7.6 Modeling with trigonometric equations  (Page 2/14)

We will solve these problems according to the models.

1. $y=2\sin \left(\frac{1}{4}x\right)$ involves sine, so we use the form
   $$y=A\sin \left(Bt+C\right)+D$$

   We know that $\left|A\right|$ is the amplitude, so the amplitude is 2. Period is $\frac{2\pi }{B},$ so the period is

   $$\begin{array}{l}\frac{2\pi }{B}=\frac{2\pi }{\frac{1}{4}}\hfill \\ =8\pi \hfill \end{array}$$

   See the graph in [link] .

   

2. $y=\mathrm{-3}\sin \left(2x+\frac{\pi }{2}\right)$ involves sine, so we use the form
   $$y=A\sin \left(Bt-C\right)+D$$

   Amplitude is $\left|A\right|,$ so the amplitude is $\left|-3\right|=3.$ Since $A$ is negative, the graph is reflected over the x -axis. Period is $\frac{2\pi }{B},$ so the period is

   $\frac{2\pi }{B}=\frac{2\pi }{2}=\pi$

   The graph is shifted to the left by $\frac{C}{B}=\frac{\frac{\pi }{2}}{2}=\frac{\pi }{4}$ units. See [link] .

   

3. $y=\cos x+3$ involves cosine, so we use the form
   $$y=A\cos \left(Bt\pm C\right)+D$$

   Amplitude is $\left|A\right|,$ so the amplitude is 1. The period is $2\pi .$ See [link] . This is the standard cosine function shifted up three units.

   

The amplitude is $|-4|=4.$ The period is $\frac{2\pi }{\omega }=\frac{2\pi }{\pi }=2.$ (Recall that we sometimes refer to $B$ as $\omega .)$ One cycle of the graph can be drawn over the interval $\left[0,2\right].$ To find the key points, we divide the period by 4. Make a table similar to [link] , starting with $x=0$ and then adding $\frac{1}{2}$ successively to $x$ and calculate $y.$ See the graph in [link] .

Recall that amplitude is found using the formula

Thus, the amplitude is

The data covers a period of 12 months, so $\frac{2\pi }{B}=12$ which gives $B=\frac{2\pi }{12}=\frac{\pi }{6}.$

The vertical shift is found using the following equation.

Thus, the vertical shift is

So far, we have the equation $y=13.3\sin \left(\frac{\pi }{6}x-C\right)+\mathrm{55.8.}$

To find the horizontal shift, we input the $x$ and $y$ values for the first month and solve for $C.$

We have the equation $y=13.3\sin \left(\frac{\pi }{6}x-\frac{2\pi }{3}\right)+\mathrm{55.8.}$ See the graph in [link] .