Chapter 6.6, Problem 31AYU

Monthly Temperature The given data represent the average monthly temperatures for Indianapolis, Indiana.

(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 models the data.

(c) Draw the sinusoidal function found in part $\left(b\right)$ 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.

To determine

To find:

a. Draw a scatter diagram of the data.

Answer to Problem 31AYU

a.

Explanation of Solution

Given:

$y=A\sin \left(\omega x-\varphi \right)+B$

Calculation:

a. Draw a scatter diagram of the data.

To determine

To find:

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

Answer to Problem 31AYU

b. $y=23.65\sin \left(\frac{\pi }{6}x-\frac{2\pi }{3}\right)+51.75$

Explanation of Solution

Given:

$y=A\sin \left(\omega x-\varphi \right)+B$

Calculation:

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

Determine $\text{A}$, the amplitude of the function.

$\text{Amplitude }=\frac{\text{largest data value }-\text{ smallest data value}}{2}=\frac{75.4-28.1}{2}=\frac{47.3}{2}=23.65$

Determine $\text{B}$, the vertical shift of the function.

$\text{Vertical shift }=\frac{\text{largest data value }+\text{ smallest data value}}{2}=\frac{75.4+28.1}{2}=\frac{103.5}{2}=51.75$

Determine $\omega$, It is easier to find the horizontal stretch factor first. Since the period of the function is $T=12$. Because $T=\frac{2\pi }{\omega }=12$. Therefore $\omega =\frac{2\pi }{12}=\frac{\pi }{6}$.

Now superimpose the graph of $y=23.65\sin \left(\frac{\pi }{6}x\right)+51.75$ on the scatter diagram. See Figure, where it is clear that the graph still needs to be shifted horizontally.

Determine the horizontal shift of the function by using the period of the data. we must shift the graph of the function.

$y=23.65\sin \left(\frac{\pi }{6}x-\varphi \right)+51.75$

At $x=1$, $y=28.1$

$28.1=23.65\sin \left(\frac{\pi }{6}-\varphi \right)+51.75$

$-23.65=23.65\sin \left(\frac{\pi }{6}-\varphi \right)$

$\sin \left(\frac{\pi }{6}-\varphi \right)=-1$

$\frac{\pi }{6}-\varphi =-\frac{\pi }{2}$

$\varphi =\frac{2\pi }{3}$

$y=23.65\sin \left(\frac{\pi }{6}x-\frac{2\pi }{3}\right)+51.75$

To determine

To find:

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

Answer to Problem 31AYU

c.

Explanation of Solution

Given:

$y=A\sin \left(\omega x-\varphi \right)+B$

Calculation:

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

To determine

To find:

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

Answer to Problem 31AYU

d. $y=24.25\sin \left(0.493x-1.927\right)+51.61$

Explanation of Solution

Given:

$y=A\sin \left(\omega x-\varphi \right)+B$

Calculation:

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

$y=24.25\sin \left(0.493x-1.927\right)+51.61$

To determine

To find:

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

Answer to Problem 31AYU

e.

Explanation of Solution

Given:

$y=A\sin \left(\omega x-\varphi \right)+B$

Calculation:

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