One of our goals today is to start graphing the six trigonometric parent functions and to think about the important characteristics of the graphs. This will lead us to the transformations that we will be working on later in this unit. If students have a strong foundation with these parent functions, the transformations should be a breeze.
Since my students have studied these functions in the past, today's lesson will refresh the concepts. We are not discovering these graphs for the first time, so my approach today assume my students come with a little bit of background information.
Give students this worksheet and have them see how many of the six parent functions they can graph. They should be working with their table groups. I don't want them to use the graphing calculator for this task because I want them to be thinking conceptually about these functions and reasoning through them
I will have the class work to the point where their thought process can be something like this:
I am graphing the secant function and I know that secant equals 1/cos(x). If I want to find the asymptote, it would be where the denominator is equal to zero, so I just have to find where cos(x) equals zero. I know that the cosine is equal to zero at 90 and 270 degrees.
Too often my students will just memorize these important aspects of the graph, and I want them instead to be able to reason about them. It is important to model this thought process in front of the class. This can be done by the teacher or a student who demonstrates the he/she can explain it clearly. Otherwise, the class might not pick it up on their own.
If a group gets stuck on the tangent graph, for example, you can ask them to rewrite the function as y = sinx/cosx. Then you can draw upon our experiences with rational functions to think about the x-intercepts and asymptotes. The same thought process will work here - we set the numerator and denominator equal to zero and find those critical values. A sign diagram will also help them sketch the general shape of these functions.
In my experience students usually do pretty well with the sine and cosine graphs. So, during the discussion of the parent functions, I pay particular attention to the graphs of secant, cosecant, tangent, and cotangent. You may not have to go through every single graph, but it is important to go through one of secant or cosecant, and one of cotangent or tangent.
While discussing the graphs of these functions, it is important model the thought process that I alluded to in the Launch and Explore section. It would be great to find a student who is using this approach while students work at the beginning of class. If you don't find a student who has used this thought process, you can model it for the class your self.
One common error is to graph the secant function as the inverse of the cosine function, meaning it is reflected over the line y = x. Students know that the secant and cosine are multiplicative inverses, so they may extend this to say that they are inverse functions. If you see a handful of students do this, talk about it as a class and come to terms with the thinking that is leading to this conclusion.
When completing the chart on the task page, students may show difficulty writing down the x-intercepts since there are infinitely many. It is important to talk about why there are infinitely many. Many students will just list the first few values and an ellipsis (e.g. 90 degrees, 270 degrees, 450 degrees, ...). Getting them to write them as an expression such as 90 + 180n, where n is an integer, will help to establish structure once we start to solve trigonometric equations and want to find every answer.
The transformations of sine and cosine functions have always been problematic for my students. There are a few issues that seem to reoccur year after year. In the video below I discuss two issues that are difficult for students - the coefficient of x and the phase shift.
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On the worksheet, students are to list the transformations they notice and discuss how they will affect the graph of the parent function. We will not have enough time to discuss these together as a class, but students should be able to start on these with their groups. Stress that you want them to check their answers on a graphing calculator so that if there is a misconception with the phase shift, students can immediately start thinking about why the graph of y = cos(2x-60°) is not shifted 60 degrees to the right.
Getting students to think about the difference of these two situations will lead us to the graphs of trig functions that include all of the possible transformations (like: y = 2 + 5cos(2(x-30°)). Clearing up this confusing topic today will make tomorrow's lesson go smoothly.