In Exercises 31–34, determine the amplitude of each function. ...

Question

In Exercises 31–34, determine the amplitude of each function. Then graph the function and y = cos x in the same rectangular coordinate system for 0 ≤ x ≤ 2π. y = 2 cos x

Accepted Answer

Welcome back. I am so glad you're here. We are asked to find the amplitude of the given function and plot it along with Y equals the cosine of X on the same rectangular coordinate system for the domain of X from 0 to pi and our given function is Y equals 13 cosine X. So we have two functions that we need to graph. The first one, Y equals the cosine of X. That's our base cosine function. There's nothing being done to that one. It's just Y equals the cosine of X and then we have Y equals 13 cosine X. That one's very similar. The only difference is that, that cosine is being multiplied by 13. We recognize that both of these are in the format of Y equals a cosine X. And for our Y equals the cosine of X, the A term being multiplied by cosine is unwritten, there's nothing there. So the A term or A is equal to one and for Y equals 13 cosine X, the A term, the one being multiplied by cosine is 13. So here A equals 13. But other than that, these are pretty much the same so we can graph both of them at once. We can kind of work on both of them. They're very similar. And then we'll see what the similarities and differences between them are. We can start off drawing part of our graph. We know that we're restricted in the domain of X from 0 to 2 pi. So that tells us that we can draw our Y axis, the vertical y axis pretty far to the left because we're not really using any part, any negative part of the X axis. And then the X axis will be right in the middle of the Y axis, the horizontal line. And we know that for the right hand side of our X axis, it's only going up as high as two pi. So we can label two pi here, we don't know the scale of our Y axis and we don't know what, how to divide up our X axis yet. So let's figure that out. Now, we recall from previous lessons that when we are graphing a cosine function, we're going to start off figuring out the amplitude, the period and things like that. So let's start that for both of these, some things will be the same and some will be very um similar but not exactly the same. So for Y equals the cosine of X for the amplitude, the amplitude, as we recall from previous lessons is equal to the absolute value of A. And here we said that A equals one. So that's the absolute value of one which equals one for the amplitude, for Y equals 13 cosine X. Again, that's the absolute value of A. We said that A is 13. So the absolute value of 13 is 13, that was one of the things we were asked to find the amplitude of our given function. It is 13. All right. When we are graphing a fun a cosine function, we also want to figure out the period. Now, when it's in the format Y equals a cosine XX, the period is going to be two pi both of these are in that format Y equals a cosine X. So the period for our cosine functions is going to be two pi and that makes sense, we're graphing for one period. Now, what about where do we begin for each of these? So a cosine function we recall from previous lessons. A cosine function is going to begin at zero A here for Y equals cosine X A is one. So it's going to begin at 01 for our Y equals 13 cosine X, it begins at zero A and A equals 13. So it's going to begin at zero 13. Now, we can figure out we know the domain for both of these is going to be from 0 to pi what about the range we know from previous lessons that the range for our Y equals the cosine of X is going to be from the amplitude below to the amplitude above the X axis. Our amplitude is one. So that's going to be from negative one to positive one. And likewise for our Y equals 13 cosine X, it's going to be the amplitude below to the amplitude above the X axis. The amplitude is 13. So we're going from negative 13 to positive 13. And that helps us label our Y axis because we're going up to 13 above and down to below, we can label our Y axis. We can put above the origin, we can put a positive five, a positive 10 and a positive 15. And then below the origin, we can put a negative five, a negative 10 and a negative 15. And then for the X axis, we are going to be looking for the key X positions, the key X values, those are going to be our X intercepts our maximums and our minimums. And we find those by taking quarter periods. So for a quarter period, we're taking a period divided by four. The period for both of these is two pi so two pi divided by four equals pi divided by two. So we can mark the quarter periods for both of these as pi divided by two. So a long hour X axis from the origin will go to the right label our first unit as pi divided by two, adding another pie divided by two will give us pie, adding another pie divided by two will be three pi divided by two. And then we have our two pie. So now let's work on step two for graphing our cosine function which will be to find those key X values. As we said, our X intercepts and maximums and minimums. And here we're only going through one period. So we said that Y equals cosine of X begins at 01, the X value at 01 is zero. And then for our second X value, this is where we add the quarter periods. So we've already discerned the X sub two is pi divided by two. Then adding another quarter period is pi adding another quarter period. Pi divided by two is three pi divided by two. And then finally, we get to two pi and we're going to have the exact same key X points, key X values for our Y equals 13 cosine XX sub one is going to be zero because that one begins at 0 13, X sub two, we're adding a quarter period. So that's pi divided by two, X sub three at a quarter period. That's PX sub four is three pi divided by two and X sub five is two. Pi the last step before we draw the graphs is we find our associated Y values with the key X values. So for Y sub one, that's going to be where this begins for Y equals the cosine of X, we said it begins at 01. So why sub one is one? Now we know this about the cosine function. But you can always take the X value, plug it into the original function and then what you get is going to be the corresponding Y value. But knowing what's going on with the cosine function, the first one is going to be a maximum when A is positive. So we had 01 as our first point for Y sub two, that's going to be an X intercept. So the Y value here is zero for Y sub three because Y sub one was a maximum, Y sub three is going to be a minimum that's going to be the amplitude below the X axis. So the amplitude here is one. So it's going to be a negative one. Y sub four goes back up to an X intercept. So Y sub four is zero and Y sub five is going to be one period later. We're back to where we started with the Y values. It's a maximum, it's the amplitude above the X axis. The amplitude is one so that Y sub five is one, right? Very similarly four. Excuse me for our Y values four, Y equals 13 cosine X. We are going to analyze it the same way, but we're going to get different values here because we have a different amplitude. We said that Y equals 13 cosine X begins at 13, the Y value there is 13, that's a maximum, that's the amplitude above the X axis Y sub two is going to now be an X intercept as well. So that is zero, Y sub three is going to be the amplitude below the X axis. That's 13 below. So we'll be negative 13. That one's a minimum. Y sub four is back up to an X intercept that's zero and Y sub five is back to where we started for the amplitude above the X axis, which is a positive 13. And again, you could have gotten any of those values by plugging in the associated X values into our function Y equals 13 cosine X. OK. Now let's plot these. So for Y equals cosine X, the first point is 01 from the origin will go up one for the next point pi divided by 20 from the origin. We're going to the right pi divided by two. The next point is pi negative one from the origin. We go to the right pi and down one, the next point is three pi divided by 20 from the origin. We go to the right three pi divided by two. And the last point is two pi one from the origin, we go to the right two pi and up one. Now we can try to connect those as smoothly as possible. And we have our Y equals the cosine of X function from 0 to 2 pi now we can graph Y equals 13 cosine X. The first point is 0 13 from the origin will go up 13. The next point is pi divided by 20 from the origin. We'll go to the right pi divided by two. The next point is pi negative 13 from the origin. We go to the right pi and down 13, the next point is three pi divided by 20 from the origin. We go to the right three pi divided by two. And then finally, our last point is two pi 13 from the origin. We go to the right two pie and up 13 and we try to connect those as smoothly as possible. And there we have our Y equals cosine X. All right. There's our graph. Well done. We'll catch you on the next one.

In Exercises 31–34, determine the amplitude of each function. Then graph the function and y = cos x in the same rectangular coordinate system for 0 ≤ x ≤ 2π. y = 2 cos x

Key Concepts

Amplitude of a Trigonometric Function

Graphing the Cosine Function

Comparing Multiple Functions on the Same Graph

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