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

Aug 14, 2018

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

$y = 2 \cos \left(3 \left(x - \frac{\pi}{4}\right)\right) \in 2 \left[- 1 , + 1\right] = \left[- 2 , + 2\right]$

The period $= \frac{2 \pi}{3}$

The amplitude = 2

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

Phase shift $= \frac{\pi}{4}$

Crest level: $y = 2$

Trough level: $y = - 2$

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

$= \frac{1}{3} \left(2 k + 1\right) \frac{\pi}{2} + \frac{3}{4} \pi = k \frac{\pi}{3} + \frac{5}{12} \pi = \left(4 k + 5\right) \frac{\pi}{12}$,

$k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

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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