1. Amplitude, Period, and Phase Shift

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.

3. Section 4.5: Figure 4.49, Key Points in the Sine and Cosine Curves

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. Amplitude
Period: 2π/b
Phase Shift: Left (+)
Right (-)
Vertical Shift
Up (+)
Down (-)

6. The Graph of y = AsinB(x - C)
The graph of y = A sin B(x – C) is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to x = C. The number C is called the phase shift.
amplitude = | A|
period = 2 /B. y
y = A sin Bx
Amplitude: | A| x
Starting point: x = C
Period: 2/B
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

7. If |a| > 1, the amplitude stretches the graph vertically.
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 x
y = 2 sin x
y = sin x
y = sin x
y = – 4 sin x
reflection of y = 4 sin x
y = 4 sin x
Amplitude

8. If b > 1, the graph of the function is shrunk horizontally.
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
For b  0, the period of y = a cos bx is also
If 0 < b < 1, the graph of the function is stretched horizontally. y x
period:
period: 2
If b > 1, the graph of the function is shrunk horizontally. y x
period: 4
period: 2
Period of a Function

9. 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 x
y = sin (–x)
Use the identity sin (–x) = – sin x
y = sin x
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 x
Use the identity cos (–x) = cos x
y = cos (–x)
y = cos (–x)
Graph y = f(-x)

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. 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:

12. 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 3 -3
y = 3 cos x 2  x y x
(0, 3)
( , 3)
( , 0)
( , 0)
( , –3)
Example: y = 3 cos x

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)

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 - )