Trig F07 O’Brien
How to Graph y  a csc (bx  c)  d and y  a sec (bx  c)  d
1.
Write the guide function. For y  a csc (bx  c)  d the guide function is y  a sin (bx  c)  d and
for y  a sec (bx  c)  d the guide function is y  a cos (bx  c)  d .
2.
Identify a, b, c, and d.
3a.
b.
Find the amplitude of the guide function: amp = |a|. Note: Cosecant and secant do not have amplitude.
Determine if there has been an x-axis reflection: if a is negative, there has been an x-axis reflection.
4a.
Find the period: period =
b.
5.
6.
7.
2π
.
b
1
 period .
4
Find the phase shift by setting the argument, bx – c, equal to zero and solving for x:
c
c
c
c
P.S. =
right (if we see bx – c and
is positive) or P.S. =
left (if we see bx + c and
is negative).
b
b
b
b
Find the vertical shift: |d| up if d is positive or |d| down if d is negative.
Find the x-increment: x-increment =
Identify the five key points of the guide function:
c 
for sine:  , d  (x1 + x-inc, d+a) (x2 + x-inc, d)
b 
c

for cosine:  , d  a 
b

(x1 + x-inc, d)
(x3 + x-inc, d – a)
(x2 + x-inc, d – a)
(x4 + x-inc, d)
(x3 + x-inc, d)
(x4 + x-inc, d + a)
8.
Plot the five key points of the guide function and connect them with a smooth, continuous, dotted curve, then
extend the graph to two full periods.
9.
Draw the given function using these guidelines:
Anywhere the guide function crosses its horizontal axis (i.e., when y = d), the given function has a vertical asymptote.
Anywhere the guide function has a minimum, the given function has a maximum and we draw a parabola shaped
curve opening down. Anywhere the guide function has a maximum, the given function has a minimum, and we draw
a parabola shaped curve opening up.
Example 1: y = 2 csc (x – 3π) – 1
1.
Guide function: y = 2 sin (x – 3π) – 1
2.
a=2
3.
amp of sine = |2| = 2
4.
period =
5.
phase shift = 3π or
6.
vertical shift = 1 down
7.
 6π

, 1
5 key points of the sine: 
2


b=1
c = 3π =
2π
 2π
1
6π
2
d = –1
There is no x-axis reflection.
x-increment =
6π
right
2
1
2π π
 2π 

4
4
2
x  3π  0  x  3π 
 7π 
 , 1
 2

 8π

 , 1
2


6π
2
 9π

 ,  3
2


 10π

, 1

2


1
Trig F07 O’Brien
8.
9.
Example 2: y = –
π
1

sec  x   + 3
2
4

π
1

cos  x   + 3
2
4

1.
Guide function: y = –
2.
a=–
3.
amp of cosine = 
4.
period =
5.
phase shift =
6.
vertical shift = 3 or
7.
 π 11 
5 key points of the cosine:   ,

 2 4
8.
1
4
b=1
c=–
π
2
1
1
=
4
4
2π
 2π
1
d=3=
There is an x-axis reflection.
x-increment =
π
left
2
12
4
x
1
2π π
 2π 

4
4
2
π
π
0  x 
2
2
12
up
4
 12 
 0,

4

 π 13 
 ,

2 4 
 2π 12 
,


4
 2
 3π 11 
 ,

 2 4
9.
2