How Do You...Trigonometric Graphs

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."

Objective

Students understand how values of the unit circle are periodic and are connected with the appearance of a sin and cos graph. Students understand the ideas of amplitude and period. Students understand that it is possible for an identical graph to be produced by a sin and cos function with appropriate use of transformations. Students understand that tan has a different period by default.

Progression

There's a lot to do here and I find this to be one of my favorite topics. We move from values on the unit circle to more generic methods for sketching the graph of sin and cos functions. We make some science connections with sound waves.

First grab some trig graph paper and have students get out a unit circle and some markers. Have them systematically plot the values of sin for the whole circle and connect the dots. Do the same for cos and then tan.

Define the terms amplitude and period. Some of this will be familiar from prior science classes. I interchange period with wavelength and frequency. As a further connection, we talk about the "speed" of the function and define this base form as "normal" speed.

Together, I show them how you can plot the graph of modified sin/cos/tan functions if you know something about how amplitude and period frame the picture. We plot a few examples together. Students often find it weird that sin 2x has a shorter period. There is a tendency to start talking about arbitrary opposites a la "oh well, we do the opposite of the modification, therefore 2x implies x / 2" hand waving. I found it really useful to continue that metaphor of speed. The function sin 2x completes its run two times "faster" than the "normal" we defined previously. For something to be faster, the elapsed time must be shorter. Therefore, one wave takes up less distance along the x-axis. This becomes really handy when you want to talk about sin 2x/3, where the whole opposite operation thing gets ugly to describe. Rather, at 2/3 the speed, we should expect waves to be longer. Throwing in stuff like -3 sin x or sin x + 2 doesn't cause much confusion. I don't do horizontal transformations just yet.

Next they work through two activities to get more practice. It's easy to go fast through this topic and wind up with lots of confusion. I made a concerted effort to slow down and offer lots of opportunities for students to discuss this with each other.

After the activities, they completed some individual practice.

Activities

First I have to make some connections between pictures and the equations associated with them. I hand a out a set of 16 unidentified graphs (Answer Key). Students are given an iPad and desmos to determine how the pictures were made.

Next students play a little game drawing randomly generated trig functions. Students are given whiteboards, markers, cards, and 1 die. Students flip a card which describes something about the amplitude, period, and/or transformation of a function. Students roll the die. A 1/2 means sin, 3/4 cos, and 5/6 tan. Students must draw the appropriate function that has the properties of the card. I wrote a more extensive description at the time.

Next students are given a chance to design some trig functions of their own, labeling the graphs appropriately. They design 3 sin and 3 cos functions. They must demonstrate amplitude changes, slower periods, and faster periods in whatever combination they choose. Students create the graphs with desmos on an iPad, print them out and annotate.

Last, we make some connections with sound waves. How much I spend with this depends on where I am in the semester. Last year we were right up against the end, so the science stuff was just a couple side discussions. You could flesh it out more as I have in the past, or start with this. I use a tone generator to talk about how we perceive amplitude and period with sound. I also have a set of blinking lights that are fun to talk about too. You can go really deep into harmonics and springs and things if you have time to fill in the physics background.

Assessment

This traditionally hits at the end of the first semester. Assessment was pretty straightforward. I'm looking at their ability to sketch sin/cos/tan functions by interpreting amplitude and period correctly spread across two SBG topics. I ask some conceptual questions as a third topic.

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