Amplitude, Period, Phase Shift - Sec. 6.2
The amplitude of the functions y = Asin x and y = Acos x is the absolute value
of A, or A .
Example 1: State the amplitude of the function y = 3cos x.
Amplitude = 3
We will graph this function in class and compare it to the graph
of y = cos x.
The period of the functions y = sin kx and y = cos kx is
the function y = tan kx is
180˚
.
k
360˚
. The period of
k
Example 2: State the period of the function y = sin 4x.
period =
360˚
= 90˚
4
We will graph this function in class and compare it to the graph
of y = sin x.
c
k
The phase shift of the function y = Asin(kx + c) is − . If c > 0, the shift is
to the left. If c < 0, the shift is to the right. This definition applies to all
of the trig fuctions.
Example 3: State the phase shift of the function y = tan(x - 45˚).
phase shift = −
c
−45˚
=−
= 45˚
k
1
Since c < 0, the shift is the the right.
We will graph this function in class and compare it to the graph
of y = tan x.
Example 4: State the amplitude, period, and phase shift for the function
y = -4sin(2x - 90˚).
amplitude = −4 = 4
period =
360˚
= 90˚
4
phase shift = −
c
−90˚
=−
= 45˚ (shift is to the right)
k
2
Example 5: Find the amplitude, period, and phase shift of the function
y = 2tan(3x + 270˚)
amplitude = none (the tan and cot graphs have no amplitude)
180˚
= 60˚
3
c
270˚
phase shift = − = −
= −90˚ (shift is to the left)
k
3
period =
We can write an equation for a trig function if we are given the amplitude,
period, and phase shift.
Example 6: Find the possible equations of a cosine function with amplitude 3,
period 90˚, and phase shift 45˚.
The form of the equation will be y = Acos(kx + c).
First find A:
amplitude = 3 means that A = 3, so A = 3 or -3.
Now, find k when the period is 90˚.
period =
360˚
k
360˚
= 90˚
k
90˚k = 360˚
k= 4
Then, find c for a phase shift of 45˚.
phase shift = −
c
= 45˚
k
c
− = 45˚
4
c = −180˚
c
k
−
The possible equations are:
y = 3cos(4x - 180˚) or y = -3cos(4x - 180˚)