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Chapter 4: Problem 43

Sketch the graph of the function. (Include two full periods.) $$y=\cos \frac{x}{2}$$

Short Answer

Expert verified
The graph of the function \(y=\cos (\frac{x}{2})\) features a stretched cosine wave with a period of \(4\pi\), repeating this pattern for two full cycles in the sketch.

Step by step solution

01

Identification of Basic Function Pattern

First, identify the nature of the cosine function. A typical cosine function \(y=\cos(x)\) starts at a maximum, decreases to a minimum, and then increases back to the maximum. This repeats every \(2\pi\) units.
02

Determine Impact of Modification

Next, evaluate the effect of the \(\frac{x}{2}\) term. This modifies the period of the function, stretching it by a factor of 2. As such, instead of completing a full cycle every \(2\pi\) units, the function \(y=\cos(\frac{x}{2})\) completes a full cycle every \(4\pi\) units.
03

Sketch Preliminary Graph

Having determined the pattern of the basic function and the impact of the modification, draw the preliminary sketch. Begin at a maximum at x=0, descend to a minimum at x=2蟺, ascend to the maximum at x=4蟺 and continue this pattern for one more period up to x=8蟺.
04

Complete and Review Graph

Complete the sketch by continuing the pattern for a second full period. Confirm that the graph begins and ends at a maximum, decreases to a minimum halfway between these two points, and repeats this pattern every \(4\pi\) units.

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