As students come into class the following question is displayed on the board:
How can we sketch y = tan x by hand?
Many students get out calculators to see the graph of tangent. When we discuss the graph, many students will use a finger to demonstrate the look of tangent. The use of gesture tells me that they lack some of the terminology needed and they are still figuring things out. As I listen to their explanations, I am trying to choose a student to draw his/her idea on the board.
Once we have a visual to look at on the board, we will delve deeper into what the students know by discussing questions like:
Once we have a productive set of ideas to consider, I will use Desmos to graph_y=_tan_x and display the graph on the board. First, we will discuss the asymptotes. I'll ask, "Why does the graph of tangent have asymptotes while the graphs of sine and cosine do not?" I mention sine and cosine intentionally, since this may help with their explanation (tanx = sinx / cosx). I want my students to understand and to be able to explain:
We will continue to refer to these questions as my students work together to develop a method to graph the tangent function.
Since the asymptotes will be an important consideration, I will focus on ensuring that my students understand the behavior of the tangent function near the asymptotes:
Eventually, to make tangent seem a little less unique I will ask about y=cotangent x for comparison. I'll ask, "Will cotangant's graph have similar features as tangent? Why?"
Once the students are beginning to formulate their own ideas, we'll work together to identify the key features. We'll discuss the same features we explored for sine and cosine:
The students also look a the graph from Desmos to see how the curve is placed and shaped. I may talk about how the graph changes concavity (Here I usually say curvature and not get into concavity too deeply I will say that when we talk about the curvature of a graph we call this the concavity). We discuss where we will have the period start.
When discussing these graphs, it is often "obvious" to my students that the period should be from asymptote to asymptote. If you decide that the period is from asymptote to asymptote the graph of y=tan x is at the "midline" in the middle period . Otherwise students can say the period begins at the midline.
Throughout the discussion I continue to sketch a graph on the board.
I ask students to find they when x=pi/4 for y=tan x. Instead of doing only tangent I graph cotangent at this time. I ask "What function will have a graph similar to tangent? Why? What are the differences?" I am looking for students to see that the asymptotes will be at different x values and that the graph concavity and direction is a little differnt (from left to right it goes high to low instead of low to high) Since the asymptotes are different the point where the graph crosses the midline is different. Students explain why they think these differences exist. Some will say it is because Cotangent is the reciprocal of tangent so whenever tangent is 0 cotangent is undefined others will talk about cotangent is the ratio of x/y so when y is 0 it is undefined. As students talk about the difference we sketch the graph.
Once students see how to graph one period of the basic graph I move to transformations I ask How will the graph change if the equation is y=2 tan x or y=3 tan x? I want students to understand that the 2 and 3 will stretch the graph so the value of the function at x=pi/4 will change. When students graph tangent and cotangent we graph the asymptotes the midpoint and the point half way between the midline point and the asymptote (in the basic function this is at pi/2 for tangent).
As students work on graphing I ask:
Students work on a graph with a transformed tangent and cotangent equation. Students work in small groups and then share results with the class.
I assign a few problems for students to practice graphing (#44, 50, 58 on page 327 of Larson's "Precalculus with Limits" book). This will allow students a opportunity to practice the ideas we discussed today.
As an exit slip I ask students to sketch a graph of y=3 tan (x-pi/3) by hand. Some students may not be ready for this problem, so I will suggest that students list what they know about the graph and formulate a question about what they did not understand. With concepts as detailed as graphing trigonometric functions, I need my students to be proactive with identifying what they know and what they need help with.
My goal in graphing with paper and pencil is not to make the students advanced at this skill. With technology today doing a lot of paper pencil graphs seems to be too much. Students need to understand how the graphs look and how the transformations work. By graphing a few functions the students are better able to analyze graphs and information.
After this unit most students will be able to explain the shape of the graph and the relationship with the key features such as midline, asymptotes, maximums and minimums.
When the functions are discussed in later units students should be able to sketch a quick graph maybe not to scale by doing the pencil paper if students need a more complicated graph they can use the online tools we used for this unit.