Graphing Transformations of Tangent, Cotangent, Secant, and Cosecant Functions

1. Transforming the Graphs of the Secant and Cosecant Functions

Recall that meaning that if the value of is known, then is its reciprocal.

Consider now this table of values.

	0
	1				0	-1	0	1
	1			2	undef.	-1	undef.	1

Since we can build the graph of the secant function from the cosine function, as shown in the figure below.

Properties of the graph:

• There are vertical asymptotes at the same values where has its x-intercepts.
• The domain of is the set of all real numbers excluding odd multiples of
• The range of is
• The period is
• Since the graph is not bounded, there is no amplitude.

Recall that meaning that if the value of is known, then is its reciprocal.

Consider now this table of values.

	0
	0				1	0	-1	0
	undef.	2			1	undef.	-1	undef.

Since we can build the graph of the cosecant function from the sine function, as shown in the figure below.

Properties of the graph:

• There are vertical asymptotes at the same values where has its x-intercepts.
• The domain of is the set of all real numbers excluding integer multiples of
• The range of is
• The period is
• Since the graph is not bounded, there is no amplitude.

Now, let’s turn our attention to graphing transformations of the secant and cosecant functions, whose equations have the form and

• The value of is not related to amplitude since the secant and cosecant graphs are not bounded. This value simply determines vertical stretch or compression, and a reflection over the x-axis if
• Assuming the period is
• The phase shift is
• The vertical shift is d units, which means the midline is

As you can see, except for amplitude, the process for transforming secant and cosecant graphs is similar to transforming graphs of the sine and cosine functions.

watch

In this video, we’ll walk through the transformations required to graph

try it

Consider the graph of

Determine the period, phase shift, and vertical shift.

Period:
Phase shift: units to the left
Vertical shift: down 2 units

Write equations of two vertical asymptotes.

Two asymptotes of the graph of are and

Since this graph is shifted to the left units, the asymptotes are as well.

Then, subtract from each given value to find where the vertical asymptotes are:

Thus, the equations of two vertical asymptotes are and

Note: There are many other asymptotes.

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2. Transforming the Graphs of the Tangent and Cotangent Functions

Before getting to the graphs of the tangent and cotangent functions, the figure below shows the unit circle with angles and

Assuming that recall that when the terminal side of angle intercepts the unit circle at the point

Then, for the angles and in the figure, we have and

In general, this means that when is defined, then has the same value.

big idea

Since this means that the period of the tangent function is

Note that the value of the tangent function is the slope of the line containing the points and

watch

In this video, we’ll investigate the behavior of the tangent function as its input increases, particularly around values for which the tangent function is undefined.

Recall also that and that when x is any odd multiple of Then, is undefined when x is any odd multiple of

The graph of the function is shown below.

Properties of the graph:

• There are vertical asymptotes at every value of x that is an odd multiple of
• The domain of is the set of all real numbers excluding multiples of
• The range of is
• The period is
• Since the graph is not bounded, there is no amplitude.

Also from the unit circle, recall that when the terminal side of angle intercepts the unit circle at the point

Since these are reciprocal values of the tangent function, the basic cotangent function also has a period of meaning that as long as is defined.

Since the cotangent function is undefined for all values of where or where k is an integer.

All of this information considered, below is the graph of Even though it is not marked with a dashed line, there is also a vertical asymptote at

Properties of the graph:

• There are vertical asymptotes at where k is an integer.
• The domain of is the set of all real numbers excluding integer multiples of
• The range of is
• The period is
• Since the graph is not bounded, there is no amplitude.

Now, let’s turn our attention to graphing transformations of the tangent and cotangent functions, whose equations have the form and

• The value of is not related to amplitude since the tangent and cotangent graphs are not bounded. This value simply determines vertical stretch or compression, and a reflection over the x-axis if
• Assuming the period is
• The phase shift is
• The vertical shift is d units, which means the midline is

As you can see, except for amplitude and period, the process for transforming tangent and cotangent graphs is similar to transforming graphs of the sine and cosine functions.

watch

In this video, we’ll walk through the transformations required to graph

try it

Consider the graph of

Determine the period, phase shift, and vertical shift.

The period is

There is no number added or subtracted in the angle, which means there is no phase shift.

The vertical shift is up one unit. We know this from the constant term.

Write equations of two vertical asymptotes.

The asymptotes correspond to where the angle is equal to etc.

Therefore, set and to get two of the vertical asymptotes. Solving each equation gives and

Note: There are many other asymptotes.

SOURCE: THIS WORK IS ADAPTED FROM PRECALCULUS BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/BOOKS/PRECALCULUS/PAGES/1-INTRODUCTION-TO-FUNCTIONS