Q. 30 - Precalculus Enhanced with Graphing Utilities

Question

The data on the next page represent the average monthly temperatures for Washington, D.C.

(a) Draw a scatter diagram of the data for one period.

(b) Find a sinusoidal function of the form $y=A \sin (\omega x-\varphi )+B$ B that models the data.

(c) Draw the sinusoidal function found in part (b) on the scatter diagram.

(d) Use a graphing utility to find the sinusoidal function of best fit.

(e) Graph the sinusoidal function of best fit on a scatter diagram of the data.

Step-by-Step Solution

Answer

(a) The scatter diagram for the data of one period is: 

(b) The sinusoidal function for the given data is $y=22.7\sin (\frac{\mathrm{\pi }}{6}x-\frac{2\mathrm{\pi }}{3})$

(c) The graph of the sinusoidal function found in part (b) Is: 

(d) The sinusoidal function of best fit by using the graphing utility is $y=22.7\sin (\frac{\mathrm{\pi }}{6}x-\frac{\mathrm{\pi }}{2})+58.042$

(e) The sinusoidal function of best fit on the scatter diagram is: 

Part (a) Step 1. Given

A table of information

To draw a scatter diagram of the data for one period.

Part (a) Step 2. Plot the points in the graph

The scatter plot for the data is:

Part (b) Step 1. To find the amplitude A, vertical shift B of the function

Amplitude$=\frac{Largest value-Smallest value}{2}$

$$\begin{gathered} =\frac{80-34.6}{2} \\ =22.7 \end{gathered}$$

Vertical shift$=\frac{Largest value+Smallest value}{2}$

$$\begin{gathered} =\frac{80+34.6}{2} \\ =57.3 \end{gathered}$$

Part (b) Step 2. Determine ω

The data repeats for every $12$ months, so

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

The function is $y=22.7\sin \left(\frac{\mathrm{\pi }}{6}x\right)+57.3$

Step 3. Determine horizontal shift

To determine the horizontal shift, we use the period $T=12$ and divide the interval $(0,12)$ into four sub intervals of length $12÷4=4$

$(0,3),(3,6),(6,9),(9,12)$

The sine curve is increasing on the interval $(0,3)$ and is decreasing on the interval $(3,9)$ so a local maximum occurs at $x=3$. The data indicate that
a maximum occurs at $x=7$, so we must shift the graph of the function  $4$units to the right by replacing $x$  by $x-4$

Part (b) Step 3. Plot the function

So we obtain the function, $y=22.7\sin \left(\frac{\mathrm{\pi }}{6}\right(x-4)+57.3$

So the function is $y=22.7\sin (\frac{\mathrm{\pi }}{6}x-\frac{2\mathrm{\pi }}{3})+57.3$

Part (c) Step 1. Draw the sinusoidal function

The graph of the sinusoidal function $y=22.7\sin (\frac{\mathrm{\pi }}{6}x-\frac{2\mathrm{\pi }}{3})+57.3$ is:

Part (d) Step 1. Finding function by using graphing utility

By entering the data in the graphing utility, we found the sinusoidal function as, 

$y=22.7\sin (\frac{\mathrm{\pi }}{6}x-\frac{\mathrm{\pi }}{2})+58.042$

Part (e) Step 1: Plot the function in the scatter diagram

Plot the function in the graph.