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 = x + (1/10) sin 30x

Accepted Answer

Welcome back, everyone. In this problem, we want to sketch the graph of Y E equals 5 multiplied by the cosine of 15 X minus qua quarter of pi plus 2. And here we have a graph on which we will plot. Note that the window should give a picture of the overall behavior of the function. Now, before we plot, OK, let's try to rewrite our function in the general form because that can help us to tell features of our graph. Recall That the general form of the sine function. Y is equal to A cost multiplied by the cost of B X minus C plus D. Now, in this case, for a function here, Y equals 5, cost of 15 X minus quarter of pi plus 2, we can factor out 15 from the argument of our function to put it in the general form. So now, when we do that, we should get Y to be equal to 5 multiplied by the cosine of. 15 multiplied by x minus. A 60 of pay. Plus 2. And know by the general form this implies that a equals 5 B equals 15. C equals a 60th of. And d equals 2. Now what do all of these values tell us about our graph? Well, for our midline, the equation of our midline will be Y equals D. So in this case, this means our midline is Y equals 2. OK? So now, for a function, we can draw a horizontal line at the point where Y equals 2. And I'll just put that as a dotted line to show that it's the midline, OK? So that's what he was able to tell us. Next, the amplitude is the absolute value of a. So in this case, our amplitude will be the absolute value of A, which means it's the absolute value of 5. So that means our graph oscillates 5 units above and below the midline. Therefore, our maximum value is going to be equal to the value of our midline 2 plus our maximum value of 52 plus A, which equals 7. And our minimum value will be equal to our midline 2 minus A. So that's 2 minus 5, which equals -3. Thus, 77 is the maximum value of our graph or maximum value of Y, while negative 3 is the minimum value of Y. Next, B can help us to find the period of our graph because the period is equal to 2 pi divided by b. So in this case it's going to be 2 pi divided by 15. In other words, we have points along the x axis are going to be spaced by 2/15 of pi to indicate 1 full cycle. And now finally our phase shift is equal to C. So this tells us then that the phase shift is a 6060 of pi. In other words, our graph Y is shifted 1/60 of pi. To the right. Now that we have all of these features, we just need to determine the 5 key points for us to plot one full cycle of our sign sign function within the viewing window. And that means we're gonna find our maximum and minimum points and then or where our midlines are for our function, OK? So for our graph, let me put this in red to the side, 4 or graph, OK. Starting with our maximum. At the maximum Oh, which is the starting point, X is going to be equal to a 60 of pi, OK? And Y is gonna be equal to 7, as we said earlier. Next, for midline at the descending point. OK, so it's just right, descending here. Then X is going to be equal to a 60 of pi plus a quarter of the period, so a quarter of 2 pi divided by 15, which is going to be equal to 1/20 of pi. OK. So X equals 1/20 of pi at the descending midline and that Y will be equal to 2. Next, we know that for our minimum. OK, then X will be equal to a 60 of pi plus a half of 2 pi divided by 15. In other words, we are halfway through the cycle. That will give us 7/60 of pi, OK? Andy will be equal to -3. OK, so minimum Y equals -3 like we had said earlier. Next for our midline ascending. OK. Then X is gonna be equal to a 60 of pi plus 3/4 of Y. And that. We'll simplify the 7/60 of pi. And now I realized I, I made an error here. Let me just fix it. Where the X value for or minimum instead should actually be 1125th of pie. My apologies there. So that's 1 12th of pi. And now going back to our ascending midline, its x value is 7/60 of pi, which means that Y equals 2. Finally, Our last maximum value. No, this one is completing the cycle. OK. Is going to be equal to the X value is gonna be a 60 of pi plus 1 multiplied by 2/150 of pi. And that gives us 3/20 of pi, while y is gonna be equal to 7, the maximum value. Know that we have these X and Y values for our graph, we can go ahead and plot. So now when we go back to our graph, do you remember all of those values that we found? Well, remember that we said that a 60 of 1/60 of pi, which is gonna be here, and 7 are going to correspond. So we'll have that point up here, OK? Where Y equals 7. OK. Next, we know that we have 1/20 of pi that corresponds with Y equals 2, so that point will be right here. OK. And then 1/12 of i or minus 3. So now we go down to -3 and 1/12 of pi. It's gonna be a little bit outside of, or a little bit greater than 0.25. OK. So that's a 1/2 of -3. Then we have 760 of 5 and 2, which should be right here. OK. So you know what, let me put that on the outside. 760 of pie and 2. And then finally, we are going to have 3 20ths of pie, which is a little bit greater than 0.45 and 7. OK, so that one is going to be right here. And let me put this one above. Know that we have these points, we can plot our curve. So starting from our midline, you know our curve is gonna come in here, then go down here, OK. And it's gonna look something like that. And it's just a sketch. So thus, based on our graph, this is a sketch of Y. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Graphing Functions

Periodicity and Amplitude

Linear and Trigonometric Combination

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