Chapter 7, Problem 5CRT

(a)

To determine

To find: The amplitude, period and phase shift of function $f\left(x\right)=-2\sin \left(2x-\frac{\pi }{2}\right)$ .

Answer to Problem 5CRT

The amplitude, period and phase shift of function are 2, $\pi$ and $\frac{\pi }{4}$ respectively.

Explanation of Solution

Given information:

The function is $f\left(x\right)=-2\sin \left(2x-\frac{\pi }{2}\right)$ .

Calculation:

Rewrite the above function as, $f\left(x\right)=-2\sin \left(2\left(x-\frac{\pi }{4}\right)\right)+0$ .

Compare given function with the form $f\left(x\right)=a\sin \left(b\left(x+c\right)\right)+d$ to find amplitude, period and phase shift, where $a$ is amplitude, $b$ and $c$ is phase shift.

The amplitude is 2.

Calculate the period as follows.

  $\begin{array}{c}\text{Period}=\frac{2\pi }{b}\\ =\frac{2\pi }{2}\\ =\pi \end{array}$

The phase shift is $\frac{\pi }{4}$ to the right side.

Therefore, the amplitude, period and phase shift of function are 2, $\pi$ and $\frac{\pi }{4}$ respectively.

(b)

To determine

To sketch: The graph of one complete period of $f\left(x\right)=-2\sin \left(2x-\frac{\pi }{2}\right)$ .

Answer to Problem 5CRT

The graph is shown below.

Explanation of Solution

Given information:

The function is $f\left(x\right)=-2\sin \left(2x-\frac{\pi }{2}\right)$ .

Calculation:

The graph is as follows.