Chapter 13.8, Problem 3CYP

To determine

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

Answer to Problem 3CYP

The amplitude, period, phase shift and vertical shift of the function is $2,4\pi ,\frac{\pi }{2}totheleft\&2unit\mathrm{s}down$ .

Explanation of Solution

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

Concept Used:

In this, use the concept of the general equation for cosine $y=a\cos b(\theta +h)+d$ , where a is amplitude, $\frac{2\pi }{\left|b\right|}$ is period, h is the phase shift and if the h is positive , then h units to the right and if h is negative then h units to the left and d is the vertical shift if d is positive it shifts to up and if it is negative , it shift to down.

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

In this, $a=2$ , so it’s amplitude is $2$ .

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

From the equation it is find that h is $-\frac{\pi }{2}$ with negative sign, so the phase shift is $\frac{\pi }{2}$ to the left.

Again, From the equation it also find that d is $-2$ with negative sign, so vertical shift is $2$ units down.

The graph of the function is given below:

  

Conclusion:

The amplitude, period, phase shift and vertical shift are $2,4\pi ,\frac{\pi }{2}totheleft\&2unit\mathrm{s}down$ .