Chapter 1.6, Problem 12E

(a)

To determine

To find: The period of the function $y=2\sin \left(4x+\pi \right)+3$ .

Answer to Problem 12E

The period of the function is $\frac{\pi }{2}$ .

Explanation of Solution

Given information:

The given function is $y=2\sin \left(4x+\pi \right)+3$ .

Calculation:

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

Consider the given function.

  $y=2\sin \left(4x+\pi \right)+3$

So, the period of the given function is:

  $\begin{array}{c}\frac{2\pi }{B}=4\\ B=\frac{2\pi }{4}\\ =\frac{\pi }{2}\end{array}$

Therefore, the period of the function is $\frac{\pi }{2}$ .

(b)

To determine

To find: The domain of the function $y=2\sin \left(4x+\pi \right)+3$ .

Answer to Problem 12E

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

Explanation of Solution

Given information:

The given function is $y=2\sin \left(4x+\pi \right)+3$ .

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 $y=2\sin \left(4x+\pi \right)+3$ 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 $y=2\sin \left(4x+\pi \right)+3$ .

Answer to Problem 12E

The range of the function is $\left[1,5\right]$ .

Explanation of Solution

Given information:

The given function is $y=2\sin \left(4x+\pi \right)+3$ .

Calculation:

The general equation for the sine function can be written as:

  $y=a\sin \left(bx-c\right)+d$

The amplitude is $\left|a\right|$ and the midline is $y=d$ . The maximum value is $d+\left|a\right|$ and the minimum $y$ value is $d-\left|a\right|$ .

For the given function $y=2\sin \left(4x+\pi \right)+3$ , amplitude is $\left|a\right|=2$ and vertical shift is $d=3$ .

The minimum value of $y$ can be calculated as:

  $\begin{array}{c}{y}_{\min }=d-\left|a\right|\\ =3-2\\ =1\end{array}$

The maximum value of $y$ can be calculated as:

  $\begin{array}{c}{y}_{\max }=d+\left|a\right|\\ =3+2\\ =5\end{array}$

Therefore, the range of the function is $\left[1,5\right]$ .

(d)

To determine

To graph: The function $y=2\sin \left(4x+\pi \right)+3$ .

Explanation of Solution

Given information:

The given function is $y=2\sin \left(4x+\pi \right)+3$ .

Graph:

To graph a function $y=2\sin \left(4x+\pi \right)+3$ , 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 $y=2\sin \left(4x+\pi \right)+3$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2}\boxed{\text{SIN}}\boxed{4}\boxed{\text{X}}\boxed{+}\boxed{2\text{nd}}\boxed{^}\boxed{)}\boxed{+}\boxed{3}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=2\sin \left(4\text{X}+\pi \right)+3$

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.