/*! 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 41 Sketch the graph of the function... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 41

Sketch the graph of the function. (Include two full periods.) $$y=\frac{1}{3} \cos x$$

Short Answer

Expert verified
The graph of \(y = \frac{1}{3}\cos x\) is a cosine wave with an amplitude of \(\frac{1}{3}\) and a period of \(2 \pi\). The graph completes two full periods at \(4\pi\).

Step by step solution

01

Identify the Amplitude of the Function

In \(y=a \cos x\), 'a' represents the amplitude of the function. It is the maximum distance from the maximum value \(a\) to the minimum value \(-a\). Looking at the function \(y=\frac{1}{3} \cos x\), it can be seen that the amplitude is \(\frac{1}{3}\).
02

Identify the Period of the Function

By default, the period of the standard cosine function \(\cos x\) is \(2\pi\). This means it repeats every \(2\pi\) units. The given function \(y=\frac{1}{3} \cos x\) has the same period because there are no transformations which would change the period.
03

Draw the Graph

Draw a graph of the function \(y=\frac{1}{3}\cos x\). Mark the maximum and minimum points on the y-axis as \(\frac{1}{3}\) and \(-\frac{1}{3}\) respectively. Also divide x-axis into four equal sections to indicate the essential points in the period, which are 0, \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\). Place the dots on the graph and connect the dots using a smooth curve. Remember to include two full periods of the function so continue the nodes and the graph for another period.

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