Chapter 13.8, Problem 30PPS

To determine

Find the amplitude, period, phase shift and vertical shift for each function. Also, draw the graph of the function.

Answer to Problem 30PPS

The amplitude, period , phase shift and vertical shift of the function is $\frac{1}{2},2\pi ,phaseshift\frac{\pi }{2}totheright\&\text{vertical shifts}:4unit\mathrm{s}up.$

Explanation of Solution

Given: It is given in the question that the function is $y=\frac{1}{2}\sin (\theta -\frac{\pi }{2})+4$ .

Concept Used:

In this, use the concept of the general equation for sine $y=a\sin b(\theta -h)+d$ , where a is amplitude, $\frac{2\pi }{\left|b\right|}$ is period and d is the vertical shift and h is the phase shift.

Calculation: Here, the function is $y=\frac{1}{2}\sin (\theta -\frac{\pi }{2})+4$ .

In this, $a=\frac{1}{2}$ , so it’s amplitude is $\frac{1}{2}$ .

Now, the value of b is $1$ , so the period is: $\frac{2\pi }{\left|b\right|}=\frac{2\pi }{\left|1\right|}=4\pi$ .

The phase shift of the function $y=a\sin b(\theta -h)+d$ is h units to the right if h is positive or h units to the left if h is negative.

In this case, $h=\frac{\pi }{2}$ so the phase shift is $\frac{\pi }{2}$ to the right.

From the equation it is seen that $d=4$ so the vertical shifts is four units up.

The graph of the function is given below:

  

Conclusion:

The amplitude, period , phase shift and vertical shift are $\frac{1}{2},2\pi ,phaseshift\frac{\pi }{2}totheright\&\text{vertical shifts}:4unit\mathrm{s}up.$