Graph each function over a one-period interval. See Example 3....

Question

Graph each function over a one-period interval. See Example 3.

y = (3/2) sin [2(x + π/4)]

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph the following function consider only one period. Our function is Y equals three halves multiplied by the sine of now, all in parentheses, two multiplied by the quantity of X plus pi divided by three. Then we have a blank graph. We have a vertical Y axis, a horizontal X axis. They come together the origin, the domain for what's drawn for our X axis is from 0 to 5 pi divided by four. And that's in increments of pi divided by four. And the range for what's drawn per hour, Y axis is from negative two to positive two. And that's in increments of one. All right. So we've got Y equals three halves times the sine of two multiplied by the quantity of X plus pi divided by three. And that's given to us in the format of Y equals C plus A multiplied by the S of B multiplied by the quantity of X minus D. So we can compare our given function with the traditional format of assign function and identify our ABC and D and use those to help us graph our function. So comparing here we have nothing being added to our sine function. So C is equal to zero, we have three halves being multiplied by our sine function. So A is equal to three halves that has to do with our amplitude. We recall from previous lessons that the amplitude is the absolute value of A. And then comparing further inside of the sine function in the parentheses that B is what's being multiplied by our X. And here B is equal to two that has to do with our period. We recall that the period is given by two pi divided by B for a sine function C we said that was zero C relates to a vertical shift. So there is no vertical shift here. And then finally, d what's being subtracted from X inside of our sine function here we have X plus pi divided by three. So D is going to be a negative pi divided by three. That's what's being subtracted. A negative pi divided by 3d has to do with a phase shift. So we said that A has to do with amplitude and the absolute value of A is our amplitude. So if A is three halves, the absolute value of three halves is three halves, our amplitude is three halves, our period is two pi divided by BB is two. So we have two pi divided by two which simplifies to pi we have a period of pi we have no vertical shift C equals zero and our phase shift because it is a negative pi divided by three, it means we're going to the left pi divided by three units. Now, if you know what a sign function looks like and can get all of those three shifts down, that's great. But another way we can do this now is to create an XY table, figure out some key X values for five key points For our sine function, plug those X values into our function. Figure out the corresponding Y values and then plot those points. So typically, if there's no phase shift, we would start with an X value of zero. But we do have a phase shift to the left pi divided by three units. So we're going to start with an X value of negative pi divided by three. However, if you recall, that's not on our given graph. So what we'll do is we'll add 1/4 of our period. Our period was pi. So 1/4 of that is going to be pi divided by four. We're going to add 1/4 of the period to get the next key X value. And we'll see if that's on our chart. So negative pi divided by three plus pi divided by four is negative pi divided by 12. Nope, that one's still not on our chart yet. So we can add another pi divided by four to that. And we'll get pi divided by six that one's there. All right, pi divided by six plus pi divided by four is five pi divided by 12. Add pi divided by four. To that, we get two pi divided by three, add another pie divided by four, we get 11 pie divided by 12. And now to get one full period with five key points. So we're going to have the last value at seven pi divided by six. Now we can plug those X values into our function. And I'm going to start with the negative pi divided by 12. So we can see what's happening to the left of our Y axis, to the left of X equals zero. So plugging that into a function, we'd have three halves multiplied by the sign of and your calculator gives you parentheses. So then it'll be two multiplied by another set of parentheses. Negative pi divided by 12 plus pi divided by three, close both sets of parentheses hit enter and you'll get a Y value of three halves or 1.5. So we can see that to the left of our Y axis, which isn't shown we are going to be up above the X axis at about 1.5. So I'm just going to draw something there. So we know what direction we're coming from. All right. Now, plugging in pi divided by six for our X value, you get a corresponding Y value of zero. Plugging in five pi divided by 12 we get a negative three halves or a negative 1.5 plugging in the X value of two pi divided by three, we get a Y value of zero plugging in the X value of 11 pi divided by 12. We get a Y value of three halves or 1.5 and the last one on here so that we get a full period mapped out plug in seven pi divided by six for the X value and you get a Y value of zero. So now we can plot these points noting that our X axis is divided up into um increments of pi divided by four. And we're not exactly on the pi divided by four parts. So you have to be converting these into 12th for the most part to figure out where you're going to be. So the first point we're going to plot is pi divided by 60. So from the origin, we're going to the right pi divided by six units, which is two thirds of the way to pi divided by four. The next point we're plotting is at five pi divided by 12, negative three halves and five pi divided by 12 is going to be about two thirds of the way between pi divided by four and pi divided by two and then down 1.5 units. The next point is two pi divided by 30. That is two thirds of the way between pi divided by two and three pi divided by four. Next is 11 pi divided by 12, 3 halves. 11 pi divided by 12 is two thirds of the weight in between three pi divided by four and pi and then up 1.5. And the final unit on here is at the final 0.7 pi divided by 60 and seven pi divided by six is just two thirds of the way between pi and five pi divided by four. So now from that, we can start at the Y axis, we're coming down and then we'll pass through that point at pi divided by 60. And then you can draw these connecting as smoothly as possible through all your points. And there is our three halves sine of two multiplied by the quantity of X plus pi divided by three. Well done. Well, catch you on the next one.

Graph each function over a one-period interval. See Example 3.
y = (3/2) sin [2(x + π/4)]

Key Concepts

Sine Function

Transformations of Functions

Period of a Function

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