Yesterday's drawbridge task was a good way to present students with a context where an inverse trigonometric function is necessary – where we know two of the side lengths of a right triangle and we want to know the angle measure. The big idea from yesterday was just to identify that the input and output have switched and to identify the need for an inverse.
To start today's discussion, think back to the graph from the Drawbridge Task and ask students why the graph stops when we look at it on Desmos or a graphing calculator. Discuss the fact that if we do not restrict the domain of original sine function for the drawbridge graph, then there will be an infinite number of outputs for each input. In the video below, I discuss how this relates to the drawbridge task.
Inverse functions are tough for students. The symbolism is unusual and the process is abstract. In recognition of this, today we will start by reviewing a procedure that is more accessible to students – finding the inverse of a quadratic function.
I will start by giving students the worksheet and asking them to work on the first page in their groups. As they work I will pay particular attention to whether or not students see a problem with the inverse function of the quadratic when they solve algebraically. I want them to notice that since it has a plus or minus in it, the inverse is not a function. Each input produces two outputs. After students have found this inverse, I will remind them of the horizontal line test and why we have to restrict the domain of the original function in order for the inverse to be a function. For example, if the domain of the inverse is not restricted, the input x = 17 produces the ambiguous output f'(17) = 4 or -4.
When I go through the answers from the first page and I make a big deal about Question 5 and how the domain and range switch from the original function to the inverse.
For the next phase of today's exploration, students are going extend their knowledge of the inverse from quadratic to trigonometric functions as they work on Inverse Trig Functions. Just like we did on the front page, students will start by highlighting a portion of the original function that does pass the horizontal line test, and then find the domain range of that highlighted piece. Since there are many different pieces of the graph that will pass the horizontal line test, I ask students to draw in pencil to begin with and then go back and highlight in a different color once the class finalizes their intervals.
As students are working, I watch out for students who highlight a portion of the graph that passes the horizontal line test but does not encompass every possible input and output (this is especially tempting for the secant and cosecant graphs).
After students complete this portion and discuss as a class, I ask them to:
As the students graph, I remind the students that the inverse function will be a reflection over the line y = x. This fact usually helps them to draw the sketches. I usually go through this first sketch as a class so that students can get a feel for how to think through the process of sketching the inverse.
After this graph has been completed, as a class we practice using the sketch to find the y-value of a ratio of 1/2. I point out that the input is a ratio of side lengths and the output is the angle measure. I often do one more graph together, particularly a tricky one such as inverse secant.
Instructional Note: Today I noticed that many of my students only highlighted a portion of a top branch of the graph and neglected the negative y-values. Keep an eye out for that. After completing the inverse secant together, cut them loose and have them work on the remaining graphs.
To finish today's lesson, I have students present their inverse graphs for the remaining four trig functions. Then I ask them again why we had to restrict the graphs in order to produce the inverse. I also review the stipulations for choosing the highlighted portion one more time.
I conclude by saying that the highlighted portion must:
1. pass the horizontal line test,
2. include every possible y-value of the original function, and
3. use the easiest numbers that are closest to zero.