Graph each function over a two-period interval.
y = sin ...

Question

Graph each function over a two-period interval.

y = sin [2(x + π/4) ] + 1/2

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph the following function, consider two periods. Our function is Y equals the sign of and then open bracket two multiplied by the quantity of X plus pi divided by six. Close parentheses, close bracket plus 3/4. Then we're given a blank graph. It has 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 7 pi divided by three. And it's marked in increments of pi divided by three and the Y axis, the range for what's drawn is from negative one to positive three and it's marked in increments of one. All right, we recall from previous lessons that the sine function is given to us with the formula where Y equals a multiplied by the sine of open brackets. B multiplied by the quantity of X minus D closed parentheses, close brackets plus C. And now we can just look at our given function and identify our A, our B, our C and our D from that. And that will help us to graph. So a what's being multiplied by the sign, there's nothing written. So that means our A value is one that is typical. So there's no change there. A tells us about our amplitude and about any reflection over the X axis are B value. What's being multiplied inside of the sine function and multiplied to the X value here is two, the B value has to do with the period because we know the period for a sine function is two pi divided by B, the C value here. What's being added at the end? Sometimes it's added at the beginning, but here it's added at the end is a positive 3/4. The C value tells us if there is a vertical shift. And here because we have a positive 3/4 we have a vertical shift up 3/4 of a unit and our D value what's being subtracted directly from that X here D is going to be a negative pi divided by six because it, we have a positive X plus pi divided by six. So because it's X minus D, that's going to be a negative pi divided by six, that indicates a phase shift, our D value does. And because it's a negative pi divided by six, it's a phase shift left pi divided by six units. So that's what our ABC and D tell us. We can identify specifically what the period is. We say that the period is two pi divided by B while our B values two. So two pi divided by two is equal to pi. And that helps us understand we're graphing two periods how far we go? So if a period is pi, we have one period from zero to pi along the x axis and another from pi to two pi and we want to divide up our period into four equal parts to identify our five important points to help us graph this sine function. And so if we take pi and divide that into four equal parts, we'll have pi divided by four will be the increment that we're adding. So typically, we would now create an XY table and we would typically start with a value of zero for our first X value. Plug that in to our equation, come up with a corresponding Y value and we'd have a point to plot. However, this one has a phase shift, it's gone to the left pi divided by six units. So one of our key values isn't going to be along the Y axis at X equals zero. So we're going to have our first key value to the left of that at a negative pi divided by six note. However, that negative pi divided by six isn't on our given graph. So we're going to use that as a starting point, but we'll have to graph an extra point because that one isn't on our given graph. We're still adding pi divided by four to each X value in order to get our next four key points for graphing. So if we take negative pi divided by six and add pi divided by four, that's going to be pi divided by 12. Adding pi divided by four to that, we get pi divided by three. Add pi divided by four to that, we get seven pi divided by 12 and add pi divided by four to that, we get five pi divided by six. There's one period. Now we'll make another quick chart XY table and we'll add pi divided by 4 to 5 pie divided by six. We get 13 P divided by 12. Add pi divided by four, we get four pi divided by three. Add pie divided by four. We get 19 pie divided by 12. Add pie divided by four, we get 11 pie divided by six and add pi divided by four and we'll get 25 pi divided by 12. That's our last point that we are going to need because that negative pi divided by six doesn't get graphed. Now, we're going to plug in our X values into our given function and that will give us our corresponding Y values. I'll show you for the first one. So for the X value of negative pi divided by six, plug into your calculator sign of two multiplied by you can put another parentheses. X value is negative pi divided by six. Then add pi divided by six. That's going to be zero in there, close the two sets of parentheses and add 3/4. Well, the sign of zero plus 3/4 just equals 3/4. So that's our first point, but we can't plot it. Now, we're going to continue to plug in X values in for X or plug our values in for X in the calculator. Get the corresponding Y values. If you need to pause your video and plug those into the calculator, go for it. I've already done that. So for pi divided by 12 X value, the corresponding Y value is seven divided by four. For pi divided by three, the Y value is 3/4 for seven pi divided by 12. It's negative 1/4 for five pi divided by six. It's back up to 3/4 for 13 pi divided by 12. We're at seven fourths for four pi divided by 3, 3/4 19 pi divided by 12, negative 1/4 11 pi divided by 6, 3/4 and 25 pi divided by 12 is seven fourths. All right, there's two full periods that we can graph. Now, the only thing about our given graph instead of being in increments of pi divided by four, they're in increments of pi divided by three. So it's almost like if you, if you're into music, it's like we've got triplets. We're trying to figure out here and we're in 44 time. So the first one that we can graph is going to be pi divided by 12, 7 4th and that is just after zero. So from the origin, we're going to the right pi divided by 12 units, which isn't much, it's about 1/4 of pi divided by three. And then we're going up to seven fourths, seven fourths is almost two. So we'll put it about here. Now, keeping in mind that we have a point that we're not graphing to the left of that at negative pi divided by six 3/4. So from the origin, it'll be coming or from the y axis, it will be coming up to that point. That one is a maximum. All right, the next point that we're graphing is pi divided by 3, 3/4. Next point is seven pi divided by 12, negative 1/4. And that one is going to be just a little bit before the two pi divided by three. Next 0.5 pi divided by 6 3/4. Five pi divided by six is about halfway between two pi divided by three and pi the next point is 13 pi divided by 12, 7 4th and 13 pi divided by 12 is just after pi next 0.4 pi divided by 3, 3/4. That 14 pi divided by three is on here. Next 0.19 pi divided by 12, negative 1/4 19 pi divided by 12 is going to be just before five pi divided by three. Next 0.11 pi divided by 6, 3/4 that 111 P divided by six is right between five pi divided by three and two pi. And the last point on here from 25 pi divided by 12 7 4th, 25 pi divided by 12 is just past our seven or no. Excuse me, it's just past two P OK. So now connecting these as smoothly as possible in our sine curve, hopefully yours is connected more smoothly and beautifully than mine. But there we go, we have our graph of Y equals the sign of two multiplied by the quantity of X plus pi divided by six plus 3/4. Well done. We'll catch you on the next one.

Graph each function over a two-period interval.
y = sin [2(x + π/4) ] + 1/2

Key Concepts

Amplitude and Vertical Shift

Period and Frequency

Phase Shift

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