How do you find the amplitude, period and phase shift for $y = 2 cos (x/2 - Pi/6)$ ?

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Answer 1 · 2015-07-29T15:09:44

$y = 2cos (x/2 - pi/6)$

Amplitude: (-2, 2)
Period of cos x is $2pi$ --> period of $x/2$ is $4pi.$
Phase shift: $-pi/6$

Answer 2 · 2015-07-29T15:34:13

There are several steps.

The form of this equation is

$y = Acos(Bx+C)" "$ or $" "y=Acos(Bx-C)$

(Some textbooks use the first, others use the second.)

The Amplitude is $abs(A)$

Period can be found by $(2pi)/B$

Phase Shift is found by solving:
$Bx+C = 0" "$ (or $Bx-C = 0" "$ depending on textbook.)

For $y = 2cos(x/2-pi/6)$ , note that we can write this as:

$y = 2cos(1/2x-pi/6)" "$ (so it is clear that $B = 1/2$ )

Amplitude: $" " 2$

Period: $" "4pi$

Found by simplifying $(2pi)/(1/2) = (2pi)/1 * 2/1 = 4pi$

Phase Shift: $" " pi/3$

Found by solving: $x/2-pi/6 = 0$ so $x/2 = pi/6$ and $x = (2pi)/6 = pi/3$

How do you find the amplitude, period and phase shift for y = 2 cos (x/2 - Pi/6)y = 2 cos (x/2 - Pi/6)?