Phase shift and amplitude of trigonometric

Amplitude, Period, &
Phase Shift
6.2 Trig Functions
3 ways we can change our graphs

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
6. The cycle repeats itself indefinitely in both directions of the
x-axis.
Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
1. The domain is the set of real numbers.
5. Each function cycles through all the values of the range
over an x-interval of .

2
2. The range is the set of y values such that .
1
1 

 y

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.
0
-1
0
1
0
sin x
0
x
2

2
3

2

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
2
3


 2



2
2
3

2

2
5
1

1
x
y = sin x

Graph of Tangent Function: Periodic
θ tan θ
−π/2 −∞
−π/4 −1
0 0
π/4 1
π/2 ∞
0 θ
tan θ
−π/2 π/2
One period: π
3π/2
−3π/2
Vertical asymptotes
where cos θ = 0



cos
sin
tan 

Graph of Cotangent Function: Periodic
θ tan θ
0 ∞
π/4 1
π/2 0
3π/4 −1
π −∞
3π/2
−3π/2
Vertical asymptotes
where sin θ = 0



sin
cos
cot 
π
-π −π/2 π/2
cot θ

Cosecant is the reciprocal of sine
One period: 2π
π 2π 3π
0
−π
−2π
−3π
Vertical asymptotes
where sin θ = 0
θ
csc θ
sin θ

Secant is the reciprocal of cosine
One period: 2π
π 3π
−2π 2π
−π
−3π 0
θ
sec θ
cos θ
Vertical asymptotes
where cos θ = 0

y
1
1

2

3

2
x
 
3

2

 
4
Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [-π,4] on your x-axis
max
x-int
min
x-int
max
3
0
-3
0
3
y = 3 cos x
2

0
x 2

2
3
(0, 3)
2
3
( , 0)
( , 0)
2


2
( , 3)

( , –3)

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.
2
3
2

4
y
x
4


2

y = –4 sin x
reflection of y = 4 sin x y = 4 sin x
y = 2sin x
2
1
y = sin x
y = sin x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
y
x

 
2

sin x
y 

period: 2

period:
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 .
For k  0, the period of y = a cos kx is also .
If 0 < k < 1, the graph of the function is stretched horizontally.
If k > 1, the graph of the function is shrunk horizontally.
y
x

 
2
 
3 
4
cos x
y 

period: 2
2
1
cos x
y 

period: 4
For k  0, the period of y = a tan kx is .

y
x

2

y = cos (–x)
Use basic trigonometric identities to graph y = f(–x)
Example 1: Sketch the graph of y = sin(–x).
Use the identity
sin(–x) = – sin x
The graph of y = sin(–x) is the graph of y = sin x reflected in
the x-axis.
Example 2: Sketch the graph of y = cos(–x).
Use the identity
cos(–x) = – cos x
The graph of y = cos (–x) is identical to the graph of y = cos x.
y
x

2

y = sin x
y = sin(–x)
y = cos (–x)

2
y
2

6

x
2


6
5
3

3
2
6

6

3

2

3
2
0
2
0
–2
0
y = –2 sin 3x
0
x
Example: Sketch the graph of y = 2 sin(–3x).
Rewrite the function in the form y = a sin kx with k > 0
amplitude: |a| = |–2| = 2
Calculate the five key points.
(0, 0) ( , 0)
3

( , 2)
2

( ,-2)
6

( , 0)
3
2
Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x
period: 2 
2
3
=

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.
amplitude = | A|
period = 2 /K.
x
y
Amplitude: | A|
Period: 2/B
y = A sin Kx
Starting point: x = -C/K
The Graph of y = Asin(Kx - C)

Example
Determine the amplitude, period, and phase shift of y = 2sin(3x-)
Solution:
Amplitude = |A| = 2
period = 2/K = 2/3
phase shift = -C/K = /3 to the right

Example cont.
 y = 2sin(3x- )

Amplitude
Period: 2π/k
Phase Shift:
-c/k
Vertical
Shift

17
State the periods of each function:
1.
2.
4π or 720°
π/2 or 90°

18
State the phase shift of each function:
1.
2.
Right phase shift 45°
Left phase shift -90°

19
State the amplitude, period, and phase shift of each function:
1.
2.
3.
4.
5.
6.
A = 4, period = 360°,
Phase shift = 0°
A = NONE, period = 45°,
Phase shift = 0°
A = 2, period = 180°,
Phase shift = 0°
A = 4, period = 720°,
Phase shift = 0°
A = NONE, period = 90°,
Phase shift = π/2 Right
A = 3, period = 360°,
Phase shift = 90° Right

20
State the amplitude, period, and phase shift of each function:
1.
2.
A = 10, period = 1080°,
Phase shift = 900° Right
A = 243, period = 24°,
Phase shift = 8/3°

21
Write an equation for each function described:
1.) a sine function with amplitude 7, period 225°, and
phase shift -90°
2.) a cosine function with amplitude 4, period 4π, and phase
shift π/2
3.) a tangent function with period 180° and phase shift 25°

Graph each function:
1.) 2.)

Phase shift and amplitude of trigonometric

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