Chapter 6, Problem 13CR

To determine

The sinusoidal function for the following graph.

  

Answer to Problem 13CR

Solution:

The equation of the graph is $y=3\cos \left(\frac{\pi }{6}x\right)$

Explanation of Solution

Given information:

The graph of the sinusoidal function:

  

Given graph is sinusoidal with amplitude $3$

The graph has characteristic of a cosine function, since the maximum value $3$ occurs at point $x=0$

So, the equation can be viewed as a cosine function $y=A\cos \left(\omega x-\varphi \right)+D$ with $A=3$

The cycle starts at the point $x=-6$ and ends at the point $x=6$

Thus, the period of cycle is $6-\left(-6\right)=12$

Therefore, period $=\frac{2\pi }{\omega }=12\Rightarrow \omega =\frac{\pi }{6}$

There is no phase shift and vertical translation.

Thus, $\varphi =0,D=0$

Thus, the equation of the graph is $y=3\cos \left(\frac{\pi }{6}x\right)$