Graphs of Tangent and
Cotangent Functions
Tangent and Cotangent
Look at:
¾Shape
¾Key points
¾Key features
¾Transformations
2
Graph
Set window
Domain: -2π to 2π
x-intervals: π/2
(leave y range)
Graph
y = tan x
3
Graph of the Tangent Function
sin x
To graph y = tan x, use the identity tan x =
.
cos x
At values of x for which cos x = 0, the tangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = tan x
1. domain : all real x
π
x ≠ nπ + (n ∈ Ζ )
π
2
2. range: (–∞, +∞)
3. period: π
4. vertical asymptotes:
π
x = nπ + (n ∈ Ζ )
2
− 3π
2
3π
2
x
−π
2
2
period: π
4
Graph
y = tan x and y = 4tan x in the same window
What do you notice?
y = tan x and y = tan 2x
What do you notice?
y = tan x and y = -tan x
What do you notice?
5
Graph
Set window
Domain: 0 to 2π
x-intervals: π/2
(leave y range)
Graph
y = cot x
6
Graph of the Cotangent Function
cos x
cot
x
=
To graph y = cot x, use the identity
.
sin x
At values of x for which sin x = 0, the cotangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = cot x
y = cot x
1. domain : all real x
x ≠ nπ (n ∈ Ζ )
2. range: (–∞, +∞)
3. period: π
4. vertical asymptotes:
x = nπ (n ∈ Ζ )
vertical asymptotes
x
3π
−
2
−π π
−
2
x = −π
π
π
2
x=0
x =π
3π
2
2π
x = 2π
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Graph Cotangent
y = cot x and y = 4cot x in the same window
What do you notice?
y = cot x and y = cot 2x
What do you notice?
y = cot x and y = -cot x
What do you notice?
y= cot x and y = -tan x
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Key Steps in Graphing Tangent
and Cotangent
Identify the key points of your basic graph
1. Find the new period (π/b)
2. Find the new beginning (bx - c = 0)
3. Find the new end (bx - c = π)
4. Find the new interval (new period / 2) to divide
the new reference period into 2 equal parts to
create new x values for the key points
5. Adjust the y values of the key points by applying
the amplitude (a) and the vertical shift (d)
6. Graph key points and connect the dots
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