Q. 13 - Precalculus Enhanced with Graphing Utilities

Question

Find a sinusoidal function for the following graph.

Step-by-Step Solution

Answer

 The sinusoidal function for given graph is :$y = 3\cos \left(\frac{x}{2}\right) or 3\sin \left(\frac{\mathrm{\pi }}{2}-\frac{x}{2}\right)$

Step 1. Given information

Step 2. Assume the graph equation

This shown graph shows the characteristics of the cosine function, it is symmetric about the y-axis

so let's assume the graph like

$y= A \cos \left(\omega x\right)$

from figure 

amplitude $\left|A\right|= 3$

period $T = 12$

Step 3. calculating the ω

We know that $T = \frac{2\mathrm{\pi }}{\omega }$

$$\begin{gathered} 12 =\frac{2\mathrm{\pi }}{\omega } \\ \omega = \frac{\mathrm{\pi }}{6} \end{gathered}$$

so,the cosine function  is,for amplitude  3 and$\omega =\frac{\mathrm{\pi }}{6}$ is

$y= 3\cos \left(\frac{\mathrm{\pi }}{6}x\right) or 3\sin \left(\frac{\mathrm{\pi }}{2}-\frac{\mathrm{\pi }}{6}x\right)$