Chapter 4.4, Problem 80E

Solution

To graph: Find sinusoidal function and graph function with scatter plot.

Model the temperature T as a sinusoidal function of time, using 20 as the minimum value

and 69 as the maximum value. The average monthly Fahrenheit temperatures in Helena, MT, fo 2012 are shown in the table below ( , etc.):

Month	1	2	3	4	5	6	7	8	9	10	11	12
Temp	20	26	34	43	53	62	69	67	55	45	32	21

Given information

The given table:

Month	1	2	3	4	5	6	7	8	9	10	11	12
Temp	20	26	34	43	53	62	69	67	55	45	32	21

Explanation:

Consider the given information.

Choose cosine function.

  $T=a\cos \left(b\left(t+h\right)\right)+k$

Where $a$ is determining amplitude of graph and h , is phase shift k is vertical shift, b is constant and t is time (month).

Maximum T is 68 and minimum is 20.

So,

  $\begin{array}{c}\text{Amplitude}=\frac{M-m}{2}\\ =\frac{68-20}{2}\text{Substitute}M=68,m=20\\ A=24\end{array}$

Hence, amplitude is 24.

Vertical shift is

  $\begin{array}{c}\text{vertical}\text{shif}t=\frac{M-m}{2}\\ =\frac{68+20}{2}\text{Substitute}M=68,m=20\\ K=44\end{array}$

Period of the function is 12 months.

  $\begin{array}{c}\text{period}=\frac{2\pi }{\left|b\right|}\\ 12=\frac{2\pi }{\left|b\right|}\\ b=\frac{2\pi }{12}\\ b=\frac{\pi }{6}\end{array}$

Since, it is cosine function, so maximum should at t = 0 but this data is maximum at t = 7, so phase shift is $h=-7$

Now apply the formula $T=a\cos \left(b\left(t+h\right)\right)+k$

  $y=24\cos \left(\frac{\pi }{6}\left(t-7\right)\right)+44$

Now draw function on the scatter plot.

To plot the scatter plot of T as a function of t take t on the x − axis and T on the y −axis and plot ordered pairs like $\left(1,20\right),\left(2,26\right)\mathrm{....}\left(12,21\right)$ .