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Stalc the domain and range of Graph y arccos anusate 48 Graph y sin rand of the function_ the function 51. For what value of x does sln sin x Use 50. Graph one cycl...

Question

Stalc the domain and range of Graph y arccos anusate 48 Graph y sin rand of the function_ the function 51. For what value of x does sln sin x Use 50. Graph one cycle ofy tan) xand state the domaln graphing calculator to approximate the WCr: and range of the function; 52 For whal value 0f x does cOS cos 4 Use graphing calculator Lo approximate the answer

stalc the domain and range of Graph y arccos anusate 48 Graph y sin rand of the function_ the function 51. For what value of x does sln sin x Use 50. Graph one cycle ofy tan) xand state the domaln graphing calculator to approximate the WCr: and range of the function; 52 For whal value 0f x does cOS cos 4 Use graphing calculator Lo approximate the answer



Answers

In Problems 35-58, graph each function. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function.
$$ y=4 \cos x $$

So here, asked Graff. Why is Michael to our cause and effect? So the inverse, of course, that's already have here graft, white clothes and exit. It's important to know because you would realize that. Why was Arkle science? It is not a function because if you draw a horizontal line through this there too many intersections. So the investors basically wth e x value swishing the why about in the library so should exploit. So if you trust us around, basically the function would end up looking like this. Sort of if he had it like this, like a few twisted around or it's like in the opposite off and s so if you draw a vertical line or if you drill cars onto my make me. But if you drew a vertical line, you'd see that they're to me intersections, so this necessary wouldn't be a function. So what happened? Institutions have aside, Is that too limited to mean thio? Just negative one and one. So we have it would be you limited sum. So they said that this here be the domain. So what we have to do to graft this now would be to find us So here we have, um, 01 So, basically for the inwards who just committed to 10 to swish it around here. This is pie or two zeros. Pirate ooze about 157. So want, but 57 0 you to switch. Arrive. You get zero in one point due soon, then here three pi over too. Um, you say, don't worry about three badges of pies. Stupid one for So this would be. Now, if this is this point, here is 3.141 So we just switched around. You get maybe one and 314 about. So now we can graft these values so that he have 10 So this is 1 57 pirated. So I'm just been around a year. So this is 10 and then we have 10 and one. He said so that would be around here somewhere. I see your warrant do seven. And then we have people one for and won. So abuse and we're here. So we have 3.14 and negative one. So it would be NATO one and do bowling for. So we have negative one here and appear 71 for simplicity. Native one and one for Graffin from She's on long enough. So we have that and then we collect it. Sort of flake, um, and s without e ends. So this is the graph off. Why is equal to this? Is the bath wise? We are co sign. That's so the domain that we know now, what do you mean is negative 1 to 1? Because if it wasn't, then it wouldn't be a function. And the range would have to be The range would have to be zero to pie because, as you can see here, right here, Thio here would be zero.

So her problem 40 we have H. F. Say to yeah which is equal to negative five thirds sign. You know, over to with this function of mind we want to plot values that we have so we can put in um theater of zero and get 92 5. Make them flat. Um However to yeah pie and get zero and then two pi give us a negative positive five. With these guys in mind we see we're going to have this sinus white old pattern that's going to continue so it's going to have all these points involved. Um And then we see the domain is going to be all real numbers but the range will always be between positive five and negative five.

Question 40 asked you to grab the function H of data equals negative five co sign of data over to, um then use a graphing utility to verify that graph as well as find the domain and range. Um, so then first, what we're gonna do is plug in values of data to get h of data, which would be on our y axis. Um, so first, starting off at negative three pi is zero negative. Two pi is positive. Five negative pi is zero zero would be negative. Five positive pi would be zero two pi would be five and three pi zero. Um, so now putting this on our grass going to positive three pi on the X axis here and negative three pi Ah, and then on the y axis were just ranging from negative five 25 Now plotting those points Negative three pi zero negative two pi five pie back to zero. Negative. Five is at zero. Hi. Would be zero two pi back up to positive five and three pi back down to zero. So just connect those points like a regular co sign function should look like so now, looking at our domain That would be all real numbers for which our function is defined. Um, and any X value will give us a defined function, so that would be negative. Infinity to infinity is our domain. Our range would be those associated y values are Function never goes below negative five and above five. But it does include those. So bracket negative five 25 would be a range, um, and then verifying that on a graphing utility. I used a t i 84. Go to your y equals set. Why? 12 negative five co sign of X over two. Go to your window. I set my X men as negative three pi x max. Two. Positive through pie. Why men? Two negative five and why? Max 25 And you should get a graph that's identical to this. And that's your answer for question. 40

So in this problem, we have five minus three Sign of shoe X, which I'm not crazy about the formats. I'm gonna rewrite that. Why equals negative three sign of two X plus five. Now it's much easier to see that my amplitude is three. I'm reflecting it over the X access. I'm shifting it up. Five units. The midline would be Y goes five. My frequency is too meaning that I'm gonna be able to complete two full cycles in a standard cycle. So in order to graph this, I am going to work with the parent function in order to help us to find the pattern. And then it will be easy to grab the parent function. Here is the sign of acts in this particular case. Remember that the sign of X starts on the line ends on the line 00 pi over 21 py 03 pi over to negative one, etcetera, etcetera, etcetera Can we're just looking for the path. Everything on the outside affects the output. I'm stretching and vertically by a factor of three, reflecting it over the X axis and then shifting at up five units. So I multiply all of my output values by negative three and add five that will get us our new location of all of our Y values. Zero times negative 30 plus five is five negative. Three plus five is chu zero plus five is 53 plus five is eight, etcetera, etcetera, etcetera. We can see that our highest point will end up occurring at eight. Our lowest value will occurred to our mid line's gonna occur at 512345 And we already knew that. Based on this value right here, labeling my ex access, we're gonna have to work a little bit more on our values. Here on the inside of my function, it says, Multiply all the exes by two. But since it's on the inside of my function, I'm going to divide by two into the opposite operation while climbed by. 1/2 will accomplish the same task that will give me zero hi over four. And that's actually enough right there, because I'm gonna continue to add by pi over for every time. Two pi over four is the same as pi. Over two. The reply over four etcetera, etcetera, etcetera Those will give me my ex values so zero pi over four two pi over four three pi over four four pi over four five pi over four six pi over four seven pi over four eight pi over four. So you'll see that it's gonna take to pie to complete two cycles whereas it used to could take two pied a complete only one cycle. Now remember, Sign is going to start on the line. Go up middle down middle unless it's the opposite which it is here. So it's going to start on the middle. It's gonna go down and these points over here will help you as well. It's gonna go down to its lowest point back to its middle up to its highest point back to its middle, low, middle, high, middle. And you'll see that we have an upside down sign. Kurt, you can see that the maximums occur at three pi over 48 and seven pi over 48 The minimums occur at pi over 4 to 5 pi over 42 etcetera, etcetera, etcetera. The mid lines at Y equals five. We have our amplitude of three above and below the midline, right? No X intercepts the domain and the range here, the domain. The X values go from negative infinity to infinity, the Y values. You can see those from the graph here. Go from a positive to up to a positive eight. They will never go outside of that range.


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