6.2 Trig Functions
Amplitude, Period, &
Phase Shift
3 ways we can change our graphs
Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
1. The domain is the set of real numbers.
2. The range is the set of y values such that  1  y  1.
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
5. Each function cycles through all the values of the range
over an x-interval of 2 .
6. The cycle repeats itself indefinitely in both directions of the
x-axis.
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2
Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0

2
2
sin x
0
2
1
0
-1
0
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = sin x
y
3

2



1

2
2

3
2
2
5
2
x
1
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3
Graph of Tangent Function: Periodic
Vertical asymptotes
where cos θ = 0
sin 
tan  
cos 
tan θ
θ
tan θ
−π/2
−∞
−π/4
−1
0
0
π/4
1
π/2
∞
−3π/2
−π/2
0
π/2
One period: π
3π/2
θ
Graph of Cotangent Function: Periodic
Vertical asymptotes
where sin θ = 0
cos 
cot  
sin
cot θ
θ
tan θ
0
∞
π/4
1
π/2
0
3π/4
−1
π
−∞
−3π/2
-π
−π/2
π/2
π
3π/2
Cosecant is the reciprocal of sine
csc θ
−3π
Vertical asymptotes
where sin θ = 0
θ
0
−2π
−π
π
One period: 2π
2π
3π
sin θ
Secant is the reciprocal of cosine
Vertical asymptotes
where cos θ = 0
sec θ
θ
−3π
−2π
−π
0
π
One period: 2π
2π
3π
cos θ
Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [-π,4] on your x-axis
x
y = 3 cos x
y

(0, 3)
2
1

0
3
0

-3
x-int
min
2
max
3
2
0
2
3
x-int
max
(2, 3)

1 
( , 0)
2
2
3
( , –3)
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2
( 3 , 0)
2
3
4 x
8
The amplitude of y = a sin x (or y = a cos x) is half the distance
between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| < 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis.
y
4
y = 2sin x

2
y=
1
2

3
2
2
x
sin x
y = – 4 sin x
reflection of y = 4 sin x
y = sin x
y = 4 sin x
4
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9
The period of a function is the x interval needed for the
function to complete one cycle.
For k  0, the period of y = a sin kx is 2 .
k
For k  0, the period of y = a cos kx is also 2 .
k

For k  0, the period of y = a tan kx is
.
k

If k > 1, the graph of they function
is shrunk horizontally.
y  sin 2x
period: 2

period: 
y  sin x x




2
If 0 < k < 1, the graph of the function is stretched horizontally.
y
y  cos x
1
y  cos x
period: 2
2 
2
3
4

x
period: 4
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Use basic trigonometric identities to graph y = f (–x)
Example 1: Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in
the x-axis.
y = sin (–x)
y
Use the identity
sin (–x) = – sin x
y = sin x
x

2
Example 2: Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x.
y
Use the identity
x
cos (–x) = – cos x

2
y = cos (–x)
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11
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin kx with k > 0
y = 2 sin (–3x) = –2 sin 3x
Use the identity sin (– x) = – sin x:
2  2
period:
amplitude: |a| = |–2| = 2
=
3
k
Calculate the five key points.
x
0
y = –2 sin 3x

6
3
2
2
3
–2
0
2
0
(  , 2)
2
6


6
3
(0, 0)
2


0
y


(  ,-2)
2
2
3

2
5
6

x
(  , 0) 2
3
( , 0)
3
6
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The Graph of y = Asin(Kx - C)
The graph of y = A sin (Kx – C) is obtained by horizontally shifting the graph
of y = A sin Kx so that the starting point of the cycle is shifted from x = 0 to
x = -C/K. The number – C/K is called the phase shift.
y
amplitude = | A|
period = 2 /K.
y = A sin Kx
Amplitude: | A|
x
Starting point: x = -C/K
Period: 2/B
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13
Example
Determine the amplitude, period, and phase
shift of y = 2sin(3x-)
Solution:
asin( kx  c)
Amplitude = |A| = 2
period = 2/K = 2/3
phase shift = -C/K = /3 to the right

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14
Example cont.
• y = 2sin(3x- )
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15
asin( kx  c)  d
Amplitude

Period: 2π/k
Phase Shift:
-c/k
Vertical
Shift
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16
State the periods of each function:
1.

2.
1
y  cos x
2
y  tan 2x
4π or 720°
π/2 or 90°

17
State the phase shift of each function:
1.
y  tan(  45)
Right phase shift 45°

2.
y  sin( 2x 180)
Left phase shift -90°

18


State the amplitude, period, and phase shift of each function:
1.
y  4sin 
4.
A = 4, period = 360°,
Phase shift = 0°
2. y 10tan 4

A = NONE, period = 45°,
Phase shift = 0°

3. y  2cos2
A = 2, period = 180°,
Phase shift = 0°

y  4 sin

2
A = 4, period = 720°,
Phase shift = 0°
5.
y  tan(2x   )
A = NONE, period = 90°,
Phase shift = π/2 Right
6.
y  3cos(  90)
A = 3, period = 360°,
Phase shift = 90° Right
19
State the amplitude, period, and phase shift of each function:
1.


y 10sin 12   300
A = 10, period = 1080°,
Phase shift = 900° Right

2.

y  243sin(15  40)
A = 243, period = 24°,
Phase shift = 8/3°
20
Write an equation for each function described:
1.) a sine function with amplitude 7, period 225°, and
phase shift -90°
8

y  7sin   144
5

2.) a cosine function with amplitude 4, period 4π, and phase
shift π/2

1
 
y  4 cos   
2
4 
3.) a tangent function with period 180° and phase shift 25°

y  tan  25
21
Graph each function:
1.)
y  3sin 
2.)
y  cos(  30)

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