Graphing the nonsinusoidal trigonometric functions Day 1 of 2

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:

• What is the period of the tangent function?
• How is the graph of tangent different than the graphs of sine and cosine?
• Is the tangent line continuous?
  

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:

1. Why the domain of tangent is not all real numbers...
2. What causes the behavior of the plot as it approaches the points where it is undefined

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:

• What is happening to tangent as it gets close to the asympotote? (We'll use the trace function on Desmos to see that the graph gets extremely large or small)
• How many asymptotes will tangent have?

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:

• Do tangent and cotangent graphs have amplitudes? Why or why not?
  
• What does the parameter "a" in the y=a tan b(x-c)+d do to the graph?
• How will the vertical shift affect the graph?
• What is the period of y=tan x or y=cot x?
• How will we find the period if it is not pi such as y=2tan x?

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:

• What should we identify first? Students usually identify all the key features I make sure the students do not write the amplitude we will say the graph has a vertical stretch of 2.
• What should we do not to make the graph? Students usually want to label the x-axis and locate the asymptotes.
• What points should we graph? Students state that want the point on the midline and then the point that is halfway between the midline point and the asymptote.

Students work on a graph with a transformed tangent and cotangent equation. Students work in small groups and then share results with the class.