Chapter 10.5, Problem 1GP

Solution

To state: A function for the sinusoid.

  

The resultant values are $y=2\cos 3x$ .

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 2 and minimum value is -2.

So the Amplitude is:

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

The vertical shift is:

  $\begin{array}{c}\text{Vertical shift}\left(k\right)=\frac{M+m}{2}\\ =\frac{2+\left(-2\right)}{2}\\ =0\end{array}$

From the graph, the period of the sinusoid is $\frac{2\pi }{3}$ . Then $\frac{2\pi }{b}=\frac{2\pi }{3}$ .

  $\begin{array}{c}\frac{2\pi }{b}=\frac{2\pi }{3}\\ 6\pi =2\pi b\\ b=3\end{array}$

Since the graph does not cross the midline at $y=0$ and is maximum at $x=0$ , the graph will be a cosine curve with no horizontal shift. Thus the value of h will be 0.

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

The function will become:

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

Therefore, the resultant sinusoid is $y=2\cos 3x$ .