Question

The data on the next page represent the average monthly temperatures for Washington, D.C. (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. $$ \begin{array}{|lc|} \hline \text { Decade, } x & \text { Major Hurricanes, } \boldsymbol{H} \\ \hline 1921-1930,1 & 17 \\ 1931-1940,2 & 16 \\ 1941-1950,3 & 29 \\ 1951-1960,4 & 33 \\ 1961-1970,5 & 27 \\ 1971-1980,6 & 16 \\ 1981-1990,7 & 16 \\ 1991-2000,8 & 27 \\ 2001-2010,9 & 33 \\ \hline \end{array} $$ (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. $$ \begin{array}{lc|} \hline \text { Month, } \boldsymbol{x} & \begin{array}{c} \text { Average Monthly } \\ \text { Temperature, }^{\circ} \mathrm{F} \end{array} \\ \hline \text { January, } 1 & 36.0 \\ \text { February, } 2 & 39.0 \\ \text { March, } 3 & 46.8 \\ \text { April, } 4 & 56.8 \\ \text { May, } 5 & 66.0 \\ \text { June, } 6 & 75.2 \\ \text { July, } 7 & 79.8 \\ \text { August, } 8 & 78.1 \\ \text { September, } 9 & 71.0 \\ \text { October, } 10 & 59.5 \\ \text { November, } 11 & 49.6 \\ \text { December, } 12 & 39.7 \end{array} $$

Short answer

The sinusoidal function is approximately \( y = 21.9 \sin\left( \frac{\pi}{6} x - \frac{\pi}{2} \right) + 57.9 \).

Step-by-step solution

Organize the data

The average monthly temperatures for Washington, D.C. are given. Assign each month a number from 1 to 12 (January = 1, February = 2, and so on).

Draw a scatter plot

On graph paper or using graphing software, plot the points where the x-coordinate is the month number and the y-coordinate is the corresponding average temperature. Points to plot are: (1, 36.0), (2, 39.0), (3, 46.8), (4, 56.8), (5, 66.0), (6, 75.2), (7, 79.8), (8, 78.1), (9, 71.0), (10, 59.5), (11, 49.6), (12, 39.7).

Find a sinusoidal function

To model the data as a sinusoidal function, we need the form: \[ y = A \sin(\omega x - \phi) + B \]Estimate values for A, B, \(\omega\), and \(\phi\):- A is the amplitude, calculated as half the difference between the maximum and minimum temperatures.- B is the vertical shift, calculated as the average of the maximum and minimum temperatures.- \(\omega\) is the frequency, usually \(2\pi/12\) because there are 12 months.- \(\phi\) is the horizontal shift and can be approximated by observing where the sine function should start.

Calculate A and B

Maximum temperature: 79.8 (July)Minimum temperature: 36.0 (January)Amplitude (A): \[ (79.8 - 36.0)/2 = 21.9 \]Vertical shift (B): \[ (79.8 + 36.0)/2 = 57.9 \]

Determine \(\omega\) and \(\phi\)

Since there are 12 months, \(\omega = 2\pi/12 = \pi/6\).The phase shift \(\phi\) can be assumed to align the peak of the sinusoidal function with the hottest month (July, month 7): \[ \phi = \pi/2 \]

Write the sinusoidal equation

Combine the values into the sinusoidal function: \[ y = 21.9 \sin\left(\frac{\pi}{6} x - \frac{\pi}{2}\right) + 57.9 \]

Graph the sinusoidal function

On the same scatter plot, draw the sinusoidal function obtained in the previous step. Ensure it goes through the critical points and fits the overall shape of the scatter plot.

Use a graphing utility for best fit

Use a graphing calculator or software to find the best fit sinusoidal function. Input the data points and let the utility calculate the most accurate sinusoidal model.

Graph the best fit function

Plot the best fit sinusoidal function from the graphing utility on the same scatter plot as the original data points. Compare this graph with the manually derived sine function.

Key concepts

scatter plots

A scatter plot is a type of graph used to display values for typically two variables. In this exercise, we use it to represent the average monthly temperatures in Washington, D.C. Scatter plots help visualize the distribution and trends over time.

To create a scatter plot for this data, each month is assigned a number from 1 to 12. January is 1, February is 2, and so on. Then, the average monthly temperature is plotted against each month number. For example, January has an average temperature of 36°F, so the point (1, 36) is plotted.

Key steps in drawing a scatter plot include:

• Labeling the x-axis with the months and the y-axis with temperature values.
• Plotting each data point corresponding to the month and its temperature.
• Clearly marking each point to help visualize the trend.

Once the scatter plot is drawn, it's easier to observe how temperatures change over the months and to identify any cycles or patterns.

trigonometric modeling

Trigonometric modeling involves using trigonometric functions like sine and cosine to approximate and describe cyclical behaviors, such as temperature changes. These functions are useful because they inherently account for periodicity and wave-like patterns.

In this case, we use a sinusoidal function: \( y = A \sin(\omega x - \phi) + B \)
This function includes several important parameters:

• Amplitude (A): Indicates the peak deviation from the central value.
• Vertical shift (B): Represents the average value around which the function oscillates.
• Angular frequency (\( \omega \)): Reflects how many cycles occur over a specific period, often related to the period's length.
• Phase shift (\( \phi \)): Indicates how the function is shifted horizontally.

Creating a good model involves estimating these parameters to fit the given data as closely as possible, ensuring representation of real-world cycles.

temperature data analysis

Analyzing temperature data typically involves identifying patterns and changes over a period. Monthly temperature data offers a clear view of seasonal variations and long-term trends.

For Washington, D.C., the average monthly temperatures reflect warmer summers and colder winters. By plotting the data and analyzing it, we can often predict general trends:

• Identify the hottest month (July) and the coldest month (January).
• Calculate peak-to-peak differences to determine the temperature range.
• Use these trends to inform or validate the sinusoidal model parameters.

This analysis reveals valuable insights into how temperature shifts over the year and helps in creating more accurate models for forecasting.

amplitude and phase shift

Amplitude and phase shift are critical concepts in trigonometric modeling of periodic functions.

Amplitude (A): Measures the height of the wave. It is half the difference between the maximum and minimum values. For example, if the highest temperature is 79.8°F and the lowest is 36°F, the amplitude is \( (79.8 - 36.0)/2 = 21.9 \).

Phase Shift (\( \phi \)): Represents a horizontal shift. It determines where the cycle starts on the x-axis. In our example, the phase shift aligns the peak of the sine function with the hottest month (July), meaning \( \phi \) is set so that at x = 7, the sine function achieves its maximum. This shift ensures the model accurately represents the real timing of temperature peaks and troughs.

Correctly determining these parameters allows for more precise trigonometric models that fit the observed data closely. It also helps in predicting future values based on past observed behavior.