Chapter 1.3, Problem 18E

To determine

To sketch: The function $y=\cos x$ by hand without plotting any points and then graph of the function $y=3-2\cos x$.

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 $y=2\cos x$ by stretching the graph of $y=\cos x$ vertically by a factor of 2. Thus, the graph of $y=2\cos x$ is shown below in Figure 2.

From Figure 2, it is observed that Figure 1 is stretchedvertically by a factor of 2.

Then, draw the graph of $y=-2\cos x$ by reflecting the graph of $y=2\cos x$ about the origin. Thus, the graph of $y=-2\cos x$ is shown below in Figure 3.

From Figure 3, it is observed that Figure 2 is reflected about the origin.

To draw the graph of $y=3-2\cos x$, shift the graph $y=-2\cos x$ by 3 units upward. Thus, the graph of $y=3-2\cos x$ is shown below in Figure 4.

Observe that Figure 4is obtainedfrom Figure 1 in such a way that it is stretched vertically by a factor of 2, reflect the graph about the origin and then shift 3 units upward.