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Adapt the method of Example 6 to determine the amplitude, period, and phase shift for the given function. Graph the function over one period indicating the $x$ -intercepts and the coordinates of the highest and lowest points on the graph.

(a) $y=3 \cos (\pi-2 x)$

(b) $y=\frac{5}{2} \sin (-4 \pi x-\pi)$

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Numerade Educator

Missouri State University

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So, for part A, we have the equation. Why is equal to three co sign off? Hi minus two x. This equation can be regarded as why is he called a three course sign minus two x less pie. And then we can fact the negative, too. So this would be bicycle to three. Co sign off Negative too. And then x minus high over to. And since co sign is an even function, therefore, full sign off. Negative two times X minus pi over two is the same as the graph off three Coast night off two times X minus. So now this is the graphic is gonna drops. The fourth thing is, we have an aptitude off. Three. Next thing we have the opinion which would be to pie divided like two, and that's it too high. And then we have a face shift off. Hi over to So these will help us draw this craft. So be cool and forced to fall Set up our axes, the amplitude history. So we draw this line here and why is it going to three and then another line act wisely, co two negative. So take care of the consortium for next to Peter despite. So we're gonna divide the Ph into 40 foot parts pi over four I or two. But he pied over four and then high. Now the graph has a face shit off. Kyra, to what that means is that it's starting at the 0.30 We will. 03 We were starting point pi over 23 and going to the right. We will draw D, uh, Coast a car and then we can't that curve to the left side. And now you can actually see that the X intercepts There are two X intercepts, and those X intercepts are at X is equal to pi over four. And also Act X is equal to three pi over four. So that's apart. Now let's take a look at part B. So for part B, we have Why is equal to five over To sign off Negative four pi X minus high and in the same way you say, Why's he going to five or two sigh negative for pipe and then X and then this would be a plus 1/4. So now, since a sign is an odd function, therefore this negative sign because in the front, and this becomes negative. Five over to site off four pi times X plus 1/4. So looking at the equation for thing we see is that the amplitude is 5/2. The P AED is two pi divided by four pi. That's equal to 1/2. And then there's a face shift negative wanna work for. So we're gonna keep these things in mind, as we call it the grass. So we set up our axes, you have 5/2, and then we have negative 5/2. So these are going to be the boundaries off our Y values. And then the Peter is 1/2 so introduced. If I take 1/2 into four parts, so we would have it's and then we have 1/4 and then we have three. It's and then the F 1/2. Now, we also have a face shift off one fort, so you just go negative one for so you just go over in the next direction. So there we have it. Now our graph is ready. You start here, and it has appeared 1/2 So the graph looks something like this. So there is DeGraff off. Fight over to sign. Not this. We did not take care of the negative sign here. So this car right here is why is equal to fight over? To sign off for pie Times X plus one over for now, in order to draw the graph or negative sign, all we need to do is take this graph and reflect it over the X axis. So let's do that. So you gonna set up our axes once more? So I have a line here at five. Over to Manila. Line here at negative five more to and then we have so and then we have one of these 1/4 he needs and 1/2. And now we're gonna reflect this over the X axis. So therapy go. And now here we have to grapple bicycle to negative fight over to sign four pi times X plus one before the X intercepts you and actually see there are three x intercepts and those X intercepts are ecstasy. Put 20 at X is equal to 1/4, and at X is equal to 1/2