Graphing Sine and Cosine Functions (Day 2 of 2)
Lesson 6 of 13
Objective: SWBAT graph by hand sine and cosine graphs with transformations
Today we'll continue our work with graphing sinusoidal graphs. Today we will focus on horizontal (or phase) shifts. When the class begins I'll give my students a task that asks them to explain the difference between y=sin x and several different transformations of this function.
I expect many of my students to quickly state that the first problem has an amplitude of 3. When they make this observation, I will try to bring them back to the prompt by asking, "How is this different than y=sin x?" The students will need to revise their explanation to state that the maximum is at 3 instead of 1. They know that the 3 represents the amplitude, but I want students to answer the question directly, rather than implicitly (MP6). Once students see how to answer the questions the other problems go smoothly.
When students discuss Questions 3 and 4, some will say the frequency has changed while others will talk about the period. I will listen to these conversations to make sure that they are on track. If necessary I will refer back to the frequency activity. I'll try to ensure that students understand that "2" in Question 3 refers to 2 cycles over a distance of 2 pi radians. "1/2" in Question 4 means that one-half of a cycle will be completed over the distance of 2 pi radians. I believe that if a student understands that only 1/2 of a cycle is in 2pi, they will be better able to visualize the function and create a graph more accurately.
Graphing horizontal shifts
Now that students have seen how to graph when the following parameters are changed, we are ready to discuss a phase change or horizontal shift:
I will start by writing the function y=2sin(x-pi/2) on the board. I will ask my students to identify the key features of the graph of the function.
My students will undoubtedly use technology on calculator or phone to help them. To help understand the horizontal shift, I will ask "If the position of the graph is moving to the right, what is happening? " We will approach the idea of a phase shift slowly. I expect we'll start with an understanding like, "a period starts at pi/2." From here we will look at a graph together to discuss and to verify students observations.
As students gain a general sense of how to proceed with a phase shift, I will give them some some problems to work. My plan for this work is flexible, but I plan for students to share their results with the class. As students work, I will ask questions such as:
- Have you identified the key features?
- Where will the cycle start?
- What is the distance between the start and end of the cycle?
- How are you going to divide the interval? What are the x values in the interval you have marked?
The final two problems will be difficult for my students. The first function includes all of the transformations that we have discussed. Students sometimes are confused by the fact that the vertical shift is at the beginning of the expression. But, it is important that students see different ways to write the equations so that they do not get the misconception that the vertical shift is always the last number in the equation.
The final question is written differently than other problems I have posed. Some students will not at first realize that there is a difference. Others will immediately realize that the expression (3x-pi) needs to have a 3 factored out, but are unsure of how to do this. I plan to let the students identify the key features. Then, I'll put the graph on the board. As I look at the graph I will ask my students if their graph looks the same. It usually takes a minute or so for my students to realize that the horizontal shift is not what they determined. Once they reconize this fact, I'll ask why that might be. I may need to remind students that our general equation is y=a sin b(x-c)+d and then question whether the equation is of the same form.
Once students see that it is not exactly the same we discuss how to factor the 3 out of (3x-pi) to get 3(x-pi/3). From there student see how the horizontal shift is pi/3 to the right. I explain that many books and instructors write the general equation as y=a sin (bx-c)+d. We then discuss how the only difference with the different forms is finding the horizontal shift.
For homework I assign page 327 #44, 50, 58 from Larson's Precalculus with Limits. There are a lot of problems in the book that the students can do. I picked 3 just to check for understanding. These will be turned in to me tomorrow to assess student understanding.
I will ask my student to write out the process they used to graph a function as an Exit Slip. I will analyze their responses to see where students are struggling. Students will have different processes. I am interested in seeing how they explain their identification of the key features. I am also hoping for a clear explanation of how to use the key features to sketch a graph.