Question

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

Answer 1

See explanation and graph

Explanation:

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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