Unit 7, Lesson 16
Features of Trigonometric Graphs (Part 1)
Algebra 2
16.1
Warm-up
Instructional Routines
MLR5: Co-Craft QuestionsMaterials
NoneActivity Narrative
The purpose of this Warm-up is to elicit the idea that the period of a trigonometric graph can vary, which will be useful when students examine the graph of the position of a point on a wheel in a later activity.
This activity uses the Co-Craft Questions math language routine to advance reading and writing, as students make sense of a context and practice generating mathematical questions.
Launch
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Introduce the context of wavelength of musical notes. Use Co-Craft Questions to orient students to the context and to elicit possible mathematical questions.
Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation, before they compare questions with a partner.
Activity
NoneHere are pictures of sound waves for two different musical notes:
Student Response
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Building on Student Thinking
Activity Synthesis
Invite several partners to share one question with the class. Record their responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “periods” and “amplitudes” of the graphs. If the period and amplitude of the graphs do not come up during the conversation, ask students to discuss these ideas.
Tell students that the amplitude of a sound wave relates to how loud the sound is (larger amplitude means a louder sound). The period of the sound wave is related to its pitch. More highs and lows in the wave, that is a shorter period, means a higher pitch. Later in this lesson, they will investigate the period of a trigonometric function in a different context.
Lesson Synthesis
The purpose of this synthesis is to give students an opportunity to put together the different aspects of transforming periodic functions that they have learned so far. Since students will work more with period in the following lesson, the focus here on the horizontal scale factor is purposefully straightforward.
Begin by asking students to identify these features for the function :
- Midline ()
- Amplitude (2)
- Horizontal translation ( to the right)
Next, ask them to sketch the graph, and then show them a picture to verify their thinking. For an additional challenge, invite students to make their sketch without referring to a unit circle or any graphs of the sine or cosine functions.
Finally, display a graph of :
- “How is this graph alike and different from the graph of ?” (It has the same wavy shape, but the graph of repeats twice between 0 and instead of just once for .)
- “How is the graph of alike and different from the graph of ?” (It has the same wavy shape but the graph of repeats three times between 0 and , while the graph of repeats only twice.)
Student Lesson Summary
We can find the amplitude and midline of a trigonometric function using the graph or from an equation. For example, let’s look at the function given by the equation . We can see that the midline of this function is 2 because of the vertical translation up by 2. This means the horizontal line goes through the middle of the graph. The amplitude of the function is 3. This means that the maximum value it takes is 5, 3 more than the midline value, and the minimum value it takes is -1, 3 less than the midline value. The horizontal translation is to the left, so instead of having, for example, a minimum at , the minimum is at . Here is what the graph looks like:
Another type of transformation is one that affects the period, and that is when a horizontal scale factor is used. For example, let's look at the equation , where the variable, , is multiplied by a number. Here, 2 is the scale factor affecting . When , we have so the graph of this cosine equation starts at , just like the graph of . When , we have , so the graph of goes through two full periods in the same horizontal span that it takes to complete one full period, as shown in their graphs.
Notice that the graph of has the same general shape as the graph of (same midline and amplitude) but the waves are compressed together. And what if we wanted to give the graph of cosine a stretched appearance? Then we could use a horizontal scale factor between 0 and 1. For example, the graph of has a period of .