In Exercises 35–42, determine the amplitude and period of each...

Question

In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = -4 cos 1/2 x

Accepted Answer

Welcome back. I am so glad you're here. We're asked to find the amplitude and the period of the given function and to sketch its graph for one period, our given function is Y equals negative cosine of 1/4 X. And we are given a graph. It has a vertical Y axis, a horizontal X axis. They come together at the origin. The range for our Y axis is from negative 20 to 20. And the domain for our X axis is from 0 to pi, all right. So taking a look at our function, we recognize that we have a function in the format of Y equals a cosine of BX where A is being multiplied by our cosine. And here A is equal to negative 19. It's not a negative here, negative 19. And our B is being multiplied by the X here B equals 1/4. And so when we have this format, it's very easy to figure out our amplitude in period. Our amplitude, as we recall from previous lessons is equal to the absolute value of A. So we're taking here the absolute value of negative 19 which is a positive 19 So our amplitude is 19. Now for the period that's equal to two pi divided by B, we said that B is equal to 1/4. So we have two pi divided by 1/4 which is the same as two pi multiplied by four, which is eight pi. So our period is eight pi all right. Now, that is very helpful in sketching our graph. We just need to know a few more things following the steps from the textbook and previous lesson videos. Step one, we would figure out the amplitude, the period, whether there's a phase shift. And when it's in this format, Y equals a cosine BX, there is no phase shift. So that means for cosine it's going to begin at zero A. So we said that A is negative 19. So this is going to begin at zero, negative 19. So that's very helpful. That's where this is going to begin. Now, for step two, for graphing this, we want to figure out our five key X values. So we'll write X sub one, X sub two, X sub three, X sub four and X sub five. Those are going to be our X intercepts our maximums and minimums. And so what we do is for cosine, we said that X sub one is zero with no phase shift. So X sub one is zero and to find the subsequent X values, we're going to add quarter periods. So we take our period divide it by four to get a quarter period. We said the period is eight pi so eight pi divided by four that simplifies 22 pi so zero plus two pi to get our X sub two is two pi then add another two pi we get four pi for X sub three, X sub four is going to be six pi and X sub five will be eight pi one period and we're graphing one period. So that's great. Now, step sub or excuse me, step three is going to be to find the corresponding Y values. So Y sub one, Y sub two, Y sub three, Y sub four and Y sub five, as we said, cosine is going to begin at zero A and A here is negative 19. So Y sub one is going to be negative 19. So that's going to be a minimum where the amplitude below the X axis and then it's going to come up and we're going to have an X intercept. So Y sub two is going to be zero. You can always get these Y values by plugging in the corresponding X values into our original function. You'll get, when you plug zero in for X, you'll get a Y value of negative 19. When you plug two pi in for X, you'll get a Y value of zero. But by just understanding what's going on with our cosine function, you can graph it without having to plug all of that. In all right, we said that our second point is an X intercept. That means our third point because the first one was a minimum is going to be a maximum, the maximum is going to be the amplitude above the X axis. So that's going to be a positive 19 Y sub four another X intercept. So that's zero and Y sub five, that's one full period later, we're back to where we started at negative 19. Now, the next step is to just graph these points and connect them. So our first point is zero, negative 19 from the origin will go down 19 units. The next point is two pi zero from the origin. We'll go to the right two pi the next point is four pi 19 from the origin. We'll go to the right four pi and up 19, the next point is six pi zero from the origin. We'll go to the right six pi and the final point is eight pi negative 19 from the origin. We'll go to the right eight pi and down 19. Now we try to connect these points as smoothly as possible. Yours might be smoother than mine. Oh my, that's just not even, that's not smooth at all. So hopefully yours is smooth and that's one period of our cosine function. So I'll highlight the amplitude. We said the amplitude was 19. The period is a pie and then we have our graph well done. We'll catch you on the next one.

In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = -4 cos 1/2 x

Key Concepts

Amplitude of a Trigonometric Function

Period of a Trigonometric Function

Graphing One Period of a Cosine Function

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