Chapter 4.4, Problem 46E

Solution

To state: The transformation required to obtain the graph of $y=\frac{3}{4}\sin \frac{x}{5}$ from a basic trigonometric graph.

The graph will stretch horizontally by $5$ and shrink vertically by $\frac{3}{4}$ from the graph of $y=\sin x$ .

Given information:

The given function is $y=\frac{3}{4}\sin \frac{x}{5}$ .

Explanation:

Consider the below given information:

If $\left|b\right|>1$ then there is a horizontal shrink by the factor of $\frac{1}{\left|b\right|}$ .

If $\left|b\right|<1$ then there is a horizontal stretch by the factor of $\frac{1}{\left|b\right|}$ .

If $\left|b\right|<0$ then there is a reflection across the y- axis.

Now consider the given function: $y=\frac{3}{4}\sin \frac{x}{5}$

Compare the function $y=\frac{3}{4}\sin \frac{x}{5}$ with the standard function $y=a\sin bx$ .

Then the values are $a=\frac{3}{4}\text{ and }b=\frac{1}{5}$ .

Here $a$ is the amplitude of the function and its value is $\frac{3}{4}$ .

Thus the graph of $y=\frac{3}{4}\sin \frac{x}{5}$ is vertical shrink of graph of $y=\sin x$ by a factor of $\frac{3}{4}$ .

Now find the period of the function.

  $\text{Period}=\frac{2\pi }{\left|b\right|}$

Substitute the values $b=\frac{1}{5}$ in the formula:

  $\begin{array}{c}\text{Period}=\frac{2\pi }{\left|\frac{1}{5}\right|}\\ =10\pi \end{array}$

Thus the graph of $y=\frac{3}{4}\sin \frac{x}{5}$ is vertical shrink of graph of $y=\sin x$ by a factor of $\frac{3}{4}$ and horizontal stretch by a factor of $5$ .