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 2πx

Accepted Answer

Welcome back. I am so glad you're here. We're asked to find the amplitude and period of the given function and to sketch its graph for one period, our given function is Y equals 17 cosine of six PX. And then we're given a graph. We have 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 positive 20. And the domain for our X axis is from negative 0.5 to 0.7. All right, looking at our function here, we see we have a function in the format of Y equals a multiplied by the cosine of BX where A is what's being multiplied by our cosine. And here A is equal to 17 and B is what's multiplied by our X. And here our B is equal to six pi and this is very helpful for our amplitude in period. Our amplitude is equal to the absolute value of A A is 17. So we're talking about the absolute value of 17, which is 17. So our amplitude is a positive 17. Now, for the period, we find that by taking two pi divided by B two pi divided by B, we said B is six pi so two pi divided by six pi that simplifies to be one third. So our period here is one third. Well, the next step is to sketch that and the first step of sketching is to find the amplitude in the period. So that helps. Now, we also want to figure out if there's a phase shift. But when it's in the format Y equals a cosine of BX, there is no phase shift, there's no C term. So there is no phase shift. So a cosine function will begin when there's no phase shift at zero A and we said that A is 17. So this is going to begin at 0 17. So that's helpful in the next few steps. Step two is we're going to find our five key X values. So for X sub one, the starting point for X, we said this begins at 0 17 with no phase shift for cosine. So our X sub one, our first X value is zero. Now our X sub twos and X sub threes and so on, those are going to be found by adding quarter periods. So a quarter period is taking our period. So here one third and dividing it by four and one third, divided by four is the same as one third multiplied by 1/4. This equals 1/12. So quarter periods being added to each X value to get the subsequent ones. Zero plus 1/12 is 1/12. So our X sub two is 1/12 1 12th plus 1/12 is 1/6. So our X sub three is 1/6 1/6 plus 1/12 is 1/4. So our X of four is 1/4 and 1/4 plus 1/12 is one third. So our X sub five is one third. Perfect, because it's supposed to be one period away. We're only graphing for one period. So we can stop there. Now, our third step is to find the Y values that correspond with the X values that we just found. And there are a couple ways to do this, not Y sub five Y sub three. So a couple of ways to do this, we can take our X values and plug them into our original function and that will give us our corresponding Y values. But another way to do it is just to understand what's going on with our cosine function. We said that the cosine function will begin at zero A A is 17. So the Y value for our initial point is 17. Now that is the amplitude above the X axis, then this cosine function will come down and will have an X intercept. So the second point, our Y sub two value will be zero for the X intercept. Now it's continuing down and it's going to be a minimum next because Y sub one was a maximum, Y sub three is a minimum and that's going to be the amplitude below the X axis. So a negative 17, then we'll come back up for another X intercept. So Y sub four is zero and Y sub 51 period later is back to where we started with a positive 17. That maximum the amplitude above the X axis. All right. Next we graph. So the first point is 0 17. From the origin, we go up 17 units and our next point is 1/12 0. So from the origin, we're going to the right 1/12 and then we are not going up or down 1/12. That's something like this. All right. No, I think it's more to the left of it. OK. We'll call that here 1/12. All right. X our, our third point is 1/6 negative 17. So from the origin, we go to the right 1/6 of a unit and down 17 units. So that's something like this. Our next point is 1/4 0. From the origin, we go to the right, 1/4 maybe there. And from the next 0.1 3rd, 17, we go to the right one third of a unit and then we go up 17 units. So up here going to redraw one of these approximately there. Ok. So we've got these points and now it's just a matter of connecting them so we can connect the points in order from left to right, as smooth as possible. And there we go, we have our graph of Y equals cosine six PX. 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 2πx

Key Concepts

Amplitude of a Trigonometric Function

Period of a Trigonometric Function

Graphing One Period of a Cosine Function

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