Problem

For the following exercises, graph two full perio…

02:58

Problem 10 Medium Difficulty

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum $y$ -values and their corresponding $x$ -values on one period for $x>0 .$ Round answers to two decimal places if necessary.
$$
f(x)=2 \cos x
$$

Answer

$\begin{aligned} \operatorname{Max}(x, y) &=\left(\frac{\pi}{2}, 4\right) \\ &=(1.57,4) \\ \operatorname{Min}(x, y) &=\left(\frac{3 \pi}{2},-4\right) \\ &=(4.71,-4) \end{aligned}$

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Chapter 6

Periodic Functions

Section 1

Graphs of the Sine and Cosine Functions

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Video Transcript

Hi, everyone. So today we're gonna be looking at the function F of X is equal to two co sign of AKs and just analyzing it. So before analyzing, we should graft this function first. It is the same as a normal co sign of X function, just with the vertical scale of two evidence by this to outside of the coastline right there. All that means is that instead of every single point on this function will just be two times higher or stretched by two vertically than it was before. So, just like a normal function coastline function, we can start at the top and then at zero, it'll be at two down, zero negative, too. Back up, on and right, there is the end of the first period of the graph, meaning that after that point, this graph it would restart again, or the cycle restart. So this exact same shape is gonna hook her for the second part of this for the second period of the graph that we're going to draw, which will end right there. That's the second period. So now we have two periods of the scrap, which is probably enough to kind of analyze what it looks like so we can look at the fact that the period of this craft is gonna be the same as a normal coastline of X period, which is to pie, because after two pi, that cycle restarts. So we know our period here to pie pie. There you go. The mid line is once puts this graph in half this graph in half horizontally and here we can see that that line is still kind of at a y equals zero. And the amplitude of this graph is how high it goes away from the midline. You know, the peak of these peaks in the coast line and we can see that that's that, too, because of the vertical scale of two way up here, it's too high. Star amplitude is too, uh, one more thing. Now we want to look at where the maximum value on this graph is, and because it's a coastline function, we can see that the message will be at that first pig which would be right over. They're So it starts at zero. So our maxim value will be at zero, and that's gonna look like zero. The coastline of zero we know is one and two times one will get us to, So it's a maximum value for a minimum value. We can see that that will be way down here, all right, the bottom peak. And that will happen at pi, the same as a normal coastline function. And the coastline of pie gets us negative one and negative. One times two is negative, too. And for those of you who enjoy decimals, that's 3.14 Come on. Negative, too. And there you have it. That's a simple analysis of the function. F of X is equal to two co sign of ex. Thanks for watching.

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