/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 59 Sketch the graph of the function... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 59

Sketch the graph of the function. (Include two full periods.) $$y=\frac{2}{3} \cos \left(\frac{x}{2}-\frac{\pi}{4}\right)$$

Short Answer

Expert verified
The graph of the function \(y=\frac{2}{3} \cos \left(\frac{x}{2}-\frac{\pi}{4}\right)\) has the following characteristics: it oscillates between \(\frac{2}{3}\) and \(-\frac{2}{3}\), has a period of \(\pi\), and phase shift of \(\frac{\pi}{4}\) to the right. The graph resembles the wave of a normal cosine function but is stretched vertically by a factor of \(\frac{2}{3}\), compressed horizontally by a factor of 2, and shifted to the right by \(\frac{\pi}{4}\) units.

Step by step solution

01

鈥� Identify all transformations

Start off by identifying the transformations applied to the base cosine function. In this case, \(y=\frac{2}{3} \cos \left(\frac{x}{2}-\frac{\pi}{4}\right)\) has all of the following transformations: \n\n1. Vertical Stretch: It's seen from the factor \(\frac{2}{3}\) multiplied to the cosine function. This stretches or dilates the graph vertically by a factor of \(\frac{2}{3}\). \n2. Horizontal Compression: The term \(\frac{x}{2}\) within the cosine function compresses the graph horizontally by a factor of 2. This halves the period of the cosine function from \(2\pi\) to \(\pi\).\n3. Horizontal Shift: The term \(\frac{\pi}{4}\) within the cosine function shifts the graph to the right by \(\frac{\pi}{4}\) units.
02

鈥� Determine the amplitude, period, and phase shift

The amplitude of the function is the absolute value of the coefficient of the cosine term which is \(\left|\frac{2}{3}\right|=\frac{2}{3}\). The period is determined as \(T=\frac{2\pi}{|b|} = \frac{2\pi}{1/2}=\pi\). The phase shift is \(\frac{\pi}{4}\) to the right.
03

鈥� Plot the function

Now it's time to plot the function. Start with the phase shift by marking a point at \(x=\frac{\pi}{4}\) on the x-axis. Then, determine the new period and mark the x-values for one complete cycle of the function starting from the phase shift. Draw the cosine wave keeping in mind the amplitude of \(\frac{2}{3}\). Finally, repeat this process to draw two full periods of the function.

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