Amelia Jamison VALLIVUE HIGH SCHOOL, CALDWELL, ID
Algebra II : Unit #9 - Trigonometric Functions : Lesson #11

Transformations of Sine and Cosine Graphs

Objective: Students will be able to use transformations of functions to graph and write equations for sine and cosine functions.
Standards: HSF-IF.C.7e HSF-BF.B.3 MP2 MP3 MP8
Subject(s): Math
60 minutes
1 Warm-up and Homework Review - 5 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative specifically explains this lesson’s Warm up-Transformations of sine and cosine, which asks students to determine which of two students is correct in finding the period of a cosine function.

I also use this time to correct and record the previous day's Homework.

Making Sense of Period
Developing a Conceptual Understanding

I think that horizontal shifts and period changes confuse students the most when dealing with graphs of sine and cosine.  With period, so often students want to say that the 2 in f(ϴ)=sin(2ϴ) makes the period twice as long which seems to intuitively makes sense.  If we have f(ϴ)=2sin(ϴ), this is twice as tall.  In fact, this has the same inverse operation effect as a horizontal shift so this was one approach I used when I taught this lesson.  Verbally, I also told my students that this meant twice as many curves in a 2π interval.  For those that struggled with the inverse operation concept, this seemed to help. 

I was pleased with the responses from this warm up.  I was expecting more students to be mistaken but most of the responses I received were like Warm Up 1 and Warm Up 2.  With today’s lesson that puts it all together, I hope that we will further solidify their knowledge of period and avoid this error.  

2 Review of Stretch/Shrink Tranksformations - 7 minutes

In the previous lesson, students learned about both period and amplitude which can also be expressed as the vertical and horizontal stretch/shrink of a periodic function.  The first task students are given in this lesson is to determine which transformations they already know.  I have the students discuss this together and then call on several to share their conclusions.   It is important to refer them back to the translations that they have seen all year long as this connection will immensely improve the speed they grasp these transformations as well as their confidence.

They should connect the period to a horizontal stretch/shrink and the amplitude to a vertical stretch/shrink.  If not, I either guide the class there through some leading questions or share this information directly.  A possible leading question would be "When I change the amplitude, what type of transformation does this produce?"  I then give them an opportunity to write these transformations in their notes along with a graph example.  I chose to graph the transformations by hand each time rather than just have a picture of them in the PowerPoint because I can model the important features and help solidify the basic graph structure in the head's of those students who are still struggling to graph them. 

I don't spend too long in the section as they have already seen this and the time will be better spent in other places.

3 Vertical and Horizontal Translations - 10 minutes

The vertical translation is an easy one.  This is a good time to talk about the domain and range.  I ask them to talk in pairs and determine whether the domain and/or range has changed with this translation. 

The horizontal translation is more involved.   A possible way to deal with this shift is to locate the x-intercepts first and then draw the rest of the graph around them.  Again, modeling is going to be key here.

4 Putting it all Together - 10 minutes

In this section, I will give the students an opportunity to practice using this new knowledge.  The first several problems just ask the students to list the transformations.  I have the students do them and then check their work with their partner before we discuss as a class.  The next couple of examples ask the students to actually graph the function.  Again, I give the students an opportunity to try each problem first and then model it for them.  

5 How Many Equations Can We Write? - 15 minutes

This section provides the most critical thinking for this entire lesson.  Students are used to math having only one answer.  They are asked to write an equation from a graph that has some transformations.  The students work in pairs to come up with an equation (Math Practice 2).  I ask for volunteers to share their equation.  Hopefully, there are several different equations.  If not, I add a couple myself.  The goal is to get several on the board.  Next ask your students which is the right one? (Math Practice 3)  They discuss in pairs and then we identify acceptable solutions as a class. 

Once we have concluded that there is more than one possible answer, ask the students to discuss in pairs how many different equations we could come up with (Math Practice 8).   There are theoretically an infinite number.  A good extension question at this point would be to ask the students why there are so many different equations.  

The next example offers another graph with some transformations and asks the students to identify as many different equations that work as they can.  We list them on the board once they have had a reasonable amount of time to work.

6 Homework - 0 minutes

The first four problems of this Homework  asks students to list the transformations from an equation.  The next six have the students graph two periods of a sine or cosine function given the equation.  I did not include any transformations with both a horizontal translation and a horizontal dilation.  This type of problem could be used as an extension problem if you want to take this lesson farther.  The final four problems ask the students to write one sine and one cosine equation given a graph (Math Practice 8).  

This assignment was created with Kuta Software, an amazing resource for secondary mathematics teachers.

7 Exit Ticket - 3 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.  

This Exit Ticket asks students to list the transformations of the cosine function.