Finding a Viewing Window

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Question

Finding a Viewing Window

In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.

y = 3 cos 60x

Accepted Answer

Welcome back, everyone. In this problem, we want to sketch the graph of Y E equals 4 of 20X minus 1/3 of pi. Note that the window should give a picture of the overall behavior of the function. Now, if we're going to plot our function on our graph, it would be best to understand the components of our function and how it relates to our graph. What do we know about the sign function? We'll recall that the standard form of assign function is a sign of B multiplied by X minus C plus D. OK, where each of these A, B, C, and D, each of these values help us to figure out different features of our graph. So let's first transform our function and then try to figure out what each part means. So for our function, Y equals 4 of 20 X minus 13 of pi. OK. If we factor 020 from the argument, OK, then we're gonna get Y to be equal to 4 multiplied by 4 multiplied by the sign of 20 multiplied by X minus 1/60 of pi. And no, this tells us then that A equals 4, B equals 20, C equals a 60 of pi, OK, and D equals 0. There's no constant being added to our sine function. So now what does all of this mean? Well, first, the midline, OK, is given by Y equals D. So in this case, the midline of our function, D is gonna be equal to 0, OK? So that means we can draw a horizontal line at Y equals 0. So, I should have, instead of saying D equals 0, we know D is 0. Let me say here that Y equals 0. So let me put that in blue, OK. I thought about it. I'd actually write this in blue as well, OK. So this is the midline. Let me just make this a dotted line here, OK? Now, what else? Well, we also know that the amplitude is the absolute value of A, OK. So this means then that the amplitude of our function, OK. The maximum value of our function. Is 4, OK. The absolute value or amplitude is 4. So that means the graph oscillates 4 units above and below the minimum and below the midline. In other words, the max value of our function is 4, and the minimum value of our function is -4. Now, next we can start to think about the period of our function. And recall that, let me come to the left. The period is equal to 2 pi divided by B. So B helps us to find a period. Thus this tells us then that our period. Is going to be equal to. OK, our period is going to be equal to 2 pi divided by B which is 20, which is gonna simplify to 10th of pi. So hence we mark along the x axis, mark points along the x axis that are spaced by 11th of pi to indicate one full cycle. And now finally C helps us to find our phase shift because the phase shift. Is equal to C. So this implies then that the phase shift of our graph is equal to 1/60 of pi. Now that we have all this information, we can determine 5 key points to help us plot. From the starting midline to the ending midline, OK? And then we're gonna also think about the maximum and minimum points, OK, along our cycle. So no for one full cycle. OK. At the starting midline. Starting midline, OK, where it's ascending, X is going to be equal to a 60 of pi, and when X is a 60 of pi, then Y or function will be equal to 0. Next, for our maximum value, OK. Remember that we said that Y would be equal to 4 and X is going to be equal to 60 of pi plus 250 of pi or quarter of the way through our cycle, and this is going to be equal to 1/24 of pi, OK. So at our maximum value, Y equals 4 and X equals 124th of pi. Next, for our minimum value, OK. Then X is, well, before we get to our minimum, we'll get to our midline. My apologies. And now at our midline descending, then X is gonna be equal to a 60 of pi plus a 1/2 of 10 of pi, which is gonna be equal to 1/15 of pi and Y equals 0. Now for the minimum, OK, then X will be equal to a 60 of pi. Plus 3/4 of a 10 of pie. Which would be equal to 11 pi divided by 120, OK. And why, as we said earlier, the minimum value is -4. And then finally, let me write this to the right because I don't have any space for our ending midline. Then X is gonna be equal to a 60 of pi plus 10th of pi, which equals 7/60 of pi. While Y would be equal to 0, OK? So now, what does this tell us? Know that we have these 5 key points, it can help us to draw a continuous, smooth and rounded curve through them to complete one full cycle of the sine wave. So now when we go back to our graph, OK. Then like we said. We know that when X equals a 60 of pi, Y equals 0. OK, so uh we can put that here. Let me put that in red. So that would be somewhere here. OK, are we X equals 0.5, sir. Uh, when X equals 124th of pi, which is going to be between 0.10 and 0.15, then X will be 4. OK, so we'll have that point here. When X equals a 15th of pi, which is just a little bit outside of 0.2, Y is 0. When X equals 11 pi divided by 120, which is a little bit close to 0.3, then Y equals -4. And then when X equals 7/60 of pi, which is a little bit outside of 0.35, Y equals 0. And again, we, we know here that at negative for what our value will be. So now, if we go ahead and plot, then our curve, OK, let me try to draw this as smooth as possible. Our curve should look something like this, OK? What I should have been doing was putting the points as I was going along, but our curve is going to look something like that. I know this is going to be the graph of Y equals F of X. Well, it's just Y, the graph of Y, which is for sin of 20 X minus 1/3 of pi. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Periodicity of Trigonometric Functions

Amplitude of Trigonometric Functions

Graphing Software Viewing Window

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