Precalculus
HS Mathematics
Unit: 04 Lesson: 02
Graphing Sinusoidal Functions KEY
The following diagrams can be used to help you position the graphs of sinusoidal functions, based on
the values of a, b, c, and d in the functions.
f ( x )  a sin(b( x  c ))  d
Period:
P
f ( x )  a cos(b( x  c ))  d
2
b
Period:
P
2
b
y=d+a
y=d+a
a
a
y=d
y=d
y=d–a
y=d–a
x = c + 21 P
x = c + 21 P
x=c
x=c
x=c+P
x=c+P
Fill in the information, and graph each sinusoidal function.
1)
f ( x )  3 sin(2x )  1
d = -1
a=3
c=0
b=2
P=
Period: P = 
2
3
-1
-4
/2

0
2)
f ( x )  3 sin(x  4 )  5
d=5
a=3
c = /4
b=1
P = 2
Period: P = 2
8
3
5
2
/4
©2012, TESCCC
07/25/12
5/4
9/4
page 1 of 4
Precalculus
HS Mathematics
Unit: 04 Lesson: 02
Graphing Sinusoidal Functions KEY
3)
f ( x )  4 cos(21 ( x  ))  2
d=2
a=4
c=
b = 1/2
P = 4
Period: P = 4
6
4
2
-2

4)
f ( x )  5 cos(31 ( x  ))  2
d=2
a = -5
c = -
b = 1/3
3
5
P = 6
Period: P = 6
7
5
2
-3
-
2
5
Watch for negative a!
5)
f ( x )  3 sin(x )  6
d=6
a=3
c=0
b=
P=2
Period: P = 2
9
3
6
3
0
2
4
The x-axis is scaled in numeric radian form, i.e. /2 = 1.57,  ≈
3.14, 2 = 6.28.
©2012, TESCCC
07/25/12
page 2 of 4
Precalculus
HS Mathematics
Unit: 04 Lesson: 02
Graphing Sinusoidal Functions KEY
6)
f ( x )  6 cos(5 ( x  3))  2
d=2
a=6
c=3
b = /5
P = 10
Period: P = 10
8
6
2
-4
8
3
13
The x-axis is scaled in numeric radian form, i.e. /2 = 1.57,  ≈
3.14, 2 = 6.28.
7)
f ( x )  5 cos(23 ( x  0.5))  7
d=7
a=5
c = 0.5
P=3
b = 2/3
Period: P = 3
12
5
7
2
0.5
2
3.5
The x-axis is scaled in numeric radian form, i.e. /2 = 1.57,  ≈
3.14, 2 = 6.28.
8)
f ( x )  2.5 sin(4 ( x  2))  3.5
d = 3.5
a = -2.5
c = -2
b = /4
P=8
Period: P = 8
6
2.5
3.5
1
-2
The x-axis is scaled in numeric radian form, i.e. /2 = 1.57,  ≈
3.14, 2 = 6.28.
©2012, TESCCC
07/25/12
2
6
Watch for negative a!
page 3 of 4
Precalculus
HS Mathematics
Unit: 04 Lesson: 02
Graphing Sinusoidal Functions KEY
Determine two functions for each sinusoidal function based on the given graph or table.
9)
10)
The x-axis is scaled in numeric radian form,
i.e. /2 = 1.57,  ≈ 3.14, 2 = 6.28.
11)
The x-axis is scaled in numeric radian form,
i.e. /2 = 1.57,  ≈ 3.14, 2 = 6.28.
d = 35
Period = 6
d=5
Period = 4
a = 25
b = /3
a = 15
b = /2
Phase shift for cosine: c = 0.5
Phase shift for cosine: c = 2
 f(x) = 25 cos (/3 (x – 0.5)) + 35
 f(x) = 15 cos (/2 (x – 2)) + 5
Phase shift for sine: c = -1
Phase shift for sine: c = 1
 f(x) = 25 sin (/3 (x – -1)) + 35
 f(x) = 15 sin (/2 (x – 1)) + 5
x
/6
/3
/2
2/3
5/6

7/6
f(x)
2
8
2
-4
2
8
2
12)
x
-3
2
7
12
17
22
27
32
Maximum
Minimum
f(x)
11
7
3
7
11
7
3
7
Maximum
Minimum
d=2
Period = 2/3
d=7
Period = 20
a=6
b=3
a=4
b = /10
Phase shift for cosine: c = /3
Phase shift for cosine: c = -3
 f(x) = 6 cos (3 (x – /3)) + 2
 f(x) = 4 cos (/10 (x – -3)) + 7
Phase shift for sine: c = /6
Phase shift for sine: c = 12
 f(x) = 6 sin (3 (x – /6)) + 2
 f(x) = 4 sin (/10 (x – 12)) + 7
©2012, TESCCC
07/25/12
page 4 of 4