Dr. Neal, WKU
MATH 117
Sine/Cosine Graphs
Each trigonometric function can be graphed as a function of x , where the variable x is
always in radians. In particular, the functions y = sin x and y = cos x are defined for all x
and are cyclic. That is, the shape of the graph repeats itself over the periods
. . . . !4" # x # !2"
!2" # x # 0
0 ! x ! 2"
4! " x " 6! . . . etc.
2! " x " 4!
which are simply wrap-arounds of the unit circle.
The Sine Graph
One cycle of the sine graph occurs as x ranges from 0 to 2π (radians), and the shape of
!
3!
the graph can be molded around the values occuring at x = 0, , π,
, and 2π.
2
2
x
0
!
2
y = sinx
0
1
!
3!
2
2!
0
–1
0
1
1
!
2
!
3!
2
2!
–2!
2!
–1
–1
One Cycle of y = sinx
Two Cycles
Multiplying by a negative reflects the graph about the x -axis and multiplying by a
constant “ a ” increases the range to – a ≤ y ≤ a rather than –1 ≤ y ≤ 1.
4
3
2!
!
!/2
–3
–4
One Cycle of y = 4sinx
Amplitude = 4
One Cycle of y = !3 sinx
Amplitude = |–3| = 3
2!
Dr. Neal, WKU
Phase Shift
Given the graph of y = f (x) , we can shift the graph to the right by c units with the
function y = f (x ! c) . We shift to the left c units with the function y = f (x + c) .
3
y = x2
–2
y = (x ! 3)2 shifts x
to the right by 3
2
y = (x + 2)2 shifts x
to the left by 2
2
Similarly, we can shift the sine graph to the left or right by a certain angle with the
function y = a sin(x ± c) .
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
Example 1. Graph one cycle each of the following functions. Label five points on the x axis that show where the roots and the peaks occur.
"
!%
(a) y = 2 sin $ x + '
#
3&
#
"&
(b) y = !3 sin% x ! (
$
4'
"
, where one cycle begins. One cycle ends
3
"
" 6" 5"
after length 2! which is at ! + 2" = ! +
. So now divide the cycle length of
=
3
3 3
3
2! !
!
2! into 4 equal pieces of length
from the
= . To do so, we must add lengths of
4
2
2
"
2"
3!
starting point of ! . That is, start at !
and start adding
.
3
6
6
Solution. (a) The sine graph is shifted to !
2"
6
"
2"
! =!
3
6
Start at !
Add
3!
6
3!
6
4! 2 !
=
6
3
Add
!
6
Add
7!
6
3!
6
2
2
"
!
!
2!
7!
5!
3
6
3
6
3
–2
–2
"
!%
y = 2 sin $ x + '
#
3&
3!
6
10 ! 5!
=
6
3
Add
Dr. Neal, WKU
!
, where one cycle begins. Now divide the
4
2!
2!
cycle length of 2! into 4 equal pieces of length
, and add these lengths of
from
4
4
!
the starting point of .
4
(b) The negative sine graph is shifted to +
Start at
!
4
Add
3!
4
!
4
2!
4
Add
5!
4
2!
4
2!
4
7!
4
2!
4
9!
4
Add
Add
3
!
3!
5!
7!
9!
4
4
4
4
4
–3
#
"&
y = !3 sin% x ! (
$
4'
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
The Cosine Graph
One cycle of the cosine graph occurs as x ranges from 0 to 2π (radians), and the shape
!
3!
of the graph can be molded around the values occuring at x = 0, , π,
, and 2π.
2
2
x
0
!
2
y = cosx
1
0
!
3!
2
2!
–1
0
1
1
1
!
2
7!
2
!
3!
2
–1
2!
"
!
2
!
2!
–1
One Cycle of y = cosx
Two Complete Cycles
Dr. Neal, WKU
5
!
"
2
5"
2
!
!
–2
2!
–5
One Cycle of y = !5 cos x
Amplitude = |–5| = 5
y = 2cos x
Amplitude = 2
Example 2. Graph one cycle each of the following functions. Label five points on the x axis that show where the roots and the peaks occur.
#
"&
(a) y = 3 cos% x ! (
$
6'
#
(b) y = !4 cos% x +
$
"&
(
8'
!
, where one cycle begins. Divide the
6
2! 3!
3!
cycle length of 2! into 4 equal pieces of length
, and add these lengths of
=
4
6
6
!
from the starting point of .
6
Solution. (a) The cosine graph is shifted to +
Start at
!
6
!
6
3!
6
4! 2 !
=
6
3
Add
Add
7!
6
3!
6
3!
6
10 ! 5!
=
6
3
Add
3
!
2!
7!
5!
13!
6
3
6
3
6
–3
#
"&
y = 3 cos% x ! (
$
6'
3!
6
13!
6
Add
Dr. Neal, WKU
"
, where one cycle begins. Divide the
8
2! 4!
4!
cycle length of 2! into 4 equal pieces of length
, and add these lengths of
=
4
8
8
"
from the starting point of ! .
8
(b)
The negative cosine graph is shifted to !
Start at !
!
"
8
"
8
Add
3!
8
4!
8
4!
8
7!
8
4!
8
11!
8
Add
Add
4
!
"
3"
7"
11"
15 "
8
8
8
8
8
–4
#
y = !4 cos% x +
$
"&
(
8'
4!
8
15!
8
Add