Chapter 1.3, Problem 24E

To determine

To sketch: The function $y=\cos x$ by hand without plotting any points and then graph the function $y=\left|\cos \pi x\right|$.

Explanation of Solution

The standard graph of the function $y=\cos x$ is roughly drawn and shown below in Figure 1.

Then, draw the graph of $y=\cos \pi x$ by compressing the graph $y=\cos x$ horizontally by $\pi$ units. Thus, the graph of $y=\cos \pi x$ is shown below in Figure 2.

From Figure 2, it is observed that Figure 1 is shrunken horizontally by a factor of $\pi$ units.

Now, to draw the graph of $y=\left|\cos \pi x\right|$, reflect the entire part of the graph $y=\cos \pi x$ that appears below the x-axis about the origin. Thus, the graph of $y=\left|\cos \pi x\right|$ is shown below in Figure 3.

Observe that Figure 3 is obtained from Figure 1 in such a way that it is horizontally compressed by a factor of $\pi$ units and then reflected the portions appeared below the x-axis about the x-axis.