The 30-year average monthly temperature, ^∘ F, for each month of the year for Washington, D.C.,

step 1 I'm going to enter a table of data into the decimal scrapping calculator where x is in months, o

The 30-year average monthly temperature, ^∘ F, for each...

The 30-year average monthly temperature, $^{\circ} \mathrm{F}$, for each month of the year for Washington, D.C., is given in Table 3 ( World Almanac). (A) Using 1 month as the basic unit of time, enter the data for a 2 -year period in your graphing calculator and produce a scatter plot in the viewing window. Choose $0 \leq y \leq 80$ for the viewing window. (B) It appears that a sine curve of the form $$ y=k+A \sin (B x+C) $$ will closely model these data. The constants $k, A,$ and $B$ are easily determined from Table 3 as follows: $A=(\max y-\min y) / 2$, $B=2 \pi /$ Period, and $k=\min y+A .$ To estimate $C,$ visually estimate to one decimal place the smallest positive phase shift from the plot in part A. After determining $A, B, k,$ and $C,$ write the resulting equation. (C) Plot the results of parts $\mathrm{A}$ and $\mathrm{B}$ in the same viewing window. (An improved fit may result by adjusting your value of $C$ slightly.) (D) If your graphing calculator has a sinusoidal regression feature, check your results from parts $\mathrm{B}$ and $\mathrm{C}$ by finding and plotting the regression equation.

The 30-year average monthly temperature, $^{\circ} \mathrm{F}$, for each month of the year for Washington,
D.C., is given in Table 3 ( World Almanac).
(A) Using 1 month as the basic unit of time, enter the data for a 2 -year period in your graphing calculator and produce a scatter plot in the viewing window. Choose $0 \leq y \leq 80$ for the viewing window.
(B) It appears that a sine curve of the form
$$
y=k+A \sin (B x+C)
$$
will closely model these data. The constants $k, A,$ and $B$ are easily determined from Table 3 as follows: $A=(\max y-\min y) / 2$, $B=2 \pi /$ Period, and $k=\min y+A .$ To estimate $C,$ visually estimate to one decimal place the smallest positive phase shift from the plot in part A. After determining $A, B, k,$ and $C,$ write the resulting equation.
(C) Plot the results of parts $\mathrm{A}$ and $\mathrm{B}$ in the same viewing window. (An improved fit may result by adjusting your value of $C$ slightly.)
(D) If your graphing calculator has a sinusoidal regression feature, check your results from parts $\mathrm{B}$ and $\mathrm{C}$ by finding and plotting the regression equation.

Key Concepts

Scatter Plots

Scatter plots are graphical representations of data points on a coordinate system, where each point represents an observation. They are used to visualize the distribution and pattern within the data, aiding in the identification of trends, cycles, or periodic behavior.

Sinusoidal Regression

Sinusoidal regression is the process of fitting a sine or cosine function to a set of data points. It involves estimating the function parameters that minimize the difference between the observed data and the model, often using computational tools like graphing calculators to automate and refine the fitting process.

Vertical Shift

The vertical shift, or midline, is a constant that moves the sine wave up or down along the vertical axis. It sets the central value around which the function oscillates, and is typically determined by the average of the maximum and minimum data values.

Phase Shift

The phase shift represents the horizontal displacement of the sine wave from its standard position. It adjusts where the cycle begins along the horizontal axis, ensuring the waveform aligns with the timing of the observed data peaks and troughs.

Period

The period of a sine function is the interval over which the function completes one full cycle. It is related to the coefficient in the function's argument, with the period calculated as 2? divided by the absolute value of that coefficient. This concept is essential for matching the function's cycle with the recurring pattern in the data.

Amplitude

Amplitude is the measure of the maximum deviation from the midline (or average value) of a sinusoidal function. It is calculated as half the difference between the maximum and minimum values observed in the data and reflects the intensity or magnitude of the oscillation.

Sinusoidal Modeling

Sinusoidal modeling uses sine or cosine functions to represent periodic phenomena. It focuses on capturing the repeating patterns observed in data by identifying parameters such as amplitude, period, phase shift, and vertical shift, which determine the shape and position of the sine wave.

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