Q. 54 - Precalculus Enhanced with Graphing Utilities

Question

The following data represent the average monthly temperatures for Phoenix, Arizona.

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

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

Step-by-Step Solution

Answer

The sinusoidal function for the given data is $y=20\sin (\frac{\mathrm{\pi x}}{6}+4.1)+70$.

Part (a) Step 1. A scatter plot for the data of one period is

Part (b) Step 1. A sinusoidal function that fits the data is

From the scatter plot,

$$\begin{gathered} y_{max}=90 \\ {y}_{min}=50 \end{gathered}$$

This gives

Amplitude, $$\begin{gathered} A=\frac{90-50}{2} \\ =20 \end{gathered}$$

Period$=12$

Vertical shift$=y_{min}+A$

$$\begin{gathered} B=50+20 \\ =70 \end{gathered}$$

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

$\varphi =-4.2$

So the sinusoidal function is $y=20\sin (\frac{\mathrm{\pi x}}{6}+4.2)+70$

Part (c) Step 1. The graph of sinusoidal function obtained over scatter plot is

Part (d) Step 1. Using graphing utility to find line of best fit.

Desmos software can be used to find the line of best line and it will give $y=20\sin (\frac{\mathrm{\pi x}}{6}+4.1)+70$.

Part (e) Step 1. The graph of sinusoidal function of best fit on scatter plot is