Chapter 13.8, Problem 25PPS

To determine

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

Answer to Problem 25PPS

The amplitude, period and vertical shift and the equation of the midline of the function is $\frac{1}{3},2\pi ,\text{vertical shifts}:7unit\mathrm{s}up\&y=7$ .

Explanation of Solution

Given: It is given in the question that the function is $y=\frac{1}{3}\sin \theta +7$ .

Concept Used:

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

Calculation: Here, the function is $y=\frac{1}{3}\sin \theta +7$ .

In this, $a=\frac{1}{3}$ , so it’s amplitude is $\frac{1}{3}$ .

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

From the equation it is seen that $d=7$ so the vertical shifts is seven units up .

Also, $d=7$ then the midline is $y=7$ .

The graph of the function is given below:

  

Conclusion:

The amplitude, period and vertical shift and the equation of midline are $\frac{1}{3},2\pi ,\text{vertical shifts}:7unit\mathrm{s}up\&y=7$ .