Graphing Tangent and Cotangent Functions ANSWER KEY

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?

They values repeat. They are cyclical.

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

There is a vertical asymptote.

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

The period is

Can you predict what the graph will look like?

Use your graphing calculator to graph and sketch the graph here:

Where are the asymptotes?

They occur every for all rational numbers, n.

Do tangents have amplitudes? Why or Why not?

No. Tangent graphs go on forever in the vertical direction and so there can be no “height”.

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?

They values repeat. They are cyclical.

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

There is a vertical asymptote.

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

The period is

Can you predict what the graph will look like?

Use your graphing calculator to graph and sketch the graph here:

Where are the asymptotes?

They occur every for all rational numbers, n.

Do cotangents have amplitudes? Why or Why not?

No. Cotangent graphs go on forever in the vertical direction and so there can be no “height”.

How does this graph relate to the graph of ?

The x-intercepts of the graph of become asymptotes in the graph of . The asymptotes of the graph become x-intercepts in the graph of . The points that have y-coordinates of 1 and -1 have the same x-coordinates, respectively, in the graphs of both and . Essentially, the graphs "face" opposite directions : to the right and to the left.

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?

The graphs "stretch" away from the x-axis.

b) What happens when 0 < A < 1?

The graphs "shrink" toward the x-axis.

c) When -1 < A < 0?

The graphs "flip" about the x-axis and "shrink" toward the x-axis.

d) What about when A = -1?

The graphs "flip" about the x-axis.

e) And when A < -1?

The graphs "flip" about the x-axis and "stretch" away from the x-axis.

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?

The graphs "squeeze" horizontally.

b) What happens when 0 < B < 1?

The graphs "stretch" horizontally.

c) When -1 < B < 0?

The graphs "flip" about the x-axis and "stretch" horizontally.

d) What about when B = -1?

The graphs "flip" about the x-axis.

e) And when B < -1?

The graphs "flip" about the x-axis and "squeeze" horizontally.

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

Period =

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?

The graph shifts C units to the left.

b) What about when C is negative?

The graph shifts C units to the right

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

Then the shift is given by C/B.

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?

The graph shifts D units up.

b) What about when D is negative?

The graph shifts D units down.

Practice:

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

a)	b)

none, pd = , ps = left, vs = 3 up	v. stretch = 5, pd = , ps = left, vs = 0

c)	d)

none, pd = , ps = right, vs = 5 down	v. stretch = 3, pd = , ps = 0, vs = 1 up

2. Without the aid of a calculator, graph each function over the interval (Remember to graph by stages if necessary).

a)	b)

c)	d)