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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 . xcos
xsinxtan
Notice that the denominator is cos x. This indicates a relationship between a tangent graph and a cosine graph.
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This is the graph for y = cos x.
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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 . xsin
xcosxcot
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.
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For tangent and cotangent graphs, the distance between any two consecutive vertical asymptotes represents one complete period.
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y = tan x.
y = cot x.
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One complete period is highlighted on each of these graphs.
For both y = tan x and y = cot x, the period is π. (From the beginning of a cycle to the end of that cycle, the distance along the x-axis is π.)
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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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y = a tan b (x - c).
For a tangent graph which has no vertical shift, the equation for the graph
can be written as
For a cotangent graph which has no vertical shift, the equation for the graph
can be written as
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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y = a tan b (x - c) y = a cot b (x - c)
Unlike sine or cosine graphs, tangent and cotangent graphs have no maximum or minimum values. Their range is (-∞, ∞), 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.
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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 - ∞.
As each section nears the asymptote on its right, the y-values approach + ∞.
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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 + ∞.
As each section nears the asymptote on its right, the y-values approach - ∞.
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Notice the behavior of y = cot x.
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This is the graph for y = tan x.
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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.
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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
.b
period
Conversely, when you already know the period of a tangent or cotangent graph, b can be determined by
.period
b
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A complete period (including two consecutive vertical asymptotes) has been highlighted on the tangent graph below.
The distance between the asymptotes in this graph is .
Therefore, the period of this graph is also .
3x
3x
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For all tangent graphs, the period is equal to the distance between any two consecutive vertical asymptotes.
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period
b
We will let a = 1, but a could be any positive value since the graph has not been reflected about the x-axis.
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2Use , the period of this tangent graph, to calculate b.
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31 ba
An equation for this graph can be written as xy2
3tan1
or . xy2
3tan
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A complete period (including two consecutive vertical asymptotes) has been highlighted on the cotangent graph below.
The distance between the asymptotes is .
Therefore, the period of this graph is also .
0x 4x
4
For all cotangent graphs, the period is equal to the distance between any two consecutive vertical asymptotes.
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864202468
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.4
1
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period
b
We will let a = 1, but a could be any positive value since the graph has not been reflected about the x-axis.
4Use , the period of this cotangent graph, to calculate b.
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11 ba
An equation for this graph can be written as
or .
xy4
1cot1
864202468
xy4
1cot
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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.
We designated the vertical asymptote on the y-axis (at x = 0) as the key asymptote.
x = 0
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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.
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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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2
2
tan xy
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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.
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For this graph, c = because the key asymptote shifted left to .
An equation for this graph can be written as or
2
2
2cot xy
.2
cot
xy
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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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periodb
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041 cba
The tangent equation for this graph can be written
as or .)0(4tan1 xy xy 4tan
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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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periodb
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c
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841
cba
The cotangent equation for this graph can be written
as .
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4cot xy
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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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3tan3
2
xy
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2 cba
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.6
3tan3
2
xy
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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.6
3tan3
2
xy
b = 3
3
bperiod
Use b to calculate the period. Remember that the period is the distance
between vertical asymptotes.
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.6
3tan3
2
xy
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c
This phase shift means the key point has shifted spaces
to the right. It’s x-coordinate is . Also, notice that the key point is an x-intercept.
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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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3tan3
2
xy
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.
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.6
3tan3
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xy
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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. 6
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x-intercept
Half of the period
Vertical asymptote
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.6
3tan3
2
xy
Continue to add or subtract half of the period, , to determine the
labels for additional x-intercepts and vertical asymptotes. 6
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Vertical asymptoteHalf of the period
x-intercept
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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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x4cot3y
8c4b3a
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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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x4cot3y
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b = 4
4bperiod
Use b to calculate the period. Remember that the period is the distance
between vertical asymptotes.
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x4cot3y
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8c
This phase shift means the key asymptote has shifted
spaces to the left. The equation for this key asymptote is
.
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8x
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x4cot3y
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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.
The distance from an asymptote to the x-intercepts on either side of it is equal to half of the period.
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Sometimes a tangent or cotangent graph may be shifted up or down. This is called a vertical shift.
y = a tan b (x - c) +d.
The equation for a tangent graph with a vertical shift can be written as
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.
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x4cot3y
dcxbcotay
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x4cot3yThe 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.3
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The graph now represents .
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This concludesTangent and Cotangent
Graphs.