Tangent and Cotangent Graphs

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Tangent and Cotangent Graphs

• Reading and Drawing
• Tangent and Cotangent Graphs

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This is the graph for y tan x.
This is the graph for y cot x.
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One definition for tangent is .
Notice that the denominator is cos x. This
indicates a relationship between a tangent graph
and a cosine graph.
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To see how the cosine and tangent graphs are
related, look at what happens when the graph for
y tan x is superimposed over y cos x.
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In the diagram below, y cos x is drawn in gray
while y tan x is drawn in black. Notice that
the tangent graph has horizontal asymptotes
(indicated by broken lines) everywhere the cosine
graph touches the x-axis.
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One definition for cotangent is .
Notice that the denominator is sin x. This
indicates a relationship between a cotangent
graph and a sine graph.
This is the graph for y sin x.
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To see how the sine and cotangent graphs are
related, look at what happens when the graph for
y cot x is superimposed over y sin x.
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In the diagram below, y sin x is drawn in gray
while y cot x is drawn in black. Notice that
the cotangent graph has horizontal asymptotes
(indicated by broken lines) everywhere the sine
graph touches the x-axis.
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y tan x.
y cot x.
For tangent and cotangent graphs, the distance
between any two consecutive vertical asymptotes
represents one complete period.
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y tan x.
One complete period is highlighted on each of
these graphs.
y cot x.
For both y tan x and y cot x, the period is
p. (From the beginning of a cycle to the end of
that cycle, the distance along the x-axis is p.)
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For y tan x, there is no phase shift.
The y-intercept is located at the point (0,0).
We will call that point, the key point.
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A tangent graph has a phase shift if the key
point is shifted to the left or to the right.
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For y cot x, there is no phase shift.
Y cot x has a vertical asymptote located along
the y-axis. We will call that asymptote, the key
asymptote.
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A cotangent graph has a phase shift if the key
asymptote is shifted to the left or to the right.
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For a cotangent graph which has no vertical
shift, the equation for the graph can be written
as
For a tangent graph which has no vertical shift,
the equation for the graph can be written as
y a tan b (x - c).
y a cot b (x - c).
c indicates the phase shift, also known as the
horizontal shift.
a indicates whether the graph reflects about the
x-axis.
b affects the period.
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Unlike sine or cosine graphs, tangent and
cotangent graphs have no maximum or minimum
values. Their range is (-8, 8), so amplitude is
not defined. However, it is important to
determine whether a is positive or negative.
When a is negative, the tangent or cotangent
graph will flip or reflect about the
x-axis.
y a tan b (x - c) y a cot b (x - c)
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Notice the behavior of y tan x.
Notice what happens to each section of the graph
as it nears its asymptotes. As each section
nears the asymptote on its left, the y-values
approach - 8. As each section nears the asymptote
on its right, the y-values approach 8.
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Notice the behavior of y cot x.
Notice what happens to each section of the graph
as it nears its asymptotes. As each section
nears the asymptote on its left, the y-values
approach 8. As each section nears the asymptote
on its right, the y-values approach - 8.
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This is the graph for y tan x.
y - tan x
Consider the graph for y - tan x In this
equation a, the numerical coefficient for the
tangent, is equal to -1. The fact that a is
negative causes the graph to flip or reflect
about the x-axis.
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This is the graph for y cot x.
y - 2cot x
Consider the graph for y - 2 cot x In this
equation a, the numerical coefficient for the
cotangent, is equal to -2. The fact that a is
negative causes the graph to flip or reflect
about the x-axis.
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y a tan b (x - c) y a cot b (x - c)
b affects the period of the tangent or cotangent
graph. For tangent and cotangent graphs, the
period can be determined by
Conversely, when you already know the period of a
tangent or cotangent graph, b can be determined by
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A complete period (including two consecutive
vertical asymptotes) has been highlighted on the
tangent graph below.
For all tangent graphs, the period is equal to
the distance between any two consecutive vertical
asymptotes.
The distance between the asymptotes in this graph
is . Therefore, the period of this
graph is also .
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Use , the period of this tangent graph,
to calculate b.
We will let a 1, but a could be any positive
value since the graph has not been reflected
about the x-axis.
An equation for this graph can be written as
or .
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A complete period (including two consecutive
vertical asymptotes) has been highlighted on the
cotangent graph below.
For all cotangent graphs, the period is equal to
the distance between any two consecutive vertical
asymptotes.
The distance between the asymptotes is
. Therefore, the period of this graph is also
.
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Use , the period of this cotangent graph,
to calculate b.
We will let a 1, but a could be any positive
value since the graph has not been reflected
about the x-axis.
An equation for this graph can be written as or
.
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y tan x has no phase shift.
We designated the y-intercept, located at (0,0),
as the key point.
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y cot x has no phase shift.
x 0
We designated the vertical asymptote on the
y-axis (at x 0) as the key asymptote.
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If the key point on a tangent graph shifts to the
left or to the right,
or if the key asymptote on a cotangent graph
shifts to the left or to the right,
that horizontal shift is called a phase shift.
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y a tan b (x - c)
c indicates the phase shift of a tangent graph.
For a tangent graph, the x-coordinate of the key
point is c.
For this graph, c because the key point
shifted spaces to the right. An equation
for this graph can be written as .
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y a cot b (x c)
c indicates the phase shift of a cotangent
graph. For a cotangent graph, c is the value
of x in the key vertical asymptote.
For this graph, c because the key
asymptote shifted left to . An
equation for this graph can be written as
or
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Graphs whose equations can be written as a
tangent function can also be written as a
cotangent function.
Given the graph above, it is possible to write an
equation for the graph. We will look at how to
write both a tangent equation that describes this
graph and a cotangent equation that describes the
graph. The tangent equation will be written as
y a tan b (x c). The cotangent equation will
be written as y a cot b (x c).
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For the tangent function, the values for a, b,
and c must be determined.
This tangent graph has reflected about the
x-axis, so a must be negative. We will use a
-1.
The period of the graph is .
The key point did not shift, so the phase shift
is 0. c 0
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The tangent equation for this graph can be
written as
or .
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For the cotangent function, the values for a, b,
and c must be determined.
This cotangent graph has not reflected about the
x-axis, so a must be positive. We will use a 1.
The period of the graph is .
The key asymptote has shifted spaces to
the right , so the phase shift is .
Therefore, .
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The cotangent equation for this graph can be
written as
.
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It is important to be able to draw a tangent
graph when you are given the corresponding
equation. Consider the equation Begin by
looking at a, b, and c.
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The negative sign here means that the tangent
graph reflects or flips about the x-axis. The
graph will look like this.
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b 3
Use b to calculate the period. Remember that the
period is the distance between vertical
asymptotes.
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This phase shift means the key point has shifted
spaces to the right. Its x-coordinate is
. Also, notice that the key point is an
x-intercept.
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Since the key point, an x-intercept, is exactly
halfway between two vertical asymptotes, the
distance from this x-intercept to the vertical
asymptote on either side is equal to half of the
period.
The period is half of the period is
. Therefore, the distance between the
x-intercept and the asymptotes on either side is
.
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We can use half of the period to figure out the
labels for vertical asymptotes and x-intercepts
on the graph. Since we already determined that
there is an x-intercept at , we can add half
of the period to find the vertical asymptote to
the right of this x-intercept.
Vertical asymptote
x-intercept
Half of the period
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Continue to add or subtract half of the period,
, to determine the labels for additional
x-intercepts and vertical asymptotes.
x-intercept
Vertical asymptote
Half of the period
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It is important to be able to draw a cotangent
graph when you are given the corresponding
equation. Consider the equation Begin by
looking at a, b, and c.
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The positive sign here means that the cotangent
graph does not reflect or flip about the
x-axis. The graph will look like this.
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b 4
Use b to calculate the period. Remember that the
period is the distance between vertical
asymptotes.
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This phase shift means the key asymptote has
shifted spaces to the left. The equation for
this key asymptote is .
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The distance from an asymptote to the
x-intercepts on either side of it is equal to
half of the period.
The period is half of the period is
. Therefore, the distance between asymptotes
and their adjacent x-intercepts is . This
information can be used to label asymptotes and
x-intercepts.
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Sometimes a tangent or cotangent graph may be
shifted up or down. This is called a vertical
shift.
The equation for a tangent graph with a vertical
shift can be written as
y a tan b (x - c) d.
The equation for a cotangent graph with a
vertical shift can be written as
y a cot b (x - c) d.
In both of these equations, d represents the
vertical shift.
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• A good strategy for graphing a tangent or
  cotangent function that has a vertical shift
• Graph the function without the vertical shift
• Shift the graph up or down d units.
• Consider the graph for .
• The equation is in the form where
  d equals
• 3, so the vertical shift is 3.

The graph of was drawn in the
previous example.
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To draw , begin with the graph for
.
Draw a new horizontal axis at y 3. Then
shift the graph up 3 units.
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The graph now represents
.
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This concludesTangent and Cotangent Graphs.