Average Monthly Temperature The average monthly temperature (in ^∘ F ) in Seattle, Washington,

step 1 Hello, this is instructor Nicole. I've copied your data table But without the months because whe

Average Monthly Temperature The average monthly temperature...

Average Monthly Temperature The average monthly temperature (in ${ }^{\circ} \mathrm{F}$ ) in Seattle, Washington, is shown in the table. $$ \begin{array}{c|c||c|c} \text { Month } & \text { 'F } & \text { Month } & \text { 'F } \\ \hline \text { Jan } & 42 & \text { July } & 66 \\ \hline \text { Feb } & 43 & \text { Aug } & 66 \\ \hline \text { Mar } & 47 & \text { Sept } & 61 \\ \hline \text { Apr } & 50 & \text { Oct } & 53 \\ \hline \text { May } & 56 & \text { Nov } & 45 \\ \hline \text { June } & 61 & \text { Dec } & 41 \end{array} $$ (a) Plot the average monthly temperature over a two-year period, letting $x=1$ correspond to January of the first year. Do the data seem to indicate a translated sine graph? (b) The highest average monthly temperature is $66^{\circ} \mathrm{F},$ and the lowest average monthly temperature is $41^{\circ} \mathrm{F}$. Their average is $53.5^{\circ} \mathrm{F}$. Graph the data together with the line $y=53.5 .$ What does this line represent with regard to temperature in Seattle? (c) Approximate the amplitude, period, and phase shift of the translated sine wave. (d) Determine a function of the form $f(x)=a \sin [b(x-d)]+c,$ where $a, b, c,$ and $d$ are constants, that models the data. (e) Graph $f$ together with the data on the same coordinate axes. How well does $f$ model the given data? (f) Use the sine regression capability of a graphing calculator to find the equation of a sine curve that fits these data (over a two-year interval).

Average Monthly Temperature The average monthly temperature (in ${ }^{\circ} \mathrm{F}$ ) in Seattle, Washington, is shown in the table.
$$
\begin{array}{c|c||c|c}
\text { Month } & \text { 'F } & \text { Month } & \text { 'F } \\
\hline \text { Jan } & 42 & \text { July } & 66 \\
\hline \text { Feb } & 43 & \text { Aug } & 66 \\
\hline \text { Mar } & 47 & \text { Sept } & 61 \\
\hline \text { Apr } & 50 & \text { Oct } & 53 \\
\hline \text { May } & 56 & \text { Nov } & 45 \\
\hline \text { June } & 61 & \text { Dec } & 41
\end{array}
$$
(a) Plot the average monthly temperature over a two-year period, letting $x=1$ correspond to January of the first year. Do the data seem to indicate a translated sine graph?
(b) The highest average monthly temperature is $66^{\circ} \mathrm{F},$ and the lowest average monthly temperature is $41^{\circ} \mathrm{F}$. Their average is $53.5^{\circ} \mathrm{F}$. Graph the data together with the line $y=53.5 .$ What does this line represent with regard to temperature in Seattle?
(c) Approximate the amplitude, period, and phase shift of the translated sine wave.
(d) Determine a function of the form $f(x)=a \sin [b(x-d)]+c,$ where $a, b, c,$ and $d$ are constants, that models the data.
(e) Graph $f$ together with the data on the same coordinate axes. How well does $f$ model the given data?
(f) Use the sine regression capability of a graphing calculator to find the equation of a sine curve that fits these data (over a two-year interval).

Key Concepts

Sine Regression

Sine regression is a statistical technique that uses computational tools to fit a sine curve to observed data by optimizing the parameters of the sine function. This method is particularly useful when the periodic behavior of the data is complex, allowing for an empirical approach to modeling real-world cyclical phenomena.

Sinusoidal Transformations

Sinusoidal transformations involve changing the basic sine function through adjustments in amplitude, period, phase shift, and vertical shift. This process tailors the sine function to the specific characteristics of the data, making it possible to closely align the model with observed periodic trends.

Vertical Shift (Midline)

The vertical shift, often referred to as the midline, is the baseline around which the sine function oscillates. It is determined by the average value of the maximum and minimum observed data and represents the central or average level of the phenomenon being modeled, such as a long?term average temperature.

Sine Function Modeling

Sine function modeling involves using sine functions of the form f(x) = a sin(b(x - d)) + c to represent periodic phenomena. This approach is particularly effective for data that exhibit repeating patterns over regular intervals, making it a natural choice for modeling cyclical behaviors such as seasonal temperature changes.

Period

The period of a sine function is the length of the interval required for the function to complete one full cycle of its repetitive pattern. It is a key parameter that describes the frequency of the cyclic behavior, making it critical for accurately modeling natural cycles like seasonal variations.

Amplitude

Amplitude refers to the height of the wave measured from the midline to a peak (or trough), representing half the total range of the oscillation. It quantifies the extent of variation in a periodic phenomenon from its central value, and is essential for understanding how much the data deviate from the average.

Phase Shift

Phase shift is the horizontal translation of the sine graph that aligns the function’s peaks, troughs, and other critical points with the appropriate values on the x-axis. It is used to synchronize the mathematical model with real-world timing of events, ensuring that the model accurately reflects where the cycle begins.

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