Average Monthly Temperature The average monthly temperature (in ^∘ F ) in Phoenix, Arizona, is

step 1 Hello, this is instructor Nicole. So first of all, the average annual temperature, well if you t

Average Monthly Temperature The average monthly temperature...

Average Monthly Temperature The average monthly temperature (in ${ }^{\circ} \mathrm{F}$ ) in Phoenix, Arizona, is shown in the table. $$ \begin{array}{c|c||c|c} \text { Month } & \text { 'F } & \text { Month } & { }^{\circ} \text{F} \\ \hline \text { Jan } & 56 & \text { July } & 95 \\ \hline \text { Feb } & 60 & \text { Aug } & 94 \\ \hline \text { Mar } & 65 & \text { Sept } & 88 \\ \hline \text { Apr } & 73 & \text { Oct } & 77 \\ \hline \text { May } & 82 & \text { Nov } & 64 \\ \hline \text { June } & 91 & \text { Dec } & 55 \\ \hline \end{array} $$ (a) Predict the average annual temperature. (b) Plot the average monthly temperature over a two-year period, letting $x=1$ correspond to January of the first year. (c) Determine a function of the form $f(x)=$ $a \cos [b(x-d)]+c,$ where $a, b, c,$ and $d$ are constants, that models the data. (d) Graph $f$ together with the data on the same coordinate axes. How well does $f$ model the given data? (e) 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 Phoenix, Arizona, is shown in the table.
$$
\begin{array}{c|c||c|c}
\text { Month } & \text { 'F } & \text { Month } & { }^{\circ} \text{F} \\
\hline \text { Jan } & 56 & \text { July } & 95 \\
\hline \text { Feb } & 60 & \text { Aug } & 94 \\
\hline \text { Mar } & 65 & \text { Sept } & 88 \\
\hline \text { Apr } & 73 & \text { Oct } & 77 \\
\hline \text { May } & 82 & \text { Nov } & 64 \\
\hline \text { June } & 91 & \text { Dec } & 55 \\
\hline
\end{array}
$$
(a) Predict the average annual temperature.
(b) Plot the average monthly temperature over a two-year period, letting $x=1$ correspond to January of the first year.
(c) Determine a function of the form $f(x)=$ $a \cos [b(x-d)]+c,$ where $a, b, c,$ and $d$ are constants, that models the data.
(d) Graph $f$ together with the data on the same coordinate axes. How well does $f$ model the given data?
(e) 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

Regression Analysis

Regression analysis is a statistical process used to estimate the relationships among variables. When applied to periodic data, sine or cosine regression techniques help in fitting a function to the observed data points, minimizing the differences between the model and the actual data. This approach provides a mathematical model that can predict future behavior based on historical trends.

Vertical Shift

Vertical shift is the amount by which a periodic function is moved up or down on the graph. This shift usually corresponds to the average value of the data, ensuring that the oscillations of the periodic function occur around the correct central value. It is a key component in accurately modeling the overall trend of the data.

Phase Shift

Phase shift is the horizontal translation of a periodic function and determines where the cycle begins along the time axis. It is crucial when modeling data that does not start at a standard position in its cycle. Adjusting the phase shift in a cosine or sine function allows the model to align correctly with the observed data.

Graphical Representation

Graphical representation involves plotting data points and the corresponding mathematical model on a coordinate system. This visual depiction is essential for comparing the model with the actual data, assessing the model’s accuracy, and communicating the underlying trends effectively. It is a crucial tool in both exploratory data analysis and in validating model assumptions.

Period and Frequency

The period is the length of one complete cycle of a periodic function, while frequency is the number of cycles that occur in a unit of time. In temperature modeling, knowing the period (commonly one year) is essential to accurately represent the timing and repetition of seasonal changes. The relationship between period and frequency is given by the equation b = 2?/Period in models using trigonometric functions.

Periodic Functions

Periodic functions are mathematical functions that repeat their values at regular intervals, known as periods. They are particularly useful in modeling phenomena that are cyclical in nature, such as seasonal temperatures. Understanding periodic functions helps in capturing the inherent cyclical behavior of real-world data.

Average (Mean) Calculation

This concept involves finding the arithmetic mean of a set of numerical values. In the context of temperature over months, the average annual temperature is determined by summing the monthly temperatures and dividing by the number of months. It is a fundamental statistical measure that helps summarize a large data set into a single representative value.

Amplitude

Amplitude represents the half of the vertical distance between the maximum and minimum values of a periodic function. In practical applications like temperature modeling, the amplitude indicates the extent of fluctuation from the average value. It is a key parameter in determining the shape and spread of the cosine or sine model.

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