Examples of Graphs of y = a sin(bx + c)
Consider y = a sin(bx + c), with b > 0.
• Amplitude: A = |a|
• Period: T =
2π
b
• Phase shift: s = −
c
b
Note that s > 0 means a shift to the right and s < 0 means a shift to the left.
Example 1. Consider y = 5 sin(2x + π3 ).
1. The amplitude is A = 5.
2. The period is T =
2π
2
= π.
π
3. The phase shift is s = − π/3
2 = −6.
Let’s sketch the graph over one period.
y
5
−16
1
12
π
π
1
3
−5
Gilles Cazelais. Typeset with LATEX on October 3, 2011.
π
7
12
π
5
6
π
x
Example 2. Consider y = −2 sin(3x − π4 ).
1. The amplitude is A = | − 2| = 2.
2. The period is T =
2π
3 .
3. The phase shift is s = − −π/4
=
3
π
12 .
Let’s sketch the graph over one period. Observe that since a < 0, the graph of the sine
curve is reversed.
y
2
1
12
π
1
4
π
5
12
7
12
π
π
3
4
π
x
−2
Example 3. Let’s find an equation of the form y = a sin(bx + c) for the following curve.
y
3
1
−20
9
20
π
π
x
−3
By inspection we see that a = 3. The period is T =
9π
20
π
− (− 20
) = π2 . Then,
2π
π
=
=⇒ b = 4.
b
2
π
The phase shift is s = − 20
. Then,
c
π
π
π
π
− =−
=⇒ c =
b = (4) = .
b
20
20
20
5
π
An equation for the curve is then: y = 3 sin 4x +
.
5