Chapter 4.4, Problem 79E

Solution

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

Model the temperature T as a sinusoidal function of time, using 34 as the minimum value

and 79 as the maximum value. The average monthly Fahrenheit temperatures in Albuquerque, New Mexico, for 2012 are shown in the table ( , etc.):

Month	1	2	3	4	5	6	7	8	9	10	11	12
Temp	34	40	47	55	64	74	79	76	69	57	44	35

Given information

The given table

Month	1	2	3	4	5	6	7	8	9	10	11	12
Temp	34	40	47	55	64	74	79	76	69	57	44	35

Graph:

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 79 and minimum is 36.

So,

  $\begin{array}{c}\text{Amplitude}=\frac{M-m}{2}\\ =\frac{79-36}{2}\text{Substitute}M=79,m=36\\ A=21.5\end{array}$

Hence, amplitude is 21.5.

Vertical shift is

  $\begin{array}{c}\text{vertical}\text{shif}t=\frac{M-m}{2}\\ =\frac{79+36}{2}\text{Substitute}M=79,m=36\\ K=57.5\end{array}$

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

  $y=21.5\cos \left(\frac{\pi }{6}\left(t-7\right)\right)+57.5$

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 $\left(1,36\right),\left(2,41\right)\mathrm{....}\left(12,36\right)$ .