Question

The following data represent the average monthly temperatures for Indianapolis, Indiana. $$ \begin{array}{|lc|} \hline & \text { Average Monthly } \\ \text { Month, } \boldsymbol{x} & \text { Temperature, }^{\circ} \mathrm{F} \\\ \hline \text { January, } 1 & 28.1 \\ \text { February, } 2 & 32.1 \\ \text { March, } 3 & 42.2 \\ \text { April, } 4 & 53.0 \\ \text { May, } 5 & 62.7 \\ \text { June, } 6 & 72.0 \\ \text { July, } 7 & 75.4 \\ \text { August, } 8 & 74.2 \\ \text { September, } 9 & 66.9 \\ \text { October, } 10 & 55.0 \\ \text { November, } 11 & 43.6 \\ \text { December, } 12 & 31.6 \\ \hline \end{array} $$ (a) Draw a scatter plot of the data for one period. (b) Find a sinusoidal function of the form $$ y=A \sin (\omega x-\phi)+B $$ that models the data. (c) Draw the sinusoidal function found in part (b) on the scatter plot. (d) Use a graphing utility to find the sinusoidal function of best fit. (e) Graph the sinusoidal function of best fit on a scatter plot of the data.

Short answer

1. Plot the data on a scatter plot. 2. Estimate and write a sinusoidal function. 3. Plot this function. 4. Use graphing software to find the best fit function. 5. Plot the best fit function.

Step-by-step solution

Construct the Scatter Plot

Plot the given average monthly temperatures against the months on a graph. Let the x-axis represent the months (from 1 to 12) and the y-axis represent the temperatures in degrees Fahrenheit. Mark each data point based on the given values.

Understand the Sinusoidal Function

A sinusoidal function has the form \(y = A \sin (\omega x - \phi) + B\). Here, \(A\) is the amplitude, \(\omega\) determines the frequency, \(\phi\) is the phase shift, and \(B\) is the vertical shift.

Estimate Initial Parameters

From the scatter plot, determine the following: - Amplitude \(A\) is half the difference between the maximum and minimum temperatures. - Vertical shift \(B\) is the average of the maximum and minimum temperatures. - Frequency \(\omega\) can be approximated to \(\frac{2\pi}{12}\). - Phase shift \(\phi\) can be estimated visually from the plot.

Write the Estimated Sinusoidal Function

Using the estimated parameters, write the function \(y = A \sin (\omega x - \phi) + B\). For example, if the amplitude is 23, \(\omega = \frac{2\pi}{12}\), phase shift \(\phi = -\frac{\pi}{2}\), and vertical shift \(B\) is 53.75, the function would be \(y = 23 \sin \left(\frac{2\pi}{12}x + \left(-\frac{\pi}{2}\right)\right) + 53.75\). Adjust these initial estimates based on your scatter plot.

Plot the Sinusoidal Function

On the same graph as the scatter plot, draw the sinusoidal function using the estimated parameters. Compare how well this function fits the data points.

Use Graphing Utility for Best Fit

Use a graphing calculator or software to input the data points and let the tool find the best fit sinusoidal function. Tools like Desmos, GeoGebra, or a graphing calculator can be used for this task.

Write the Best Fit Sinusoidal Function

Based on the graphing utility, write down the equation of the sinusoidal function that best fits the data. This function will have its coefficients finely tuned for a better fit.

Plot the Best Fit Function

On the same graph, draw the sinusoidal function obtained from the graphing utility. Observe how closely it matches the scatter plot data points.

Key concepts

Scatter Plot

To begin modeling the average monthly temperatures data for Indianapolis, we first construct a scatter plot. A scatter plot is a type of graph that displays individual data points based on two variables. In this case, one variable is the month (on the x-axis), and the other is the temperature (on the y-axis). To create the scatter plot, plot each month's average temperature at the corresponding point on the graph. For example, for January (month 1), plot the point at 28.1°F. This visual representation helps us observe any patterns or trends in the data, such as periodicity or seasonality, which are crucial when fitting a sinusoidal function.

Data Fitting

Data fitting involves finding a mathematical function that best describes the relationship between the data points in our scatter plot. In this exercise, we are particularly interested in finding a sinusoidal function because temperature data often follows a seasonal (cyclical) pattern. The fitting process starts by estimating the parameters of our sinusoidal function, usually of the form: \( y = A \, \sin(\omega x - \phi) + B \). We need to determine four parameters: amplitude \(A\), frequency \(\omega\), phase shift \(\phi\), and vertical shift \(B\). These parameters are initially estimated based on the scatter plot and then refined using graphing utilities to achieve the best fit. This process ensures the chosen function accurately represents the observed data pattern.

Sinusoidal Functions

A sinusoidal function, such as \( y = A \, \sin(\omega x - \phi) + B \), is particularly useful for modeling periodic data like the average monthly temperatures. Here is what each parameter represents:

• Amplitude \(A\): This is the peak deviation from the central value. In our case, it is half the difference between the maximum and minimum temperatures.
• Frequency \(\omega\): This determines how often the sinusoidal curve repeats over a cycle. For temperature data, it is often based on a 12-month cycle, so \(\omega\) can be approximated as \( \frac{2\pi}{12} \).
• Phase shift \(\phi\): This parameter shifts the curve horizontally. Visually inspecting the scatter plot gives an initial estimate of this shift.
• Vertical shift \(B\): This is the centerline around which the sinusoidal curve oscillates. It is the average of the highest and lowest temperatures.

By understanding and correctly estimating these parameters, we can create a sinusoidal model that closely fits our data.

Graphing Utilities

Graphing utilities are essential tools for accurately fitting a sinusoidal function to data. These tools can automatically fine-tune the parameters of your estimated function for a closer fit. Popular graphing utilities include software like Desmos or GeoGebra and graphing calculators. Here's how to use them:

• Input the data points from your scatter plot.
• Select the option to fit a sinusoidal function.
• The utility will compute the best-fit parameters for \(A\), \(\omega\), \(\phi\), and \(B\).

Once the best-fit function is found, it can be graphed alongside the original data points. This visual comparison helps to verify how well the function models the data. Utilizing these graphing tools simplifies the fitting process and ensures a more accurate representation of the temperature trends.