Amplitude, Phase Shift and Period

Q:  Consider the function y = -5 cos (2x – .1*pi)

a)  Identify the amplitude

b)  Identify the phase shift

c)  Identify the period

Answer:

The following formulas / concepts will work for sin and cos graphs:

y = a cos (b*(x – h) ) + k

Note:  a, b, h and k are just numbers that will affect the graph.

The amplitude is |a|

The phase shift if 2*pi / |b|

The phase shift is “h”

The vertical shift is “k”

A big thing to notice:  the “b” value is factored out in this formula.

So…. Let’s start with our actual example: y = -5 cos (2x – .1*pi)

Identify who is a, b, h, and k.  Notice, the b is not factored out, so let’s do that firt:

y = -5 cos (2x – .1*pi)

y = -5 cos (2*(x – .05*pi))

[if you multiply the 2 back through, you get the same as the original]

Now,

a = -5

b = 2

h = .05*pi

k = 0 (there is no + k at the end of the problem)

a)  The amplitude is |a| = |-5| = 5

b)  The phase shift is h = .05*pi (the graph is shifted .05*pi units to the right)

c)  The period is 2*pi/|b| = 2*pi/|2| = pi