Lesson 4-6
Graphs of Secant and
Cosecant
Get out your graphing calculator…
Graph the following
y = cos x
y = sec x
What do you see??
2
Graph of the Secant Function
1
sec
x

The graph y = sec x, use the identity
.
cos x
At values of x for which cos x = 0, the secant function is undefined
and its graph has vertical asymptotes.
y
y  sec x
Properties of y = sec x
1. domain : all real x

x  k  (k  )
2
2. range: (–,–1]  [1, +)
3. period: 2
4. vertical asymptotes:

x  k  k   
2
4
y  cos x
x



2
2

3
2
2
5
2
3
4
3
First graph:
• y = 2cos (2x – π) + 1
Then try:
• y = 2sec (2x – π) + 1
4
Graph
Graph the following
y = sin x
y = csc x
What do you see??
5
Graph of the Cosecant Function
1
To graph y = csc x, use the identity csc x 
.
sin x
At values of x for which sin x = 0, the cosecant function
is undefined and its graph has vertical asymptotes.
y
Properties of y = csc x
4
y  csc x
1. domain : all real x
x  k k  
2. range: (–,–1]  [1, +)
3. period: 2
4. vertical asymptotes:
x  k k  
where sine is zero.
x



2
2

3 2
2
5
2
y  sin x
4
6
First graph:
• y = -3 sin (½x + π/2) – 1
Then try:
• y = -3 csc (½x + π/2) – 1
7
Key Steps in Graphing Secant
and Cosecant
1.
2.
3.
4.
5.
6.
7.
8.
Identify the key points of your reciprocal graph (sine/cosine),
note the original zeros, maximums and minimums
Find the new period (2π/b)
Find the new beginning (bx - c = 0)
Find the new end (bx - c = 2π)
Find the new interval (new period / 4) to divide the new
reference period into 4 equal parts to create new x values for
the key points
Adjust the y values of the key points by applying the change in
height (a) and the vertical shift (d)
Using the original zeros, draw asymptotes, maximums become
minimums, minimums become maximums…
Graph key points and connect the dots based upon known
shape
8
Graphs of Tangent and
Cotangent Functions
Tangent and Cotangent
Look at:
Shape
Key points
Key features
Transformations
10
Graph
Set window
Domain: -2π to 2π
x-intervals: π/2
(leave y range)
Graph
y = tan x
11
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

x  n  n   
2
2. range: (–, +)
3. period: 
4. vertical asymptotes:

x  n  n   
2

2
 3
2
3
2
x

2
period: 
12
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?
13
Graph
Set window
Domain: 0 to 2π
x-intervals: π/2
(leave y range)
Graph
y = cot x
14
Graph of the Cotangent Function
cos x
To graph y = cot x, use the identity cot x 
.
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  n n  
2. range: (–, +)
3. period: 
4. vertical asymptotes:
x  n n  
vertical asymptotes

3
2
 

2
x  
x0

 3
2
2
x 
x
2
x  2
15
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
16
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
17