How do you use the amplitude and period to graph #y = 2 cos 3 (x - (pi/4))#?

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Answer 1 · 2018-08-14T10:46:29

See explanation and graph

In wave forms, the amplitude and period are the important

structural parameters of a wave,

Period gives the periodic pattern.

The amplitude decides the limits for the periodic rise to crest, level

and fall to trough level.

Here, the cosine wave equation is

#y = 2 cos (3 ( x - pi/4 )) in 2 [ -1, + 1 ] = [ - 2, + 2 ]#

The period #= (2pi)/3#

The amplitude = 2

There is no vertical shift. So, the axis is #y = 0#.

Phase shift #= pi/4#

Crest level: #y = 2#

Trough level: #y = - 2#

Periodic x-intercepts: x = {zero of cos ( 3 ( x - pi/4 ))}

#= 1/3 (2 k + 1 )pi/2 + 3/4pi = kpi/3 + 5/12pi = ( 4k + 5 )pi/12#,

#k = 0, +-1, +-2, +-3, ...#

See graph depicting all these aspects.
graph{(y-2 cos (3 ( x - pi/4 )))(y-2)(y+2)(x+pi/4)(x-5/12pi)=0[-4 4 -2 2]}

The period marked in the graph is #x in [ - pi/4, 5/12pi ], at

alternate zeros of y.

The graph is on uniform scale.

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