Temperature The table shows the normal daily high...

Temperature The table shows the normal daily high temperatures for Miami $M$ and Syracuse $S$ (in degrees Fahrenheit) for month $t,$ with $t=1$ corresponding to January. (Source: National Oceanic and Atmospheric Administration) $$ \begin{array}{|c|c|c|c|c|c|c|}\hline t & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline M & {76.5} & {77.7} & {80.7} & {83.8} & {87.2} & {89.5} \\ \hline S & {31.4} & {33.5} & {43.1} & {55.7} & {68.5} & {77.0} \\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|c|c|}\hline t & {7} & {8} & {9} & {10} & {11} & {12} \\ \hline M & {90.9} & {90.6} & {89.0} & {85.4} & {81.2} & {77.5} \\ \hline S & {81.7} & {79.6} & {71.4} & {59.8} & {47.4} & {36.3} \\ \hline\end{array} $$ (a) A model for Miami is $$ M(t)=83.70+7.46 \sin (0.4912 t-1.95) $$ Find a model for Syracuse. (b) Use a graphing utility to plot the data and graph the model for Miami. How well does the model fit? (c) Use a graphing utility to plot the data and graph the model for Syracuse. How well does the model fit? (d) Use the models to estimate the average annual temperature in each city. Which term of the model did you use? Explain. (e) What is the period of each model? Is it what you expected? Explain. (f) Which city has a greater variability in temperature throughout the year? Which factor of the models determines this variability? Explain.

Temperature The table shows the normal daily high temperatures for Miami $M$ and Syracuse $S$ (in degrees Fahrenheit) for month $t,$ with $t=1$ corresponding to January. (Source: National Oceanic and Atmospheric Administration)
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline t & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline M & {76.5} & {77.7} & {80.7} & {83.8} & {87.2} & {89.5} \\ \hline S & {31.4} & {33.5} & {43.1} & {55.7} & {68.5} & {77.0} \\ \hline\end{array}
$$
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline t & {7} & {8} & {9} & {10} & {11} & {12} \\ \hline M & {90.9} & {90.6} & {89.0} & {85.4} & {81.2} & {77.5} \\ \hline S & {81.7} & {79.6} & {71.4} & {59.8} & {47.4} & {36.3} \\ \hline\end{array}
$$
(a) A model for Miami is
$$
M(t)=83.70+7.46 \sin (0.4912 t-1.95)
$$
Find a model for Syracuse.
(b) Use a graphing utility to plot the data and graph the model for Miami. How well does the model fit?
(c) Use a graphing utility to plot the data and graph the model for Syracuse. How well does the model fit?
(d) Use the models to estimate the average annual temperature in each city. Which term of the model did you use? Explain.
(e) What is the period of each model? Is it what you expected? Explain.
(f) Which city has a greater variability in temperature throughout the year? Which factor of the models determines this variability? Explain.

Key Concepts

Sinusoidal Functions

Sinusoidal functions are periodic functions that model phenomena with repeating patterns over time, such as seasonal changes. They are characterized by parameters that control the amplitude, period, phase shift, and vertical shift. These models help in understanding, visualizing, and forecasting patterns that repeat in a predictable cycle.

Amplitude

Amplitude refers to the distance from the midline (average value) to the maximum (or minimum) value of the periodic function. It indicates the degree of variation in the data and is critical in determining the extent of fluctuation in phenomena such as temperature changes throughout the year.

Period

The period of a sinusoidal function is the length of time required for the function to complete one full cycle. It is determined by the coefficient of the variable inside the sine (or cosine) function. Understanding the period helps in matching the model to the cyclic nature of the phenomenon being studied, such as a year’s seasonal cycle.

Phase Shift

Phase shift is the horizontal displacement of the sinusoidal function. It represents how much the cycle is shifted left or right on the time axis relative to a standard sine or cosine function. This parameter is essential for aligning the model with the start of the cycle observed in real data.

Vertical Shift

Vertical shift is the upward or downward movement of a sinusoidal function, which adjusts the midline of the model. In the context of phenomena like temperature, the vertical shift typically represents the average or baseline level around which the cyclic variations occur.

Model Fitting

Model fitting involves adjusting the parameters of a function so that the curve closely matches observed data points. In the study of periodic behaviors, graphing utilities are often used to visually assess how well a sinusoidal model represents the actual data, allowing for refinements in amplitude, period, phase shift, and vertical shift.

Average Annual Value Estimation

The estimation of the average annual value from a sinusoidal model is derived from the vertical shift of the function. This coefficient represents the constant component of the model and reflects the overall average level of the phenomenon across its cycle.

Variability in Data

Variability refers to how much the data deviates from the average value over a cycle. In sinusoidal models, the amplitude is the factor that directly determines this variability. A higher amplitude indicates greater fluctuation from the average value, which can be crucial in understanding and comparing different datasets.

Transcript

00:01 Okay, we're going to go ahead and talk about developing a sign model to represent some data.

00:10 And so we know that a sign function is typically given by a, where a is what is called the amplitude, is the distance from some horizontal line to the highest point or the distance from the horizontal line to the lowest point.

00:29 B is known as the period and we know that a period or is part of a period so period is that it actually equal to 2 pi divided by that b value c is what is called a phase shift or a horizontal shift and d is known as a vertical shift and in real life applications is also known as the average value.

01:11 And this is going to give that horizontal line that is the distance from that horizontal line or average value to the highest point is going to be defined as your amplitude.

01:24 Okay.

01:25 And so our table represents months from january being one to december being 12.

01:32 Of some average temperatures or temperatures in the state of maine, i believe, or miami, average temperatures from january to december for miami versus the temperatures in syracuse.

01:50 And these are given in degrees fahrenheit.

01:52 And we were told that the sign function that represents miami is given by 83 .7.

02:03 Plus 7 .46 times sine of 0 .4912t minus 1 .95.

02:10 And we are asked to find the model that represents the syracuse temperature.

02:18 Okay.

02:19 So if we actually look at this, one way to do it is in order to actually find the amplitude, the amplitude is going to be equal to the highest minus the lowest value.

02:36 This is just going to be a quick and easy way divided by two.

02:41 Okay.

02:43 And so if i actually look at my syracuse, i notice that my lowest temperature is 31 .4.

02:50 My highest temperature, if i scan through here, is 81 .7.

02:55 So this is going to be 81 .7 minus 31 .4.

03:00 And we're going to divide by 2.

03:02 And so that is going to be 81 .7 minus 31 .4 and we're going to divide by two.

03:11 And so that is going to be 25 .15 sign and then we're going to have some values in here.

03:20 Okay.

03:20 Now, what do i need to add to 25 .15? so we're going to have my highest value is equal to, some value plus the 25 .15.

03:39 And we know the highest value is 81 .7 minus 25 .5 .15.

03:46 And that is going to give me this x value or what is also called my vertical shift or my average values.

03:53 So if i take 81 .7 and subtract 25 .15 from it, i get 56 .55.

04:02 Okay, so now if i take 56 .55 and subtract 25 .15, i should get my lowest value of 31 .4.

04:15 So that's a quick check on how to do that.

04:18 Now i want to know what my b value is, and i know my b is my period.

04:24 And we would expect our period to be 12 months, right? so we would expect our period to be 12.

04:31 And so that should equal to 2 .5 .5.

04:34 Divided by b and so b is going to equal pi over six okay and if we want to put it in terms of decimals we do pi divided by six so that's point five two three six so this is zero point five two three six t and now i need some shift value and so what i do know for a trig function that if i do pi over six times, and now my highest value occurs at seven, and that should equal pi over two, right? because i know typically a sign function the highest point occurs at pi over two.

05:31 So if i do that, i know c is actually going to be equal.

05:36 So if i do seven, so if i do pi over two, and i subtract 7 pi, i get negative 2 .0944.

06:00 So this should be minus 2 .0944.

06:08 So that is now my model that i think represents the syracuse temperatures.

06:18 So those are the first things we want to do.

06:25 And now what we want to do is go ahead and graph our data.

06:32 So we actually want to plot the points for both miami and syracuse, and we actually want to graph the two equations that we modeled the temperatures for.

06:45 So i'm going to change to a online graphing utility.

06:49 And then i've kind of got to...

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