pi t}{30}\right)\) where time \(t\) is measured in minutes after 8:00 a.m.

  1. What is the diameter of the Ferris wheel? Explain how you know.
  2. How long does it take the Ferris wheel to make a complete revolution? Explain how you know.
  3. Give at least three different times when the passenger seat modeled by \(f\) is at its lowest point. Explain how you know.
  4. Sketch a graph of the height of the seat on the Ferris wheel for at least two full revolutions.

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 a graph of a wave where the amplitude is not constant but rather decreases over time. Write an equation which could match this graph.

Graph of a wave where amplitude decreases over time. Horizontal axis, t. Period = 6 pi. 

 

 

 

 

Student Response

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

Anticipated Misconceptions

If students are unsure how to interpret the expression inside the sine function, ask them to consider what value they would use for \(t\) if they want the input expression to be 0, \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), or \(2\pi\).

Activity Synthesis

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 \(260\sin(\frac{2\pi t}{30})\) will go between -260 and +260 for different inputs. This means the diameter of the wheel is 520 feet.
  • The expression \(\frac{2\pi t}{30}\) gives the period. When \(t\) goes from 0 to 30, this expression goes from 0 to \(2\pi\) so the Ferris wheel makes one full revolution in 30 minutes meaning its period is 30 minutes.
  • The function has its lowest value when \(260\sin(\frac{2\pi t}{30})=\text-260\). This is when \(\sin(\frac{2\pi t}{30})=\text-1\), which happens when the input to sine is \(\frac{3\pi}{2}\). We can calculate the value of \(t\) by solving \(\frac{2\pi t}{30}=\frac{3\pi}{2}\), which is when \(t=22.5\).
  • Since the period of the wheel is 30 minutes, then 22.5 minutes after 8 a.m. and every 30 minutes after that passenger seat is at its lowest point.

As students share, highlight how each of these features can be seen on a displayed graph of \(y=f(t)\).

Lesson Synthesis

Lesson Synthesis

Arrange students in groups of 2. The purpose of this discussion is for students to compare graphs two at a time where one graph is a transformation of the other. Display each of the given pairs one at a time. Give students brief quiet think time and then time to share their answer with a partner before selecting students to share their responses.

  • “How does the graph of \(f(\theta) = \sin(\theta-5) + 2\) compare to the graph of \(g(\theta) = \sin(\theta)\)?” (It is translated up two units and right 5 units.)
  • “How does the graph of \(f(\theta) = 2\sin(\theta)-5\) compare to the graph of \(g(\theta) = \sin(\theta)\)?” (It has twice the amplitude and is translated down 5 units. Or, it has a midline at -5, a maximum at -3, and a minimum at -7.)
  • “How does the graph of \(f(\theta) =5 \sin(2\theta)\) compare to the graph of \(g(\theta) = \sin(\theta)\)?” (It has 5 times the amplitude and half the period.)
  • “How does the graph of \(f(\theta) = 5\sin(2\pi \theta)\) compare to the graph of \(g(\theta) = \sin(\theta)\)?” (It has 5 times the amplitude and a period of 1 instead of \(2\pi\).)

If time allows, invite students to come up with their own pairs and challenge their partners to identify the transformations.

16.4: Cool-down - Turn, Turn, Turn (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Here is a point \(P\) on a wheel.

circle divided into 12 equivalent parts. radius = 1 foot. point p on circle circumference.

Imagine the height \(h\) of \(P\) in feet relative to the center of the wheel after \(t\) seconds is given by the equation \(h= \sin(2\pi t)\). When \(t = 1\), that is after 1 second, the wheel will be back where it started. It will return to its starting position every second.

What if \(h= \sin(4\pi t)\) was the equation representing the height of \(P\) instead? What does this equation tell us about how long it takes the wheel to complete a full revolution? In this case, when \(t = 1\) the wheel has made 2 complete revolutions, so it makes one complete revolution in 0.5 seconds. Here are the graphs of these two functions.

graph of y = sine of 2 pi t. amplitude = 1, period = 1 second. graph of y = sine of 4 pi t. amplitude = 1, period = 0 point 5 seconds.

Notice that the midline is the \(t\)-axis for each function and the amplitude is 1. The only difference is the period, how long it takes each function to complete one full revolution.

Now let's consider the height of a point on a wheel with a radius of 11 inches, the center of the wheel 11 inches off the ground, and completing two revolutions per second.

Graph of a function.

Notice that the midline and amplitude are 11. An equation defining this graph is \(h = 11\sin\left(4\pi t\right)+11\) where \(h\) is the height, in inches, of the point on the wheel after \(t\) seconds.