Chapter 1.6, Problem 14E

(a)

To determine

To find: The period of the function $y=2\sin \left(2x+\frac{\pi }{3}\right)$ .

Answer to Problem 14E

The period of the function is $\pi$ .

Explanation of Solution

Given information:

The given function is $y=2\sin \left(2x+\frac{\pi }{3}\right)$ .

Calculation:

The period of a function is given by $\frac{2\pi }{B}$ .

Consider the given function.

  $y=2\sin \left(2x+\frac{\pi }{3}\right)$

So, the period of the given function is:

  $\begin{array}{c}\frac{2\pi }{B}=2\\ B=\pi \end{array}$

Therefore, the period of the function is $\pi$ .

(b)

To determine

To find: The domain of the function $y=2\sin \left(2x+\frac{\pi }{3}\right)$ .

Answer to Problem 14E

The domain of the function is $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $y=2\sin \left(2x+\frac{\pi }{3}\right)$ .

Calculation:

The domain of a function is the set of all input values such that the function is defined.

The sine function is defined for all real values of $x$ . So, the domain of the function $y=2\sin \left(2x+\frac{\pi }{3}\right)$ is the set of all real values of $x$ .

Therefore, the domain of the function is $\left(-\infty ,\infty \right)$ .

(c)

To determine

To find: The range of the function $y=2\sin \left(2x+\frac{\pi }{3}\right)$ .

Answer to Problem 14E

The range of the function is $\left[-2,2\right]$ .

Explanation of Solution

Given information:

The given function is $y=2\sin \left(2x+\frac{\pi }{3}\right)$ .

Calculation:

It is known that the sine function lies between $-1$ and 1. So,

  $-1\le \sin \left(2x+\frac{\pi }{3}\right)\le 1$

The amplitude of the given function is $2$ , multiply by $2$ on both side.

  $-2\le 2\sin \left(2x+\frac{\pi }{3}\right)\le 2$

Therefore, the range of the function is $\left[-2,2\right]$ .

(d)

To determine

To graph: The function $y=2\sin \left(2x+\frac{\pi }{3}\right)$ .

Explanation of Solution

Given information:

The given function is $y=2\sin \left(2x+\frac{\pi }{3}\right)$ .

Graph:

To graph a function $y=2\sin \left(2x+\frac{\pi }{3}\right)$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=2\sin \left(2x+\frac{\pi }{3}\right)$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2}\boxed{\text{SIN}}\boxed{2}\boxed{\text{X}}\boxed{+}\boxed{(}\boxed{2\text{nd}}\boxed{^}\boxed{÷}\boxed{3}\boxed{)}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=2\sin \left(2x+\left(\pi /3\right)\right)$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of both the equations as shown below.

  

Figure (1)

Interpretation: From the graph it can be noticed that two periods of the function are shown in the window.