# Inside Out and Upside Down - Modeling with Inverse Functions

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## Objective

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!

5 minutes

## Put into Action

40 minutes

Graphing (25 min)

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.