Graphing Sine and Cosine Functions (Day 1 of 2)
Lesson 5 of 13
Objective: SWBAT graph by hand sinusoidal graphs with transformations
After looking at graphs and the transformations I am ready to have students draw sketches of trigonometric functions.
The students have looked at the graphs of sine and cosine over the last several days and are beginning to remember how the look. I begin today by putting the graph of y=sin x on the board and ask:
- If I want to make a sketch of y=sin x by hand what is the fewest points I would need to make the graph?
- What are the points?
Student's discuss this for a couple of minutes. I then ask students to tell the class what points they would graph. I record the answers and put up a rectangular graph to put the points on the graph.
I do have some students that want to make a table of values. I try to start the conversation with a student that wants a table of values. I will ask "What values will you put in the table? If you don't have a calculator what values will you put in the table?" I try to get students to consider the special angles we discussed in the last unit. I then ask "So how many points will you need?"
As the class discusses the table I will have students say we can do it with fewer points so I continue the discussion with:
- What points would you graph?
- Why would you graph those points?
- What points are important to get the shape of the graph?
- If we only plot those points would someone understand the shape?
As the students discuss they will eventually determine the first and last point of the period the max and min point then the middle point. I ask "Once you have these points how would you continue extend the graph to the right and left?" Since students know the graph is periodic they should understand continuing the pattern of the point placement.
Graphing y=sin x
I now give students graph paper on which the x-axis is labeled. I want to begin graphing without changes in the period or horizontal shifts so this graph paper help students see how to label the x-axis.
I put up problems for students to graph. I begin with changes in the amplitude and then move to horizontal shifts. Students work on the problems for a few minutes and then share the graphs with the class. Questions are answered by students some questions that may occur are "Why did you start with -2 on the third graph?" Students can explain this to each other.
Once students understand the amplitude and vertical shift I put a problem up with a change in period. I let the students ponder this graph for a couple of minutes and then ask "What questions do you have about graphing this function?" Many students expect me to just start showing them how to graph but I hearing the question they have will help me focus on what I need to clarify. I usually get responses like "I don't know what to do with the 2. Is the 2 the amplitude? Or I know that the 2 changes the period but how do I show that on the graph?"
As students respond others may start answer the questions. I make sure that students document what information they have on the key features. I ask:
- What is the amplitude?
- Why is the 2 not the amplitude?
- What is the vertical shift?
- You said that that the period is changed? How did we find the period in the previous lesson?
- Does the scale of the x-axis always need to be what is labeled on your graphs?
- What points did we decide we needed to graph?
- What does the period tell use about a cycle? (here I am wanting that the period is the distance for one cycle)
As I ask these questions I am documenting responses on the board. Once my students realize you can change the scale of the x-axis, I help them determine how to scale the axis. I begin by plotting the endpoints for the period and then discuss how we need three points in the interval. This is a great review of how to divide a length into 3 equal sections. (For students that struggle with the idea of thirds I discuss finding the midpoint of the total length and then divide the 2 halves into 2 equal sections by using the midpoint again.)
Once students have the axis labelled we sketch the graph. After doing a graph together I give students more problems to try on their own. As I move around I clarify how to label the axis.
After most students have finished making a graph by hand, I will bring the class back together for a discussion. I plan to ask:
- How many of you found this tedious? Why?
- Did any of you think of a method to shorten the process?
- If you wanted to graph 1 period, what would be the fewest numbers of points you would need to plot to get a sketch of y=sinx? Why?
Students determine they need the maximum, minimum and midline points to graph. I really like when the students say the midline since this will help when we graph transformed functions. If students say the x-intercepts I ask why the x-intercepts. At this point students explain that these are on the midline of y=sin x.
As class ends I explain that we will use what they noticed with graphing to graph functions that are transformed. Any students that have not graphed y = cos x and compared the graph to y = sin x are asked to complete this task as homework.