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The scatter plot on the next page displays average monthly temperatures for Death Valley, California. (The averages were obtained using daily maximums.) Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. (No table is given; you will need to rely on the scatter plot and make estimates.)

(TABLE CANNOT COPY)

$$y=25\sin\big(\frac{\pi}{6}t-\frac{2\pi}{3}\big)+90$$

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for this plot of the tempo of the average temperatures per month in Death Valley, California We don't have a table but will still be able to use this. Plus on these points to write a periodic function in the form y equals a sign of BT minus seats plus D in order to get an approximation of these temperatures per month. So the only based on previous problems, the only information that we need really is maximum minimum and the period. So our maximum value seems to be here with the coordinate of seven. And let's call that 115th. This is this plot is not completely accurate. It's based on the plot that's in your textbook. Um, so we'll call that about 115. We have a second maximum, but this is the only maximum in one period. We don't need toe list, both of them. Our minimum is down here with coordinates of one. And let's call that 65 and finally our period. Well, this is monthly, so it goes from January all the way through to December. So that's and then it repeats again with January all the way through to December. So our period is 12 because we do have 12 months on this plot. Well, we have 24 but we have 12 months per period on this plot. So that's actually all the information that we need from this plus. So let's remove this, and we'll come back to it later after we find our equations. So let's start by finding a so a is equal to the maximum minus the minimum Y values divided by two. So it's going to be 115 minus 65 divided by 2 50 divided by two, which is equal to 25. So it is going to be equal to 25. Next for B B is based on the periods of the period is equal to 12 and the period is always equal to two pi divided by B. So we'll have 12 b equals two pi, which dividing both sides by 12 will get B equals pi over six. Now we're going to find the sea, which is the shift variable of our equation, some in in a normal sign curve that doesn't have any shift. If it has a period of 12 that means we should expect a maximum at 1/2 that at 1/4 of this period. So at three we have a maximum at seven those So that's add shifting it by four, adding four to make it equal to seven. So our see value is going to be equal to four times I would be variable so four times pi over six. Now let's see what we have so far of this equation we have y equals A, which is 25 time sign of be just pi over six times T minus sea, which is two pi over three plus de, which is still unknown. So lets you find this. Let's find the D variable, which is the vertical shift, every confined. But by plugging in one of thes very one of these sets of points over here, let's choose 1 65 just cause putting one equal. The tea will be a little easier than putting seven in for teens. So let's have 65 people's 25 time sign of pi over six minus four pi over six plus de so inside of this sign, we have pi over six minus four pi over six, which is going to be equal to negative 1/2 negative sign of negative 1/2 is negative one. So 65 equals 25 times negative one plus de. And then, if we add 25 to boat size, will get 90 people's D. So our solution curve is going to be equal to why equals 25 for a time sign of B, which is pi over six times t minus to pry over three plus de, which is 90. So let's grow down a bit and look at what this looks like on the plot that we had. So I plotted this onto Dez most over the plot that we already had, and here it is, and it actually does match up fairly well with these temperatures that were given to us.