2. 2
Objectives
• I can determine amplitude,
period, and phase shifts of trig
functions
• I can write trig equations given
specific period, phase shift, and
amplitude.
4. 4
Radian versus Degree
• We will use the following to graph or write
equations:
– “x” represents radians
– “” represents degrees
– Example: sin x versus sin
5. 5
sin ( )
a b x ps d
Amplitude
Period:
2π/b Phase Shift:
Left (+)
Right (-)
Vertical Shift
Up (+)
Down (-)
7. 7
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 = 2 sin x
2
1
y = sin x
y = sin x
8. 8
y
x
2
sin x
y
period: 2
2
sin
y
period:
The period of a function is the x interval needed for the
function to complete one cycle.
For b 0, the period of y = a sin bx is .
b
2
For b 0, the period of y = a cos bx is also .
b
2
If 0 < b < 1, the graph of the function is stretched horizontally.
If b > 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
9. 9
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)
10. 10
Example
Determine the amplitude, period, and phase shift of
y = 2sin (3x - )
Solution:
First factor out the 3
y = 2 sin 3(x - /3)
Amplitude = |A| = 2
period = 2/B = 2/3
phase shift = C/B = /3 right
11. 11
Find Amplitude, Period, Phase Shift
• Amplitude (the # in front of the trig. Function
• Period (360 or 2 divided by B, the #after the trig function
but before the angle)
• Phase shift (the horizontal shift after the angle and inside
the parenthesis)
• y = 4sin y = 2cos1/2 y = sin (4x - )
Amplitude:
Phase shift:
Period:
4 2 1
NA NA )
(
4
Right
360
720
2
12. 12
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 [0, 2] into four equal parts. Find the five key
points; graph one cycle; then repeat the cycle over the interval.
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)
13. 13
Writing Equations
• Write an equation for a positive sine curve with an amplitude
of 3, period of 90 and Phase shift of 45 left.
• Amplitude goes in front of the trig. function, write the eq.so
far:
• y = 3sin
• period is 90. use P =
•
• rewrite the eq.
• y = 3 sin4
• 45 degrees left means +45
• Answer: y = 3sin4( + 45)
4
90
360
,
90
360
B
so
B
14. 14
Writing Equations
• Write an equation for a positive cosine curve with an
amplitude of 1/2, period of and Phase shift of right .
• Amplitude goes in front of the trig. function, write the eq.so
far:
• y = 1/2cos x
• period is /4. use P =
•
• rewrite the eq.
• y = 1/2cos 8x
• right is negative, put this phase shift inside the parenthesis
w/ opposite sign.
• Answer: y = 1/2cos8(x - )
8
4
1
2
,
4
2
B
so
B
4
![12
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 [0, 2] into four equal parts. Find the five key
points; graph one cycle; then repeat the cycle over the interval.
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)](https://image.slidesharecdn.com/triggraphamplitudeandperiod-250330054117-a8796bf5/85/trigonometry-amplitude-and-period-explanation-12-320.jpg)
