Inside Out and Upside Down - Modeling with Inverse Functions

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SWBAT explain that the inverse trigonometric functions can only be graphed over a restricted domain because of the periodic nature of the sine, cosine and tangent functions.

Big Idea

Turn graphs around and see what happens. Graphing the inverse trig functions can be fun if you play with the graphs a bit!

Set the Stage

5 minutes

           There is a video narrative in my resources that further explains my pedagogy for this section of the lesson.  You will want to make sure your students have their graphing calculators and/or be ready to project the graphs for your class. In addition, I use individual whiteboards for this lesson, but you could also use plain paper or notecards.  I open the lesson with a question about the graphs of sine, cosine and tangent functions and their transformations. Most students are fairly comfortable with these graphs by now, but as a refresher I ask them to graph “f(x) = sin θ”, then predict how changing to “f(x) = 3sinθ” would change the graph. (MP7) I have them think-pair-share then ask for responses.  As we review the responses as a class, I expect most of my students to be on target, since this is review.  I note which students are still struggling with these transformations so I can give them additional individual or small-group help later. I ask them to go back to their “f(x) = sin θ” graph and pose one of the true challenge questions of the day, “What do you think the graph of the inverse of the sine function would look like?” (MP2) I ask the students to sketch their idea on a whiteboard (I have a classroom set of 12” x 12” whiteboards we cut from shower board), then post the whiteboards in the front of the room.  This is always an interesting gallery, because there are so many different graphs presented.  I take a photo of the gallery with my flipcam or iPad so we have a more permanent record, then explain that today we will be looking at the graphs of all the inverse trigonometric functions.

Put into Action

40 minutes


Graphing (25 min)

For this part of the lesson, you will want to have the Graphing Inverse Trig functions handout and graph paper available.  I number the handout copies in advance to be ready for the second part of the activity, but you can also just have your students count off  or assign them by rows. You may also want copies of the unit circle with trig function values if your students don’t already know them well.  I have my students take out their copies of the unit circle with trig values labeled.  I remind them of the T-tables they’ve made before to graph an unfamiliar function and make a large T-table on the whiteboard.  I now pass out the Inverse Trig Functions handout and suggest that we begin by listing several sin values we know.  My students generally start with the easy ones like sin 0 and sin π/2, but I eventually fill the table on the board while they complete the table on their handout.  When we have filled the first column I ask which axis this would represent for a graph of sine, cosine or tangent.  Someone generally remarks that it’s the x-axis because it’s the unit circle “unrolled”, which is one of the ways I’ve described it before.  I then tell my students to work with their right shoulder partner to figure out the values for the second column, reminding them that they can use their unit circle for help. (MP2)  I walk around checking for accuracy and helping as needed while everyone completes the task.  When all students are finished, I ask my students to think-pair-share with their partner about what happens to the x and y values of a given function if you want to graph the inverse. After about 2-3 minutes, (longer if they are really engaged!) I randomly call on a few students to share what their partner said.  Occasionally asking for students to share their partner’s comments encourages them to listen attentively to each other. 

When we reach a consensus that the x and y values are reversed, I tell them that today they will be graphing the inverse of the sine function and ask what they need to do with the T-table they’ve created to make that work.  When someone responds with the correct answer, I ask my students to label the first column of their T-table as “Y” and the second column as “X”.  I then ask them to make x and y axes on their graph paper, with the x axis near the bottom of the short side of the page and the y axis near the center of the long side.  You can see an example of a completed graph in my resources. For the next part of this activity, I tell my students they will be working independently to accurately graph the inverse of the sine function.  I remind them to label their axes and to be careful about scale before beginning to plot points, then walk around while they’re working to help and redirect as needed. (MP5, MP6)  This part takes a while, especially for those students who tend to be very precise. For those students who work more quickly or need additional challenge I ask them to consider what the inverse of the sine function is called, and/or what the inverse graphs of other trig functions would look like.  When everyone is done, I ask the students to compare their graphs to the predictions they made at the beginning of class using the whiteboards.  I ask them to reflect on why they made their original graph the way they did and how this activity has changed their perspective.  


Practice (10-15 min)

For this part of the lesson you will want to have additional graph paper available for students who need a “fresh sheet”.  I tell my students that they will now have the opportunity to expand their understanding of inverse trig functions to the graphs of cosine and tangent.  I say that they will be working independently to graph the inverse of either the cosine or the tangent functions and then will be sharing their results with the class.  I tell them to look at the top of their Handout to see if they have an odd or even number.  Odds get to work with tangent and evens get to work with cosine.  I suggest that there are tables for each on the back of the handout to help organize their graphing and tell them they have about 15 minutes to complete the activity. (MP1, MP4) While students are working I walk around and look for any who need additional help or who are getting off track.  Toward the end of this time, I divide my front board into three parts with vertical lines and label the left column “inverse cosine”, the center “inverse sine” and the right “inverse tangent”.  When everyone has completed their graphs, I ask for volunteers (without telling them what they’re volunteering for) and then take about 1/3 of the class and tell them they will be posting their inverse sine graphs. You will want to try to have a balance of the three functions posted. I then tell all my students to tape inverse tangent graphs on the right side of the board, inverse sine graphs in the middle and inverse cosine graphs on the left side so that we can all see the results. 

Wrap Up + Homework

10 minutes

When all the graphs are posted, I ask my students if they see any similarities between the graphs.  If no one mentions that they all multiple y-values each for several x-values, I ask for a definition of “function”.  Generally someone remembers either a graphing definition or a table definition and since we have both available, we can see that our three inverse graphs don’t fit the pattern.  This discussion calls on prior understanding of what the definition of a function and also on their ability to look at both the table and the graph to explain or defend their position. I believe this builds on  MP7 by allowing students to use their understanding of the structure of a function to identify why their graphs are not functions and then to identify suitable restrictions on the domain. My classes are small enough to do this as a whole class, but you could also have students discuss in smaller groups and then summarize and share their explanation with the class.  I then challenge my students to identify whether limiting the domain will give a true function.  I tell them to think-pair-share with their left shoulder partner (see my strategies for assigning partners).  After a few minutes I randomly ask students to share what they’ve discussed and usually get a wide variety of suggestions. I allow brief discussion, bringing the class to a consensus that these inverse graphs indeed be functions if their domain is restricted.  I tell them that the convention is to use the domain from –π/2 to π/2 and assign them to write a brief explanation of why that would be better or worse than the restriction they chose, an application of MP3.  That is their ticket-out-the-door today or, if more time is needed, it’s homework.