# d c) (bx ```Trig F07 O’Brien
How to Graph y  a sin(bx  c)  d and y  a cos (bx - c)  d
1.
Identify a, b, c, and d.
2a.
b.
Find the amplitude: amp = |a|.
Determine if there has been an x-axis reflection: if a is negative, there has been an x-axis reflection.
3a.
Find the period: period =
b.
4.
5.
6.
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:
c 
for sine:  , d  (x1 + x-inc, d+a)
b 
c

for cosine:  , d  a 
b

7.
2π
.
b
(x2 + x-inc, d)
(x3 + x-inc, d – a)
(x2 + x-inc, d – a)
(x1 + x-inc, d)
(x4 + x-inc, d)
(x3 + x-inc, d)
(x4 + x-inc, d + a)
Plot the five key points and connect them with a smooth, continuous curve, then extend the graph to two full periods.
π

Example 1: y  2 sin x    1
3

1.
a = –2
2.
amp = |–2| = 2
3.
period = 2π
b=1
c=
π
3
d=1
There is an x-axis reflection.
x-increment =
1
2π π 3π
 2π 
 
4
4
2
6
Note: To make graphing the phase shift easier, we will use an x-increment of
π 2π
or
right
3
6
4.
phase shift =
5.
vertical shift = 1 up
6.
 2π 
5 key points:  , 1
 6

 5π

 , - 1
 6

x
π
.
6
π
π 2π
0  x  
3
3
6
 8π 
 , 1
 6

 11π 
, 3

 6

 14π 
, 1

 6

7.
1
Trig F07 O’Brien
Example 2: y =
1
cos (4x + π) – 2
3
1.
a=
1
3
b=4
2.
amp =
1
1
=
3
3
3.
period =
4.
phase shift =
5.
vertical shift = 2 or
6.
5
 2π
,  
5 key points:  
3
 8
c = –π
d = –2 = –
6
3
There is no x-axis reflection.
2π π

4
2
x-increment =
π
2π
or
left
8
4
1 π π
 
4 2 8
4x  π  0  4x   π  x  
π
2π

4
8
6
down
3
6
 π
 ,  
3
 8
7

 0,  
3

6
π
 ,  
3
8
5
 2π
,  

3
 8
7.
2
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