In Exercises 75–78, graph one period of each function.y = |2 cos ...

Question

In Exercises 75–78, graph one period of each function.y = |2 cos x/2|

Accepted Answer

Welcome back. I am so glad you're here. We're asked to draw the graph of the function for one period. Our function is Y equals the absolute value of four cosine of X divided by five. And then we have a graph, the graph has a vertical Y axis, a horizontal X axis. They come together at the origin. The domain of our graph is from negative 10 to 30 and the range of our graph is from negative 10 to 10. All right. So how do we graph cosine? Well, we recall from previous lessons that we want to look at what format it's in and we're gonna set aside the absolute value bars for a moment. And we're going to look at Y equals four cosine of X divided by five. And we see that that's in the format of Y equals a cosine of B X where A is being what's multiplied by the A is what's being multiplied by the cosine. So A here is equal to four and B is what's being multiplied by the X and that is 1/ because X divided by five is the same as X multiplied by 1/5. So we've got our A and our B and now from previous lessons and from the graphing steps for cosine in our textbook, we want to identify our amplitude, our period, our phase shift and where this function begins. So for the amplitude, that's going to be the absolute value of A, well, A is four. So the absolute value of four which is four for the period that's going to be two pi divided by B B is 1/5. So we have two pi divided by 1/5 which is the same as two pi multiplied by five. So the period is 10 pi and then for our phase shift, there is no C value, the C value is zero. So there is no phase shift. So that means for this cosine function, it is going to begin at zero A A is four. So it's be going, it's going to begin at 04. So now that we've identified all of that, what we want to do is find first. So step two of graphing first find our five key X values which are going to be the X intercepts the maximums, the minimums. So for our X sub one, as we said, this begins at 04, so the X value there is zero and for our X sub two for all of the subsequent X values, what we're going to do is add a quarter period. So that's going to be a period divided by four, the period is 10 pi so 10 pi divided by four, which simplifies 25 pi divided by two. So for X sub two, that's zero plus five pi divided by two, which is five pi divided by two. Now X sub three, we're going to add another five pi divided by two, which is going to be five pi when simplified for our X sub four, we're adding another five pi divided by two. That's going to be 15 pi divided by two. And for our X sub five, we add another pi divided by two. And that's going to be 10 pi which makes sense because that is one period. So we only have to do one period. We only need those five X fun X values. Now for step three, we're going to find the corresponding Y values for those. So there's a couple ways to do this for Y sub one, you can plug X sub one into our function here. It's very important. We go back to the original function of Y equals the absolute value of four cosine of X divided by five, you plug zero into that and you're gonna get a Y value of a positive four. Now for our X sub two and Y sub two and for our X sub four and our Y sub four for cosine those are going to be our X intercepts. So the Y values there are going to be zero, you can get that by plugging those X values into the function. But it also helps to just know what's going on with our graph. Now, for the Y sub three, typically, if the first one is the maximum, then the next one is going to be the minimum for our cosine. The Y sub one is the maximum Y sub three would be the minimum. However, this is the absolute value. So instead of being the amplitude below our X axis, it's going to be the absolute value of that amplitude. It's gonna be a positive four. So that's what makes this graph look pretty neat for our Y sub five that would be cycling back up to the maximum. So the amplitude above the X axis, so back up to four again. So everything is going to be above the X axis here because it's the absolute value of that whole thing. And so it's only going to be positive values. So now we can graph these five points and then smoothly connect them. So our first point is at 04 from the origin, we go up four units. Our next point is five pi divided by and five pi divided by two is equal to between seven and eight. So from the origin, we're going to the right five pi divided by two units and staying right there for the next 0.5 P 45 pi is equal to between 15 and 16. So from the origin, we're going to the right five pi units and then we are going up four units. The next 0.15 pi divided by 2015 pi divided by two is equal to between 23 24. So from the origin, we're going to the right 15 pi divided by two units. And then final point is 10 pie. 4, 10 pie is equal to about between 31 32. So from the origin, we're going to the right 10 pie units and up four units. And so now we want to connect these, but these connect very interestingly. Normally, we would, I'll show you what that would look like with a different color. So normally we would have a smooth curve that would then come below the X axis, but it's going to mirror that. So it smoothly curves from the first point to the second point, but then it kind of bounces. So it's going to mirror what that would look like, but it looks like it is a concave lens here. So now we're just gonna go can continue connecting points and then it kind of bounces back up above the X axis and this is what Y equals the absolute value of four cosine of X divided by five looks like well done. We'll catch you on the next one.

In Exercises 75–78, graph one period of each function.y = |2 cos x/2|

Key Concepts

Cosine Function

Amplitude and Period

Absolute Value Function

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