8. Give the equation for the sinusoidal best fit line: y = ______ Use the model to answer the following questions: 9. What will be average temperature be on each of the following dates? April 15 (x = 4): ______ June 30 (x = 6.5): ______ November 10 (x = 9): ______ 10. When will the daily average temperature be 45°F? ______ When will it be freezing? ______ 11. On what day(s) will the average temperature be the warmest? ______ 12. Do you think that this model would work for long periods of time, i.e. decades or centuries? Why or why not? 13. Could this function be used to predict the temperature at a specific time on a given day? Explain briefly. 14. What do you see as advantages to the use of this function? Disadvantages?
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(Modeling) Monthly Temperatures A set of temperature data (in ${ }^{\circ} \mathrm{F}$ ) is given in the tables for a particular location. (a) Plot the data over a two-year interval. (b) Use sine regression to determine a model for the two-year interval. Let $x=1$ represent January of the first year. (c) Graph the equation from part $(b)$ together with the given data on the same coordinate axes. $$ \begin{aligned} &\text { Average Monthly Temperature, Buenos Aires, Argentina }\\ &\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c} \text { Jan } & \text { Feb } & \text { Mar } & \text { Apr } & \text { May } & \text { Jun } & \text { Jul } & \text { Aug } & \text { Sept } & \text { Oct } & \text { Nov } & \text { Dec } \\ \hline 77.2 & 74.7 & 70.5 & 63.9 & 57.7 & 52.2 & 51.6 & 54.9 & 57.6 & 63.9 & 69.1 & 73.8 \\ \hline \end{array} \end{aligned} $$
Graphs of the Circular Functions
Translations of the Graphs of the Sine and Cosine Functions
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.
Trigonometric Functions
More General Trigonometric Functions and Models
(Modeling) Monthly Temperatures A set of temperature data (in ${ }^{\circ} \mathrm{F}$ ) is given in the tables for a particular location. (a) Plot the data over a two-year interval. (b) Use sine regression to determine a model for the two-year interval. Let $x=1$ represent January of the first year. (c) Graph the equation from part $(b)$ together with the given data on the same coordinate axes. $$ \begin{aligned} &\text { Average High Temperature, Buenos Aires, Argentina }\\ &\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c} \text { Jan } & \text { Feb } & \text { Mar } & \text { Apr } & \text { May } & \text { Jun } & \text { Jul } & \text { Aug } & \text { Sept } & \text { Oct } & \text { Nov } & \text { Dec } \\ \hline 86.7 & 83.7 & 79.5 & 72.9 & 66.2 & 60.1 & 58.8 & 63.1 & 66.0 & 72.5 & 77.5 & 82.6 \\ \hline \end{array} \end{aligned} $$
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