Trigonometric Graphs (Day 3 of 3)
Lesson 3 of 13
Objective: SWBAT discover how to find the period of a transformed trigonometric function.
Today we begin with the activity given at the end of yesterday's lesson. Student are given a function and ask to describe the key features of its graph. At this point, I expect my students to quickly find the answers until they come to the period of the function. A few may have figured out how to find the period, but most students will benefit from today's lesson.
After giving students time to accomplish what they can, I will have students share their answers on the board and we will take a moment to discuss questions the students want to ask.
- I make sure students understand that the amplitude is positive since it is a distance from the mid-line.
- I will ask students, "What does negative on the 2 tell you about the graph?"
- We also talk about how to determine the mid-line. I typically let students explain how to find the mid-line. Some will say it is the constant of the function or it is the vertical shift.
Contextual note: This section is to be brief. My goal is to get students to ask questions about the graph. I also use this as a way to demonstrate how to analyze a graph by changing the scale on the graph. I may comment that this is what mathematicians and scientist research, they make a conjecture then test to see if it is correct. If wrong they try something else.
After the discussion of the Bell Work, I plan to take a deeper look at the period of the function. I will ask my students to discuss whether or not they think that the quantity, 3, in the function definition effects the period (see bell work day 3). Next, I will put a graph of the function on the board (page 1). In the graph the x-axis has increments of 1 for the variable, t. I choose this scale because some of my students will think that the period is 3.
Next, I ask students to define the mathematical term period in their own words. It usually does not take too long for me students to arrive at the idea that the period is the distance it takes for the function to repeat. I then ask the students to determine the starting point and ending of a cycle of one period on the graph.
As they work I will ask, "Is the distance from the beginning and to the end of a cycle 3?" Many students will estimate the distance as a little more than 2. I will then change the scale on for the horizontal axis to be pi/2. We'll look at this new graph together to try to determine the length of a period. Students see that it is more than pi/2 and may ask to change the scale again. I adjust the scale for students I let students find give me values for the scale. By trial and error the class will eventually change the scale to pi/3 and find that the period is 2pi/3.
I now state that there must be an easier way to find the period than guess and check with a graph. The students of course want me to give them the answer. Other students notice that the denominator of the period is the 3 from the equation. We notice how the 2pi is the period of cosine that is not transformed. I let students know that we are going to explore this idea more to find a method to find the period.
Now that students see how I used the scale to help find the period, I have students work on Determining the Period of a Trigonometric Functions activity. The students will work in pairs while completing the activity. Students will use Desmos as they explore.
This activity gives students equations with whole number values for coefficient on the domain variable. I give my students whole numbers, since this makes it easier for the students to see the pattern. The activity begins by using f(t) and t for the variable then moves to using y and x. This is a subtle way of moving students towards the way the equations are written in the textbooks. By not making a big deal about the difference students, my students comfortably realize that the equations are representing the functions.
I move around the room as the students are exploring clarifying the process. Some students will struggle with the last equation in the table. I work with these students and start thinking out loud. I say something like:
Let's see, when the coefficient in front of the x or t is 1 the period is 2*pi. When the coefficient is 2 the period is half as long or pi. Okay. Now when the coefficient is 3 we get 2pi/3.
At this point some students will notice that the 3 is b and say the rule is 2*pi/coefficient. As students make this observation, I ask them to present it to their groups and see if the rest of the group can understand and confirm this rule. If, however, students are not making this breakthrough I will continue:
Oh where else do I have the 3? Let's see if I go back and take the 2pi and divide by 2 what do I get? Okay let's look at when the coefficient is 4, what could I do that is similar to what we just did?
At this point there will usually be at least one student that can try to explain what is going on. I'll let him/her offer an explanation to help the group start in the right direction.
After students have had some time to work we get back as a class to verify students are seeing how to find the period of a function. Students share their answers for question 3 and then discuss how to find the period for the other trigonometric functions. I plan to leave the last question for the students to complete as homework.
With 5 minutes left in class, I plan to take the students into the realm of a real world application. I will begin by telling the students that over the course of the year, the daily high temperature (on average) for Independence, Missouri, is 88 degrees. The minimum high temperature is 38 degrees. Then, I will ask:
Based on your experience with the weather in Missouri, do you think that daily high temperatures modeled by using a trigonometric function? How can we use this information to start writing a model for the graph?
I will have students work in groups for a few minutes before turning their ideas in to me. This exit slip gives me a chance to see if the students are able to imagine how they might use a trig function to describe a real world situation. If students are stuck, I will hint that they might want to start by considering whether there are numbers that make sense for amplitude and the mid-line at this point.
The temperature context will show up again when we write equations for real world situations.
Once questions are answered students work on Problems_6_and_7. These 2 questions will be graded. I will move around the room and answer questions when needed. With 5 minutes left I give the students an exit slip that asks each group to determine the amplitude, vertical shift, horizontal shift, period, frequency and mid-line. Groups turn in one copy of their answer before they leave.
As the groups work I walk around and note groups that are struggling or obvious misconceptions students have. These are items I will discuss with individual students or with the class the next day.