PreCalculus Class Notes TFG5 Sine and Cosine Graphs:
Amplitude, Frequency, Period, Vertical Shift
Basic Graphs of Sine and Cosine
GSP Trigonometry Tracers: Sin/Cos tab
y = cos x
y = sin x
y
y
1
−2π 3π
2
−π
1
π
2
π
2
π
3π
2
−2π 3π
2π x
π
2
π
2
−π
2
−1
π
3π
2
2π x
−1
domain: all real numbers
range: –1 ≤ y ≤ 1
domain: all real numbers
range: –1 ≤ y ≤ 1
GSP Trig6abcd, select sin or cos, move a, b, d only
Reflections
y
y
1
Reflection over
the x-axis
1
O
π
2π
x
O
π
−1
−1
y = −sinx
x
y = −cosx
y
y
1
2
O
Reflection over
the y-axis
2π
π
−1
2π
x
1
O
π
y = −sinx
−1
−2
Odd function
y = cosx
Even function
2π
x
Basic Transformations
The general form of a sine or cosine function is y = a sin ( bx ) + d or y = a cos ( bx ) + d .
y
y
2
1
Vertical dilation
(a)
O
|a|
π
O
π
x
2π
x
2π
−1
−2
y = −2cosx
|a|
y = asinx
y
Maximum
Maximum
y
1
1
Zero
Zero
Horizontal
dilation
(b)
O
O
2π
b
Zero
2π
b
x
Zero
Zero
Minimum
−1
−1
Minimum
y = sinbx
x
y = cosbx
y
6
y = cos(x)-3y
5
x
π/2
4
2
−1
Vertical shift
(d)
3
−2
2
2
−3
1
O
−4
π
y = 2sinx + 3
2π
x
π
3π/2
2π
y
Period =
2π
b
|a|
d
O
x
y = asin(bx) + d
y
Period =
2π
b
|a|
d
x
O
y = acos(bx) + d
Characteristics of Sine and Cosine Graphs
amplitude
|a|
How far above and below the midline the graph extends
(if a < 0, the graph is reflected over its midline).
angular frequency
b
How many complete cycles of the graph occur in any interval
of length 2π.
period
2π
b
interval between
key points
Period
4
The horizontal distance between the maximum, midline, and
minimum points.
vertical shift
d
How far the midline of the graph has been shifted up or down
The length of one complete cycle (period) of the graph.
Example: State the amplitude, frequency, period and vertical shift. Then sketch each graph.
y = 5 + 3sin ( 2 x )
y = −2 cos 4 x + 1
Amplitude
Frequency
Period
Vertical shift
Key points
1
⋅ period
4
Sketch y = 5 + 3sin ( 2 x )
8
y
7
6
5
4
3
2
1
O
−1
π
2π
4
4
3π
4
4π
4
5π
4
6π
4
7π
4
8π
4
x
Sketch y = −2 cos 4 x + 1
y
3
2
1
O
−1
π
8
2π
8
3π
8
4π
8
5π
8
6π
8
7π
8
π
x
Write an equation of the function shown in the graph below.
y
5
0
a
5
10
Period
x
b
d
Write an equation of the function shown in the graph below.
20
15
10
5
−3π
a
−2π
−π
Period
O
−5
−10
−15
−20
−25
−30
y
π
2π
3π
b
x
d