Hello wonderful mathematics people, This is Anna Cox from Kellogg Community College. Y equal a sine, the quantity BX plus C + d and Y equal a cosine. The quantity BX plus C + d actually act very similar. Starting with the original graph of sine or cosine, the absolute value of the A tells us the amplitude, or the highest and the lowest the graphs ever going to go. The new. Is the original. Which is 2π divided by the coefficient on the X term, so 2π / b. The phase shift is the negative C / b and the vertical translation is that plus D At the end for cosecant and secant, the A is actually going to tell us our range and we're going to go negative, Infinity to negative. That absolute value of A union the absolute value of A to Infinity. The reason it needs to be an absolute value is sometimes the a coefficient could be negative. That negative just tells us it's a reflection around the X axis. The period for cosecant and secant is 2π, so our new. Is going to be 2π divided by the coefficient on the X, which is that B, the phase shift negative C / b, and the vertical translation up and down by D That plus at the end for tangent and cotangent. We're going to start with the original tangent and cotangent graphs. The A is going to tell us a vertical stretch or shrink. If we think about the tangent of π force being one, then if there's an A in front, that tangent of π force is now going to be 1 * A or whatever the A was. Same thing for cotangent. Cotangent of π force is 1. So one times the A is going to be our stretch or shrink. The period for tangent and cotangent is π. So now we have π / B. The phase shift is negative C / b and the vertical translation is D. Fitting data to a sinusoidal curve Y equal a sine the quantity Omega X -, v + b. The amplitude is just the largest minus the smallest divided by two. The vertical shift is the largest plus the smallest divided by two. The period is the new. Which equals 2π divided by Omega. So Omega could be thought of as equaling 2π / t The phase shift is Omega X minus Phi equals 0. So if we add Phi to each side and divide by Omega, we get X equal Phi over Omega. So Phi over Omega is the phase shift. If we look at an example, our example has given U.S. data for the 12 months of the year. So we're going to look at the 12 months of the year. We're going to look for the highest, the highest being 56.2, the lowest being 25. So I'm going to take the highest minus the lowest and divide by two, and that's my amplitude. Then I'm going to do the vertical shift, which is the highest plus the lowest divided by two and get 40.6. We know that there are 12 months of the year and then it's going to repeat. So our new. Is 12. That equals the original period of 2π divided by Omega. So Omega is π six. Now the phase shift is just Phi divided by Omega and the phase shift is going to be 4 because on a tan or on a sine graph, remember we have to look for the middle point. So if here's my highest and here's my lowest, my lowest, and my highest, that 0 concept comes halfway between the lowest and the highest. So halfway between the lowest and the highest would be month 4. So we knew the Omega was Pi 6, so we're going to solve for the Phi multiplied by Pi 6 and we're going to get Phi equal to π / 3. So now literally we just plug in all those numbers Y equal the amplitude 15.6 sine the quantity the Phi no the Omega Pi 6X minus the Phi of 2π thirds plus the vertical translation of 40.6. Now if we wanted to actually look at that on our graphing calculators, we would put in to our graphing calculator under statistics and edit. And I've inputted the data in my list 1 and list 2 already. Then we're going to do a second quit and go back into statistics. And now we want to do a calculation. So arrow over one and the calculation. If we arrow down, we want to look for sinusoidal calculation and that should be letter C Now if I tell it I want To Do List 1 as my input, list 2 as my output, If I want to put it directly into my graph, I go to the vars key, I arrow over to Y vars and hit the function and I'm going to put it in Y1. Now when I hit enter, it comes up with the best fit. Now remember, that doesn't mean it's going to be what we found. This is what's the best fit. So we have the sign regression best fit. It gives us our ABC and D If we go into Y equal, it's put that equation in for us already. If we do a second Y equal, which is a stat plot, we're going to turn this first stat plot on. So I want to turn it on. Now if we do a zoom and a fit, zoom fit, so I'm going to arrow down. I'm going to arrow and tell. It tells me zoom fit. The computer's going to catch up any minute. Zoom fit I believe is 0 and we hit enter. Oh, and now the computer's frozen. If you hit graph at this point, you should be able to see the original graph and the little squares of the plots that we just did. Thank you and have a wonderful day.