Graphing Tangent and Cotangent Functions

Objective: To determine the properties of the tangent and cotangent functions; to graph the tangent and cotangent functions.

First let’s look at the values of tangent functions. Use your unit circle and prior knowledge to fill in the following table of values:

tan

What do you notice about the values in this table?

What does the graph look like when the tan value is undefined?

From just looking at the table, what do you think is the period of the tangent function?

Can you predict what the graph will look like?

Plot the points and use your graphing calculator to graph and check your graph.

Where are the asymptotes?

Do tangents have amplitudes? Why or Why not?

Now let’s do the same thing for the cotangent function. Use your unit circle and prior knowledge to fill in the following table of values:

cot

What do you notice about the values in this table?

What does the graph look like when the cot value is undefined?

From just looking at the table, what do you think is the period of the cotangent function?

Can you predict what the graph will look like?

Plot the points and use your graphing calculator to graph and check your graph.

Where are the asymptotes?

Do cotangents have amplitudes? Why or Why not?

17. How does this graph relate to the graph of ?

The general forms of the tangent and cotangent functions are:

and

We are going to explore the effect that each variable has on the mother functions.

I. variable “A”

1. Using your graphing calculator, Graph and on the same coordinate plane

2. Then graph and on the same coordinate plane.

3. Examine more graphs of this form if you find it necessary to be able to answer the following questions.

4. Answer the following questions about equations in the forms and :

a) What happens to the graph of and when A > 1?

b) What happens when 0 < A < 1?

c) When -1 < A < 0?

d) What about when A = -1?

e) And when A < -1?

II. variable “B”

1. Using your graphing calculator, Graph and on the same coordinate plane

2. Then graph and on the same coordinate plane.

3. Examine more graphs of this form if you find it necessary to be able to answer the following questions.

4. Answer the following questions about equations in the forms and :

a) What happens to the graph of and when B > 1?

b) What happens when 0 < B < 1?

c) When -1 < B < 0?

d) What about when B = -1?

e) And when B < -1?

f) How can you find the period of a tangent or cotangent function based on its equation?

III. variable “C”

1. Using your graphing calculator, Graph and on the same coordinate plane

2. Then graph and on the same coordinate plane.

3. Examine more graphs of this form if you find it necessary to be able to answer the following questions.

4. Answer the following questions about equations in the forms and :

a) What happens to the graph of and when C is positive?

b) What about when C is negative?

c) What happens if there is also a B variable?

IV. variable “D”

1. Using your graphing calculator, Graph and on the same coordinate plane

2. Then graph and on the same coordinate plane.

3. Examine more graphs of this form if you find it necessary to be able to answer the following questions.

4. Answer the following questions about equations in the forms and :

a) What happens to the graph of and when D is positive?

b) What about when D is negative?

Practice:

1. Determine the vertical stretching/shrinking, period, phase shift (horizontal shift), and vertical shift of each function.

a)	b)

c)	d)