Chapter 4.4, Problem 52E

Solution

To state: The transformation required to obtain the graph of $y_2=2\sin \frac{\pi x}{3}$ from the graph of $y_1=3\sin \frac{2\pi x}{3}$ .

The graph of $y_2$ is a vertical shrink of 1 and horizontal stretch of $\frac{1}{2}$ from $y_1$ .

Given information:

The given function are $y_2=2\sin \frac{\pi x}{3}\text{ and }{y}_1=3\cos \frac{2\pi x}{3}$ .

Explanation:

The amplitude will determine if the graph will stretch or shrink vertically.

If the amplitude is greater than 1, the graph stretches with respect to y- axis.

If the amplitude is less than 1, the graph shrinks with respect to y- axis.

Now consider the given functions: $y_2=2\sin \frac{\pi x}{3}\text{ and }{y}_1=3\cos \frac{2\pi x}{3}$

Compare the given functions with $y=a\cos \left(bx+c\right)d$ .

To describe how the graphs are related, compare the amplitude and period of these functions.

The amplitudes for $y_2\text{ is 2 and }{y}_1\text{ is 3}$ . It means there is vertical shrink of 1.

The period b is $y_2\text{ is }\frac{\pi }{3}\text{ and }{y}_1\text{ is }\frac{2\pi }{3}$ it means there is a horizontal shrink of $\frac{1}{2}$ .

Thus, the graph of $y_2$ can be obtained from $y_1$ by vertical shrink of 1 and horizontal stretch of $\frac{1}{2}$ .