Chapter 10.5, Problem 10Q

Solution

To state: A function for the sinusoid.

  

The resultant values are $y=3\sin \pi x-2$ .

Given information:

The given sinusoid is:

  

Formula used: The amplitude formula is $\text{Amplitude}\left(a\right)=\frac{M-m}{2}$ .

Calculation:

Consider the given graph:

  

Observe the graph, the maximum value is 1 and minimum value is -5.

So the Amplitude is:

  $\begin{array}{c}\text{Amplitude}\left(a\right)=\frac{M-m}{2}\\ =\frac{1-\left(-5\right)}{2}\\ =3\end{array}$

The vertical shift is:

  $\begin{array}{c}\text{Vertical shift}\left(k\right)=\frac{M+m}{2}\\ =\frac{1+\left(-5\right)}{2}\\ =-2\end{array}$

From the graph, the period of the sinusoid is $2$ . Then $\frac{2\pi }{\left|b\right|}=2$ .

  $\begin{array}{c}\frac{2\pi }{\left|b\right|}=2\\ 2\pi =2b\\ b=\pi \end{array}$

Since the graph crosses the midline at $y=-2$ on y -axis, the graph will be a sine curve with no horizontal shift. Thus the value of h will be 0.

Then the resultant values are $a=3,b=\pi ,h=0,k=-2$ .

The function will become:

  $\begin{array}{c}y=a\sin \left(bx-h\right)+k\\ y=3\sin \left(\pi x-0\right)-2\\ y=3\sin \pi x-2\end{array}$

Therefore, the resultant sinusoid is $y=3\sin \pi x-2$ ..