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Problem 79 Hard Difficulty

Temperature Data The normal monthly Fahrenheit temperatures in Albuquerque, NM, are shown in the table below (month $1=$ Jan, month $2=$ Feb, etc.):
Model the temperature $T$ as a sinusoidal function of time, using 36 as the minimum value and 79 as the maximum value. Support your answer graphically by graphing your function with a scatter plot.

Answer

$T=21.5 \sin \left(\frac{\pi}{6}(t-7)\right)+57.5$

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Video Transcript

All right, this question. We're gonna be modeling some signing. So functions were asking too. We're being asked to model the temperature T as a Sinus. Total function of time. Using 36 is the minimum, and 79 is the maximum. Never going to grab our function with these guys What you're given hear. So first, really, for the amplitude, the amplitude is going to be equal to half of the vertical distance during the minimums and the maximums. We're gonna calculate this with our minimum value of 36 our maximum you 79 Jake 79 minus 36. We divide that by two Megan amplitude of 21.5. So let's start drawing our son. You settle function. We're doing it as a mop. Is a model of the temperature capital T as the function of time. Little tea, sir. Right. This is a big T little city and write in our amplitude of 21 point size and I'm going to use co sign for the Maya graph because co sign starts at a maximum at X equals zero. Next, we're going to look at our phase shift. That's what comes here. So I'm saying that I need to obtain a maximum at seven x at X equal seven because you see that it gets larger, it gets to seven, and then it starts decreasing. So using my ex value of seven, I'm going to place that in my face shift. And lastly, we need to calculate our vertical translation s so you can tell that this is not going to zero. So we know it's going to be. We're adding something and moving our graph up, and we're going to catch it or face shift the opposite way. They calculate amplitude or simply way honestly, we take a R two minimums and let me take a minimum and maximum our maximum of 79 in our minimum of 36. We add them together. Do you find the half way point between them? We divided by two, and that would get us a value of 57.5 as our vertical translation up. I have run a fruit so we're adding 57.5 to our sign away. Our coastline live, and that is our final answer. That is our function of temperature as a function of tea, Sinus motor function and just check. I have made a scatter plot with this equation. I mean, see that it lines up pretty well.

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