Chapter 6.5, Problem 8CFU

To determine

To state: amplitude, period, phase shift and vertical shift of the function and then sketch its graph.

Answer to Problem 8CFU

  $\begin{array}{c}\text{Amplitude}=2\\ \text{Period}=\pi \\ \text{Phase shift}=-\frac{\pi }{2}\\ \text{Vertical shift}=-5\end{array}$

Explanation of Solution

Given information:

  $y=2\sin \left(2\theta +\pi \right)-5$

Calculation:

The equation of cosine function is as follows:

  $y=A\sin \left(k\theta +c\right)+h$...........(1)

Here $A$ is a coefficient whose absolute value is termed as amplitude, c is a constant, $k=\frac{2\pi }{\text{Period}}$ and h is a vertical shift.

Compare the given equation with equation (1) to get,

  $\begin{array}{c}A=2\\ k=2\\ c=\pi \\ h=-5\end{array}$

Now evaluate the amplitude as shown below.

  $\begin{array}{c}\text{Amplitude}=\left|A\right|\\ =\left|2\right|\left[\text{Put the value of }A\right]\\ =2\end{array}$

The period of the function will be evaluated as follows:

  $\begin{array}{c}\text{Period}=\frac{2\pi }{k}\\ =\frac{2\pi }{2}\left[\text{Put the value of }k\right]\\ =\pi \end{array}$

The phase shift of the function will be evaluated as follows:

  $\begin{array}{c}\text{Phase shift}=-\frac{c}{k}\\ =-\frac{\pi }{2}\left[\text{Put the value of }c\text{ and }k\right]\end{array}$

The vertical shift of the function is 5 units downwards i.e. $h=-5$ .

Thus, the amplitude, period, phase shift and vertical shift of the function is $2,\pi ,-\frac{\pi }{2}$ and $-5$ respectively.

The graph of the function will be as follows: