Trigonometric Graphs (Day 1 of 3)

Learning Target for graphing

This unit progresses from the unit circle to observing how the trigonometric functions can be graphed as a relation between the measure of an angle, x,  and the value of a trig function f(x). To enable students to use their existing habits, we follow a progression similar to our work with absolute value, exponential, and logarithmic functions.

I begin with the function f(t)_=_t^3 (See my Confusion between... Reflection on why I use t and f(t) in this lesson). I ask my students the following questions: 

• What variable is used to represent the values in the domain of the function?
• What name is given to describe the set of values in the range of the function?
• What is the domain of the function?
• What is the range of the function?
• How steps can we take to graph this function?  

I use a similar protocol to discuss and graph the function f(t)=ln t.

I finally move to f(t)=sin t. I ask the same questions as above. Although they are familiar with the Unit Circle and the Sine function, I expect some of my students will get stuck when they try and determine the domain and range values. They will also be unsure how to graph the function. Others will use their graphing calculators to determine the values. In this case, I expect some will try to graph f(x) = sin(t), and will receive an error message. 

After we discuss the obstacles that arise, I will explain that we are going to look carefully at how to graph f(t)=sin t and the other trigonometric functions today, so keep in mind some of the obstacles that came up so far.  

With my students wondering some of the challenges of graphing f(t) = sin t, we'll begin exploring the graphs of the six trigonometric functions. We start with an activity that uses the Desmos Graphing Calculator app. I prefer using the computer for these functions because my students are better able to see the graph than on a graphing calculator. This becomes important when there are asymptotes. Desmos is also user friendly, and can be used at home easily.

The exploration begins with students graphing each function, then determining the domain and range for each. I expect my students will begin to have questions when they graph the tangent function. I am prepared to use questioning to help  students discover the shape and characteristics of the graph of f(t) = tan t:

• What are your setting for the x-axis (making sure they are appropriate)?
• What do you see on the graph?
• Have you tried zooming out? What do you notice when you do?
• How could you write the domain for this function?

At this point I do not expect my students to write a mathematical definition of the domain using correct notation. I expect that many will say t cannot equal pi/2, 3pi/2, etc., which is a good starting point and prepares them for the remaining functions. 

Teacher's Note: As students continue they will be amazed when they see they graphs of secant and cosecant. They will continue to need some support to write the domain and the range, but they are often excited about the graphs. This fact helps them to appreciate why it is helpful to have concepts like domain and range for a function.

After students have looked at the graphs I want to make sure everyone understands how to write the domain and ranges of the functions using correct mathematical notation. First, I bring the class back together to make sure we all agree on the answers. Then we will discuss a process and a notation for describing the domain of the functions.

I start by asking my students if they found any interesting or unusual patterns. Many will report that secant and cosecant are strange, some will comment on patterns with discontinuity. If my students do not say anything, I will be prepared to ask questions of students whom I overheard discussing the graphs as they worked on the activity.

When I display the first question from the activity on the board, I ask students to volunteer responses to fill in the cells. Once answers are on the board I use questioning to probe my students ideas about the answers:

• Does it make sense that sine and cosine have a range of [-1,1]? Why? (Here I want students to realize that the radius of is always bigger than the x or y or that the hypotenuse of the triangle is larger than either leg)
• Why is tangent undefined at pi/2? 3pi/2? etc? (I ask similar question for secant, cosecant, and cotangent)
• Why is secant's and cosecant's range not between -1 and 1? (I want students to realize that when you take a reciprocal of a number between -1 and 1 you get a number greater than 1 or less than -1)

Once we have discussed the graphs of the functions, we will move on to a presentation of  how to express the domain and range using n*pi.

The definition of domain as a pattern confuses some students. In order to help, I show how 3*pi/2 is the same as pi/2+pi. I also ask if we can write -pi/2 as pi/2 + or - a number. We also go to 5*pi/2 and see that we are just adding a multiple of pi. Once students understand this I explain that in mathematics the parameter n is often used to indicate an integer, so we can say the domain of the tangent function is all real numbers t such that t does not equal pi/2+n*pi.

We then discuss how to write the domain of secant and cosecant by using the interval notation.